석사학위논문
Masterrsquos Thesis
지진 하중을 받는 사장교를 위한
수동 제어 장치의 설계 Design of Passive Control System
for Seismically Excited Cable-Stayed Bridges
이 성 진 (李 聖 振 Lee Sung Jin) 건설 및 환경공학과
Department of Civil and Environmental Engineering
한 국 과 학 기 술 원 Korea Advanced Institute of Science and Technology
2004
지진 하중을 받는 사장교를 위한
수동 제어 장치의 설계
Design of Passive Control System
for Seismically Excited Cable-Stayed Bridges
Design of Passive Control System
for Seismically Excited Cable-Stayed Bridges
Advisor Professor In-Won Lee
by
Sung-Jin Lee
Department of Civil and Environmental Engineering Korea Advanced Institute of Science and Technology
A thesis submitted to the faculty of the Korea Advanced Institute of Science and Technology in partial fulfillment of the requirements for the degree of Master of Engineering in the Department of Civil and Environmental Engineering
Daejeon Korea 2003 12 22 Approved by __________________ Professor In-Won Lee Major Advisor
지진 하중을 받는 사장교를 위한
수동 제어 장치의 설계
이 성 진
위 논문은 한국과학기술원 석사학위논문으로 학위논문 심사위원회에서 심사 통과하였음
2003 년 12 월 22 일
심사 위원장 이 인 원 (인)
심 사 위 원 윤 정 방 (인)
심 사 위 원 김 진 근 (인)
i
MCE
20023430
ABSTRACT
In this dissertation the design procedure and guidelines of lead rubber bearing
(LRB) are proposed and the effectiveness of designed LRB is investigated for seismically
excited cable-stayed bridges Furthermore additional control device ie viscous damper
(VD) is considered to improve the control performances
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because these provide structural
support base isolation damping and restoring forces in a single unit The most important
feature of the seismic isolation system for short-span bridges and buildings is lengthening
the natural period of structures However the seismic characteristics of long-span bridges
such as cable-stayed bridges are different from those of short-span bridges and buildings
and these bridges have very complex behavior in which the vertical translational and
torsional motions are often strongly coupled For these reasons it is conceptually
unacceptable for long-span bridges to use directly the recommended design procedure
and guidelines of LRB for short-span bridges and buildings Therefore new design
approach and guidelines are required to design LRB for cable-stayed bridges
Considering important responses of cable-stayed bridges the design index (DI) is
proposed to design LRB The proper LRB is selected when proposed DI is minimized or
converged for variation of properties of LRB The design results show that the damping
and energy dissipation effect of LRB are more important than the shift of the natural
이 성 진 Lee Sung Jin Design of Passive Control System for Seismically
Excited Cable-Stayed Bridges 지진 하중을 받는 사장교를 위한 수동
제어 장치의 설계 Department of Civil and Environmental Engineering
2003 55p Advisor Professor Lee In Won Text in English
ii
period of structures for cable-stayed bridges And the control performance of designed
LRB is also verified
The sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes The performance of designed LRB is not changed
significantly for different characteristic of input earthquakes and thus the robustness of
designed LRB is verified for different characteristics of earthquakes
Finally the VD is employed to obtain the additional reduction of seismic responses
because there are some responses that are not controlled sufficiently by only LRB
Additional VD can reduce the some responses such as shear at deck level of towers and
deck displacement without loss of control effects of LRB These results show that the
seismic responses of cable-stayed bridges can be controlled sufficiently by appropriate
designed passive control devices
iii
TABLE OF CONTENTS
ABSTRACT i
TABLE OF CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER 1 INTRODUCTION 1
11 Backgrounds 1
12 Literature Review 3
13 Objectives and Scopes 4
CHAPTER 2 PROPOSED DESIGN PROCEDURE OF LRB 6
21 LRB 6
211 Design Parameters of LRB 6
212 LRB Model 8
22 Proposed Design Procedure 10
CHAPTER 3 NUMERICAL EXAMPLE 13 31 Bridge Model 13
32 Design and Seismic Performance of LRB 15
321 Design Earthquake Excitations 15
322 Design of LRB 17
323 Control Performance of Designed LRB 24
33 Effect of Characteristics of Earthquakes 36
331 Effect of Frequency Contents of Earthquakes 36
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
지진 하중을 받는 사장교를 위한
수동 제어 장치의 설계
Design of Passive Control System
for Seismically Excited Cable-Stayed Bridges
Design of Passive Control System
for Seismically Excited Cable-Stayed Bridges
Advisor Professor In-Won Lee
by
Sung-Jin Lee
Department of Civil and Environmental Engineering Korea Advanced Institute of Science and Technology
A thesis submitted to the faculty of the Korea Advanced Institute of Science and Technology in partial fulfillment of the requirements for the degree of Master of Engineering in the Department of Civil and Environmental Engineering
Daejeon Korea 2003 12 22 Approved by __________________ Professor In-Won Lee Major Advisor
지진 하중을 받는 사장교를 위한
수동 제어 장치의 설계
이 성 진
위 논문은 한국과학기술원 석사학위논문으로 학위논문 심사위원회에서 심사 통과하였음
2003 년 12 월 22 일
심사 위원장 이 인 원 (인)
심 사 위 원 윤 정 방 (인)
심 사 위 원 김 진 근 (인)
i
MCE
20023430
ABSTRACT
In this dissertation the design procedure and guidelines of lead rubber bearing
(LRB) are proposed and the effectiveness of designed LRB is investigated for seismically
excited cable-stayed bridges Furthermore additional control device ie viscous damper
(VD) is considered to improve the control performances
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because these provide structural
support base isolation damping and restoring forces in a single unit The most important
feature of the seismic isolation system for short-span bridges and buildings is lengthening
the natural period of structures However the seismic characteristics of long-span bridges
such as cable-stayed bridges are different from those of short-span bridges and buildings
and these bridges have very complex behavior in which the vertical translational and
torsional motions are often strongly coupled For these reasons it is conceptually
unacceptable for long-span bridges to use directly the recommended design procedure
and guidelines of LRB for short-span bridges and buildings Therefore new design
approach and guidelines are required to design LRB for cable-stayed bridges
Considering important responses of cable-stayed bridges the design index (DI) is
proposed to design LRB The proper LRB is selected when proposed DI is minimized or
converged for variation of properties of LRB The design results show that the damping
and energy dissipation effect of LRB are more important than the shift of the natural
이 성 진 Lee Sung Jin Design of Passive Control System for Seismically
Excited Cable-Stayed Bridges 지진 하중을 받는 사장교를 위한 수동
제어 장치의 설계 Department of Civil and Environmental Engineering
2003 55p Advisor Professor Lee In Won Text in English
ii
period of structures for cable-stayed bridges And the control performance of designed
LRB is also verified
The sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes The performance of designed LRB is not changed
significantly for different characteristic of input earthquakes and thus the robustness of
designed LRB is verified for different characteristics of earthquakes
Finally the VD is employed to obtain the additional reduction of seismic responses
because there are some responses that are not controlled sufficiently by only LRB
Additional VD can reduce the some responses such as shear at deck level of towers and
deck displacement without loss of control effects of LRB These results show that the
seismic responses of cable-stayed bridges can be controlled sufficiently by appropriate
designed passive control devices
iii
TABLE OF CONTENTS
ABSTRACT i
TABLE OF CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER 1 INTRODUCTION 1
11 Backgrounds 1
12 Literature Review 3
13 Objectives and Scopes 4
CHAPTER 2 PROPOSED DESIGN PROCEDURE OF LRB 6
21 LRB 6
211 Design Parameters of LRB 6
212 LRB Model 8
22 Proposed Design Procedure 10
CHAPTER 3 NUMERICAL EXAMPLE 13 31 Bridge Model 13
32 Design and Seismic Performance of LRB 15
321 Design Earthquake Excitations 15
322 Design of LRB 17
323 Control Performance of Designed LRB 24
33 Effect of Characteristics of Earthquakes 36
331 Effect of Frequency Contents of Earthquakes 36
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Design of Passive Control System
for Seismically Excited Cable-Stayed Bridges
Advisor Professor In-Won Lee
by
Sung-Jin Lee
Department of Civil and Environmental Engineering Korea Advanced Institute of Science and Technology
A thesis submitted to the faculty of the Korea Advanced Institute of Science and Technology in partial fulfillment of the requirements for the degree of Master of Engineering in the Department of Civil and Environmental Engineering
Daejeon Korea 2003 12 22 Approved by __________________ Professor In-Won Lee Major Advisor
지진 하중을 받는 사장교를 위한
수동 제어 장치의 설계
이 성 진
위 논문은 한국과학기술원 석사학위논문으로 학위논문 심사위원회에서 심사 통과하였음
2003 년 12 월 22 일
심사 위원장 이 인 원 (인)
심 사 위 원 윤 정 방 (인)
심 사 위 원 김 진 근 (인)
i
MCE
20023430
ABSTRACT
In this dissertation the design procedure and guidelines of lead rubber bearing
(LRB) are proposed and the effectiveness of designed LRB is investigated for seismically
excited cable-stayed bridges Furthermore additional control device ie viscous damper
(VD) is considered to improve the control performances
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because these provide structural
support base isolation damping and restoring forces in a single unit The most important
feature of the seismic isolation system for short-span bridges and buildings is lengthening
the natural period of structures However the seismic characteristics of long-span bridges
such as cable-stayed bridges are different from those of short-span bridges and buildings
and these bridges have very complex behavior in which the vertical translational and
torsional motions are often strongly coupled For these reasons it is conceptually
unacceptable for long-span bridges to use directly the recommended design procedure
and guidelines of LRB for short-span bridges and buildings Therefore new design
approach and guidelines are required to design LRB for cable-stayed bridges
Considering important responses of cable-stayed bridges the design index (DI) is
proposed to design LRB The proper LRB is selected when proposed DI is minimized or
converged for variation of properties of LRB The design results show that the damping
and energy dissipation effect of LRB are more important than the shift of the natural
이 성 진 Lee Sung Jin Design of Passive Control System for Seismically
Excited Cable-Stayed Bridges 지진 하중을 받는 사장교를 위한 수동
제어 장치의 설계 Department of Civil and Environmental Engineering
2003 55p Advisor Professor Lee In Won Text in English
ii
period of structures for cable-stayed bridges And the control performance of designed
LRB is also verified
The sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes The performance of designed LRB is not changed
significantly for different characteristic of input earthquakes and thus the robustness of
designed LRB is verified for different characteristics of earthquakes
Finally the VD is employed to obtain the additional reduction of seismic responses
because there are some responses that are not controlled sufficiently by only LRB
Additional VD can reduce the some responses such as shear at deck level of towers and
deck displacement without loss of control effects of LRB These results show that the
seismic responses of cable-stayed bridges can be controlled sufficiently by appropriate
designed passive control devices
iii
TABLE OF CONTENTS
ABSTRACT i
