PASSIVE CONTROL OF HIGHWAY STRUCTURES - … · PASSIVE CONTROL OF HIGHWAY STRUCTURES: ... University of California at Irvine ... lead to the design of a safe and economical structure

Kajima - CUREe Research Project

PASSIVE CONTROL OF HIGHWAY STRUCTURES: -Use of Damped Resonant Appendages to Augment Damping in

Cable-Stayed Bridges:A Feasibility Study-

LIM

Mr. Tetsuo Takeda

Mr. Seiji Tokuyarna

Mr. Masaorni lizuka

Mr. Toshirnichi Ichinorniya

Mr. Tomohiko Arita

Mr. Katsuhisa Kanda

Mr. Kazuhiko Yamada

Mr. Hachiro Ukon

Mr. Yoshihide Okirni

Mr. Hiroaki Okamoto

Prof. Ahmed M.Abdej-Ghaffar

Prof. Samj F. Masrj

Dr. Hosam-Eddin Mi

Prof. Roberto Villaverde

Mr. Scott C. Martin

1992.8

Kajima Corporation CUREe

Kajima Corporation

CUREe;

California Universities for Research

in Earthquake Engineering

* Kajima Institute of Construction Technology

* Information Processing Center

* Structual Department, Architectual

Design Division * Civil Engineering Design Division

* Kobori Research Complex

* The California Institute of Technology

* Stanford University

* The University of California, Berkeley

* The University of California, Davis

* The University of California, Irvine

* The University of California, Los Angeles

* The University of California, San Diego

* The University of Southern California

SUMMARY REPORT CUREe-KAJIMA RESEARCH PROJECT

ON PASSIVE CONTROL OF HIGHWAY (BRIDGES) STRUCTURES

ANALYTICAL AND EXPERIMENTAL STUDIES OF THE EFFECTIVENESS OF DAMPING-AUGMENTATION DEVICES

IN CABLE-STAYED BRIDGES

OVERALL SUMMARY PROJECT OBJECTIVE & SCOPE

Report Prepared by

Professor Ahmed M. Abdel-Ghaffar, Team Leader

RESEARCH TEAMS

University of Southern California (USC); Profs. A.M.Abdel-Ghaffar S.F.Masri and Dr. Hossam E. Au

University of California at Irvine (UCI): Prof. Roberto Vilaverde Mr. Scott C. Martin

Kajima Technical Research Institute(KTRI): Mr. T. Takeda Mr. T. Ichinomiya Mr. Y.Okimi Mr. S. Tokuyama

SUMMARY

The objective of this project is to introduce recent development, in seismic-counter measure techniques to assess analytical and experimental effectiveness, feasibility and limitations of damping augmentation devices, such as lead rubber bearings, elastometric bearings and damped resonant appendages, with respect to the seismic performance of cable-stayed bridges. To achieve this objective, emphasis was placed on: (1) the results of the USC two shake tables (4 ft x4 ft with separating distance 12 ft center-to-center) tests of a 1:100 reduced scale model of an existing Japanese cable-stayed bridge; (the Yobuko Bridge) tests were carried out without the energy absorption device and with the hysteric energy dissipation device, (2) the results of damped resonant (tuned) appendages with relatively small mass and high damping ratio using a pair of small shaking tables (at UCI) to longitudinally shake the 12-ft long bridge model of the same cable stayed in Part 1, and (3) the results of computational analysis of the prototype by the Kajima research team to improve the earthquake resistance of floating type prestressed concrete cable-stayed bridges by introducing passive vibration systems such hysteretic type devices and tuned mass damper devices.

GENERAL INTRODUCTIO AND OVERVIEW

Rapid progress has been made over the past twenty years in the design techniques for cable-stayed bridges; this progress is largely due to the use of electronic computers, the development of box girders with orthotropic plate decks, and the manufacturing of high strength wires that can be used for cables. This progress has also let to increased competition among the bridge engineers in Japan, Europe and the United States. Cable-stayed bridges are now entering a new era, reaching to medium and long span lengths with a range of 1300 ft (400m) to 3000 ft (1000rn) for the center span.

Cable-stayed bridges are increasing in numbers and popularity. This, in addition to the increase in the span lengths of these flexible structures raise many concerns about their behavior under environmental dynamic loads such as wind, earthquake and service loads such as vehicular traffic-loads. From the analysis of various observational data, including ambient forced vibration test of cable-stayed bridges, it is known that these bridges have very small mechanical or structural damping (0.3% -2%). Moreover these bridges occasionally experience extreme loads, especially during a strong earthquake or in a high wind environment. For such circumstances, the response should be controlled within certain limits for serviceability (human comfort) and for safety (risk of damage of failure).

For typical span highway bridges. Modern seismic bridge codes and provisions have now been developed to the point where the basic earthquake-resistant requirements to be imposed on a "standard" bridge are specified adequately, and intelligent consideration of these requirements will lead to the design of a safe and economical structure. For new cable-stayed bridges, however, the provisions of the highway bridge seismic codes may not be applicable, and accordingly, there is an urgent need to develop general seismic design guidelines tailored especially for these bridges and based on research, experimental studies and full-scale observational data. Furthermore, due to the large displacements and member forces induced by strong ground shaking in this type of structure, energy absorption devices and special bearings should be provided at the supporting points to dissipate seismic energy, thus assuring the serviceability of the bridge.

The response of a cable-stayed bridge to applied loads is highly dependent on the manner in which the bridge deck is connected to the towers. If the deck is swinging freely at the towers, the induced seismic forces will be kept to minimum values, but the bridge may be very flexible under service loading conditions (i.e. dead loads and live loads). On the other hand, a rigid connection between the deck and the towers will result in reduced movements under service loading conditions but will attract much higher seismic forces during an earthquake. Therefore, it is extremely important to provide special bearings or devices at the deck-tower connections to absorb the' large seismic energy and reduce the response amplitudes. Good examples of these devices, which make it possible to control the natural period of vibration, are rubber-lead block bearings elastic links, spring shoes and elasto-meric bearings. These devices should be dimensioned so that they provide adequate stiffness high enough to produce acceptable performance under day-to-day service conditions, yet soft enough to prevent high seismic inertial forces from being transmitted to the towers from the deck. These devices should also constitute a multi-defense line; that is, they should be composed of different, tough structural subsystems which are interconnected by very tough structural elements.(structural fuses) whose inelastic behavior would permit the whole bridge to fmd its way of the critical range of dynamic response.

