Dissertation2007-OrtizRestrepo[1]

8/8/2019 Dissertation2007-OrtizRestrepo[1]

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Istituto Universitario

di Studi Superiori

Universit degli

Studi di Pavia

EUROPEAN SCHOOL FOR ADVANCED STUDIES IN

REDUCTION OF SEISMIC RISK

ROSE SCHOOL

DISPLACEMENT-BASED DESIGN OF CONTINUOUS CONCRETE

BRIDGES UNDER TRANSVERSE SEISMIC EXCITATION

A Dissertation Submitted in Partial

Fulfilment of the Requirements for the Master Degree in

EARTHQUAKE ENGINEERING

by

JUAN CAMILO ORTIZ RESTREPO

Supervisor: Prof. M.J.N. PRIESTLEY

June, 2006


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The dissertation entitled Displacement-Based Design of Continuous Concrete Bridges Under

Transverse Seismic Excitation, by Juan Camilo Ortiz Restrepo, has been approved in partial

fulfilment of the requirements for the Master Degree in Earthquake Engineering.

M.J.N. PRIESTLEY _

G.M. CALVI_____


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Abstract

ABSTRACT

In this work a displacement-based design procedure for multi-span reinforced concrete bridge

structures when subjected to seismic action in the transverse direction is presented. The procedure,

initially proposed by Priestley [Priestley, 1993], is reviewed and some improvements are

implemented. The design methodology is then applied to different possible bridge configurations. The

accuracy of the method in terms of reaching the target displacements under the design earthquake

level is then assessed using inelastic time-history analysis. Discussion of the appropriate level of

damping to be considered in the inelastic time-history analysis of this type of structures is provided

based in a recent a recent work developed at the ROSE School on equivalent damping for

displacement-based design applications [Grant et al., 2004].

Dynamic amplification of the deck transverse moments is investigated and compared with analytical

results using different variations of the modal superposition approach. What has been called the

Effective Modal Superposition, is then proposed as an efficient method to account for higher mode

effects on the deck transverse moment distributions.

A comparison of the direct displacement-based designand the force-based design, also assessed with

time history analysis, is carried out for the different bridges configurations. Results in terms of pier

ductility demands, displacements, deck moments and longitudinal steel reinforcement ratios are

presented and discussed.

Finally, some analyses of a Rail Bridge configuration with lower deck transversal stiffness are

presented to provide an idea of the scope and applicability of the design procedure under different

conditions to those assumed for the initial designs.

Keywords: bridges; performed-based seismic design; higher modes

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Acknowledgements

ACKNOWLEDGEMENTS

I would like to mainly thank Professor Nigel Priestley for his wise advice during the development of

this work. Also thanks to Professor G.M.Calvi, director of the ROSE School, and Lorenza Petrini and

Tim Sullivan, who were always available to help me and answer my questions. A exceptional thanks

to Juan Camilo Alvarez who was all the time helping me to save time and make this work more

proficient.

Thanks to all my friends at ROSE School. I would principally like to thank Ana Beatriz, Juan Esteban,

Juan Pablo, Carlos and Natalia, Jason and Nasha, Joao and Ana, Alex, Luca and Randolph for all the

great times we shared.

I would also like to thank my former employer in Colombia, Luis Gonzalo Meja, for his wise advises,

his example of life and his constant search to making me a better engineer and mainly a better person.

This work and this Masters are entirely dedicated to my wife, Paulina, for her great love, support and

company during this time in Italy. A special mention for my parents, Luis Javier and Gloria, and my

brother, Alejandro, who have always be sustaining and encouraging me in every project of my life.

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Index

TABLE OF CONTENTS

Page

ABSTRACT ............................................................................................................................................i

ACKNOWLEDGEMENTS....................................................................................................................ii

TABLE OF CONTENTS ......................................................................................................................iii

LIST OF FIGURES ...............................................................................................................................vi

LIST OF TABLES..................................................................................................................................x

1. INTRODUCTION.............................................................................................................................1

1.1 WHY DISPLACEMENT-BASED DESIGN?...........................................................................1

1.2 SCOPE.......................................................................................................................................2

2. FUNDAMENTALS OF DIRECT DISPLACEMENT BASED DESIGN ........................................4

3. DISPLACEMENT BASED DESIGN OF MULTI-SPAN BRIDGES..............................................9

3.1 REGULAR AND IRREGULAR BRIDGES CONFIGURATIONS .........................................9

3.2 DESIGN PROCEDURE ..........................................................................................................10

3.2.1 Design Displaced Shape ................................................................................................11

3.2.2 The Equivalent SDOF System.......................................................................................13

3.2.2.1 System Design Displacement..................................................................................13

3.2.2.2 Equivalent System Damping...................................................................................13

3.2.2.3 Pier Yield Displacement..........................................................................................16

3.2.2.4 Forces taken by Piers and Abutments .....................................................................17

3.2.2.5 Effective System Mass: ...........................................................................................18

3.2.3 Equivalent SDOF Design...............................................................................................18

3.2.4 Required Columns Strength...........................................................................................19

3.2.5 Additional notes .............................................................................................................21

3.3 APPLICATION TO DIFFERENT BRIDGE CONFIGURATIONS.......................................23

3.3.1 Bridge Information and Assumptions............................................................................23

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Index

3.3.1.1 Materials:.................................................................................................................23

3.3.1.2 Abutments: ..............................................................................................................23

3.3.1.3 Bridge Deck: ...........................................................................................................24

3.3.1.4 Piers and Cap Beam: ...............................................................................................28

3.3.2 Seismic Input .................................................................................................................28

3.3.3 Design Results ...............................................................................................................29

3.3.3.1 Series of Regular Bridges........................................................................................30

3.3.3.2 Series 7: SMM.........................................................................................................38

3.3.3.3 Series 8: SML..........................................................................................................40

3.3.3.4 Series 9: SLL...........................................................................................................40

3.3.3.5 Series 10: SSM........................................................................................................44

3.3.3.6 Series 11: SSL .........................................................................................................44

3.3.3.7 Series 12: MSL........................................................................................................44

3.3.3.8 Series 13: SSMLL(1)...............................................................................................49

3.3.3.9 Series 14: SSMLL(2)...............................................................................................50

3.3.3.10 Series 15: SSLMS ...................................................................................................51

3.3.3.11 Series 16: MSLMS..................................................................................................51

3.3.3.12 Series 17: LMSSM(1) .............................................................................................52

3.3.3.13 Series 18: LMSSM(2) .............................................................................................52

4. PERFORMANCE ASSESMENT USING TIME-HISTORY ANALYSIS....................................59

4.1 MODELING ISSUES..............................................................................................................59

4.1.1 Hysteretic Rule...............................................................................................................60

4.1.2 Damping.........................................................................................................................61

4.2 SPECTRUM-COMPATIBLE TIME HISTORIES .................................................................62

4.3 DESIGN VERSUS TIME-HISTORY RESULTS...................................................................64

4.3.1 Target Displacements and Deck Transverse Moments..................................................65

4.3.1.1 Series of Regular Bridges........................................................................................65

4.3.1.2 Series 7: SMM.........................................................................................................72

4.3.1.3 Series 8: SML..........................................................................................................72

4.3.1.4 Series 9: SLL...........................................................................................................72

4.3.1.5 Series 10: SSM........................................................................................................77

4.3.1.6 Series 11: SSL .........................................................................................................77

4.3.1.7 Series 12: MSL........................................................................................................77

4.3.1.8 Series 13: SSMLL(1)...............................................................................................83

4.3.1.9

Series 14: SSMLL(2)...............................................................................................83

4.3.1.10 Series 15: SSLMS ...................................................................................................86

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Index

4.3.1.11 Series 16: MSLMS..................................................................................................87

4.3.1.12 Series 17: LMSSM(1) .............................................................................................90

4.3.1.13 Series 18: LMSSM(2) .............................................................................................90

4.3.2 Dynamic Amplification of Deck Transverse Moments .................................................93

5. COMPARISON OF DDBD WITH THE FORCE BASED DESIGN METHOD.........................106

5.1 FORCE BASED DESIGN.....................................................................................................106