TABLE OF CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER 1 INTRODUCTION 1
11 Backgrounds 1
12 Literature Review 3
13 Objectives and Scopes 4
CHAPTER 2 PROPOSED DESIGN PROCEDURE OF LRB 6
21 LRB 6
211 Design Parameters of LRB 6
212 LRB Model 8
22 Proposed Design Procedure 10
CHAPTER 3 NUMERICAL EXAMPLE 13 31 Bridge Model 13
32 Design and Seismic Performance of LRB 15
321 Design Earthquake Excitations 15
322 Design of LRB 17
323 Control Performance of Designed LRB 24
33 Effect of Characteristics of Earthquakes 36
331 Effect of Frequency Contents of Earthquakes 36
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
지진 하중을 받는 사장교를 위한
수동 제어 장치의 설계
이 성 진
위 논문은 한국과학기술원 석사학위논문으로 학위논문 심사위원회에서 심사 통과하였음
2003 년 12 월 22 일
심사 위원장 이 인 원 (인)
심 사 위 원 윤 정 방 (인)
심 사 위 원 김 진 근 (인)
i
MCE
20023430
ABSTRACT
In this dissertation the design procedure and guidelines of lead rubber bearing
(LRB) are proposed and the effectiveness of designed LRB is investigated for seismically
excited cable-stayed bridges Furthermore additional control device ie viscous damper
(VD) is considered to improve the control performances
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because these provide structural
support base isolation damping and restoring forces in a single unit The most important
feature of the seismic isolation system for short-span bridges and buildings is lengthening
the natural period of structures However the seismic characteristics of long-span bridges
such as cable-stayed bridges are different from those of short-span bridges and buildings
and these bridges have very complex behavior in which the vertical translational and
torsional motions are often strongly coupled For these reasons it is conceptually
unacceptable for long-span bridges to use directly the recommended design procedure
and guidelines of LRB for short-span bridges and buildings Therefore new design
approach and guidelines are required to design LRB for cable-stayed bridges
Considering important responses of cable-stayed bridges the design index (DI) is
proposed to design LRB The proper LRB is selected when proposed DI is minimized or
converged for variation of properties of LRB The design results show that the damping
and energy dissipation effect of LRB are more important than the shift of the natural
이 성 진 Lee Sung Jin Design of Passive Control System for Seismically
Excited Cable-Stayed Bridges 지진 하중을 받는 사장교를 위한 수동
제어 장치의 설계 Department of Civil and Environmental Engineering
2003 55p Advisor Professor Lee In Won Text in English
ii
period of structures for cable-stayed bridges And the control performance of designed
LRB is also verified
The sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes The performance of designed LRB is not changed
significantly for different characteristic of input earthquakes and thus the robustness of
designed LRB is verified for different characteristics of earthquakes
Finally the VD is employed to obtain the additional reduction of seismic responses
because there are some responses that are not controlled sufficiently by only LRB
Additional VD can reduce the some responses such as shear at deck level of towers and
deck displacement without loss of control effects of LRB These results show that the
seismic responses of cable-stayed bridges can be controlled sufficiently by appropriate
designed passive control devices
iii
TABLE OF CONTENTS
ABSTRACT i
TABLE OF CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER 1 INTRODUCTION 1
11 Backgrounds 1
12 Literature Review 3
13 Objectives and Scopes 4
CHAPTER 2 PROPOSED DESIGN PROCEDURE OF LRB 6
21 LRB 6
211 Design Parameters of LRB 6
212 LRB Model 8
22 Proposed Design Procedure 10
CHAPTER 3 NUMERICAL EXAMPLE 13 31 Bridge Model 13
32 Design and Seismic Performance of LRB 15
321 Design Earthquake Excitations 15
322 Design of LRB 17
323 Control Performance of Designed LRB 24
33 Effect of Characteristics of Earthquakes 36
331 Effect of Frequency Contents of Earthquakes 36
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
i
MCE
20023430
ABSTRACT
In this dissertation the design procedure and guidelines of lead rubber bearing
(LRB) are proposed and the effectiveness of designed LRB is investigated for seismically
excited cable-stayed bridges Furthermore additional control device ie viscous damper
(VD) is considered to improve the control performances
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because these provide structural
support base isolation damping and restoring forces in a single unit The most important
feature of the seismic isolation system for short-span bridges and buildings is lengthening
the natural period of structures However the seismic characteristics of long-span bridges
such as cable-stayed bridges are different from those of short-span bridges and buildings
and these bridges have very complex behavior in which the vertical translational and
torsional motions are often strongly coupled For these reasons it is conceptually
unacceptable for long-span bridges to use directly the recommended design procedure
and guidelines of LRB for short-span bridges and buildings Therefore new design
approach and guidelines are required to design LRB for cable-stayed bridges
Considering important responses of cable-stayed bridges the design index (DI) is
proposed to design LRB The proper LRB is selected when proposed DI is minimized or
converged for variation of properties of LRB The design results show that the damping
and energy dissipation effect of LRB are more important than the shift of the natural
이 성 진 Lee Sung Jin Design of Passive Control System for Seismically
Excited Cable-Stayed Bridges 지진 하중을 받는 사장교를 위한 수동
제어 장치의 설계 Department of Civil and Environmental Engineering
2003 55p Advisor Professor Lee In Won Text in English
ii
period of structures for cable-stayed bridges And the control performance of designed
LRB is also verified
The sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes The performance of designed LRB is not changed
significantly for different characteristic of input earthquakes and thus the robustness of
designed LRB is verified for different characteristics of earthquakes
Finally the VD is employed to obtain the additional reduction of seismic responses
because there are some responses that are not controlled sufficiently by only LRB
Additional VD can reduce the some responses such as shear at deck level of towers and
deck displacement without loss of control effects of LRB These results show that the
seismic responses of cable-stayed bridges can be controlled sufficiently by appropriate
designed passive control devices
iii
TABLE OF CONTENTS
ABSTRACT i
TABLE OF CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER 1 INTRODUCTION 1
11 Backgrounds 1
12 Literature Review 3
13 Objectives and Scopes 4
CHAPTER 2 PROPOSED DESIGN PROCEDURE OF LRB 6
21 LRB 6
211 Design Parameters of LRB 6
212 LRB Model 8
22 Proposed Design Procedure 10
CHAPTER 3 NUMERICAL EXAMPLE 13 31 Bridge Model 13
32 Design and Seismic Performance of LRB 15
321 Design Earthquake Excitations 15
322 Design of LRB 17
323 Control Performance of Designed LRB 24
33 Effect of Characteristics of Earthquakes 36
331 Effect of Frequency Contents of Earthquakes 36
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
ii
period of structures for cable-stayed bridges And the control performance of designed
LRB is also verified
The sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes The performance of designed LRB is not changed
significantly for different characteristic of input earthquakes and thus the robustness of
designed LRB is verified for different characteristics of earthquakes
Finally the VD is employed to obtain the additional reduction of seismic responses
because there are some responses that are not controlled sufficiently by only LRB
Additional VD can reduce the some responses such as shear at deck level of towers and
deck displacement without loss of control effects of LRB These results show that the
seismic responses of cable-stayed bridges can be controlled sufficiently by appropriate
designed passive control devices
iii
TABLE OF CONTENTS
ABSTRACT i
TABLE OF CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER 1 INTRODUCTION 1
11 Backgrounds 1
12 Literature Review 3
13 Objectives and Scopes 4
CHAPTER 2 PROPOSED DESIGN PROCEDURE OF LRB 6
21 LRB 6
211 Design Parameters of LRB 6
212 LRB Model 8
22 Proposed Design Procedure 10
CHAPTER 3 NUMERICAL EXAMPLE 13 31 Bridge Model 13
32 Design and Seismic Performance of LRB 15
321 Design Earthquake Excitations 15
322 Design of LRB 17
323 Control Performance of Designed LRB 24
33 Effect of Characteristics of Earthquakes 36
331 Effect of Frequency Contents of Earthquakes 36
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
iii
TABLE OF CONTENTS
ABSTRACT i
TABLE OF CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER 1 INTRODUCTION 1
11 Backgrounds 1
12 Literature Review 3
13 Objectives and Scopes 4
CHAPTER 2 PROPOSED DESIGN PROCEDURE OF LRB 6
21 LRB 6
211 Design Parameters of LRB 6
212 LRB Model 8
22 Proposed Design Procedure 10
CHAPTER 3 NUMERICAL EXAMPLE 13 31 Bridge Model 13
32 Design and Seismic Performance of LRB 15
321 Design Earthquake Excitations 15
322 Design of LRB 17
323 Control Performance of Designed LRB 24
33 Effect of Characteristics of Earthquakes 36
331 Effect of Frequency Contents of Earthquakes 36
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
iv
332 Effect of PGA of Earthquakes 40
34 VD for Additional Passive Control System 45
341 Design of VD 45
342 Control Performance of Designed LRB with VD 47
CHAPTER 4 CONCLUSIONS 49
SUMMARY (IN KOREAN) 51
REFERENCES 53
ACKNOWLEDGEMENTS
CURRICULUM VITAE
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
v
LIST OF TABLES 31 Design properties of LRB 24
32 Controlled responses of bridge for design earthquakes 25
33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing) 27
34 Uncontrolled maximum responses for performance criteria 29
35 Performance of designed LRB under El Centro earthquake 29
36 Performance of designed LRB under Mexico City earthquake 30
37 Performance of designed LRB under Gebze earthquake 30
38 Designed properties of LRB for different frequency contents 37
39 Performance of LRB for different frequency contents under scaled El Centro
earthquake 38
310 Performance of LRB for different frequency contents under scaled Mexico City
earthquake 39
311 Performance of LRB for different frequency contents under scaled Gebze
earthquake 39
312 Design properties of LRB for different PGA 40
313 Performance of LRB for different PGA of earthquake under 10 scaled
El Centro earthquake 42
314 Performance of LRB for different PGA of earthquake under 05 scaled
El Centro earthquake 42
315 Performance of LRB for different PGA of earthquake under 036 grsquos scaled
artificial random excitation 43
316 Performance of LRB for different PGA of earthquake under 018 grsquos scaled
Artificial random excitation 43
317 Additional reduction of responses with LRB and VD 48
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
vi
LIST OF FIGURES
21 Schematic of LRB 7
22 Hysteretic curve of LRB 7
31 Schematic of the Bill Emersion Memorial Bridge 13
32 Design earthquake excitation (Scaled El Centro earthquake) 15
33 Design earthquake excitation (Artificial random excitation) 16
34 Deck weight supported by LRB 17
35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
20
36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
21
37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
22
38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
23
39 Time history of three earthquakes 25
310 Time history responses of cable-stayed bridge under El Centro earthquake 31
311 Time history responses of cable-stayed bridge under Mexico City earthquake 32