Long span prestressed concrete cable-stayed bridges are often designed as a floating structure in which the girder is not supported by bearing but suspended by only cables. The aim of the use of this type of structure is to reduce the inertial force of the girder by extending the natural period of the structure in the longitudinal direction. This structure, however, has some drawbacks too, such as larger horizontal displacements of the girder and larger bending moment of the tower

2

than in bridges where the girder and the main pier are rigidly connected.

The goal of this study is to solve these problems and improve the earthquake resistance of floating-type prestressed concrete cable-stayed bridges by introducing passive vibration control systems.

PROJECT ACCOMPLISHMENTS

1. Analytical, Three Dimensional Modeling, of the Bridge and the Devices

Analytically, three-dimensional modeling is developed for both the bridges and damping devices including the bridge geometrical large-displacement global nonlinearity and the local material and geometric nonlinearity of the energy dissipation and hysteretic type devices. The effect of multiple support seismic excitation and various modeling and design parameter factors of the bridge response are also studied, including the properties, modeling accuracy and location of the devices along he bridge superstructure. These damping devices are provided at the critical connections to dissipate seismic energy due to the large displacements and member forces induced by strong ground shaking, thus assuring the serviceability of these cable-supported bridges. Computer codes of the above were developed.

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A thorough review was perfonned of seismic design of cable-stayed bridges and how effective tune appendages with a relatively small mass and a high damping ratio can be to increase the inherent damping in cable-stayed bridges and reduce, thus, their response to earthquake excitations. The study involves the review of theoretical formulation that explains why addition of these appendages may improve the damping characteristics of a structure, and numerical and experimental tests conducted to assess the validity of this theoretical information and the extent to which they can reduce the seismic response of the bridge.

3. Excitation-Resnunse Data with the Danwers Installed and Without

3.1 USC: Experimentally, the seismic response characteristics of a 1:100-scale model of a cable-stayed bridge (similar to the PC Yobuko cable-stayed bridge in Japan) with and without supplementary hysteretic and viscoelastic dampers are studied utilizing the USC dual shake table system. Each side span (with the tower and anchor-pier) is supported on one table. The major emphasis of the experiments and measurements is placed on the effectiveness of these dampers. The shaking was in the lateral (transverse) direction, however large order of magnitudes of the longitudinal response were recorded.

3.2 UCI: In the experimental test, a 12-foot long cable-stayed bridge and an appendage consisting of a small mass, a small spring, and a small viscous damper are built and tested, without and with the appendage, on a pair of shaking tables set to reproduce specified ground acceleration records from the past earthquakes. The damping ratio of the appendages in this test is 32 per cent and its weight represents 8 percent of the weight of the bridge model. The shaking was made in the longitudinal direction of the bridge.

3

4. Parameteric Analysis

A parameteric analysis was conducted, by the Kajima research team, to determine to what extent then passive vibration control systems could reduce earthquake response and to define system requirements for effective reduction of earthquake response.

The prestressed concrete cable-stayed bridge analyzed is the Yobuko Bridge, which has the longest span (250in) in Japan. The passive vibration control systems considered are a hysteretic damper and a tuned mass damper. The hysteretic damper is something like a lead rubber damper, and it was installed at the connections of the girder with the main piers and the end piers. The tuned mass damper is a mass and spring system with damper tuned to the natural frequency of the girder, which is to be installed on the girder.

The work performed in this project (particularly the experimental work) has pointed out specific strengths and weaknesses of currently employed seismic design and verification (analysis) procedures for cable-stayed bridges. It is shown that:

1. An optimum model of the seismic performance of the bridges with these passive control devices can be obtained by balancing the reduction in forces along the bridge against tolerable displacements. Thus it is concluded that appropriate locations and hysteretic energy-dissipation properties of the devices can achieve a significant reduction is seismic-induced forces, as compared to the case with no dampers added, and relatively better control displacements. In addition, proper selection of the location of the control system can help redistribute forces along the structure which may provide solutions for retrofitting some existing bridges. However, caution should be exercised in simulating the device response for a reliable bridge structural performance. Moreover, while seismic response of the bridge can be significantly improved with added dampers, their degree of effectiveness also depends on the energy absorption characteristics of the dampers.

2. The appendages reduce, analytically the longitudinal response of the bridge deck of the analyzed bridge up to about 86 per cent. Similarly, in the experimental test the appendage reduces the longitudinal bridge deck response about 41 per cent. it is concluded, thus, that the suggested appendages may indeed be effective in reducing the response of cable-stayed bridges to seismic disturbances,and that they have the potential to be come a competitive alternative for their seismic design. In the numerical study, an actual cable-stayed bridge is modeled with finite elements and analyzed with and without the proposed appendages under different earthquake ground motions. Appendages with damping ratios of 10, 15, 20, and 30 per cent and weight that respectively represent 0.67, 1.5, 2.7, and 6.0 per cent of the total weight of the bridge are considered.

3. Analytical findings obtained from the study (by the Kajima team) can be summarized as follows:

3.1 Structure with Hysteresis Type Dampers Use of the hysteresis damper in a floating-type PC cable-stayed bridge effectively reduces the displacement of the girder and the bending moment of the tower. There exists an optimum stiffness and yielding strength of the damper, which absorbs hysteretic energy and reduces the response of the structure most effectively.

In

3.2 Tuned Mass Damper (TMD) The optimum damping factor of TMD is about 5%. Higher mass ratios are more effective. TMD with a mass ratio of about 10% effectively reduces response displacements of the girder to 80%. TMD is less effective against seismic waves with its peak in early stage because a peak of response comes before TMD works. TMD is effective even if the natural period of the structure changes because of plasticification of members during a major earthquake.

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Strong shaking in the longitudinal direction of USC shake tables.

Testing more energy dissipation devices at high amplitude ground shaking.

3 The damped resonant appendages may be an effective method to reduce the response of cable-stayed bridges to earthquake ground motions. Nonetheless, extensive further research is needed before they can be implemented into practice. Among others, additional studies are required to investigate: (a) their effectiveness in medium to large scale models, under three-dimensional ground motions, at various locations along the deck and towers of the bridge, and in bridges with significant higher and closely-spaced modes; (b) the importance of uncertainties in bridge and appendage parameters; (c) the behavior of the bridge-appendage system under excitations that load the bridge beyond its elastic range; and (d) the effectiveness of multiple single-degree-of-freedom and single multi-degree-of-freedom appendages. Likewise, additional work is needed to develop: (a) reliable, functional, and economical prototype appendages; (b) practical configurations for their installation; and (c) guidelines for the seismic design of bridges with damped resonant appendages.