5.2 TIME HISTORY ANALYSIS FOR FORCE BASED DESIGNED BRIDGES...................108

5.2.1 Hysteretic rule..............................................................................................................108

5.2.2 Damping.......................................................................................................................108

5.2.3 Spectrum-Compatible time histories............................................................................109

5.3 RESULTS COMPARISON FOR REGULAR BRIDGE CONFIGURATIONS...................109

5.3.1 Series 1 and 2 ...............................................................................................................109

5.3.2 Series 3.........................................................................................................................109

5.3.3 Series 4.........................................................................................................................113

5.3.4 Series 5.........................................................................................................................113

5.3.5 Series 6.........................................................................................................................113

5.4 RESULTS COMPARISON FOR IRREGULAR BRIDGE CONFIGURATIONS...............117

5.4.1 Series 7, 8 and 9...........................................................................................................117

5.4.2 Series 10, 11 and 12.....................................................................................................117

5.4.3 Series 13, 14, 15 and 16...............................................................................................124

5.4.4 Series 17 and 18...........................................................................................................124

6. RAIL BRIDGE..............................................................................................................................131

6.1 PREVIOUS STUDY..............................................................................................................131

6.2 DIRECT DISPLACEMENT-BASED DESIGN OF A RAIL BRIDGE................................132

6.2.1 4-span Rail Bridges......................................................................................................133


6.3 COMPARISON OF DDBD AND FBD - PERFORMANCE ASSESMENT USING TIME-

HISTORY ANALYSIS ......................................................................................................................141



7. CONCLUSIONS ...........................................................................................................................149

REFERENCES ...................................................................................................................................152

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Index

LIST OF FIGURES

Page

Figure 2. 1- Effective Stiffness ...................................................................................................4

Figure 2. 2 Design Displacement Spectra................................................................................6

Figure 3. 1- Regular and Irregular Bridges. ..............................................................................10

Figure 3. 2 - Possible transverse displacement shapes for continuous bridges ........................12

Figure 3. 3 Uniform beam simply supported on elastic springs ............................................13

Figure 3. 4 Equivalent damping for deferent Takeda Thin degrading-stiffness models .......16

Figure 3. 5 Caltrans displacement ARS curves. Soil C, M = 8.00.25 and 0.7g PGA, for

different levels of damping ...............................................................................................19

Figure 3. 6 Model of the equivalent elastic system under transverse response .....................20

Figure 3. 7 Flowchart forDirect Displacement-Based Design of MDOF-bridges................22

Figure 3. 8 Bridge Typical Transverse Section .....................................................................24

Figure 3. 9 Series of 4-span and 6-span Regular Bridges (H = 7.5 m, 10.0 m, 12.5m and

15.0 m) ..............................................................................................................................25

Figure 3. 10 Series of 4-span Irregular Bridges (H = 7.5 m, 10.0 m, 12.5m and 15.0 m) .....26

Figure 3. 11 Series of 6-span Irregular Bridges (H = 7.5 m, 10.0 m, 12.5m and 15.0 m)....27

Figure 3. 12 Extended Caltrans displacement ARS curve for soil profile C, M = 8.00.25

and 0.7g PGA....................................................................................................................28

Figure 3. 13 Interaction Diagrams for piers...........................................................................29

Figure 3. 14 Design results for bridges of Series 1. ...............................................................32





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Index


Figure 3. 20 Design results for bridges of Series 7: SMM. ...................................................41

Figure 3. 21 Design results for bridges of Series 8: SML. ....................................................42

Figure 3. 22 Design results for bridges of Series 9: SLL.......................................................43

Figure 3. 23 Design results for bridges of Series 10: SSM....................................................46

Figure 3. 24 Design results for bridges of Series 11: SSL. ....................................................47

Figure 3. 25 Design results for bridges of Series 12: MSL. ..................................................48

Figure 3. 26 Design results for bridges of Series 12, H=7.5m, with strength redistribution.49

Figure 3. 27 Design results for bridges of Series 13: SSMLL(1). .........................................53

Figure 3. 28 Design results for bridges of Series 14: SSMLL(2).. ........................................54

Figure 3. 29 Design results for bridges of Series 15: SSLMS...............................................55

Figure 3. 30 Design results for bridges of Series 16: MSLMS..............................................56

Figure 3. 31 Design results for bridges of Series 17: LMSSM (1). .......................................57

Figure 3. 32 Design results for bridges of Series 18: LMSSM (2).. ......................................58

Figure 4. 1 Typical simplified plan model of bridge used in time-history analysis. .............59

Figure 4. 2 Takeda degrading stiffness model. ......................................................................60

Figure 4. 3 Artificial time histories and associated set of spectra for different damping

levels. ................................................................................................................................63

Figure 4. 4 Artificial time histories and associated set of spectra for different damping

levels. ................................................................................................................................64

Figure 4. 5 Design Vs THA for bridges of Series 1...............................................................66




Figure 4. 9 Design Vs THA for bridges of Series 5...............................................................70Figure 4. 10 Design Vs THA for bridges of Series 6.............................................................71

Figure 4. 11 Design Vs THA for bridges of Series 7: SMM. ................................................73

Figure 4. 12 Design Vs THA for bridges of Series 8: SML. .................................................74

Figure 4. 13 Design Vs THA for bridges of Series 9: SLL. ..................................................75

Figure 4. 14 Elastic and Inelastic properties for bridges of Series 8: SML. ..........................76

Figure 4. 15 Design Vs THA for bridges of Series 10: SSM. ...............................................78

Figure 4. 16 Design Vs THA for bridges of Series 11: SSL..................................................79

Figure 4. 17 Design Vs THA for bridges of Series 12: MSL. ...............................................80

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Index

Figure 4. 18 Elastic and Inelastic properties for bridges of Series 12: MSL. ........................81

Figure 4. 19 Design Vs THA, Elastic and Inelastic properties for bridge of Series 12: SML,

H=7.5 m, with strength redistribution...............................................................................82

Figure 4. 20 Design Vs THA for bridges of Series 13: SSMLL(1). ......................................84

Figure 4. 21 Design Vs THA for bridges of Series 14: SSMLL(2). ......................................85

Figure 4. 22 Elastic and Inelastic properties for bridges of Series 14: SSMLL(2)................86

Figure 4. 23 Design Vs THA for bridges of Series 15: SSLMS............................................88

Figure 4. 24 Design Vs THA for bridges of Series 16: MSLMS. .........................................89

Figure 4. 25 Design Vs THA for bridges of Series 17: LMSSM(1)......................................91

Figure 4. 26 Design Vs THA for bridges of Series 18: LMSSM(2)......................................92

Figure 4. 27 Elastic and Inelastic properties for bridges of Series 18 LMSSM(2)................93

Figure 4. 28 Deck Moments for Series 4. ..............................................................................96



Figure 4. 31 Deck Moments for Series 13: SSMLL(1). ........................................................99

Figure 4. 32 Deck Moments for Series 14: SSMLL(2). ......................................................100

Figure 4. 33 Deck Moments for Series 15: SSLMS. ...........................................................101

Figure 4. 34 Deck Moments for Series 16: MSLMS...........................................................102

Figure 4. 35 Deck Moments for Series 17: LMSSM(1). .....................................................103

Figure 4. 36 Deck Moments for Series 18: LMSSM(2). .....................................................104

Figure 5. 1 Typical simplified plan model of bridge used in Force Based Design Analysis.