312 Time history responses of cable-stayed bridge under Gebze earthquake 33
313 Restoring force of LRB under three earthquakes 35
314 Power spectral density of three earthquakes 36
315 Design of VD 46
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
1
CHAPTER 1
INTRODUCTION
11 Backgrounds
The bridge structures tend to be constructed in longer and slender form as the
analysis and design technology are advanced in civil structures high-strengthhigh-
quality materials are developed and people hope to construct beautiful bridges Therefore
the construction and research of long-span bridges such as cable-stayed and suspension
bridges have become a popular and challenging problem in civil engineering fields
However these long-span bridges have the flexibility of their cable-superstructure system
and low structural damping For these reasons excessive internal forces and vibrations
may be induced in these structures by the dynamic loads such as strong winds and
earthquakes These large internal forces and vibrations may induce direct damages as
well as fatigue fractures of structures Furthermore these may deteriorate the safety and
serviceability of bridges Therefore it is very important to control these responses of
long-span bridges and thus to improve the safety and serviceability of these bridges under
severe dynamic loads
Many seismic design methods and construction technology have been developed and
investigated over the years to reduce seismic responses of buildings bridges and
potentially vulnerable structures Among the several seismic design methods the seismic
isolation technique is widely used recently in many parts of the world The concept of the
seismic isolation technique is shifting the fundamental period of the structure to outrange
of period containing large seismic energy of earthquake ground motions by separating
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 1 Introduction 2
superstructure and substructure and reducing the transmission of earthquake forces and
energy into the superstructure However the seismic isolation technique allows relatively
large displacements of structures under earthquakes Therefore it is necessary to provide
supplemental damping to reduce these excessive displacements
The LRB is widely used for the seismic isolation system to control responses of
buildings and short-span bridges under earthquakes because this bearing not only
provides structural support by vertical stiffness but also is excellent to shift the natural
period of structures by flexibility of rubber and to dissipate the earthquake energy by
plastic behavior of central lead core
The most important design feature of the seismic isolation system is lengthening the
natural period of structures Therefore design period of structures or isolators is specified
in the first and then the appropriate properties of isolators are determined in the general
design of seismic isolation system
However most long-span bridges such as cable-stayed bridges have longer period
modes than short-span bridges due to their flexibility Therefore these bridges tend to
have a degree of the natural seismic isolation Furthermore these bridges have a lower
structural damping than general short-span bridges and exhibit very complex behavior in
which the vertical translational and torsional motions are often strongly coupled For
these reasons it is conceptually unacceptable for long-span bridges to use directly the
recommended design procedure and guidelines of LRB for short-span bridges and
buildings Therefore new design approach and guidelines are required to design LRB
because seismic characteristics of cable-stayed bridges are different from those of short-
span bridges and buildings The energy dissipation and damping effect of LRB are more
important than the shift of the natural period of structures in the cable-stayed bridges
which are different from buildings and short-span bridges
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 1 Introduction 3
12 Literature Review
The LRB was invented by W H Robinson in 1975 and has been applied to the
seismic isolation system of bridge structures in New Zealand since 1978 [2] The LRB is
excellent to shift the natural period of structures and to dissipate the earthquake energy
Furthermore this bearing offers a simple method of passive control and is relatively easy
and inexpensive to manufacture For these reasons the LRB has been widely investigated
and used for the seismic isolation system to reduce responses of buildings and short-span
bridges in many areas of the world
Many studies have been conducted for LRB in buildings [345] as well as short to
medium span highway bridges [67] and some design guidelines are suggested for
highway bridges [6] And procedures involved in analysis and design of seismic isolation
systems such as LRB are provided by Naeim and Kelly [10]
The comprehensive study of effectiveness of LRB for cable-stayed bridges is
investigated in recently Ali and Abdel-Ghaffar [1] investigated the effectiveness of
rubber bearing and LRB and they showed that earthquake-induced forces and vibrations
could be reduced by proper choice of properties and locations of these bearings This
reduction is obtained by the energy dissipation of central lead core in LRB and the
acceptable shear strength of LRB is recommended for seismically excited cable-stayed
bridges However the recommended value by Ali and Abdel-Ghaffar do not consider
characteristics of earthquake motions Park et al [89] presented the effectiveness of
hybrid control system based on LRB which is designed by recommended procedure of
Ali and Abdel-Ghaffar [1]
However there are few studies on procedures and guidelines to design LRB for
cable-stayed bridges Wesolowsky and Wilson [10] used a seismic isolation design
approach described by Naeim and Kelly [11] to control seismically excited cable-stayed
bridges with LRB This method applied for building structures begins with the
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 1 Introduction 4
specification of the effective period and design displacement of isolators in the first and
then iterate several steps to obtain design properties of isolators using the geometric
characteristics of bearings However the effective stiffness and damping usually depend
on the deformation of LRB Therefore the estimation of design displacement of bearing
is very important and is required the iterative works Generally the design displacement
is obtained by the response spectrum analysis that is an approximation approach in the
design method of bearing described by Naeim and Kelly [11] However it is difficult to
get the response spectrum since the behavior of cable-stayed bridges is very complex
compared with that of buildings and short-span bridges Therefore the time-history
analysis is required to obtain more appropriate results
13 Objectives and Scopes
The purpose of this study is to suggest the design procedure and guidelines for LRB
and to investigate the effectiveness of LRB to control seismic responses of cable-stayed
bridges Furthermore additional passive control device (ie viscous dampers) is
employed to improve the control performance
First the design index (DI) and procedure of LRB for seismically excited cable-
stayed bridges are proposed Important responses of cable-stayed bridge are reflected in
proposed DI The appropriate properties of LRB are selected when the proposed DI value
is minimized or converged for variation of properties of design parameters In the design
procedure important three parameters of LRB (ie elastic and plastic stiffness shear
strength of central lead core) are considered for design parameters The control
performance of designed LRB is compared with that of LRB designed by Wesolowsky
and Wilson approach [10] to verify the effectiveness of the proposed design method
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 1 Introduction 5
Second the sensitivity analyses of properties of LRB are conducted for different
characteristics of input earthquakes to verify the robustness of proposed design procedure
In this study peak ground acceleration (PGA) and dominant frequency of earthquakes are
considered since the behavior of the seismic isolation system is governed by not only
PGA but also frequency contents of earthquakes
Finally additional passive control system (VD) is designed and this damper is
employed in cable-stayed bridge to obtain the additional reduction of seismic responses
of bridge since some responses (ie shear at deck shear of the towers and deck
displacement) are not sufficiently controlled by only LRB
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
6
CHAPTER 2
PROPOSED DESIGN PROCEDURE OF LRB
21 LRB
211 Design Parameters of LRB
Typical earthquake accelerations have dominant periods of about 01 ~ 10 sec and
the general buildings and continuous bridges have fundamental period about 03 ~ 06 sec
[23] The basic concept of the seismic isolation system is lengthening the fundamental
period of the structures to outrange of period containing the large seismic energy of
earthquake motion by flexibility of isolators and dissipating the earthquake energy by
supplemental damping
Because the LRB offers a simple method of passive control and are relatively easy
and inexpensive to manufacture this bearing is widely employed for the seismic isolation
system for buildings and short-span bridges The LRB is composed of an elastomeric
bearing and a central lead plug as shown in figure 21 Therefore this bearing provides
structural support horizontal flexibility damping and restoring forces in a single unit
The force-displacement hysteresis curve of this bearing (elastic-plastic behavior) is
shown in figure 22 For weak winds and braking forces the LRB remains stiff due to the
central lead core However for strong winds and earthquakes this behaves like rubber
bearing because the central lead core is yielded In this figure Ke Kp and Keff are elastic
plastic and effective stiffness of LRB respectively Qy is shear strength of central lead
core Fy and Fu are yielding and ultimate strength of LRB Xy and Xd are yielding
displacement of central lead core and design displacement of LRB respectively
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 2 Proposed Design Procedure of LRB 7
Rubber
Lead Core Steel Lamination
Figure 21 Schematic of LRB
Fy
Fu
Qy
Kp
Keff
Xy Xd
Ke
Figure 22 Hysteretic curve of LRB
The LRB shifts the natural period of structures by flexibility of rubber and dissipates
the earthquake energy by plastic behavior of central lead core Therefore it is important
to combine the flexibility of rubber and size of central lead core appropriately to reduce
seismic forces and displacements of structures In other words the elastic and plastic
stiffness of LRB and the shear strength of central lead core are important design
parameters to design this bearing for the seismic isolation design
In the design of LRB for buildings and short-span bridges the main purpose is to
shift the natural period of structures to longer one Therefore the effective stiffness of
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 2 Proposed Design Procedure of LRB 8
LRB and design displacement at a target period are specified in the first Then the proper
elastic plastic stiffness and shear strength of LRB are determined using the geometric
characteristics of hysteric curve of LRB through several iteration steps [1011] Generally
the 5 of bridge weight carried by LRB is recommended as the shear strength of central
lead core to obtain additional damping effect of LRB in buildings and highway bridges
[6]
However most long-span bridges such as cable-stayed bridges tend to have a degree