5

CUREe-KAJIMA RESEARCH PROJECT

PASSIVE CONTROL OF HIGHWAY BRIDGES

USE OF DAMPED RESONANT APPENDAGES TO AUGMENT DAMPING IN CABLE- STAYED BRIDGES: A FEASIBILITY STUDY

Roberto Villaverde and Scott C. Martin

Department of Civil Engineering

University of California, Irvine

July 15, 1991 - July 14, 1992

ABSTRACT

A study is carried out to investigate how effective tuned appendages with a

relatively small mass and a high damping ratio can be to increase the inherent

damping in cable-stayed bridges and reduce, thus, their response to earthquake

excitations. The study involves the review of a theoretical formulation that

explains why the addition of these appendages may improve the damping characteris-

tics of a structure, and numerical and experimental tests conducted to assess the

validity of this theoretical formulation and the extent to which they can reduce the

seismic response of cable-stayed bridges. In the numerical study, an actual cable-

stayed bridge is modeled with finite elements and analyzed with and without the

proposed appendages under different earthquake ground motions. Appendages with

damping ratios of 10, 15, 20, and 30 per cent and weights that respectively

represent 0.67, 1.5, 2.7, and 6.0 per cent of the total weight of the bridge are

considered. In the experimental test, a 12-foot long cable-stayed bridge and an

appendage consisting of a small mass, a small spring, and a small viscous damper are

built and tested, without and with the appendage, on a pair of shaking tables set

to reproduce specified ground acceleration records from past earthquakes. The

damping ratio of the appendage in this test is 32 per cent and its weight represents

8 per cent of the weight of the bridge model. In the numerical test, it is found

that the appendages reduce the longitudinal response of the bridge deck of the

analyzed bridge up to about 86 per cent. Similarly, in the experimental test the

appendage reduces the longitudinal bridge deck response about 41 per cent. It is

concluded, thus, that the suggested appendages may indeed be effective in reducing

the response of cable-stayed bridges to seismic disturbances, and that they have the

potential to become a competitive alternative for their seismic design.

11

TABLE OF CONTENTS

ABSTRACT

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

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1: INTRODUCTION .......................... 1

1.1 Background ........................... i

1.2 Historical Review ........................ 3

1.3 Object and Scope ........................ 5

1.4 Organization .......................... 6

Chapter 2: THEORETICAL BASIS ........................ 7

2.1 Introduction ........................... 7

2.2 Natural Frequencies and Damping Ratios of Systems Without

Classical Damping ........................ 7

2.3 Damping Ratios and Natural Frequencies of Structure-Appendage

Systems ............................ 10

2.4 Parameters of Effective Resonant Appendages .......... 13

Chapter 3: NUNERICAL STUDY ........................ 15

3.1 Introductory Remarks ...................... 15

3.2 Analyzed Bridge ........................ 15

3.3 Finite Element Model ...................... 15

3.4 Bridge Dynamic Properties ................... 16

3.5 Parameters and Location of Resonant Appendage ......... 16

3.6 Dynamic Properties of Bridge-Appendage System ......... 18

3.7 Earthquake Ground Motions ................... 19

3.8 Results ............................ 19

d

111

Chapter 4: EXPERIMENTAL STUDY

21

4.1 Introduction

21

4.2 Model Description ....................... 21

4.2.1 Structural Model .................... 21

4.2.2 Model Analytical Modal Properties

22

4.2.3 Appendage Model

23

4.2.4 Analytical Modal Properties of Bridge-Appendage System

24

4.3 Equipment

24

4.3.1 Introductory Remarks .................. 24

4.3.2 Accelerometers ..................... 24

4.3.3 HP3562A Dynamic Signal Analyzer

25

4.3.4 Shaking Tables ..................... 25

4.3.5 Modal 3.0 SE System

26

4.4 Experimental Dynamic Properties

26

4.4.1 Introductory Remarks .................. 26

4.4.2 Bridge Model ...................... 26


27

4.5 Experimental Set-Up 27

4.6 Base Acceleration Time Histories ................ 28

4.7 Experimental Results ...................... 28

4.7.1 Introduction ...................... 28

4.7.2 Test with no Phase Lag ................. 29

4.7.3 Test with Phase Lag 30

4.7.4 Test with Shock-Type Base Motion ............ 30

4.7.5 Test with Slightly Out-of-Tune Appendage ........ 31

4.7.6 Test with Tuning According to Den Hartog's Formula . 31

4.7.7 Test with appendage glued to bridge deck

32

iv

Chapter 5: SUMMARY AND CONCLUSIONS 34

5.1 Summary 34

5.2 Conclusions 34

5.3 Feasibility Assessment ..................... 35

5.4 Recommendations for Future Research 37

REFERENCES 38

ACKNOWLEDGMENTS .............................. 40

V

LIST OF TABLES

Table Page

3.1. Properties of structural elements in simplified model of Yobuko Bridge ............................... 41

3.2. Parameters of appendages in simulation study ............ 41

3.3. Maximum longitudinal displacements of bridge deck end without and with appendage .............................. 42

3.4. Maximum longitudinal displacements of tower top without and with appendage ............................... 42

4.1. Characteristics of bridge model components ............. 43

4.2. Prestressing tensions in cables of bridge model ........... 43

4.3. Design parameters of appendage model ................ 44

4.4. Parameters cf appendage experimental model ............. 44

4.5. Maximum longitudinal accelerations of bridge deck end and left tower top without and with appendage ................... 44

4.6. Maximum vertical accelerations of bridge deck center without and with appendage .............................. 45

4.7. Maximum longitudinal accelerations of bridge end without and with appendage in test with phase lag .................. 45

4.8. Maximum longitudinal accelerations of bridge deck center without appendage and with appendage slightly out of tune ......... 46

4.9. Parameters of appendage model tuned with Den Hartog's formula . . . 46

4.10. Maximum longitudinal accelerations of bridge deck center without and with appendage in test with tuning according to Den Hartog's formula 47

4.11. Maximum longitudinal accelerations of bridge deck center without appendage and with appendage glued to bridge deck .......... 47