.........................................................................................................................................106

Figure 5. 2 Acceleration Spectrum for Soil Type C (M = 8.0+-0.25). .................................107

Figure 5. 3 Typical simplified plan model of bridge used in time-history analysis. ...........108

Figure 5. 4 Comparison of DDBD, FBD and THA for bridges of Series 1.........................110Figure 5. 5 Comparison of DDBD, FBD and THA for bridges of Series 2.........................111

Figure 5. 6 Comparison of DDBD, FBD and THA for bridges of Series 3.........................112




Figure 5. 10 Comparison of DDBD, FBD and THA for bridges of Series 7: SMM. ..........118

Figure 5. 11 Comparison of DDBD, FBD and THA for bridges of Series 8: SML. ...........119

Figure 5. 12 Comparison of DDBD, FBD and THA for bridges of Series 9: SLL. ............120

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Index

Figure 5. 13 Comparison of DDBD, FBD and THA for bridges of Series 10: SSM. .........121

Figure 5. 14 Comparison of DDBD, FBD and THA for bridges of Series 11: SSL............122

Figure 5. 15 Comparison of DDBD, FBD and THA for bridges of Series 12: MSL. .........123

Figure 5. 16 Comparison of DDBD, FBD and THA for bridges of Series 13: SSMLL(1).125

Figure 5. 17 Comparison of DDBD, FBD and THA for bridges of Series 14: SSMLL(2).126

Figure 5. 18 Comparison of DDBD, FBD and THA for bridges of Series 15: SSLMS......127

Figure 5. 19 Comparison of DDBD, FBD and THA for bridges of Series 16: MSLMS. ...128

Figure 5. 20 Comparison of DDBD, FBD and THA for bridges of Series 17: LMSSM(1).

.........................................................................................................................................129

Figure 5. 21 Comparison of DDBD, FBD and THA for bridges of Series 18: LMSSM(2).

.........................................................................................................................................130

Figure 6. 1 Typical transverse section of Rail Bridge. .........................................................132

Figure 6. 2 Design results for Rail Bridges of Series 2. ......................................................134

Figure 6. 3 Design results for Rail Bridges of Series 8: SML. ............................................135

Figure 6. 4 Design results for Rail Bridges of Series 12: MSL. ..........................................136

Figure 6. 5 Design results for Rail Bridges of Series 5. ......................................................138

Figure 6. 6 Design results for Rail Bridges of Series 14: SSMLL2. ...................................139

Figure 6. 7 Design results for Rail Bridges of Series 18: LMSSM2. ..................................140

Figure 6. 8 Comparison of DDBD, FBD and THA for Rail Bridges of Series 2. ...............142

Figure 6. 9 Comparison of DDBD, FBD and THA for Rail Bridges of Series 8: SML......143

re 6. 10 Comparison of DDBD, FBD and THA for Rail Bridges of Series 12: MSL.

Figu

..........146

Figu

.........................................................................................................................................147

Figure 6. 13 Comparison of DDBD, FBD and THA for Rail Bridges of Series 18:LMSSM2.........................................................................................................................148

.144

Figure 6. 11 Comparison of DDBD, FBD and THA for Rail Bridges of Series 5. ...

re 6. 12 Comparison of DDBD, FBD and THA for Rail Bridges of Series 14: SSMLL2.

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Index

LIST OF TABLES

Page

Table 3.1 - Material Properties for Design. ..............................................................................23

Table 3.2 Substitute SDOF parameters for bridges of Series 1 to 6. .......................................31

Table 3.3 Substitute SDOF parameters for bridges of Series 7 to 12......................................39

Table 3.4 Substitute SDOF parameters for bridge of Series 12, H=7.5m, with strength

redistribution. ....................................................................................................................49

Table 3.5 Substitute SDOF parameters for bridges of Series 13 to 18. ....................................50

Table 6.1 Substitute SDOF parameters for 4-span Rail Bridges. ..........................................133

Table 6.2 Substitute SDOF parameters for 6-span Rail Bridges. ..........................................137

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Chapter 1. Introduction

1.INTRODUCTION

Seismic design is currently going through a transitional period. Most of the seismic codes to

date utilize force-based seismic design, or what can also be called strength-based design

procedures. However, it is now widely recognized that force and damage are poorly

correlated and that strength has lesser importance when designing for earthquake loading than

for other actions. These, together with other problems and inconsistencies with force-based

design, [Priestley, 2003], have led to the development of more reliable seismic design

methodologies under the framework of what has been termed Performance-Based Seismic

Design (PBSD). PBSD represents basically the philosophy of designing a structure to perform

within a predefined level of damage under a predefined level of earthquake intensity.

1.1 WHY DISPLACEMENT-BASED DESIGN?

It is known that displacements correlate much better with damage than forces do. Hence, if

the design objective is to control the damage under a given level of seismic excitation it is

reasonable to attempt to design the structures using as input the desired displacements to be

sustained under the design seismic intensity.

One of the more rational and relevant approaches that has been developed over the past 10

years is the Direct Displacement-Based Design, which characterizes the structure to bedesigned by a single degree of freedom representation of performance at peak displacement

response. The objective is to design a structure which would achieve, rather than be bounded

by, a given performance limit state under a given seismic intensity [Priestley, 1993 and

Priestley, 2003]. The method utilizes the Substitute Structure approach developed by Gulkan

and Sozen [Gulkan and Sozen, 1974] to model the inelastic structure as an equivalent elastic

single-degree-of-freedom (SDOF) system. The concepts of the methodology will be presented

first in this work and its application to multi-span bridge structures discussed in detail

subsequently.

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1.2 SCOPEThe objectives of this project are to introduce possible improvements to the direct

displacement-based design procedure for the design of multi-span bridges for regular andirregular bridges configurations, initially proposed by Priestley [Priestley, 1993 and Priestley,

2003] and subsequently studied by Alvarez Botero [Alvarez Botero, 2004]; and to assess the

accuracy of the method in terms of reaching the target displacements under the design

earthquake level. The latter is done by carrying out inelastic time-history analyses for a series

of bridge structures designed using the direct displacement-based design methodology.

Additionally, the issue of dynamic amplification of deck transverse moments is investigated

and an effective method to consider this phenomenon for bridges designed using direct

displacement-based design is proposed.

A comparison between the direct-displacement based design, DDBD, and the force-based

design, FBD, is done.

Finally, a parametric study of a Rail Bridge with a low deck transversal stiffness also form

part of the investigation and is aimed to assess the applicability of the procedure under diverse

design constrains.

Chapter 2 provides the basic concepts behind the direct displacement-based design procedure

and its general application.

Chapter 3 deals with the application of the method to the specific case of multi-spanreinforced concrete bridges with continuous deck, and flexible lateral supports at abutments.

Important issues regarding the consideration of the sources of energy dissipation and the

calculation of the system damping are discussed. An iterative design procedure is introduced.

Design results for 72 different bridges are presented and discussed.

Chapter 4 presents the results of the assessment of the method in terms of reaching the target

displacements when the designs are subjected to spectrum-compatible acceleration time

histories. Description of the models used is made and a brief discussion on the seismic input

for the inelastic time-history analysis is presented. Higher-mode effects on deck transverse

moments are investigated.

Chapter 5 presents the comparison of the method with the current generally used code force-

based design method in terms of reaching the target displacements when the designs are

subjected to spectrum-compatible acceleration time histories. Description of the force-base

design models is made and a short discussion on the seismic input for the inelastic time-

history analysis is presented. Deck transverse moments are also investigated. Final design

results for both methods, DDBD and FBD, are presented in terms of pier diameter, design

moments and longitudinal reinforcement ratios.

Chapter 6 deals with the application of the method using a Rail Bridge with low decktransversal stiffness. The DDBD methodology is applied to 36 different bridges and then

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assessed with inelastic time-history analysis. Finally results for direct displacement-based

design, DDBD, and force-based design, FBD, are presented.

Finally, some conclusions are presented in Chapter 7.

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Chapter 2 Fundamentals of Direct Displacement-Based Design

2.FUNDAMENTALS OF DIRECT DISPLACEMENT BASEDDESIGN

Direct Displacement-Based Design is an approach in which, contrary to current Force-BasedDesign practice, forces are obtained for a desired performance level and based on inelastic

response of the system. The objective is to design a structure which would achieve, rather

than be bounded by, a given performance limit state under a given seismic intensity [Priestley,

2003]. The procedure is based in the Substitute Structure approach developed by Gulkan and

Sozen [Gulkan and Sozen, 1974], which models the inelastic structure as an equivalent elastic

single degree of freedom (SDOF) system. The SDOF is represented by an effective stiffness

(See Figure 2.1), mass and damping. The aim of the design procedure is to obtain the base

shear from a given target displacement and the level of ductility that can be estimated from

the structural and element geometries.