of natural seismic isolation and have lower structural damping than general short-span
bridges Furthermore the structural behavior of these bridges is very complex Therefore
increase of damping effect is expected to be important issue to design the LRB for cable-
stayed bridges In other words the damping and energy dissipation effect of LRB may be
more important than the shift of the natural period of structures in the cable-stayed
bridges which are different from buildings and short-span bridges For these reasons the
design parameters related to these of LRB may be important for cable-stayed bridges
212 LRB Model The nonlinear behavior of LRB is described by Bouc-Wen model [1213] which is a
nonlinear differential equation This model represents the bilinear hysteric behavior
sufficiently The restoring force of LRB is formulated as equation (1) that is composed of
linear and nonlinear terms as
zXXXX yrrr eeLRB KααKf )(1)( minus+=amp (1)
where α are the elastic stiffness and the ratio of plastic-elastic stiffness of LRB rX
and rXamp are the relative displacements and velocities of nodes at which bearings are
installed respectively z are the yield displacement of central lead core and the
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 2 Proposed Design Procedure of LRB 9
dimensionless hysteretic component satisfying the following nonlinear first order
differential equation formulated as equation (2)
)(1 n1n zXzzXXX
z rrry
ampampampamp βγ minusminus=minus
iA (2)
where Ai γ β and n are dimensionless constants that affect the hysteretic behavior of
model Ai = n =1 and γ=β=05 are usually used to describe the bilinear curve of LRB and
these values are adopted in this study
Finally the equation describing the forces produced by LRB is formulated as
equation (3)
LRBftimes= LRBLRB GF (3)
where GLRB is the gain matrix to account for number and location of LRB
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 2 Proposed Design Procedure of LRB 10
22 Proposed Design Procedure
The objective of seismic isolation system such as LRB is to reduce the seismic
responses and keep the safety of structures Therefore it is a main purpose to design the
LRB that important seismic responses of cable-stayed bridges are minimized Because the
appropriate combination of flexibility and shear strength of LRB is important to reduce
responses of bridges it is essential to design the proper elastic-plastic stiffness and shear
strength of LRB
The proposed design procedure of LRB is based on the sensitivity analysis of
proposed DI for variation of properties of design parameters (ie Ke Kp and Qy) In this
study the DI is suggested considering five responses defined important issues related to
earthquake responses and behavior of cable-stayed bridges [14] as shown in equation (4)
These responses are base shear and overturning moment at tower supports (R1 and R3)
shear and bending moment at deck level of towers (R2 and R4) and longitudinal deck
displacement (R5) For variation of design parameters the DI and responses are obtained
In the sensitivity analysis controlled responses are normalized by the maximum response
of each response And then these controlled responses are normalized by the maximum
response
sum=
=5
1i maxi
i
RR
DI i=1hellip5 (4)
where Ri is i-th response and Rimax is maximum i-th response for variation of properties of
design parameters
The appropriate design properties of LRB are selected when the DI is minimized or
converged In other words the LRB is designed when five important responses are
minimized or converged The convergence condition is shown in equation (5)
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 2 Proposed Design Procedure of LRB 11
ε)(le
minus +
j
1jj
DIDIDI (5)
where DIj and DIj+1 is j-th and j+1-th DI value for variation of properties of design
parameter In this study the tolerance (ε) is selected as 001 considering computational
efficiency However designerrsquos judgment and experience are required in the choice of
this value
Using the proposed DI the design procedure of LRB for seismically excited cable-
stayed bridges is proposed as follows
Step 1 Choice of design input excitation (eg historical or artificial earthquakes)
Step 2 The proper Kp satisfied proposed design condition is selected for variation of
Kp (Qy and Ke Kp are assumed as recommended value)
Step 3 With Kp obtained in step 2 the proper Qy is selected for variation of Qy (Ke
Kp is assumed as recommended value)
Step 4 With Kp and Qy obtained in step 2 and 3 the proper Ke Kp is selected for
variation of Ke Kp
Step 5 Iterate step 2 3 and 4 until the designed properties of LRB are unchanged
Generally responses of structures tend to be more sensitive to variation of Qy and Kp
than to variations of Ke [2] therefore it is usually reasonable to select Qy and Kp ahead of
Ke to design LRB In this study Kp is determined in the first During the sensitivity
analysis of Kp properties of the other design parameters are assumed to generally
recommended value The Qy is used to 9 of deck weight carried by LRB recommended
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 2 Proposed Design Procedure of LRB 12
by Ali and Abdel-Ghaffar [1] And the Ke Kp of LRB is assumed to 10 [21]
After the proper Kp of LRB is selected Qy is obtained using the determined Kp and
assumed Ke Kp of LRB (10) Finally with the determined Kp and Qy of LRB the Ke Kp of
LRB is selected Generally design properties of LRB are obtained by second or third
iteration because responses and DI are not changed significantly as design parameters are
varied This is why assumed properties of LRB in the first step are sufficiently reasonable
values
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
13
CHAPTER 3
NUMERICAL EXAMPLE
31 Bridge Model
The bridge model used in this study is the benchmark cable-stayed bridge model
shown in figure 31 Dyke et al [15] developed the benchmark cable-stayed bridge model
and this model is three-dimensional linearized evaluation model that represents the
complex behavior of the full-scaled benchmark cable-stayed bridge This bridge was
developed to investigate the effectiveness of various control devices and strategies under
the coordination of the ASCE Task Committee on Benchmark Control Problems This
bridge is composed of two towers 128 cables and additional fourteen piers in the
approach bridge from the Illinois side It have a total length of 12058 m and total width
of 293 m The main span is 3506 m and the side spans are the 1427 m in length
Figure 31 Schematic of the Bill Emersion Memorial Bridge [15]
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 14
In the uncontrolled bridge the sixteen shock transmission devices are employed in
the connection between the tower and the deck These devices are installed to allow for
expansion of deck due to temperature changes However these devices are extremely stiff
under severe dynamic loads and thus behavior like rigid link
The bridge model resulting from the finite element method has total of 579 nodes
420 rigid links 162 beam elements 134 nodal masses and 128 cable elements The
stiffness matrices used in this linear bridge model are those of the structure determined
through a nonlinear static analysis corresponding to the deformed state of the bridge with
dead loads [16] Then static condensation model reduction scheme is applied to the full
model of the bridge to obtain a 419-DOF (degree of freedom) reduced-order model The
damping matrices are developed by assigning 3 of critical damping to each mode
which is consistent with assumptions made during the design of bridge
Because the bridge is assumed to be attached to bedrock the soil-structure
interaction is neglected The seismic responses of bridge model are solved by the
incremental equations of motion using the Newmark-β method [17] one of the popular
direct integration methods in combination with the pseudo force method [17]
A detailed description of benchmark control problem for cable-stayed bridges
including the bridges model and evaluation criteria can be found in Dyke et al [15]
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 15
32 Design and Seismic Performance of LRB
321 Design Earthquake Excitations In the design of LRB using the proposed procedure input excitations are selected in
the first However the choice of critical design ground excitations for structures is not an
easy task Several possible ground motions should be considered based on the earthquake
history of the site statistical data and other geological evidence In this study one
historical and one artificial earthquake are used as the design earthquakes for numerical
design example
The first design earthquake is historical El Centro earthquake (the North-South
component recorded at the Imperial Valley Irrigation District substation in El Centro
California during the Imperial Valley California earthquake of May 18 1940) El Centro
earthquake has the general energy distribution and is widely used for studies of seismic
hazards The original peak ground acceleration (PGA) of 0348 grsquos is scaled to 0360 grsquos
which is design PGA of Emerson Memorial Bridge Figure 32 shows the time history and
power spectral density of scaled El Centro earthquake
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15
Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensi
ty
a) Time history b) Power spectral density
Figure 32 Design earthquake excitation (Scaled El Centro earthquake)
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 16
The second design earthquake is artificial random excitation that is stationary
random process with a spectral density function defined by the Kanai-Tajimi spectrum
[18] stated as equations (6) and (7)
222
22
)(4])(1[])(41[
)(ggg
ggSωωζωω
ωωζω
+minus
+= S0 (6)
S0)14(
0302 +
=gg
g
ζπωζ (7)
where gζ and ωg are dominant damping coefficient and frequency in the site area and
S0 is the power spectral intensity respectively In this study gζ =03 and ωg =373 rads
are used to generate the artificial random excitation [19] The time history and power
spectral density of artificial random excitation are shown in figure 33 The PGA of
earthquake is also scaled to 036 grsquos
0 10 20 30 40 50Time (sec)
-4
-2
0
2
4
Acc
eler
atio
n (m
s2 )
0 5 10 15 20
Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
a) Time-history b) Power spectral density
Figure 33 Design earthquake excitation (Artificial random excitation)
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 17
In the design of LRB these design ground motions are applied in the longitudinal
direction of bridge and act at simultaneously at all supports
322 Design of LRB The LRB for cable-stayed bridge is designed by using the proposed design
procedure stated in chapter 22 The LRB is employed between the deck and pierbent
connection of bridge in longitudinal direction as shown in figure 31
In the design of LRB Kp of LRB is selected in the first and it is varied from 03W
Where W is deck weight supported by LRB and the value of W is shown in figure 34 Ali
and Abdel-Ghaffar [20] adopted the approach that properties of parameters are taken as a
portion of the superstructure weight supported by bearings The W is considered as the
lumped mass since the behavior of bridges is generally governed by the motions of
longitudinal direction And in the first design step the other properties of LRB are
assumed to conventionally recommended values Qy is assumed to 009W recommended
by Ali and Abdel-Ghaffar [1] and Ke Kp is assumed to 10 [21]
Pier 1 (Bent 1)
26400 KN
26400 KN26400 KN
26400 KN
7638 KN
7638 KN
7638 KN
7638 KN
Deck
Pier 2 Pier 3 Pier 4
Figure 34 Deck weight supported by LRB (Lumped mass)
After Kp of LRB is obtained Qy and Ke Kp of LRB are designed in order At this