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vi

LIST OF FIGURES

Figure Page

3.1. Elevation of simplified bridge model in simulation study ...... 48

3.2. Simplified model of pier and tower of bridge in simulation study 49

3.3. Natural frequency and mode shape in first mode of Yobuko bridge 50

3.4. Natural frequency and mode shape in second mode of Yobuko bridge 51

3.5. Natural frequency and mode shape in third mode of Yobuko bridge 52

3.6. Natural frequency and mode shape in fourth mode of Yobuko bridge 53

3.7. Natural frequency and mode shape in fifth mode of Yobuko bridge 54

3.8. Natural frequency and mode shape in third mode of Yobuko bridge with 10 per cent damping appendage ..................... 55

3.9. Natural frequency and mode shape in fifth mode of Yobuko bridge with 10 per cent damping appendage ..................... 56

3.10. Natural frequency and mode shape in third mode of Yobuko bridge with 15 per cent damping appendage .................... 57

3.11. Natural frequency and mode shape in fifth mode of Yobuko bridge with 15 per cent damping appendage .................... 58

3.12. Natural frequency and mode shape in third mode of Yobuko bridge with 20 percent damping appendage .................... 59


3.14. Natural frequency and mode shape in second mode of Yobuko bridge with 30 per cent damping appendage .................... 61


3.16. First ten seconds of N-S ground acceleration record of May 18, 1940, El Centro earthquake ........................ 63

3.17. E-W ground acceleration record at Foster City of October 17, 1989, Loma Prieta earthquake ....................... 64

3.18. E-W ground acceleration record at UC Santa Cruz of October 17, 1989, Loma Prieta earthquake .......................65

3.19. Response spectra for 0, 1, 5, 10 and 20 per cent damping of 1940 El Centro accelerogram .........................66

3.20. Response spectra for 0, 1, 5, 10 and 20 per cent damping of 1989 Foster City................................67

vii

3.21. Response spectra for 0, 1, 5, 10 and 20 per cent damping of 1989 UC Santa Cruz accelerogram .......................68

3.22. Displacement response of deck end of Yobuko bridge with no appendage under the 1940 El Centro earthquake .................69

3.23. Displacement response of deck end of Yobuko bridge with 10 per cent damping appendage under the 1940 El Centro earthquake .........70

3.24. Displacement response of deck end of Yobuko bridge with 15 per cent damping appendage under the 1940 El Centro earthquake ........71



3.27. Displacement response of tower top of Yobuko bridge with no appendage under the 1940 El Centro earthquake .................74

3.28. Displacement response of tower top of Yobuko bridge with 10 per cent damping appendage under the 1940 El Centro earthquake ........75




3.32. Displacement response of deck end of Yobuko bridge with no appendage under the 1989 Foster City earthquake ................79

3.33. Displacement response of deck end of Yobuko bridge with 10 per cent damping appendage under the 1989 Foster City earthquake .......80




3.37. Displacement response of tower top of Yobuko bridge with no appendage under the 1989 Foster City earthquake ................84

3.38. Displacement response of tower top of Yobuko bridge with 10 per cent damping appendage under the 1989 Foster City earthquake .......85

viii




3.42. Displacement response of deck end of Yobuko bridge with no appendage under the 1989 U.C. Santa Cruz earthquake ..............89

3.43. Displacement response of deck end of Yobuko bridge with 10 per cent damping appendage under the 1989 U.C. Santa Cruz earthquake .....90




3.47. Displacement response of tower top of Yobuko bridge with no appendage under the 1989 U.C. Santa Cruz earthquake ..............94

3.48. Displacement response of tower top of Yobuko bridge with 10 per cent damping appendage under the 1989 U.C. Santa Cruz earthquake .....95




3.52. Relative displacement of 20 per cent damping appendage in Yobuko bridge under the 1940 El Centro earthquake .............99

3.53. Relative displacement of 20 per cent damping appendage in Yobuko bridge under the 1989 Foster City earthquake ............100

3.54. Relative displacement of 20 per cent damping appendage in Yobuko bridge under the 1989 U.C. Santa Cruz earthquake ..........101

4.1. Configuration and dimensions of experimental bridge model ......102

4.2. Section A-A' through experimental bridge model ...........103

4.3. Side view of abutments and detail of deck end to abutment connection 104

4.4. Analytical first natural frequency and mode shape mode of experimental model ...............................105

ix

t

4.5. Analytical second natural frequency and mode shape mode of experimental model................................ 106

4.6. Analytical third natural frequency and mode shape mode of experimental model............................... 107

4.7. Analytical fourth natural frequency and mode shape mode of experimental model ............................... 108

4.8. Approximate dimensions of appendage model .............. 109

4.9. Photograph of resonant appendage model ............... 110

4.10. Photograph of damper in appendage model .............. 111

4.11. Analytical first natural frequency and mode shape mode of experimental model with resonant appendage .................... 112

4.12. Analytical second natural frequency and mode shape mode of experimental model with resonant appendage .................. 113

4.13. Analytical third natural frequency and mode shape mode of experimental model with resonant appendage .................... 114

4.14. Analytical fourth natural frequency and mode shape mode of experimental model with resonant appendage .................. 115

4.15. Analytical fifth natural frequency and mode shape mode of experimental model with resonant appendage .................... 116

4.16. Natural frequency, damping ratio, and mode shape in first mode of experimental mode .......................... 117

4.17. Natural frequency, damping ratio, and mode shape in second mode of experimental mode .......................... 118

4.18. Natural frequency, damping ratio, and mode shape in third mode of experimental model ......................... 119

4.19. Experimental bridge model on shaking tables ............ 120

4.20. Experimental bridge model with attached appendage ......... 121

4.21. Schematic equipment arrangement for determination of dynamic properties ................................ 122

4.22. Schematic equipment arrangement for response analysis ....... 123

4.23. Attachment of appendage to experimental bridge model ........ 124

4.24. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.5 times El Centro earthquake ....... 125

4.25. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.5 times El Centro earthquake ......... 126

Ki

4.26. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.7 times Foster City earthquake ......127

4.27. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.7 times Foster City earthquake ........128

4.28. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.4 times UC Santa Cruz earthquake .....129

4.29. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.4 times UC Santa Cruz earthquake .......130

4.30. Longitudinal acceleration response of tower top of bridge model without appendage under 0.5 times El Centro earthquake .......131

4.31. Longitudinal acceleration response of tower top of bridge model with appendage under 0.5 times El Centro earthquake ...........132