Figure 2. 1- Effective Stiffness

Since the substitute structure is elastic, its response to a particular ground motion, and hence

the response of the actual structure, can be determined form the elastic response spectrum for

the appropriate level of damping. For a SDOF system the design displacement, d, for the

performance level under consideration, can be based either on material strain limits or code-

specific drift limits. The yield displacement, y, can be estimated from simplified relations

for the yield curvature, y, [Priestley, 2003] and the displacement ductility calculated as:

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d

y

=

(2.1)

Equivalent viscous damping can then be estimated as the sum of elastic and hysteretic

damping, using some relations depending of the displacement ductility, , and structure

period Teff.

eff e hyst = + (2.2)

The hysteretic component, hyst, can be computed using the equation (2.3) [Grant et al, 2005]

which is depends of the equivalent period, Teq.

(2.3)

( )1 1

1 1hyst dbeq

aT c

= + +

Where a,b, c and dare constants values that depend of the hysteretic model assumed, and isthe displacement ductility. For the Takeda Thin degrading-stiffness-hysteretic rule, which is

commonly used to represent ductile reinforced concrete columns response, these values are a

= 0.215, b = 0.642, c = 0.824, d= 6.444 [Grant et. al., 2005]

The elastic component, el, is assumed to be 5% of the critical damping but some correction

factor must be applied for the assumption of initial-stiffness or tangent-stiffness damping (Seedeeper discussion in Grant et al, 2004). The correction factor for the elastic component can

then be computed using eq. (2.4).

= (2.4)

Where is the displacement ductility and depend on the hysteretic rule used and the elasticdamping assumption. For the Takeda Thin degrading-stiffness-hysteretic rule, using tangent-

stiffness elastic damping, is equal to -0.378.

As equation (2.3) is period dependent, an iterative procedure should be implemented to obtain

the hysteretic damping (See Grant et. al., 2005 for detailed process). Alternatively, as the

period dependency of equation (2.3) is generally insignificant for periods greater than 1.0

seconds using the Takeda Thin Model [Grant et al, 2005], and as will be unusual for normal

bridges to have effective periods less than 1.0 seconds, it will generally be conservative to

ignore the period dependency in design, and the simplified equation (2.5) can be used instead

of equation (2.2).

10.05 0.444eff

= +

(2.5)

Once the design displacement has been defined and the corresponding damping estimatedfrom the expected ductility demand, the effective period at maximum response, Teff, can be

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read directly from the displacement spectrum, reduced for the corresponding level of

damping, as shown in Figure 2.2. The effective stiffness, Keff, of the equivalent SDOF can

then be determined from the period equation of a SDOF oscillator:

2

2

4 effeff

eff

KT

= (2.6)

Where Meff represent the effective mass of the structure participating in the fundamental

mode of vibration. Having the effective stiffness, the design lateral force can be readily

obtained using Equation (2.7).

B eff V K d= (2.7)

Figure 2. 2 Design Displacement Spectra

For a SDOF system the procedure ends here, the design lateral force is the corresponding base

shear of the system, and adequate strength must be then provided. Capacity design procedures

are used to ensure shear strength exceeds maximum possible shear correspondent to flexural

over-strength in the plastic region. However, for a MDOF system, the next step in the design

process is the distribution of the design lateral force, VB, throughout the structure and a

subsequent structural analysis under the distributed seismic forces.

When the design method is applied to a MDOF system, the main issues are the definition of

the Substitute Structure and the determination of the design displacement. However, the

substitute structure can be easily defined by assuming a displaced shape for the real structure.

This displaced shape is that which corresponds to the inelastic first-mode at the design level

of seismic excitation. Representing the displacement by the inelastic rather than the elastic

first-mode shape is consistent with characterizing the structure by its secant stiffness tomaximum response [Priestley et al, 2006]. During the last years, research efforts have been

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focused on the definition of design displaced shapes for different structural systems. The

design displacement of the substitute structure depends also on the limiting displacement of

the critical member, C, which in turn depends on the strain or code-drift limit for theperformance level under consideration. For bridge structures, the critical member will

normally be the shortest column.

Having defined the displacement of the critical member and the design displacement shape

the displacements of the individual masses can be obtained using Equation (2.8).

ci i

c

=

(2.8)

Where is the design displaced shape, i.e. the fundamental inelastic mode shape. Having nowthe actual design displacement pattern, the system design displacement is computed using

Equation (2.9), which is based on the requirement that the work done by the equivalent SDOF

system is equivalent to the work done by the MDOF force system, [Calvi, et al., 1995].

( )( )

2

i i

d

i i

m

m

=

(2.9)

To fully define the equivalent SDOF system an effective mass needs to be computed. The

effective mass, Meff, is defined as the mass participating in the fundamental inelastic mode of

vibration. Being consistent with the work equivalence between the two systems, the effectivemass can be obtained using Equation (2.10).

( ) ( )

( )

2

2

i i i i

eff

d i i

m mM

m

= =

(2.10)

The equivalent SDOF system is now fully defined. Using Equations (2.6) and (2.7) the total

design lateral force is obtained. This shear force must be distributed as design forces to the

various discretized masses of the structure, in order that the design moments for potential

plastic hinges can be established. Assuming essentially sinusoidal response at peak response,

the base shear should be distributed in proportion to mass and displacement at the discretizedmass locations. Thus the design force at mass i is given by Equation (2.11), [Priestley et al,

2006].

( )( )

i i

i B

i i

mF V

m

=

(2.11)

The subsequent analysis under the distributed seismic forces is straightforward; however,

careful consideration of member stiffnesses to be used in the analysis is required. In order to

be compatible with the substitute structure concept, member stiffnesses should be

representative of effective secant stiffnesses (See Figure 2.1) at the design displacementresponse.

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Particulars of the Direct Displacement-Based Design approach and its application to several

structural systems can be found in [Priestley et al, 2006]. In the next chapter of this work

application of the methodology to continuous RC bridges is presented in more detail.

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Chapter 3 Displacement Based Design of Multi-Span Bridges

3.DISPLACEMENT BASED DESIGN OF MULTI-SPANBRIDGES

Direct Displacement-Based Design is an approach in which, contrary to current Force-BasedDesign practice, forces are obtained for a desired performance level and based on inelastic

response of the system. The objective is to design a structure which would achieve, rather

than be bounded by, a given performance limit state under a given seismic intensity [Priestley,

2003]. The procedure is based in the Substitute Structure approach developed by Gulkan and

Sozen [Gulkan and Sozen, 1974], which models the inelastic structure as an equivalent elastic

single degree of freedom (SDOF) system. The SDOF is represented by an effective stiffness

(See Figure 2.1), mass and damping. The aim of the design procedure is to obtain the base

shear from a given target displacement and the level of ductility that can be estimated from

the structural and element geometries.

3.1 REGULAR AND IRREGULAR BRIDGES CONFIGURATIONSAs previous studies were done in bridges with regular configurations [Alvarez Botero, 2004],

in this dissertation aRegular Bridge will be defined as a bridge in which the structure center

of mass, CM, coincides with the structure center of strength, CV. In this case the translational

modes of vibration rule the seismic response and the rotational ones are not excited and

consequently do not participate in the seismic response of the structure.

AnIrregular Bridges will be defined as a bridge in which the structure center of mass, CM,

do not coincides with the structure center of strength, CV. In this case the seismic response is

a combination of the translational and rotational modes of vibration.

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Figure 3. 1- Regular and Irregular Bridges.

Certainly, as the method is based in the shape of the first inelastic mode shape, its efficiency

will depend of the similarities between the fundamental elastic and inelastic mode shapes for

both,Regular Bridge and Irregular Bridge. In the cases in which the first fundamental elastic

and inelastic mode shapes are very different, care must be taken. Previous research [Alvarez

Botero, 2004] has shown that depending of the seismic level considered, the parabolic

inelastic mode shape can or can not be developed, and the bridge maximum displacements,

and consequently its behaviour, can be still dominated by the response in the elastic range.