time Qy and Ke Kp of LRB are varied from 003W and from 5 The proposed DI is
computed by considering five important responses of cable-stayed bridge [14] The peak
and norm responses of the bridge are reflected in DI for scaled El Centro earthquake and
artificial random excitation respectively
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 18
The properties of LRB are selected by proposed procedure and the tolerance (ε) is
selected to 001 considering economical efficiency The first and second design iteration
of LRB sensitivity of DI and responses for variation of design properties of LRB are
shown as figure from 35 to 38 for design earthquakes
Even though cable-stayed bridge is originally flexible structure and possess a natural
seismic isolation seismic responses of bridge can be reduced by installing the proper
LRB In general the base shear and moment at towers and bending moment at deck level
of towers are significantly reduced as the base isolation system is adopted However the
negative effect such as large displacement and shear at deck level of towers may be
induced by unsuitable properties of LRB The base shear and moment at towers are
always less than those of uncontrolled system in all properties of LRB and less sensitive
However the deck displacement is more sensitive to the variation of properties of LRB
than the other responses Therefore the deck displacement shear and moment at deck
level of towers are recommended as target responses to design LRB for cable-stayed
bridges
Generally as the properties of LRB are larger responses of cable-stayed bridge are
reduced However some responses and DI are increased or not reduced when excess
properties of each parameter are employed In other words there is not improvement of
seismic reduction in the excess stiffness and shear strength of LRB As the Kp and Qy are
increased the responses and reduction rates of responses are decreased Variation of
responses and DI for Ke Kp is generally less sensitive than the other design parameters
Therefore Kp and Qy are important parameters that may influence the behavior of cable-
stayed bridges among the design parameters because the plastic behavior and energy
dissipation of LRB is more important than the elastic behavior for seismically excited
cable-stayed bridges In the other hands the Ke Kp of LRB is not important parameter
relatively to design LRB Therefore Ke Kp of LRB is recommended to use the general
value (Ke Kp=9~11) and it is not necessary to consider seriously in the design of LRB for
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 19
seismically excited cable-stayed bridges Furthermore the peak and norm responses for
the scaled El Centro earthquake and the artificial random excitation varied with a similar
trend for each design parameters
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 20
03 04 05 06 07 08 09KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=08W Ke Kp =10)
5 6 7 8 9 10 11 12KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=08W Qy=011W)
Figure 35 Sensitivity of responses and DI under scaled El Centro earthquake (1st iteration)
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 21
03 04 05 06 07 08 09 10 11KpW
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=011W Ke Kp =12)
003 004 005 006 007 008 009 01 011 012 013QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=11W Ke Kp =12)
5 6 7 8 9 10KeKp
00
02
04
06
08
10
12
14
16
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
c) Ke Kp (Kp=11W Qy=013W)
Figure 36 Sensitivity of responses and DI under scaled El Centro earthquake (2nd iteration)
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 22
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =10)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =10)
5 6 7 8 9 10 11KeKp
00
05
10
15
20
25
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design index
c) Ke Kp (Kp=12W Qy=009W)
Figure 37 Sensitivity of responses and DI under artificial random excitation (1st iteration)
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 23
03 04 05 06 07 08 09 10 11 12KpW
00
05
10
15
20
25
30
35
Eva
luat
ion
crite
ria
25
30
35
40
45
50
Design
index
a) Kp (Qy=009W Ke Kp =11)
003 004 005 006 007 008 009 010 011QyW
00
05
10
15
20
25
30
Eval
uatio
n cr
iteri
a
25
30
35
40
45
50
Design
index
Base shearDeck shearBase momDeck momDeck dispDesign Index
b) Qy (Kp=12W Ke Kp =11)
Figure 38 Sensitivity of responses and DI under artificial random excitation (2nd iteration)
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 24
Parameter
Through the proposed design procedure appropriate properties of LRB for cable-
stayed bridges are obtained for two design earthquakes In the design of LRB design
properties of LRB are obtained by second or third iteration The design result is shown in
table 31 The LRB I and II are designed for the scaled El Centro earthquake and artificial
random excitation respectively
Table 31 Designed properties of LRB
Kp (tfm) Qy (tf) Ke Kp LRB I 14W 013W 10 LRB II 12W 009W 11
The determined Kp and Qy for cable-stayed bridge are generally larger values than
those for general buildings and short-span bridges For example the Qy obtained by
proposed procedure is larger than those for general buildings and short-span bridges (ie
005W) Therefore the LRB used in seismically excited cable-stayed bridges requires
stiffer rubber and bigger central lead core size than that in general buildings and short-
span bridges In other words the damping and energy dissipation effect of LRB are more
important than the shift of the natural period of structures for seismically excited cable-
stayed bridges
323 Control Performance of Designed LRB The controlled responses and reduction rates of cable-stayed bridges are shown in
table 32 for design earthquakes The controlled responses are normalized maximum
uncontrolled responses in several parts This shows that the use of LRB offers a potential
advantage for the seismic design of cable-stayed bridge Specially the base shear and
moment at towers and bending moment at the deck level of towers are appropriately
reduced in the bridges installed LRB However the shear at deck level of towers is not
reduced efficiently compared with the other responses The peak responses under scaled
LRB
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 25
El Centro earthquake and the norm responses under artificial random excitation show
similar reduction trends in the cable-stayed bridge with LRB I and II
Table 32 Controlled responses of bridge for design earthquakes
Responses LRB I LRB II R1 base shear at towers (kN) 165times104 (0327) 217times103 (0545) R2 shear at deck level of towers (kN) 450times103 (0932) 526times102 (0836) R3 base mom at towers (kNm) 319times105 (0300) 369times104 (0448) R4 mom at deck level of towers (kNm) 868times104 (0381) 120times104 (0675) R5 longitudinal deck displacement (m) 989times10-2 (0980) 119times10-2 (1415) ( ) Controlled responseUncontrolled responses
The control performance under other historical earthquakes is investigated to verify
the effectiveness of LRB designed by proposed method Three historical earthquakes
provided in the benchmark problems are considered in this study i) 1940 El Centro NS
recorded at Imperial Valley (0348grsquos) ii) 1985 Mexico City recorded at Galeta de
Campos (0143grsquos) iii) 1999 Turkey Gebze NS recorded at Kocaeli (0265grsquos) The time
history of three earthquakes is shown in figure 39
0 25 50 75 100Time (sec)
-4
-2
0
2
4
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-2
-1
0
1
2
Acce
lera
tion
(ms
2 )
0 25 50 75 100Time (sec)
-3
-15
0
15
3
Acce
lera
tion
(ms
2 )
a) El Centro earthquake b) Mexico City earthquake c) Gebze earthquake
Figure 39 Time history of three earthquakes
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 26
The performance of cable-stayed bridge with LRB I and II is verified by adopting
eleven evaluation criteria provided in benchmark problem [15] The first six evaluation
criteria consider the ability of the controller to reduce peak responses
=max
ob
biti
GebzeMexicoCityElCentro F
)t(FmaxJ max1
=max
od
diti
GebzeMexicoCityElCentro F
)t(FmaxJ max2
=maxob
biti
GebzeMexicoCityElCentro M
)t(MmaxJ max3
=maxod
diti
GebzeMexicoCityElCentro M
)t(MmaxJ max4
minus
=oi
oiai
ti
GebzeMexicoCityElCentro T
T)t(TmaxJ max5
=ob
bi
ti
GebzeMexicoCityElCentro x
)t(xmaxJ max6
(8-13)
where )t(Fbi is the base shear at the ith tower maxobF is the maximum uncontrolled
base shear )t(Fdi is the shear at deck level in the ith tower maxodF is the maximum
uncontrolled shear at deck level of towers )t(M bi is the base moment at the ith towers
maxobM is the maximum uncontrolled moment at the base shear of the towers )t(M di is
the moment at the deck level in the ith tower maxodM is the maximum uncontrolled
moment at the deck level of towers oiT is the nominal pretension in the ith cable aiT is
the actual tension in the cable and obx is the maximum of the uncontrolled deck
response at these locations
The second five evaluation criteria consider normed (ie RMS) responses over the
entire simulation time as follows
=)t(F
)t(FmaxJ
ob
bii
GebzeMexicoCityElCentromax7
=)t(F
)t(FmaxJ
od
dii
GebzeMexicoCityElCentromax8
=)t(M
)t(MmaxJ
ob
bii
GebzeMexicoCityElCentromax9
=)t(M
)t(MmaxJ
od
dii
GebzeMexicoCityElCentromax10
minus
=oi
oiai
i
GebzeMexicoCityElCentro T
T)t(TmaxJ max11
(14-18)
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 27
where )t(Fob is the maximum RMS uncontrolled base shear of towers )t(Fod
is the maximum RMS uncontrolled shear at the deck level of towers )t(M ob is the
maximum RMS uncontrolled overturning moment of towers )t(M od is the maximum
RMS uncontrolled moment at the deck level of towers The normed value of the
responses denoted sdot is defined as
(19)
where tf is defined as the time required for the response to attenuate
To compare the results The LRB is designed using method that Wesolowsky and
Wilson [10] are presented for cable-stayed bridges This design procedure is based on
Naeim and Kelly approach [11] that specify a design displacement and period in the first
Wesolowsky and Wilson specify the effective period of isolator considering the average
maximum deck displacement (∆) and base shear (S) of the non-isolated bridge for several
earthquakes In this study the maximum displacement (Xd) and ultimate strength (Fu) of
isolators are specified to 2∆ and 025S respectively The responses under six earthquakes
are considered (El Centro Mexico City Gebze Northridge Kobe earthquake and
artificial random excitation used in chapter 32) to obtain the average maximum deck
displacement and base shear of the non-isolated bridge The properties of designed LRB
are shown in table 33
Table 33 Designed properties of LRB (Wesolowsky and Wilson-WW bearing)
Kp (tfm) Qy (tf) Ke Kp 0627W 0075W 10
The uncontrolled maximum responses under three earthquakes for performance
int sdot=sdotft
f
dtt 0
2)(1
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 28
criteria are shown in table 34 and the simulation results are shown in tables from 35 to
37 and the time history of responses considered in DI are shown in figure from 310 to
312 for three historical earthquakes
The result shows that most responses of cable-stayed bridge are reduced sufficiently
by proper LRB For example the base shear and moment at towers and moment at deck
level of towers are remarkably diminished compared with uncontrolled system And the
deck displacement of LRB I is smaller than that of LRB II However the shear at deck
level of towers and deck displacement of LRB I and II are not be reduced sufficiently
Nevertheless these responses are generally smaller than those of WW bearing
For El Centro earthquake all controlled responses are smaller than uncontrolled