4.32. Longitudinal acceleration response of tower top of bridge model without appendage under 0.7 times Foster City earthquake ......133

4.33. Longitudinal acceleration response of tower top of bridge model with appendage under 0.7 times Foster City earthquake ..........134

4.34. Longitudinal acceleration response of tower top of bridge model without appendage under 0.4 times UC Santa Cruz earthquake .....135

4.35. Longitudinal acceleration response of tower top of bridge model with appendage under 0.4 times UC Santa Cruz earthquake .........136

4.36. Vertical acceleration response of left deck center of bridge model without appendage under 0.5 times El Centro earthquake .......137

4.37. Vertical acceleration response of left deck center of bridge model with appendage under 0.5 times El Centro earthquake .........138

4.38. Vertical acceleration response of left deck center of bridge model without appendage under 0.7 times Foster City earthquake ......139

4.39. Vertical acceleration response of left deck center of bridge model with appendage under 0.7 times Foster City earthquake ........140

4.40. Vertical acceleration response of left deck center of bridge model without appendage under 0.4 times UC Santa Cruz earthquake .....141

4.41. Vertical acceleration response of left deck center of bridge model with appendage under 0.4 times UC Santa Cruz earthquake .......142

4.42. Longitudinal acceleration response of appendage mass under 0.5 times El Centro earthquake ........................143

4.43. Longitudinal acceleration response of appendage support under 0.5 times El Centro earthquake .....................144

4.44. Longitudinal acceleration response of appendage mass under 0.7 times Foster City earthquake ........................145

xi

4.45. Longitudinal acceleration response of appendage support under 0.7 times Foster City earthquake ....................146

4.46. Longitudinal acceleration response of appendage mass under 0.4 times UC Santa Cruz earthquake ......................147

4.47. Longitudinal acceleration response of appendage support under 0.4 times UC Santa Cruz earthquake ...................148

4.48. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.5 times El Centro earthquake with phase lag of 0.4 sec .............................149

4.49. Longitudinal acceleration response of left deck end of bridge model IV with appendage under 0.5 times El Centro earthquake ...........150

4.50. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.7 times Foster City earthquake with phase lag of 0.4 sec .............................151

4.51. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.7 times Foster City earthquake ........ 152

4.52. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.4 times UC Santa Cruz earthquake with phase lag of 0.4 sec ........................... 153

4.53. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.4 times UC Santa Cruz earthquake ....... 154

4.54. Cut version of N-S 1940 El Centro ground acceleration record . . . . 155

4.55. Response spectra for 0, 1, 5, 10, and 20 per cent damping of ground acceleration record in Figure 4.54 ................. 156

4.56. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.5 times the cut version of El Centro earthquake ............................. 157

4.57. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.5 times the cut version of El Centro earthquake ................................ 158

4.58. Longitudinal acceleration response of deck center of bridge model without appendage under 0.6 times El Centro earthquake .......159

4.59. Longitudinal acceleration response of deck center of bridge model with out-of-tune appendage under 0.6 times El Centro earthquake .....160

4.60. Longitudinal acceleration response of deck center of bridge model without appendage under 0.8 times Foster City earthquake ......161

4.61. Longitudinal acceleration response of deck center of bridge model with out-of-tune appendage under 0.8 times Foster City earthquake . . . . 162

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4.62. Longitudinal acceleration response of deck center of bridge model without appendage under 0.5 times UC Santa Cruz earthquake .....163

4.63. Longitudinal acceleration response of deck center of bridge model with out-of-tune appendage under 0.5 times UC Santa Cruz earthquake . . . 164

4.64. Longitudinal acceleration response of deck center of bridge model with appendage tuned according to Den Hartog theory under 0.6 times El Centro earthquake ..........................165

4.65. Longitudinal acceleration response of deck center of bridge model with appendage tuned according to Den Hartog theory under 0.8 times Foster City earthquake ........................... 166

4.66. Longitudinal acceleration response of deck center of bridge model with appendage tuned according to Den Hartog theory under 0.5 times UC Santa Cruz earthquake ........................... 167

4.67. Longitudinal acceleration response of deck center of bridge model with appendage glued to bridge deck under 0.5 times El Centro earthquake . 168

4.68. Longitudinal acceleration response of deck center of bridge model with appendage glued to bridge deck under 0.7 times Foster City earthquake ................................ 169

4.69. Longitudinal acceleration response of deck center of bridge model with appendage glued to bridge deck under 0.4 times UC Santa Cruz earthquake ................................ 170

Chapter 1

INTRODUCTION

1.1 Background

Cable-stayed bridges have been proven to be aesthetically pleasant and

economical for medium to long spans. However, they are structural systems which

normally are extremely flexible and exhibit low damping values. As a result, the

extreme wind, earthquake, and traffic loads to which they may be subjected during

their life time may induced undesirable vibrations. For this type of structures,

therefore, the use of added damping devices or any other form of structural control

is a desirable alternative.

Several techniques have been suggested to control the response of structures

to dynamic loadir. Among the ones that are presently being actively investigated,

and in some cases implemented in actual or pilot buildings, are base isolation,

tuned mass dampers, active control, or active control in combination with any of the

other techniques, in which case the control system is referred to as a hybrid

system. All of these new techniques offer a great promise and, without doubt, will

offer viable and cost-effective solutions in the near future. Notwithstanding,

independently of how effective a control system might be, it is likely that the

design profession at large will only accept those innovations that do not represent

large departures from current, accepted practice [1]. In this sense, the most easily

accepted control system will be those that simply add to a conventional structure,

rather than those which require radical changes to the way structures are designed

and constructed.

Two forms of structural control which conform to this requirement of

simplicity are the addition of hysteretic, friction, or viscous dampers and the use

1

of tuned mass dampers. Hysteretic, friction, and viscous dampers seem to be an

effective way to reduce seismic response, but, cost aside, their main problem is

that they encumber the design procedure. The process of designing with added

damping devices is so interconnected with the design of the rest of the structure

that the total engineering effort is dramatically increased [19]. In contrast,

tuned mass dampers are conceptually easy to construct; can be considered in the

design of new, conventionally designed structures as well as old ones in need of

retrofit; can be effective to reduce both wind and traffic vibrations, do not

require radically different design procedures; do not depend on an external power

source; do not interfere with the principal vertical and horizontal load paths; can

be made to respond to small levels of excitation; and, seemingly, can be cost

effective. Furthermore, tuned mass dampers can be combined with sophisticated

schemes such as active control mechanisms to function as part of the active system

and as a back-up system in the case of the failure of the active one. Thus far,

however, there is no evidence that they can be effective to reduce the earthquake

response of cable-stayed bridges.