3.2 DESIGN PROCEDURE

The displacement-based design of multi-degree-of-freedom bridge structures is based on the

concepts presented in Chapter 2. However, some specific issues must be considered carefully

during the process. The design displacement shape is a function of the relative stiffness

between columns, abutments and the deck. Resistance to transverse seismic excitation is

mainly provided by bending of the bridge piers, which are designed to respond inelastically;

and, if the abutments provide some restraint to transverse displacements, superstructure

bending will also develop. In normal seismic design practice the bridge deck is required toremain elastic under the design level earthquake. As a consequence the seismic inertia forces

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developed in the deck are taken by two different load paths, one portion is transmitted to the

piers foundations by column inelastic bending and the remainder transmitted to the abutments

by superstructure elastic bending. The portion of load carried by each of the two different loadpaths is unknown at the start of the design process and depends strongly on the relative

effective-column and deck stiffnesses as well as on the degree of lateral restrain provided by

the abutments. Since column stiffnesses are also unknown at the start of the design process,

an iterative procedure is required.

The design procedure presented here considers the discretization of the deck mass as lumped

masses at the top of the piers and at the abutments. A portion of the column masses and the

cap beam masses can also be lumped at the top, following the recommendations given in

[Priestley, et al., 1996].

The Direct Displacement-Based Design procedure for multi-degree-of-freedom bridge

structures can be summarized in the following basic steps:

1. Determination of the design displaced shape.

2. Characterization and evaluation of the equivalent SDOF system.

3. Application of the displacement-based design approach to the SDOF system.

4. Determination of column required strengths and design.

3.2.1 Design Displaced Shape

A bridge structure composed by several columns connected to a superstructure of defined

flexibility will deform in a manner that is influenced by variations in strength, stiffness and

mass distribution. The transverse displaced shape will depend strongly on the relative column

stiffness, and more considerably, on the degree of lateral restrain provided at the abutments.

Figure 3.2 depicts two different bridge configurations and the possible transverse displacedshapes indicated for the different abutment conditions.

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(a) Uniform Height Piers (b) Irregular Height Piers

Figure 3. 2 - Possible transverse displacement shapes for continuous bridges

Some flexibility will normally be considered at the abutments; hence the actual displaced

shape will be in between the fully restrained and free profiles. Since the displaced shape

depends on the relative effective stiffness of the piers, which are unknown at start of thedesign, some iteration may be required to determine the relative displacements between the

abutments and the critical pier.

Generally a parabolic displaced shape between abutments and piers can be initially assumed

for design purposes. In this work, as in the Alvarez Botero study [Alvarez Botero, 2004], the

deck-only elastic first-mode shape has been used as a first approximation to the bridge

displacement profile. Deck first-mode deformed shape can be obtained either by solving the

eigen-problem for the deck or by using an approximate first mode shape function as the one

shown in Equation (3.1) based on a half-sine wave loading shape [Alfawakhiri et al., 2000].

( )( ) 3

1 3

sin

1

x BL

xB

+=

+(3.1)

Where:

3

A

EIB

K L= (3.2)

In Equation (3.2),Eis the deck elastic modulus,Iis the deck transverse moment of inertia,KA

is the elastic abutment stiffness, andL is the total bridge length.

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Figure 3. 3 Uniform beam simply supported on elastic springs

Evaluation of the limiting displacements for each of the piers is also required in order to

determine the target displacement pattern. The limiting displacements for the piers are

function of the performance level under consideration. The final displacement pattern is likely

to consist of only one or two of the piers reaching the limiting displacement. However,

determination of the limiting displacements for all them is required in order to identify the

critical column. Normally the shortest column governs the selection of the displacementpattern. For the purposes of this work, pier limiting displacements are based on a specified

drift limit rather than on material strain limits, which can also be used.

Once the critical member has been identified, the displaced shape is scaled in such a way that

the critical member reaches its limiting displacement using Equation (2.8). It is important to

bear in mind that the critical member can change if the assumed displaced shape is not close

enough to the actual fundamental inelastic mode of vibration for the given seismic intensity.

3.2.2 The Equivalent SDOF SystemThe required properties to characterize the MDOF system as an equivalent SDOF system are:

the system design displacement, sys, the system equivalent damping, sys, and the system

effective mass, Meff.

3.2.2.1 System Design DisplacementHaving the target displacement pattern the system design displacement is readily obtained

using Equation (2.9).

3.2.2.2 Equivalent System DampingThe equivalent system damping can be obtained from a combination of the energy dissipated

by the different mechanisms activated during the structure response to seismic excitation.Some approaches, [Kowalsky, 2002], suggest to make a weighted average of the damping

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based on the work done by the members at each degree of freedom. Another way that has

been used to weight the member damping is based on the proportion of load taken by each of

the piers and the abutments, as expressed in Equation (3.3), which gives a reasonable estimateof the actual system damping in a preliminary design.

( )i isys

i

V

V

=

(3.3)

However, the previous forms of calculating the system damping do not consider the

contribution of the energy dissipated through elastic deck bending; besides, if high

proportions of the load are transmitted to the abutments, their contribution to the system

damping is over estimated. Energy dissipation at the abutments is activated if they displace,

however, it is localized and should not have a large influence on the overall system damping,especially for relatively large bridges.

Given the above reasons, it is suggested to calculate the system damping using Equation (3.4),

which explicitly considers the contribution from the elastic deck beam-action, the abutment

displacements and inelastic pier behaviour.

( ) ( ) ( )1 1 2 2 1 2A A A A Deck A A A i ipiers

sys

i

n

V F V F F F V

V

+ + + + =

(3.4)

WhereFA1andFA2are the seismic forces applied to the degrees of freedom associated to the

abutments, and are calculated using Equation (3.5) and (3.6); VA1 and VA2 represents the

proportion of the lateral force that is taken by each abutment, while Vi represent the shear

force at the degree of freedom i.

( )1 1

1A A

A B

i i

n

mF V

m

=

(3.5)

( )

2 22

A AA B

i i

n

mF V

m

=

(3.6)

Where VBis the total lateral design force, obtained using Equation (2.7), mA1andmA2are the

masses associated to each abutment degree of freedom and A1andA2are the displacement of

each abutment.

The equivalent viscous damping for the individual column members, i, is obtained from the

relationship between displacement ductility and damping, developed for the Modified Takeda

hysteretic rule using the Equation (2.2) [Grant et al., 2005] which is reproduced forward in

the Equation (3.7) when the period dependency is considered. As was said previously, this

generates an iterative process. For a detailed process see Grant et al., 2005.

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( )

1 11 1

i

ii

i e hyst el db

eq

a

T c

= + = + +

+

(3.7)

Where the values ofa, b, c, dand were previously defined in Chapter 2.

i can also be obtained with in the simplified Equation (2.5) reproduced forward as Equation

(3.8) [Priestley et al., 2006], when the structure actual period is greater than 1.0 second.

10.05 0.444 i

i

i

= +

(3.8)

Values of column displacement ductility, , are obtained by dividing the displacements fromthe target displacement pattern by the respective yield displacements.

At this point it important to note that in previous work developed by Alvarez Botero [Alvarez

Botero, 2004], Equation (3.9) proposed by Kowalsky [Kowalsky, 2002] was used to compute

the equivalent viscous damping.

( )1 10.05

i

i

r ri

= + (3.9)

Where r is the ratio of post-elastic stiffness to the elastic stiffness, normally between 0.03 and0.05 for concrete members. A value ofr= 0.03 was used in the work done by Alvarez Botero,

and will be used herein when defining the Time History Analysis post-yielding parameters.