responses except the deck displacement of LRB II The LRB designed by proposed
method shows a better performance than that by Wesolowsky and Wilson method For
example the shear at deck level of towers is reduced about 133 (LRB I) and 92
(LRB II) the deck displacement is reduced about 376 (LRB I) and 163 (LRB II)
compared with WW bearing For Mexico City earthquake however LRB I and II are not
efficient compared with WW bearing except the deck displacement This is why the WW
bearing is designed more flexible than LRB I and II Therefore the plastic behavior of
LRB is not adequately occurred in the LRB I and II because the Mexico City earthquake is
relatively small earthquake (figure 313) Nevertheless the seismic performance of
designed LRB is sufficiently good compared with uncontrolled system For Gebze
earthquake the performance of LRB designed by proposed method is also acceptable and
better than that of WW bearing The shear at the deck level of towers and deck
displacement increase a little compared with uncontrolled responses Nevertheless these
responses did not increase seriously compared with those of WW bearing The deck
displacement of WW bearing is a little large (about 17cm) because this LRB is designed
more flexible than LRB I and LRB II
These results show that seismic responses of cable-stayed bridge can be reduced
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 29
sufficiently by only proper LRB However the deck displacement for two earthquakes is
relatively larger than that of uncontrolled system Nevertheless the increased deck
displacement (under 10cm) is still less than the allowable displacement that the deck will
be disintegrated from its end connections (30cm [14])
Table 34 Uncontrolled maximum responses for performance criteria
Evaluation Criteria El Centro Mexico City Gebze
Max base shear at towers (kN) 4878times104 1118times104 3085times104
Max shear at deck level of towers (kN) 4671times103 1525times103 3150times103 Max base mom at towers (kNm) 1027times106 1982times105 6978times105 Max mom at deck level of towers (kNm) 2205times105 8670times104 1093times105 Max longitudinal deck displacement (m) 9758times10-2 2432times10-2 7192times10-2 Norm base shear at towers (kN) 5265times103 1474times103 2609times103 Norm shear at deck level of towers (kN) 4561times102 1889times102 2312times102 Norm base mom at towers (kNm) 1163times105 3147times104 5779times104 Norm mom at deck level of towers (kNm) 2013times104 6931times103 9507times103
Table 35 Performance of designed LRB under El Centro earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03194 03116 03311 J2 max shear at deck level of towers 08821 09247 10179 J3 max base mom at towers 03024 02931 03003 J4 max mom at deck level of towers 03621 04797 06047 J5 max deviation of cable-tension 01506 01673 01898 J6 max longitudinal deck displacement 08760 11764 14048 J7 norm base shear at towers 02577 02397 02347 J8 norm shear at deck level of towers 07505 08134 09829 J9 norm base mom at towers 02756 02540 02532 J10 norm mom at deck level of towers 04007 04722 05855 J11 norm deviation of cable tension 00164 00144 00176
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 30
Table 36 Performance of designed LRB under Mexico City earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 07244 06568 05725 J2 max shear at deck level of towers 10853 10847 10403 J3 max base mom at towers 07163 06344 06134 J4 max mom at deck level of towers 03882 03904 03853 J5 max deviation of cable-tension 07160 06012 00421 J6 max longitudinal deck displacement 14439 15439 19795 J7 norm base shear at towers 05520 05067 05090 J8 norm shear at deck level of towers 07922 07600 08794 J9 norm base mom at towers 05729 05132 05314 J10 norm mom at deck level of towers 05077 05051 05051 J11 norm deviation of cable tension 00101 00830 00055
Table 37 Performance of designed LRB under Gebze earthquake
Evaluation Criteria LRB I LRB II WW bearing
J1 max base shear at towers 03432 04140 04035 J2 max shear at deck level of towers 09690 10479 12565 J3 max base mom at towers 03540 04245 04249 J4 max mom at deck level of towers 05851 06673 07526 J5 max deviation of cable-tension 00749 00987 01398 J6 max longitudinal deck displacement 09986 14947 23206 J7 norm base shear at towers 03439 03335 03491 J8 norm shear at deck level of towers 08344 09279 12867 J9 norm base mom at towers 03811 03762 04338 J10 norm mom at deck level of towers 05155 06147 09597 J11 norm deviation of cable tension 00089 00077 00103
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 31
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Bas
e sh
ear
at to
wer
s (times
104
kN)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
0 20 40 60 80 100Time (sec)
-50
-25
00
25
50
Shea
r at
dec
k le
vel o
f tow
ers (times1
03 k
N)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e m
omen
t at t
ower
s (times1
06 k
N m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Base
mom
ent a
t tow
ers
(times10
6 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
0 20 40 60 80 100Time (sec)
-24
-16
-08
00
08
16
24
Mom
ent a
t dec
k le
vel o
f tow
ers (times1
05 k
N m
)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Dec
k di
spla
cem
ent (times1
0-2
m)
0 20 40 60 80 100Time (sec)
-12
-8
-4
0
4
8
12
Dec
k di
spla
cem
ent (times1
0-2
m)
e) Deck displacement
Figure 310 Time history responses of cable-stayed bridge under El Centro earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 32
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Bas
e sh
ear
at to
wer
s (10
4 K
N)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-2
-1
0
1
2
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Base
mom
ent a
t tow
ers (
105
kN m
)
UncontrolControl
0 20 40 60 80 100Time (sec)
-20
-10
00
10
20
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
0 20 40 60 80 100Time (sec)
-90
-60
-30
00
30
60
90
Mom
ent a
t dec
k le
vel o
f tow
ers (
104 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-40
-20
00
20
40
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 311 Time history responses of cable-stayed bridge under Mexico City earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 33
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
0 20 40 60 80 100Time (sec)
-30
-20
-10
00
10
20
30
Bas
e sh
ear
at to
wer
s (10
4 kN
)
a) Base shear at towers
0 20 40 60 80 100Time (sec)
-3
-2
-1
0
1
2
3
Shea
r at
dec
k le
vel o
f tow
ers (
103
kN)
0 20 40 60 80 100Time (sec)
-36
-24
-12
0
12
24
36
Shea
r at
dec
k le
vel o
f tow
ers
(10
3 kN
)
b) Shear at deck level of towers
0 20 40 60 80 100Time (sec)
-70
-35
00
35
70
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Bas
e m
omen
t at t
ower
s (10
5 kN
m)
UncontrolControl
c) Base moment at towers
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
0 20 40 60 80 100Time (sec)
-12
-08
-04
00
04
08
12
Mom
ent a
t dec
k le
vel o
f tow
ers (
105 kN
m)
d) Moment at deck level of towers
0 20 40 60 80 100Time (sec)
-80
-40
00
40
80
Dec
k di
spla
cem
ent (
10-2
m)
0 20 40 60 80 100Time (sec)
-120
-80
-40
00
40
80
120
Dec
k di
spla
cem
ent (
10-2
m)
e) Deck displacement
Figure 312 Time history responses of cable-stayed bridge under Gebze earthquake
LRB I
LRB I
LRB I
LRB I
LRB I
LRB II
LRB II
LRB II
LRB II
LRB II
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 34
The restoring force of LRB installed at pier 2 is shown in figure from 313 The
control forces and energy dissipation of LRB are sufficiently occured in El Centro and
Gebze earthquake However in the Mexico City earthquake those of LRB are relatively
smaller than those of LRB in El Centro and Gebze earthquake This is why the plastic
behavior of LRB is not adequately occurred in designed LRB because this earthquake is
relatively small Nevertheless proper restoring forces are generated during this
earthquake and seismic responses of cable-stayed bridge are reduced
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-8000
-4000
0
4000
8000
Res
tori
ng fo
rce
(kN
)
a) Restoring force of LRB under El Centro Earthquake
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-4 -2 0 2 4Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
b) Restoring force of LRB under Mexico City Earthquake (continuded)
LRB I LRB II
LRB I LRB II
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 35
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
-10 -5 0 5 10Deformation (cm)
-6000
-4000
-2000
0
2000
4000
6000
Res
tori
ng fo
rce
(kN
)
c) Restoring force of LRB under Gebze Earthquake
Figure 313 Restoring force of LRB under three earthquakes
Results in this chapter indicate that seismic responses of cable-stayed bridges are
controlled sufficiently by only appropriate LRB And the LRB used in seismically excited
cable-stayed bridges requires stiffer rubber and bigger central lead core size than that in
general buildings and short-span bridges due to its flexibility and low structural damping
In other words the damping and energy dissipation effect of LRB is more important than
the shift of the natural period of structures for seismically excited cable-stayed bridges
LRB I LRB II
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 36
33 Effect of Characteristics of Earthquakes
331 Effect of Frequency Contents of Earthquakes The earthquakes are characterized by PGA frequency contents and duration of
earthquakes Therefore in the design of seismic isolator such as LRB the PGA as well as
the frequency contents of earthquakes may be important parameter and the performance
of isolator is affected by these characteristics of earthquakes The characteristics of
earthquakes may be estimated by the earthquake history statistical and geological data of
site However different earthquakes that not considered in design of seismic isolation
system may be excited in the design structures
In this chapter LRB is designed using proposed design procedure for several
earthquakes which have the different frequency contents to investigate the effect of
frequency contents of earthquakes Furthermore the variation of performance of designed
LRB is investigated to verify the robustness of designed LRB for different frequency
contents of earthquakes Three earthquakes used in chapter 323 are considered as design
earthquakes and the PGA of three earthquakes is scaled to 036 grsquos in order to exclude the
effect of the PGA of earthquakes The power spectral density of earthquakes is shown in
figure 314
0 5 10 15Frequency (Hz)
0
2
4
6
8
10
Pow
er sp
ectr
al d
ensit
y
0 5 10 15Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
al d
ensi
ty
0 5 10 15Frequency (Hz)
0
4
8
12
16
Pow
er sp
ectr
al d
ensi
ty
a) Scaled El Centro b) Scaled Mexico City c) Scaled Gebze
Figure 314 Power spectral density of three earthquakes
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 37
Table 38 shows properties of LRB designed by the proposed design method under
different design earthquakes The LRBEL LRBME and LRBGE are designed under scaled El
Centro Mexico City and Gebze earthquake respectively
Table 38 Designed properties of LRB for different frequency contents
Frequency (Hz) Kp (tfm) Qy (tf) Ke Kp LRBEL 15 14W 013W 10 LRBME 05 11W 012W 11 LRBGE 20 14W 016W 9
As can be seen in Table 38 Kp and Qy are more sensitive to the frequency contents
of earthquake than Ke Kp of LRB For Mexico City earthquake the earthquake energy is
concentrated in low frequency (about 05Hz) the flexible LRB is required In other words
the flexible Kp and small Qy are necessary to Mexico City earthquake On the other hands
for El Centro and Gebze earthquake the earthquake energy is concentrated in relatively
high frequency (about 15Hz and 20Hz) the stiff LRB is required compared with that for
Mexico City earthquake This indicates that as the frequency contents of earthquake are