Previous studies [24, 15] have shown that if a small damped resonant appendage

that complies with some characteristics is attached to the top of a building

structure, the inherent damping in the structure is augmented, and as a result, its

response to earthquake excitations is reduced. These studies suggest thus that, at

least in principle, the attachment of a damped resonant appendage may also be an

effective way to augment the inherent damping in bridge structures and, hence, a

convenient method to reduce their response to earthquake ground motions.

Nonetheless, since the vibrational characteristics of bridges are different from

those of buildings, analytical and experimental studies are needed to demonstrate

that these devices can indeed be effective and economically feasible for bridge

structures.

2

1.2 Historical Review

The use of a small, tuned, spring- dashpot -mass system to reduce the vibrations

of dynamic systems is first suggested by Frahm in 1909 [9]. Since then, the use of

such vibration absorbers, sometimes called tuned mass dampers, has continuously

attracted the attention of the engineering profession. It has also stimulated much

research to investigate their effectiveness under diverse load conditions.

Presently, it is generally accepted that they can be effective in reducing the

response of structural systems subjected to harmonic excitation [4, 17], a summation

of sinusoidal ground accelerations [25], and to wind forces [16]. They have been

implemented effectively to reduce wind-induced vibrations in high-rise buildings [5-

8] and even to reduce floor vibrations induced by occupant activity [18, 21, 26].

To date, however, there has not been a general agreement about their adequacy to

reduce the effects of seismic loads.

In studying the influence of a group of selected elasto-plastic vibration

absorbers in the tesponse of linear single-degree-of-freedom systems subjected to

the S21W component of the Taft, 1952, earthquake, Gupta and Chandrasekaran [10] find

that the effect of the vibration absorbers is only minimal and conclude that they

are not as effective for earthquake loads as they are for sinusoidal ones.

Similarly, Wirsching and Yao [27] analyze 5 and 10-story building models with added

vibration absorbers under a simulated non-stationary ground accelerations. The

absorber's mass is considered equal to half the mass of the floors and its damping

is varied. In contrast with the results of Gupta and Chandrasekaran, Wirsching and

Yao find that the vibration absorbers with a damping ratio of 20 per cent

effectively reduce the response of the buildings' top floors. Along the same lines,

using an optimization program to find the minimal structural response, Wirsching and

Campbell [28] calculate the parameters of optimum absorbers and demonstrate their

effectiveness in reducing the first mode response of 1, 5, and 10 story linear

S

3

structures under a Gaussian white noise excitation. In exploring the use of

vibration absorbers in nonlinear systems, Jagadish et al,. [12] study the behavior

of a 2-story structure with bilinear hysteretic characteristics, when excited by the

S69E component of Taft, 1952, ground motion. They observe that, despite the non-

linearity of the structure, an upper to lower story frequency ratio between 0.8 and

1.0 has the capability of providing vibration absorption. They conclude, thus, that

it is possible to have reductions of up to 50 per cent in the ductility demand of

the lower stories.

Performing sensitivity analysis to determine the importance of different

ground motion, structure, and absorber characteristics, Kaynia et al. [13]

investigate statistically for an ensemble of 48 real earthquakes the effect of

vibration absorbers in elastic and inelastic systems. Considering absorber damping

ratios of up to 20 per cent, they find a large statistical variability of the peak

response ration between cases with and without the appendage. They conclude that

vibration absorbers are not too effective in reducing the seismic response of tall

buildings. Along the same lines, Sladek and Klinger [20] test the response of a 25-

story building with a tuned mass damper designed according to the optimum values

suggested by Den Hartog [4] to the North-South component of the El Centro, 1940,

earthquake. They find that such an optimum mass damper makes no contribution

towards reducing the maximum top displacement of the building.

More recently, Clark [2] analyzes a 8-story shear building with tuned mass

dampers designed according to Den Hartog optimum formulas under the N-S 1940 El

Centro earthquake. He finds that if only one tuned mass damper is used, only a

reduction of 11 per cent is obtained in the peak response of the building. In

contrast, if four tuned mass dampers are applied, a reduction of 56 per cent

results. He concludes, thus, that multiple tuned mass dampers may be used to reduce

4

significantly the seismic response of tall structures. Similarly, Kitamura et al.

[14] study the behavior of a tower 125 meters in height implemented with a tuned

mass damper. The tuned mass damper is installed at the tower's top floor, has a

damping ratio of 20 per cent, and its mass represents 1/195 of the total mass of the

tower in one direction and 1/130 in the perpendicular one. Using data from a 1987

I

earthquake, 6.7 in magnitude, they find that the tuned mass damper reduces the

displacements of the tower's top floor by 15 per cent in one direction and by 13 per

cent in the other. Finally, in the only study, thus far, with a cable-stayed

bridge, Unjoh, Abdel-Ghaffar, and Masri [22] analyze a cable-stayed bridge, without

and with active and passive tuned mass dampers, under one of the ground acceleration

records from the 1979 Imperial Valley earthquake. They conclude that passive tuned

mass dampers slightly reduce the response of the bridge only when the natural

frequency of the bridge in the mode to which the damper is tuned is close to the

predominant frequency of the earthquake.

1.3 Object and Scope

The primary objectives of the study herein reported are: (1) to investigate

how effective can be the use of damped resonant appendages designed with the theory

presented in Reference 24 to reduce the seismic response of cable-stayed bridges;

(2) to establish preliminary guidelines for the selection of the parameters and

location of such appendages, and (3) to assess the feasibility of their application

to actual practice. For such a purpose, analytical and experimental tests are

conducted. In both the numerical and experimental studies, appendages are first

designed for a specific cable-stayed bridge according to a previously established

theory, and then the response of the bridge without and with the designed appendage

is calculated and compared. In the numerical study, an actual cable-stayed bridge

is modeled with finite elements and analyzed under three different earthquake

excitations. Several damping ratios for the appendage are considered in this study.

5

In the experimental test, a scale model of a cable-stayed bridge is built and tested

on a pair of shaking tables set to simulate recorded earthquake ground motions.