In Figure 3.4 the Equations (3.7), (3.8) and (3.9) are plotted. It can be seen that using

Equation (3.9) results in greater equivalent damping values than using Equations (3.7) and

(3.8) for equivalent periods greater than 0.5 seconds. Therefore in the results of the present

work, generally when any configuration bridge studied in Alvarez Botero dissertation is

analyzed, for a given ductility displacement, lesser equivalent damping values will be

obtained resulting at the end of the process in bigger values of base shear, VB, and

corresponding flexural strengths in piers.

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0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 1.0 2.0 3.0 4.0 5.0 6.0

- Displacement Ductility

eq

-EquivalentDamping

Eq. (3.7) - Grant et al - Teq = 0.25 s









Eq. (3.8) - Priestley Simplified

Eq. (3.9) - Alvarez Botero, 2004

Figure 3. 4 Equivalent damping for deferent Takeda Thin degrading-stiffness models

3.2.2.3 Pier Yield DisplacementThe yield displacement, y, for a cantilever pier is given by Equation (3.10), where yis the

yield curvature andHeis the effective pier height, that considers yield penetration, [Priestley

et al., 1996], and can be estimated using Equation (3.11).

2

3

ey y

H = (3.10)

e spH H L= + (3.11)

In Equation (3.11)Lspis the strain penetration length and is given by Equation (3.12), wherefy

is the longitudinal bar yield stress and dbl is the longitudinal reinforcement bar diameter.

0.022sp y bl L f d = (3.12)

The yield curvature for circular columns, y, can be estimated using Equation (3.13),

[Priestley, 2003], where y is the longitudinal bar yield strain, andD is the column diameter.

2.25 yy

D

= (3.13)

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3.2.2.4 Forces taken by Piers and AbutmentsNote that at this stage of the design, the forces taken by each pier and by the abutments areunknown. However, the actual values of the forces are not required and only the relative

proportion is needed in order to weight the damping contributions. An initial assumption of

the proportion of the total seismic force carried by superstructure bending, SS, and transmitted

to the abutments, has to be made, and the remaining distributed among the piers.

Seismic Force Carried by Pier Bending

Since the relative strength of the piers is a design choice, a practical alternative would be to

provide the same longitudinal steel ratio and column diameter and hence the same flexuralstrength to all the piers. This selection introduces a convenient simplification for the

calculation of the column design forces. By doing so, it is found that, if all piers achieve a

displacement ductility of at least one, the lateral force resisted by a column is approximately

inversely proportional to the pier height,H.

1F

H (3.14)

That is, ifFC(%) is the proportion of the total seismic force carried by pier bending, the

proportion resisted by a column is given by Equation (3.15), where SDF is the shear

distribution factor and is calculated using Equation (3.16).

( ) ( )%i i CV SDF F = % (3.15)

1

1i

i

i

HSDF

H

=

(3.16)

For the case in which some of the piers remain elastic, i.e. have a displacement ductility

demand less than one, the secant stiffness at maximum response is the secant stiffness at yield

displacement, that is, the cracked stiffness, and Equation (3.16) must be modified. In such acase, the pier force is proportional to the fraction of the yield displacement over the column

height for the elastic columns, as shown in Equation (3.17).

FH

(3.17)

Where < 1.

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Seismic Force Carried by the Abutments

ForRegular Bridges, in which the torsional modes do not participate in the seismic response,

each abutment will take half of the seismic force carried by superstructure bending, SS. In the

case of Irregular Bridges, the seismic force taken by each abutment can be computed using

Equations (3.14) and (3.15) based in the displacements of each abutment. As the initial

displacement assumption is based in the first deck mode shape, clearly the shear distribution

factor, SDFwill be equal for both abutments at the initial stage, but along the iteration process

will be changed according to each updated displacement pattern.

11

1 2

AA

A A

SDF SS

= +

(3.18)

22

1 2

AA

A A

SDF SS = +

(3.19)

3.2.2.5 Effective System Mass:The effective system mass Meff is defined as the mass participating in the fundamental

inelastic mode of vibration under the design earthquake level, and can be obtained using

Equation (2.10), which is rewritten below as Equation (3.20).

( )i ieff

d

mM

=

(3.20)

3.2.3 Equivalent SDOF DesignHaving characterized the equivalent SDOF system the effective period of the Substitute

Structure is obtained by entering the displacement spectrum, for the appropriate level of

damping, with the system design displacement (See Figure 3.5). The elastic displacementspectrum for the required level of damping can be obtained from the 5% damping spectrum

for normal accelerograms measured at least 10 km from the fault rupture, using Equation

(3.21), fromEC8, where T,5is the response displacement for 5% damping. Figure 3.5 shows

a displacement response spectra set from the Caltrans, Seismic Design Criteria, soil profile C,

magnitude 8.00.25, 0.7g PGA [Caltrans, 2001] reduced for different levels of damping using

Equation (3.21).

0.5

, ,5

10

5T T

= +

(3.21)

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period [s]

SpectralDisplacement[m

]

10%

15%20%

30%

5%

Figure 3. 5 Caltrans displacement ARS curves. Soil C, M = 8.00.25 and 0.7g PGA, for different levels of

damping

The effective stiffness at maximum response is then obtained using Equation (3.22), and the

SDOF base shear force using Equation (3.23).

2

2

4 effeff

eff

KT

=

(3.22)

B eff V K d= (3.23)

3.2.4 Required Columns Strength

The total lateral seismic force must be now distributed to the discretized masses by means of

Equation (2.11), rewritten below as Equation (3.24), and analysis carried out.

( )( )

i i

i B

i i

mF V

m

=

(3.24)

In order to be compatible with the Substitute Structure concept, member stiffness should be

representative of effective secant stiffness, and can be evaluated from the target displacements

pattern and the shear forces carried by each pier as:

i

is

i

VK =

(3.25)

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The shear forces at each degree of freedom can be obtained using Equation (3.26), where SDF

refers to the shear distribution factors from Equation (3.16), (3.18) and (3.19).

i iV SDF V i= (3.26)

An elastic analysis of the equivalent elastic system must now be carried out under the vector

of distributed seismic forcesFi, and using the member effective stiffnesses,Ksi, from Equation

(3.25). Refer to Figure 3.6, which shows the simplified model that requires to be analyzed for

the case of a four span bridge.

Figure 3. 6 Model of the equivalent elastic system under transverse response

From the equivalent elastic analysis a revised displaced shape of the bridge is obtained as well

as a revised proportion of load carried by superstructure bending. Iteration may be required if

the revised displaced shape is not close enough to the initial assumption. Using the updated

displaced shape and percentage of load carried by superstructure bending, SS, obtained

summing the reactions at the abutments and then dividing it by the base shear force, VB, the

process is repeated until convergence of the displacements pattern is reached. Convergence is

normally obtained with little iteration, and the final flexural strength to be provided to the

individual columns is calculated from the final values of the vector of shear forces carried by

each pier, Vi.

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3.2.5 Additional notesThe iterative procedure is illustrated in Figure 3.7; it can be easily implemented in Matlab,

Math-Cad, Excel or any other programming software. Note that the convergence criterion is

based on the displacement pattern; therefore a good initial assumption will reduce the required

number of iterations.

Application of the procedure using hand calculations is also likely to be done. Good initial

approximations to the displaced shape can be achieved using the simply supported beam

model depicted in Figure 3.3, and Equation (3.1).

The method requires initial assumptions for the displaced shape and the proportion of load

carried by superstructure bending. As pointed out before, a parabolic shape can generally be

assumed. Initial displaced shape can be based on the deck first-mode displaced shape and an

initial approximation to the proportion of load carried by superstructure bending, SS, can be

obtained by applying the vector of displacements to the beam model shown in Figure 3.3.

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Figure 3. 7 Flowchart forDirect Displacement-Based Design of MDOF-bridges.

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3.3 APPLICATION TO DIFFERENT BRIDGE CONFIGURATIONS

The Direct Displacement-Based Designapproach, as presented in Section 3.2 and

summarized in Figure 3.7 was implemented in a Matlab subroutine and applied to eighteen

different series of bridge structures, which are shown in Figures 3.9 to 3.11.