concentrated in higher range stiffer LRB is needed However the Ke Kp of LRB is not
changed seriously by the frequency contents of earthquakes
Because the seismic isolation system such as LRB may have inadaptability to the
uncertainty of earthquake the performance of designed LRB is verified for the other
earthquakes which have different frequency contents The performance and variation of
performance of designed LRB are shown in table from 39 to 311 under the other
earthquakes that have different frequency contents Generally responses of cable-stayed
bridge with LRB designed by proposed design procedure are not varied seriously for
different design earthquakes However the deck displacement is more sensitive than the
other responses For example in table 310 the deck displacement of bridge with LRBME
is increased about 462(about 156cm) Nevertheless the control performance of
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 38
designed LRB using target design earthquake is not varied significantly if other
earthquakes that have different frequency contents are excited to cable-stayed bridge
Table 39 Performance of LRB for different frequency contents
under scaled El Centro earthquake
Evaluation Criteria LRBEL LRBME LRBGE
J1 max base shear at towers 0327(0) 0331(12) 0326(-03) J2 max shear at deck level of towers 0932(0) 0949(18) 0916(-17) J3 max base mom at towers 0301(0) 0298(-10) 0314(43) J4 max mom at deck level of towers 0381(0) 0415(89) 0389(21) J5 max deviation of cable-tension 0163(0) 0175(74) 0154(-55) J6 max longitudinal deck displacement 0980(0) 1077(99) 0989(09) J7 norm base shear at towers 0261(0) 0255(-23) 0275(54) J8 norm shear at deck level of towers 0800(0) 0830(38) 0808(10) J9 norm base mom at towers 0279(0) 0271(-29) 0299(72) J10 norm mom at deck level of towers 0389(0) 0426(95) 0382(-18) J11 norm deviation of cable tension 0016(0) 0015(-63) 0017(63) ( ) Variation of performance()
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 39
Table 310 Performance of LRB for different frequency contents
under scaled Mexico City earthquake
Evaluation Criteria LRBME LRBEL LRBGE
J1 max base shear at towers 0573(0) 0594(37) 0616(75) J2 max shear at deck level of towers 1070(0) 1066(-04) 1081(10) J3 max base mom at towers 0628(0) 0563(-104) 0587(-65) J4 max mom at deck level of towers 0417(0) 0398(-46) 0387(-72) J5 max deviation of cable-tension 0119(0) 0123(34) 0134(123) J6 max longitudinal deck displacement 1917(0) 1607(-162) 1601(-165) J7 norm base shear at towers 0465(0) 0469(09) 0484(41) J8 norm shear at deck level of towers 0801(0) 0771(-37) 0759(-52) J9 norm base mom at towers 0460(0) 0466(13) 0482(48) J10 norm mom at deck level of towers 0542(0) 0501(-76) 0499(-79) J11 norm deviation of cable tension 0014(0) 0016(114) 0017(214) ( ) Variation of performance()
Table 311 Performance of LRB for different frequency contents
under scaled Gebze earthquake (ControlUncontrol)
Evaluation Criteria LRBGE LRBEL LRBME
J1 max base shear at towers 0372(0) 0398(70) 0414(113) J2 max shear at deck level of towers 0960(0) 1028(71) 1073(118) J3 max base mom at towers 0376(0) 0415(104) 0425(130) J4 max mom at deck level of towers 0583(0) 0638(94) 0684(173) J5 max deviation of cable-tension 0104(0) 0122(173) 0142(365) J6 max longitudinal deck displacement 1097(0) 1314(198) 1604(462) J7 norm base shear at towers 0338(0) 0333(-15) 0335(-09) J8 norm shear at deck level of towers 0844(0) 0872(33) 0975(155) J9 norm base mom at towers 0374(0) 0373(-03) 0381(19) J10 norm mom at deck level of towers 0535(0) 0560(47) 0651(217) J11 norm deviation of cable tension 0011(0) 0010(-91) 0011(0) ( ) Variation of performance()
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 40
332 Effect of PGA of Earthquakes In this chapter the effect of PGA of earthquakes is investigated to design LRB The
LRB is designed using proposed design procedure for several earthquakes which have
different PGA
The El Centro earthquake and the artificial random excitation are employed as
design earthquakes In the El Centro earthquake the PGA of earthquake is scaled to one
and a half of original PGA (0348grsquos and 0174grsquos) The PGA is scaled to 036grsquos and
018grsquos in the artificial random excitation case
Table 312 shows the design results of LRB under different PGA of earthquake The
LRB10EL and LRB05El are designed under 10 and 05-scaled El Centro earthquake and the
LRBAI and LRBAII are designed under 036 grsquos and 018 grsquos scaled-artificial random
excitation respectively
Table 312 Designed properties of LRB for different PGA
PGA (grsquos) Kp (tfm) Qy (tf) Ke Kp LRB10EL 0348 14W 013W 9 LRB05El 0174 10W 008W 8 LRBAI 0360 12W 009W 11 LRBAII 0180 13W 006W 12
The design result shows that the shear strength of central lead core is depend on the
PGA of two design earthquakes That is as the PGA of earthquakes increases the larger
Qy of LRB is required and thus the larger energy dissipation capacity is necessary
However Kp of LRB is depended or not on the PGA of earthquake and Ke Kp of LRB is
not sensitive to this earthquake characteristic
The variation of performance of each designed LRB is simulated for other
earthquakes which have different PGA of earthquake and results are shown in table from
313 to 316
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 41
When the relatively weak earthquake is applied to bridges with LRB designed by
strong earthquakes the control effect of LRB is reduced in responses related to base of
towers (ie base shear and moment of towers) On the other hand that of LRB increases
in responses related to deck level of towers (ie shear and moment at deck level of towers
and deck displacement) This is why LRB designed under strong earthquakes have
relatively large shear strength therefore the plastic behavior of central lead core is not
happened sufficiently and thus the effect of base isolation is not generated sufficiently In
this reason responses related to base of towers increase On the other hands the control
effect of LRB decreases in responses related to deck level of tower because designed
LRB is relatively flexible
Performance of designed LRB is not varied seriously for different PGA of
earthquakes However the deck displacement and deviation of cable tension relatively
increase by this earthquake characteristic For example in table 313 the effect of
LRB05EL for deck displacement is worsened about 443 (about 128 cm) under 10-
scaled El Centro earthquake and that of LRBAII for deck displacement is did about 614
(117cm) under 036 grsquos scaled artificial random excitation And that for deviation of
cable tension is also worsened However the increased deck displacement is still less than
the allowable displacement (30 cm [14]) and the deviation of cable tension in the cable is
remaining within allowable value [15]
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 42
Table 313 Performance of LRB for different PGA of earthquakes
under 10 scaled El Centro earthquake
Evaluation Criteria LRB10EL LRB05EL
J1 max base shear at towers 0318(0) 0312(-17) J2 max shear at deck level of towers 0887(0) 0996(123) J3 max base mom at towers 0304(0) 0300(-12) J4 max mom at deck level of towers 0365(0) 0551(511) J5 max deviation of cable-tension 0148(0) 0175(181) J6 max longitudinal deck displacement 0907(0) 1309(443) J7 norm base shear at towers 0259(0) 0238(-80) J8 norm shear at deck level of towers 0762(0) 0908(191) J9 norm base mom at towers 0278(0) 0254(-86) J10 norm mom at deck level of towers 0399(0) 0512(284) J11 norm deviation of cable tension 0015(0) 0015(-64) ( ) Variation of performance ()
Table 314 Performance of LRB for different PGA of earthquakes
under 05 scaled El Centro earthquake
Evaluation Criteria LRB05EL LRB10EL
J1 max base shear at towers 0303(0) 0347(147) J2 max shear at deck level of towers 0913(0) 0810(-113) J3 max base mom at towers 0303(0) 0368(216) J4 max mom at deck level of towers 0398(0) 0360(-96) J5 max deviation of cable-tension 0072(0) 0076(66) J6 max longitudinal deck displacement 1129(0) 0977(-135) J7 norm base shear at towers 0278(0) 0325(172) J8 norm shear at deck level of towers 0846(0) 0777(-82) J9 norm base mom at towers 0307(0) 0360(174) J10 norm mom at deck level of towers 0414(0) 0396(-42) J11 norm deviation of cable tension 0068(0) 0012(806) ( ) Variation of performance ()
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 43
Table 315 Performance of LRB for different PGA of earthquakes
under 036grsquos scaled artificial random excitation
Evaluation Criteria LRBAI LRBAII
J1 max base shear at towers 0662(0) 0660(-02) J2 max shear at deck level of towers 0785(0) 0780(-06) J3 max base mom at towers 0518(0) 0575(111) J4 max mom at deck level of towers 0678(0) 0601(-113) J5 max deviation of cable-tension 0098(0) 0063(-358) J6 max longitudinal deck displacement 1929(0) 1170(-393) J7 norm base shear at towers 0545(0) 0595(94) J8 norm shear at deck level of towers 0836(0) 0856(24) J9 norm base mom at towers 0448(0) 0524(168) J10 norm mom at deck level of towers 0676(0) 0674(-03) J11 norm deviation of cable tension 1369(0) 0008(-380) ( ) Variation of performance ()
Table 316 Performance of LRB for different PGA of earthquakes
under 018grsquos scaled artificial random excitation
Evaluation Criteria LRBAII LRBAI
J1 max base shear at towers 0638(0) 0657(30) J2 max shear at deck level of towers 0776(0) 0770(-079) J3 max base mom at towers 0534(0) 0545(20) J4 max mom at deck level of towers 0651(0) 0723(111) J5 max deviation of cable-tension 0059(0) 0099(689) J6 max longitudinal deck displacement 1389(0) 2240(614) J7 norm base shear at towers 0555(0) 0522(-60) J8 norm shear at deck level of towers 0826(0) 0823(-04) J9 norm base mom at towers 0465(0) 0438(-58) J10 norm mom at deck level of towers 0656(0) 0691(54) J11 norm deviation of cable tension 0074(0) 0013(804) ( ) Variation of performance ()
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 44
These results in this chapter show that the appropriate properties and performance of
LRB are affected by the characteristics of earthquakes However most of responses
except deck displacement and the deviation of cable tension the performance of LRB
designed by proposed procedure is not varied significantly for characteristics of input
earthquakes Furthermore even though these two responses are varied for different
characteristics of earthquake these are also remaining allowable range
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 45
34 VD for Additional Passive Control System
341 Design of VD Using the proposed design procedure the LRB is designed for seismically excited
cable-stayed bridges and the effectiveness of designed LRB is verified in chapters 32
and 33 The most responses of a seismically excited cable-stayed bridge are controlled
sufficiently using the designed LRB However some responses (ie shear at the deck
levels of towers and deck displacement) are similar or larger than those of uncontrolled
system Even though these responses may not be problem for safety and serviceability of
bridges the reduction of responses is required
It is hard to reduce these two responses adopting only LRB without increase of the
other responses as seen in figure from 35 to 38 Moreover if the LRB is inordinately
stiff the yield of central lead core is not happened and thus the energy dissipation effect
of plastic behavior of LRB is not expected Therefore in this chapter other passive
control system is employed to obtain the additional reduction of seismic responses of
cable-stayed bridge Because the energy dissipation of control devices are more important
than the period shift of structure for cable-stayed bridge the VD is considered as