1.4 Organization

The report is organized in five chapters. The theoretical formulation on

which are based the central idea of the study and the design of the considered

appendages is presented in the next chapter, Chapter 2. Chapters 3 and 4 then

present the details of and the results from the numerical and experimental studies.

The final chapter, Chapter 5, contains a summary of the study and the main

conclusion derived therefrom. Based on the results from the analytical and

experimental studies, and the experience gained during the implementation of the

damped resonant appendage for the experimental model, this chapter also contains a

critical assessment about the benefits and limitations associated with the use of

such resonant appendages in cable-stayed bridges.

Chapter 2

THEORETICAL BASIS

2.1 Introduction

This chapter presents the theoretical background on which the design of the

I proposed resonant appendages is based. The meaning of damping ratios for systems

without classical modes of vibration and the equations from which they can be

I

calculated are introduced first. Then, it is demonstrated that under certain

conditions the damping ratios in two of the modes of a system consisting of a

structure and a small appendage in resonance are approximately equal to the average

of the corresponding damping ratios of the structure and the appendage. Thereafter,

on the basis of this demonstration, it is shown that an appendage with a high

damping ratio and tuned to one of the modes of vibration of a structure may be used

to increase the damping ratio of the structure in the mode to which the appendage

is tuned to a value close to half the damping ratio of the appendage.

2.2 Natural Frequencies and Damping Ratios of Systems Without Classical Damping

As is well known, the natural frequencies and damping ratios of a system can

be determined from a free-vibration analysis and an assumption about the nature of

its damping matrix. However, since the masses, stiffnesses, and damping constants

of a structure and a small appendage are usually of different orders of magnitude,

the combined system formed by the structure and the appendage cannot be considered

to possess classical modes of vibration [23]. To obtain, therefore, the natural

frequencies and damping ratios of the structure-appendage systems under cons ider- 11

ation, it is necessary to describe first what is the meaning of these two parameters

in the case of systems without classical damping and what is the equation of motion

that can be used to determine them.

7

Let, thus, a system for which its damping matrix, [C], cannot be considered

of the Rayleigh or Caughey type be described by its free vibration equation of

motion

[M] {.'} + [C] f ,} + [ K] {x} = {O}

(2.1)

where [M] and [K] respectively represent its mass and stiffness matrix, and (x)

denotes its vector of relative displacements. It is shown by Hurty and Rubinstein

[11] that for its solution this equation need be written first in its reduced form

as

[A] f} + [ B] {q} = {O}

(2.2)

where

1 [0] [Mi 1 . - f - [Mi [0]'(qj - L} f

1

[A] - L [M] [C] J [B] - [ [0] [K]] - (x} f (2.3)

Hurty and Rubinstein [11] have also shown that solutions to this reduced equation

are of the form

{q}1 = { s}1 e, r = 1,2,... ,2N (2.4)

in which N represents the number of degrees of freedom of the system, Ar its rth

complex natural frequency, and (S)r its rth complex eigenvector, which is of the

form

fs{ r = { r 1}r {Wr} }T (2.5)

where (W) r denotes the rth complex mode shape of the system.

It may be seen, thus, that by substitution of Eqs. 2.3, 2.4, and 2.5 into Eq. 10

2.2, the homogeneous reduced equation of motion of the system may be expressed as

N.

1 [0] [M]} {?r{W}r1 - [M] [0] I f

) r {W} r l {0} r[[M] [C] {W}r f [ [0] [K] {W}r J {{o}}

(2.6)

and that from the lower half of this equation, one has that

A[M] {W}r + )r [C] {W}r + [K] {W}r = {0} (2.7)

Notice, therefore, that Eq. 2.1 is satisfied by

{x} = { W}r e' , r = 1,2,.. .,2N (2.8)

Accordingly, if Eq. 2.7 is premultiplied by the transpose of the complex

conjugate of the complex mode shape (W)r, that is, ()r, the system's free vibration

equation of motion may be written alternatively as

Am+Ac+k=o, r=1,2,...,N (2.9)

where mr*, Cr*, and kr* are real-valued generalized parameters defined as

m={w}[M] {w}; c={7}[C] {W}r ; k={i)[K] {W}r (2.10)

Equation 2.9 is thus an equation in Ar with real coefficients whose solution yields

I 2 Cr

A = - + 1 -

- 4-- (2.11) r

2in1 2N mrt mrt

which in similarity with the corresponding equation for a single degree of freedom

system may be expressed as

Ar = +r (2.12)

where

= ()11 ; = c/2cJ)rm; (2.13)

Since in terms of Eq. 2.12, Eq. 2.8, which represents the solution to Eq. 2.1,

may also be expressed as

{x}1= { W}r et (COS(iYr t+1Sfl0'r t) , r=1, 2, . . . , 2N (2.14)

one may then conclude that the parameters Wr, G0' r, and er in Eq. 2.12 respectively

represent the system's rth natural frequency, rth damped natural frequency, and rth

damping ratio, and that in the case of a system without classical damping these

parameters are determined from the solution to Eq. 2.2

2.3 Damping Ratios and Natural Frequencies of Structure-Appendage Systems

As mentioned above, a complication that arises when a structure is considered

together with a small damped resonant appendage is that the combined system formed

by the structure and the appendage cannot be considered as a system that possesses

classical modes of vibration. Nevertheless, if it is assumed that each of the two

components has by itself a damping matrix proportional to its own stiffness matrix

and, hence, classical modes of vibration, it is possible to derive an approximate

expression for the damping ratios and natural frequencies of the structure-appendage

system in terms of their independent dynamic properties. In particular for the

modes of the combined system which result from tuning one of the natural frequency

of the appendage to one of the natural frequencies of the structure, such an expres-

sion can be obtained as follows:

Consider Eq. 2.2 and consider that now this equation represents the free

vibration equation of a combined structure-appendage system. Consider further that

the matrices [MI, [C], and [K] that define [A] and [B] in Eq. 2.2 respectively

represent now the mass, damping, and stiffness matrices of such a combined system,

and similarly for the vectors (x) and () which define (q). If, however, [M], [C],

and [K] are written in terms of [M]b, [C]b, and [K]b, the mass, damping and stiffness

matrices of the structure without the appendage, and [M]a, [CIa, [K]a, the

It

10

A

corresponding matrices of the appendage by itself when the end connected to the

structure is considered fixed, Eq. 2.2 can be written alternatively as

I [A1, [0] 'I{}bl+[[B]b [0] lf{q} bl> I [Q] [1'] 11{4}b + [01 [a]aJl{}aJ [[0] [b]aj(q}aJ [[p]T [0] {q}aJ