Figure 3.9 shows the six Regular Bridges (4-span and 6-span) studied in Alvarez Botero

dissertation [Alvarez Botero, 2004], which are re-computed herein with the new damping

equation (3.7) [Grant et al., 2005]. Figures 3.10 and 3.11 shows twelve different series of

Irregular Bridge structures studied in the present dissertation.

Single column bents support the superstructure of the bridges. Pier heights were varied foreach series, makingH= {7.5 m; 10.0 m; 12.5 m and15.0 m}, resulting in 72 different bridge

designs. A single design limit state was considered and is represented by an arbitrarily chosen

drift limit of 4%.

3.3.1 Bridge Information and Assumptions

3.3.1.1 Materials:Concrete and reinforcing steel properties used for design purposes are presented in Table 3.1.

Table 3.1 - Material Properties for Design.

3.3.1.2 Abutments:Abutments are usually designed and detailed for service loads and are checked for seismic

performance. Normally, equivalent linear springs are used in structural models to simulate the

restrains of the superstructure provided by the abutments. The selection of equivalent springs

must reflect the dynamic behaviour of the soil behind the abutment, the structural componentsof the abutment and their interaction with the soil. Substantial nonlinear behaviour can be

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expected as some of the elements constituting the abutments may be subjected to significant

yielding, [Maroney and Chai, 1994]. However, for all the cases that have been included in this

work, the assumption of abutments responding elastically has been made. Nonlinearcharacterization of the abutments can be obtained from a pushover analysis and the results

incorporated to the design procedure without significant difficulties. Abutment stiffness was

then chosen to beKA = 75000 kN/m and an arbitrary value of 8% damping was associated to

their response. A limiting displacement of 100 mm is also specified for the abutments as an

additional design restriction.

3.3.1.3 Bridge Deck:Current design practice intends to avoid inelastic action in the bridge deck; therefore it is

considered to respond elastically. The superstructure engages the substructure elements in the

transverse direction with shear keys and inelastic action is intended to concentrate at the

bottom of the piers. Representative dimensions of a two-lane bridge deck are used. A typical

transverse section of the bridges is depicted in Figure 3.8, the deck transverse-moment of

inertia isIyy= 44 m4

and its torsional stiffness has been ignored. The distributed weight of the

bridge deck, including asphalt, is taken as Wdeck= 175 kN/m.

Figure 3. 8 Bridge Typical Transverse Section

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Figure 3. 9 Series of 4-span and 6-span Regular Bridges (H = 7.5 m, 10.0 m, 12.5m and 15.0 m)

25


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Figure 3. 10 Series of 4-span Irregular Bridges (H = 7.5 m, 10.0 m, 12.5m and 15.0 m)

26


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Figure 3. 11 Series of 6-span Irregular Bridges (H = 7.5 m, 10.0 m, 12.5m and 15.0 m)

27


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3.3.1.4 Piers and Cap Beam:Single circular-column bents constitute the substructure of the bridges under consideration.The superstructure is simply supported on cap beams at the top of the piers and under

transverse excitation the columns are engaged by means of shear keys, refer to Figure 3.8. As

repeatedly stated before, the inelastic action is intended to be restricted to carefully-detailed

plastic hinge regions at the bottom of the piers, and it is assumed that sufficient closely spaced

transverse reinforcement is provided to achieve a satisfactory performance.

Pier diameters of 2.0 m to 2.7 m were initially assumed based on typical bridge dimensions,

and adjusted, if required, based on the design results. Pier masses were consider in all cases,

lumping one third of the calculated mass at the top, according to Priestley, et al., 1996. Cap-

beam masses were also included, and were calculated based on the assigned dimensions foreach specific case; typical cap-beam dimensions are shown in Figure 3.8, and depth is taken

as D + 0.5 m, where D is the column diameter. Concrete unit weight is assumed to be

Wc= 25 kN/m3.

3.3.2 Seismic InputThe design seismic input is represented by the 5% damped displacement response spectrum,

ARS curve, from the Caltrans Seismic Design Criteria for soil profile C, magnitude 8.00.25,

0.7g PGA [Caltrans, 2001], reduced for the appropriate level of damping, according to

Equation (3.21).

The Caltrans displacement spectra have been cut off at a period of 4 seconds, however, since

the effective period (period at maximum response) can be, for some of the cases, above 4

seconds, the design spectrum has been extended up to 5 seconds as shown in Figure 3.12.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Period [s]

SpectralDisplaceme

nt[m]

5%

Figure 3. 12 Extended Caltrans displacement ARS curve for soil profile C, M = 8.00.25 and 0.7g PGA.

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3.3.3 Design ResultsResults obtained after applying the Direct Displacement-Based Design procedure, to the

series of bridges in Figures 3.9 to 3.11, are presented in this section. They are reported in aseries of figures and tables, showing the final design displaced shapes, the yielding and

limiting displacements for the piers, the displacement ductility demands, the required ultimate

strength and values of effective stiffness and the parameters of the substitute SDOF structure.

The critical member, for each of the cases, is identified from the design displacement pattern

plots, as the member that reaches the limiting displacement.

Pier diameter was initially estimated based on common bridge dimensions and modified,

according to preliminary design results, to obtain sections that can provide the required

strengths with reinforcement ratios between the limits, 0.5% to 4.0%, recommended inPriestley, et al., 1996, for circular piers. Columns of 2.0 m, 2.2 m, 2.5 m or 2.7 m were

specified. Figure 3.13 shows the interaction diagrams obtained for all the used pier diameters

using the minimum and maximum reinforcement ratios. Note that the aim is that all the piers

will have reinforcement ratios between the limits. However, for few cases reinforcement

ratios well up the maximum limit will be obtained and in some others reinforcement ratios

below the minimum limit would be enough to provide the required strength. For these cases

the minimum or maximum steel content criterion was ignored and the required flexural

strength, as coming from the design process, used in the time-history analyses.

Figure 3. 13 Interaction Diagrams for piers.

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3.3.3.1 Series of Regular BridgesAs was previously said, the first six Series are the 4-span and 6-span bridges studied byAlvarez Botero in his dissertation [Alvarez Botero, 2004]. Here are being re-named as Series

1, 2 and 3 those characterized for have 4-span, and Series 4, 5 and 6 those having 6-span.

The main difference in the SDOF results, shown in Table 3.2, compared with the previous

ones [Alvarez Botero, 2004], is the reductions of the equivalent system damping, sys, due to

the application of the new damping equation (3.7). This leads to an increased SDOF base

shear demand, VB, and consequently, in the MDOF system, leads to an increment in the piers

strength to achieve the target displacement requirements of the direct displacement-based

design method.

Bridges results of Series 1 to 6 are presented in Table 3.2 and Figure 3.14 to 3.19.

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Table 3.2 Substitute SDOF parameters for bridges of Series 1 to 6.

H sys d Meff Teff Keff VB SS[m] [%] [m] [ton] [s] [kN/m] [kN] [%]

7.5 14.5 0.24 2704 1.06 95267 22550 13.9

10.0 13.3 0.32 2823 1.37 59756 18990 24.7

12.5 11.4 0.40 2864 1.63 42670 17003 35.9

15.0 9.8 0.48 2900 1.87 32730 15681 47.7

7.5 13.0 0.37 2727 1.57 43517 16048 30.2

10.0 10.2 0.48 2777 1.92 29753 14402 49.2

12.5 7.9 0.60 2815 2.23 22434 13475 68.4

15.0 5.9 0.72 2848 2.52 17708 12710 89.0

7.5 11.5 0.24 2953 1.01 114819 27761 19.5

10.0 9.7 0.32 2978 1.23 78302 25148 25.5

12.5 9.1 0.40 3021 1.51 52191 20941 36.7

15.0 8.4 0.48 3062 1.78 38042 18282 48.3

7.5 11.9 0.23 3085 0.99 124482 29101 -16.9

10.0 11.1 0.31 3535 1.23 92614 28454 -16.3

12.5 12.4 0.39 4109 1.62 61924 23877 -7.2

15.0 12.1 0.47 4307 1.97 43882 20495 -2.0

7.5 14.2 0.45 3937 1.99 39353 17523 -12.7

10.0 14.0 0.58 4196 2.72 22379 12870 0.8

12.5 11.8 0.71 4316 3.41 14626 10411 14.8

15.0 9.7 0.85 4396 4.07 10464 8900 30.1

7.5 12.7 0.24 4467 1.04 163251 39649 2.5

10.0 11.7 0.33 4647 1.34 102338 33350 7.4

12.5 11.1 0.41 4805 1.64 70497 28559 10.8

15.0 10.9 0.48 4858 1.96 49889 24046 15.0

Series 1

Series 2

Series 3

Series 4

Series 5

Series 6

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H = 7.5 m

A1 P1 P2 P3 A2

H[m] - 7.50 7.50 7.50 -

D[m] - 2.00 2.00 2.00 -

Mass[ton] 357 875 965 875 357

D - 3.57 5.59 3.57 - [%] 8.00 15.35 17.29 15.35 8.00V [kN] 1571 6470 6471 6470 1571