additional passive control device and installed between the deck and pier connection of
bridge in longitudinal direction
In this study the capacity of VD is considered as 1000 kN and maximum velocity of
these devices is assumed to 1ms which is identical velocity requirement of active device
in the benchmark control problem The restoring force of VD is modeled as equation (19)
mVD VF C= (19)
where C is damping coefficient and m is velocity exponent m=03~19 is usually used in
most steel structure for seismic protection [22] In this study m=10 is used (ie linear
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 46
dampers)
In the design of VD only two responses (ie shear at the deck levels of towers and
deck displacement) are considered In addition the constraint condition stated as equation
(20) is applied That is the variation of the other three responses can not be increased
over 10 of originally controlled responses
11LRB
VDLRBResponse le
+ (20)
In the first four dampers are employed to tower and deck connection (pier 2 3) and
the variation of responses are obtained by time history analysis Next other four dampers
are employed to bent 1 and pier 4 And dampers are installed until two target responses
are increased or converged or the constraint condition is not satisfied
The results are shown figure 315 The number of dampers is selected 16 since the
variation of base shear at towers under Gebze earthquake is not satisfied the constraint
condition However the shear at the deck level of towers and deck displacement is more
reduced as the VD is applied to cable-stayed bridge and the shear at deck level of towers
is more reduced when dampers are employed in pier 2 and 3 than in bent 1 and pier 4
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
a) The variation of responses under El Centro earthquake (continuded)
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 47
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400V
aria
tion
of r
espo
nse
()
Base shearDeck shearBase momDeck momDeck disp
b) The variation of responses under Mexico City earthquake
0 4 8 12 16 20Number of dampers
-400
-300
-200
-100
00
100
200
300
400
Var
iatio
n of
res
pons
e (
)
c) The variation of responses under Gebze earthquake
Figure 315 Design of VD
342 Control Performance of Designed LRB with VD The selected VD is applied to cable-stayed bridges with designed LRB to obtain
additional reduction of responses which are not reduced sufficiently in the LRB installed
bridge The result is shown in table 317 Most of responses of cable-stayed bridge more
decrease as the VD is added to LRB Reduction of shear at the deck level of towers and
deck displacement is larger than that of the other responses However the deviation of
cable-tension is increased a little as the VD is added to LRB Nevertheless this is
remaining within a recommended range of allowable values [15]
This result shows that the seismic responses of cable-stayed bridges can be
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 3 Numerical Example 48
adequately controlled by only passive control system
Table 317 Additional reduction of responses with LRB and VD
(LRB I+VD LRB I)
Evaluation Criteria El Centro Mexico City Gebze J1 max base shear at towers 0998 0906 1093 J2 max shear at deck level of towers 0804 0809 0881 J3 max base mom at towers 0989 0877 0976 J4 max mom at deck level of towers 0819 0878 0834 J5 max deviation of cable-tension 1094 0973 1194 J6 max longitudinal deck displacement 0865 0649 0786 J7 norm base shear at towers 0946 0882 0941 J8 norm shear at deck level of towers 0877 0837 0896 J9 norm base mom at towers 0955 0865 0936 J10 norm mom at deck level of towers 0991 0899 0929 J11 norm deviation of cable tension 1082 0881 0916
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
49
CHAPTER 4
CONCLUSIONS
In this study the design procedure and guidelines for LRB are proposed for
seismically excited cable-stayed bridges and the effect of LRB designed by proposed
procedure is investigated Conclusions of this study are summarized as follows
First from the design result of LRB several design feature of LRB in cable-stayed
bridges are identified The LRB used in cable-stayed bridges requires stiffer rubber and
bigger central lead core size than that in general buildings and short-span bridges due to
their flexibility and low structural damping In other words the damping and energy
dissipation effect of LRB is more important than the shift of the natural period of
structures for seismically excited cable-stayed bridges As the properties of LRB are
larger the responses of cable-stayed bridge are reduced However there is not
improvement of performance of LRB in the excess stiffness and shear strength of LRB
Among the design parameters of LRB plastic stiffness and shear strength of central lead
core are important design parameters because the plastic behavior and energy dissipation
of LRB is more important than the elastic behavior for seismically excited cable-stayed
bridges Furthermore the seismic responses of cable-stayed bridges can be reduced
sufficiently by only LRB designed with proposed procedure
Second proper properties and performance of LRB are affected by the
characteristics of earthquakes (ie frequency contents and PGA of earthquake) As the
frequency contents of earthquake are concentrated in higher range stiffer LRB is
necessary and as the PGA of earthquake increases the larger shear strength of LRB is
required However the ratio of elastic-plastic stiffness of LRB is not sensitive to these
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Chapter 4 Conclusions 50
characteristics of earthquakes Most of responses except deck displacement and the
deviation of cable tension the performance of LRB designed by proposed procedure is
not changed significantly for different characteristic of earthquakes However these two
responses are also remaining allowable range though the performance of designed LRB is
worsened
Third the most responses of cable-stayed bridge are controlled sufficiently using the
designed LRB However some responses (ie shear at deck level of towers and deck
displacement) are similar or larger than those of uncontrolled system Even though these
responses may not be problem for bridges these responses are more reduced by
additional damping devices such as viscous damper
Finally the seismic responses of cable-stayed bridges are adequately controlled by
only appropriated design passive control system
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
요 약 문
지진 하중을 받는 사장교를 위한 수동 제어 장치의 설계
본 논문에서는 지진 하중을 받는 사장교를 위한 납고무 받침의 설계 방
법 및 기준을 제안하였고 제안된 방법에 의해 설계된 납고무 받침의 면진
성능을 확인하였다 또한 사장교의 면진 성능 향상을 위해서 추가적인 수동
제어 장치로서 점성댐퍼를 적용하였다
납고무 받침은 구조물의 수직하중 지지기능 지진 격리 효과 감쇠 효과
및 복원력을 하나의 장치로 얻을 수 있어서 단경간 교량이나 건축 구조물의
면진 장치로 널리 이용되어 왔다 이런 구조물의 면진 설계 시 가장 중요한
목적은 구조물의 고유주기를 이동시키는 것이다 하지만 사장교와 같은 장
대 교량의 동적 특징은 일반 단경간 교량이나 건축 구조물과는 다르며 또한
이런 장대 교량은 비교적 복잡한 거동을 보인다 따라서 단경간 교량이나
건축 구조물에서 사용되는 납고무 받침의 설계 방법이나 기준을 사장교와
같은 장대 교량에 바로 적용하는 것은 힘들다
본 연구에서는 사장교의 동적 거동에서 중요하게 생각하는 응답을 고려
하여 납고무 받침의 설계 지수를 제안하였다 납고무 받침은 물성치의 변화
에 따라 제안된 설계 지수가 최소가 되거나 수렴할 때의 물성치 값을 선택
하여 설계하였다 제안된 방법에 의한 설계 결과 지진 하중을 받는 사장교
에 적합한 납고무 받침은 구조물의 고유주기 이동보다는 지진 에너지 소산
기능이 더 중요함을 확인하였다 또한 설계된 납고무 받침의 면진 성능 또한
검증하였고 제안된 방법에 의해 설계된 납고무 받침의 면진 성능이 우수함
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
을 확인하였다
제안된 설계 방법에 의해 지진 하중의 특성에 따라 납고무 받침을 설계
하여 그 특성을 분석하였다 또한 이때 설계된 납고무 받침에 대하여 지진
하중의 특징에 따른 면진 성능의 강인성을 조사 하였다 설계된 납고무 받침
은 면진 성능은 다른 특성을 가지는 지진하중에서도 크게 변하지 않음을 확
인하였다
마지막으로 추가적인 지진응답의 감소를 위해 점성댐퍼를 납고무 받침이
설치된 사장교에 적용하였다 이는 사장교의 경우 면진 장치에 의한 구조물
의 고유 주기 이동보다는 지진 에너지 소산 기능이 더 중요하게 요구 되기
때문이다 점성댐퍼를 추가적으로 설치할 경우 납고무 받침만으로 충분히
제어하지 못했던 응답이 더욱 감소하였다 이는 지진 하중을 받는 사자요에
서 적절히 설계된 수동제어 장치만으로도 충분히 제어 성능을 확보 할 수
있음을 보여 주는 것이다
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
53
REFERENCE
[1] H M Ali and A M Abdel-Ghaffar ldquoSeismic Passive Control of Cable-Stayed
Bridgesrdquo Shock and Vibration Vol2 No4 pp259-272 1995
[2] R I Skinner W H Robinson and G H McVerry An Introduction to Seismic
Isolation John Wiley and Sons 1993
[3] M C Griffith I D Aiken and J M Kelly ldquoDisplacement Control and Uplift
Restraint for Base Isolated Structuresrdquo Journal of Structural Engineering Vol116
pp1135-1148 1990
[4] J M Kelly ldquoBase Isolation Linear Theory and Designrdquo Earthquake Spectra
EERI Vol6 No2 pp223-244 1990
[5] S Nagarjaiah A M Robinson and M C Constantinou ldquoNonlinear Dynamics
Analysis of 3-D Base-Isolated Structuresrdquo Journal of Structural Engineering
ASCE Vol117 No7 pp2035-2054 1991
[6] A Gobarah and H M Ali ldquoSeismic Performance of Highway Bridgesrdquo Journal
of Engineering Structures Vol10 pp157-166 1988
[7] I G Buckle and R L Mayes ldquoThe Application of Seismic Isolation to Bridgesrdquo
Proceeding of the ASCE Structural Congress Seismic Engineering Research and
Practice pp633-642 1990
[8] K S Park H J Jung and I W Lee ldquoHybrid Control Strategy for Seismic
Protection of a Benchmark Cable-Stayed Bridgerdquo Engineering Structures Vol25
pp405-417 2003
[9] K S Park H J Jung B F Spencer Jr and I W Lee ldquoHybrid Control System for
Seismic Protection of a Phase II Benchmark Cable-Stayed Bridgerdquo Journal of
Structural Control Vol27 pp231-247 2003
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Reference 54
[10] M J Wesolowsky and J C Wilson ldquoSeismic Isolation of Cable-Stayed Bridges for
Near-Filed Ground Motionsrdquo Earthquake Engineering and Structural Dynamics
Vol32 pp2107-2126 2003
[11] F Naeim and J M Kelly Design of Seismic Isolated Structures-from Theory to
Practice John Willey and Sons 1999
[12] R Bouc ldquoForced Vibration of Mechanical System with Hysteresisrdquo Proceedings
of the 4th Conference on Nonlinear Oscillation 1967
[13] Y K Wen ldquoMethod for Random Vibration of Inelastic Structuresrdquo Journal of
Applied Mechanicals Division Vol42 No2 pp39-52 1989
[14] G Turan ldquoActive Control of a Cable-Stayed Bridge against Earthquake
Excitationsrdquo Thesis for the Degree of Doctoral of Philosophy in Civil Engineering
in University of Illinois at Urbana Champaign 2001
[15] S J Dyke J M Caicedo G Turan L A Bergman and S Haque ldquoPhase I
Benchmark Control Problems for Seismic Response of Cable-Stayed Bridgesrdquo
Journal of Structural Engineering Vol29 No7 pp857-872 2003
[16] J Wilson and W Gravelle ldquoModeling of a Cable-Stayed Bridge for Dynamic
Analysisrdquo Earthquake Engineering and Structural Dynamics Vol20 pp707-721
1991
[17] A K Chopra Dynamics of Structures- Theory and Application to Earthquake
Engineering 2nd Edition Prentice Hall 2001
[18] F R Rofooei A Mobarake and G Ahmadi ldquoGeneration of Artificial Records with
a Nonstationary Kanai-Tajimi Modelrdquo Engineering Structures Vol23 pp827-837
2001
[19] B F Spencer Jr S J Dyke and H S Depskar ldquoBenchmark Problems in Structural
Control part I-active tendon systemrdquo Earthquake Engineering and Structural
Dynamics Special Issue on Benchmark Problems Vol27 No11 pp1127-1137
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29
Reference 55
1998
[20] H M Ali A M Abdel-Ghaffar ldquoModeling of Rubber and Lead Passive-Control
Bearings for Seismic Analysisrdquo Journal of Structural Engineering Vol121
pp1134-1144 1995
[21] W H Robinson ldquoLead Rubber Hysteretic Bearings Suitable for Protecting
Structures during Earthquakesrdquo Earthquake Engineering and Structural Dynamics
Vol10 pp593-604 1982
[22] D Lee and P Taylor ldquoViscous Damper Development and Future Trendsrdquo The
Structural Design and Tall Buildings Vol10 pp311-320 2001
[23] 이철희 구봉근 전규식 이병진 ldquo납 면진받침을 이용한 교량의 면진설
계rdquo 한국지진공학회 1999년도 춘계학술대회 논문집 pp161-168 1999
[24] 박규식 정형조 B F Spencer Jr 이인원 ldquo수동 능동 반능동 및 복합 시
스템을 이용한 사장교의 지진응답 제어rdquo 한국지진공학회 논문집 제 7권
제 1호 pp17-29