M [kN*m] - 48526 48529 48526 -

Keff[kN/m] 75000 33737 21564 33737 75000

H = 10.0 m

A1 P1 P2 P3 A2

H[m] - 10.00 10.00 10.00 -

D[m] - 2.50 2.50 2.50 -

Mass[ton] 357 908 997 908 357

D - 3.53 5.39 3.53 - [%] 8.00 15.29 17.15 15.29 8.00V [kN] 2349 4767 4768 4767 2349

M [kN*m] - 47672 47679 47672 -

Keff[kN/m] 75000 18156 11910 18156 75000

H =12.5 m

A1 P1 P2 P3 A2

H[m] - 12.50 12.50 12.50 -

D[m] - 2.50 2.50 2.50 -

Mass[ton] 357 918 1007 918 357

D - 2.91 4.38 2.91 - [%] 8.00 14.22 16.31 14.22 8.00V [kN] 3054 3633 3633 3633 3054

M [kN*m] - 45413 45414 45413 -

Keff[kN/m] 75000 10954 7264 10954 75000

H = 15.0 m

A1 P1 P2 P3 A2

H[m] - 15.00 15.00 15.00 -

D[m] - 2.50 2.50 2.50 -

Mass[ton] 357 928 1018 928 357

D

- 2.46 3.69 2.46 -

[%] 8.00 13.20 15.51 13.20 8.00V [kN] 3745 2732 2732 2732 3745

M [kN*m] - 40976 40977 40976 -

Keff[kN/m] 75000 6820 4550 6820 75000

Displacement Pattern Limit Limit Yield

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

Figure 3. 14 Design results for bridges of Series 1.

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H = 7.5 m

A1 P1 P2 P3 A2

H[m] - 7.50 15.00 7.50 -

D[m] - 2.00 2.00 2.00 -

Mass[ton] 357 875 985 875 357

D - 5.59 2.29 5.59 - [%] 8.00 17.29 12.70 17.29 8.00V [kN] 2421 4482 2241 4482 2421

M [kN*m] - 33612 33611 33612 -

Keff[kN/m] 75000 14942 4823 14942 75000

H = 10.0 m

A1 P1 P2 P3 A2

H[m] - 10.00 20.00 10.00 -

D[m] - 2.00 2.00 2.00 -

Mass[ton] 357 882 998 882 357

D - 4.31 1.70 4.31 - [%] 8.00 16.23 10.44 16.23 8.00V [kN] 3541 2929 1465 2929 3541

M [kN*m] - 29290 29290 29290 -

Keff[kN/m] 75000 7320 2414 7320 75000

H =12.5 m

A1 P1 P2 P3 A2

H[m] - 12.50 25.00 12.50 -

D[m] - 2.00 2.00 2.00 -

Mass[ton] 357 889 1011 889 357

D - 3.51 1.36 3.51 - [%] 8.00 15.25 8.38 15.25 8.00V [kN] 4611 1702 851 1702 4611

M [kN*m] - 21278 21279 21278 -

Keff[kN/m] 75000 3403 1135 3403 75000

H = 15.0 m

A1 P1 P2 P3 A2

H[m] - 15.00 30.00 15.00 -

D[m] - 2.00 2.00 2.00 -

Mass[ton] 357 895 1025 895 357

D

- 2.95 1.13 2.95 -

[%] 8.00 14.32 6.44 14.32 8.00V [kN] 5657 557 279 557 5657

M [kN*m] - 8361 8361 8361 -

Keff[kN/m] 75000 930 312 930 75000


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Disp

lacements[m]


33


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H = 7.5 m

A1 P1 P2 P3 A2

H[m] - 15.00 7.50 15.00 -

D[m] - 2.50 2.50 2.50 -

Mass[ton] 357 928 986 928 357

D - 1.30 6.99 1.30 - [%] 8.00 7.94 18.05 7.94 8.00V [kN] 2699 5589 11178 5589 2699

M [kN*m] - 83839 83832 83839 -

Keff[kN/m] 75000 26450 37271 26450 75000

H = 10.0 m

A1 P1 P2 P3 A2

H[m] - 20.00 10.00 20.00 -

D[m] - 2.50 2.50 2.50 -

Mass[ton] 357 949 997 949 357

D - 0.97 5.39 0.97 - [%] 8.00 5.00 17.15 5.00 8.00V [kN] 3209 4621 9481 4621 3209

M [kN*m] - 92416 94811 92416 -

Keff[kN/m] 75000 16648 23713 16648 75000

H =12.5 m

A1 P1 P2 P3 A2

H[m] - 25.00 12.50 25.00 -

D[m] - 2.50 2.50 2.50 -

Mass[ton] 357 970 1007 970 357

D - 0.78 4.38 0.78 - [%] 8.00 5.00 16.31 5.00 8.00V [kN] 3842 2913 7427 2913 3842

M [kN*m] - 72830 92840 72830 -

Keff[kN/m] 75000 8415 14857 8415 75000

H = 15.0 m

A1 P1 P2 P3 A2

H[m] - 30.00 15.00 30.00 -

D[m] - 2.50 2.50 2.50 -

Mass[ton] 357 991 1018 991 357

D

- 0.65 3.69 0.65 -

[%] 8.00 5.00 15.51 5.00 8.00V [kN] 4413 1869 5717 1869 4413

M [kN*m] - 56060 85755 56060 -

Keff[kN/m] 75000 4524 9531 4524 75000


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180

Position [m]

Displacements[m]


34


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H = 7.5 m

A1 P1 P2 P3 P4 P5 A2

H[m] - 7.50 7.50 7.50 7.50 7.50 -

D[m] - 2.20 2.20 2.20 2.20 2.20 -

Mass[ton] 357 884 973 973 973 884 357

D - 0.73 4.10 6.15 4.10 0.73 - [%] 8.00 5.00 16.01 17.63 16.01 5.00 8.00V [kN] -2460 5575 7624 7625 7624 5575 -2460

M [kN*m] - 41813 57180 57185 57180 41813 -

Keff[kN/m] 75000 156289 38074 25411 38074 156289 75000

H = 10.0 m

A1 P1 P2 P3 P4 P5 A2

H[m] - 10.00 10.00 10.00 10.00 10.00 -

D[m] - 2.50 2.50 2.50 2.50 2.50 -

Mass[ton] 357 908 997 997 997 908 357

D - 1.04 3.85 5.39 3.85 1.04 - [%] 8.00 5.44 15.71 17.15 15.71 5.44 8.00V [kN] -2312 6614 6617 6618 6617 6614 -2312

M [kN*m] - 66138 66170 66175 66170 66138 -

Keff[kN/m] 75000 85920 23152 16538 23152 85920 75000

H = 12.5 m

A1 P1 P2 P3 P4 P5 A2

H[m] - 12.50 12.50 12.50 12.50 12.50 -

D[m] - 2.50 2.50 2.50 2.50 2.50 -

Mass[ton] 357 91

Documents

Dissertation2007-OrtizRestrepo[1]