(2.15)

1 [v] [T] I{q}l {o} [[T]T [o]] { q} f{ {O}}

where [AIb, [Bib, and (q), and [ala, [bla, and (q}, are defined as in Eq. 2.3, but

with the displacements and the mass, damping, and stiffness matrices of the

structure and the appendage, respectively, and where the matrices in the last two

terms of the left-hand side of the equation are simply matrices that account for the

coupling between the two subsystems. Consider now that when the structure-appendage

system is vibrating in free vibration in its rth mode, the structure and the

appendage by themselves cn be considered as vibrating under the action of an

external force whose magnitude is equal to the interaction force between the two

components and whose variation with time is given by et, where Xr denotes the rth

complex natural frequency of the structure-appendage system. In doing so, and

since the response of a system without classical damping is also given by the sum

of the response in each of its modes [11], the response of the structure and the

appendage to such an interaction force in the rth mode of the combined system may

be then written as

2Na

{q}= {S} Z1et ; {q} a = {s} ze t (2.16)

1=1 j=1

which, by considering only the dominant mode in each case, may in turn be

approximated as

(2.17)

In these equations, respectively for the structure and the appendage, Z1 and z

represent generalized coordinates, Nb and Na their total number of degrees of

I A

A

11

freedom, I and J the number that corresponds to their dominant modes, and {S)b 1 and

(S) a complex eigenvectors of the form indicated by Eq.. 2.5.

By substitution of Eqs. 2.17 into Eq. 2.15, and if the upper and lower

component equations of the latter are respectively premultiplied by the transpose

of (S)b' and ()b', the free vibration equation of motion for the structure-

appendage system may then be reduced to

{. rs4b*I + B1 + r 01 + vi r p; + TIJ 1 1z11 o (2 . 18)

rj+jjJ {0 j

where

(I)T (I)T Ar={S}b [A]b{S} ' ; BI={S}b [B]b(S}' (2.19)

a j={s} J)T [a] a {s) J) ; (2.20)

and where Q1*, v1 , Pij , and Tij* are similarly defined in terms of the coupling

matrices [Q], [P], [V] and [T].

Thus, after taking into account [11] that B*bI = - AbI A*br and b*aj = aJ a*aJ,

where AbI and Aaj respectively represent the Ith and the Jth complex natural

frequencies of the structure and the appendage, Eq. 2.18 leads to the following

eigenvalue problem:

A)I..r -AbI)+(?ro;+V) ?rP;j+T;jI0 (2.21) XP1 +Tij

I

Furthermore, if it is considered that: (a) under the assumption of components with

proportional damping, A*bI = 2iwbIM*bI and a*aJ = 2iw' aJm*aJ, where W'bI and W' aJ, and

M*bI and m*aJ, respectively represent the damped natural frequencies and generalized

masses of the structure and the appendage in their Ith and Jth modes; (b) when the

Jth natural frequency of the appendage is tuned to the Ith natural frequency of the

12

structure, the undamped natural frequencies wbj and WaJ are the same and equal to w0 ;

and (c) by assumption the mass of the appendage is much smaller than the total mass

of the structure; then such an eigenvalue problem leads to the following approximate

solution for the rth and rth + 1 complex natural frequencies of the structure-

appendage system:

A r, r+1 - (bIa,y) 0+i [(bIaJ) (2.22) 2

where bI and e.j respectively denote the damping ratios of the structure and the ap-pendage in their Ith and Jth modes; wo is the frequency that is common to the

structure and the appendage; 7IJ = m*aJ/M*bI, where m*aJ and M*bI are the Jth and Ith

generalized masses of the structure and the appendage; and 4Dk is the amplitude of

the point of the structure to which the appendage is attached in the Ith mode shape

of the structure, after it is multiplied by the structure's Ith participation

factor. In accordance with Eq. 2.22, in view of Eq. 2.12, and for relatively small

mass and damping ratios, it may be seen thus that two of the damping ratios and two

of the natural frequencies of the structure-appendage system are given approximately

by

r,r+10 r,rl-1 = br + aJ bI _ aJ) 2_Y IJ (2.23) 2 2

if Ii - aJI > Ik iyijl , and by

Ci) Ci)rz+i=(l) /YIJ_(bI_aJ)2 ; - 1

r r+1 bILJ (2.24)

0 2

if Ii - aJI I'k i-YiJl

2.4 Parameters of Effective Resonant Appendages

Equations 2.23 and 2.24 are useful to visualize the effect of installing a

damped resonant appendage on a structure. Note, for instance, that if the

13

parameters of such an appendage are such that

IbI - aJI Ik 'lijI (2.25)

then Eqs. 2.23 and 2.24 indicate that the mode of the structure whose frequency is

equal to the natural frequency of the appendage will split in two, and that each of

these two modes will have a frequency close to the tuned frequency of the absorber

(one of the frequencies will be slightly higher, the other slightly lower).

Furthermore, they indicate that the damping ratios in these two modes will be equal

to the average of those of the structure and the appendage in their tuned modes.

Accordingly, an efficient resonant appendage and its best location within the

structure can be obtained by selecting, for example, a single-degree-of-freedom

appendage with a damper that makes its damping ratio high, a mass and a location

within the structure that satisfy the above inequality, and a stiffness constant

that tunes its fundamental natural frequency to that of the structure.

4

14

Chapter 3

NUMERICAL STUDY

3.1 Introductory Remarks

To accomplish the objectives of the study, numerical simulations are conducted

with a finite element model of a full-scale cable-stayed bridge and a damped

resonant appendage designed to reduce the response of the bridge to seismic

disturbances. In these simulations, the response of the bridge to three different

earthquake ground motions, without and with the resonant appendage, is analyzed and

compared. This chapter describes the analyzed bridge, the calculation of the

parameters for the resonant appendage, the earthquake ground motions used, and the

results from the comparative analysis.

3.2 Analyzed Bridge

The cable-stayed bridge considered in the numerical simulations is described

in Figures 3.1 and 3.2. Its properties are given in Table 3.1. This bridge

represents a simplified analytical model of the Yobuko Bridge in Japan. The Yobuko

Bridge, built in May of 1989 and located in Saga Prefecture, is a three-span

continuous prestressed concrete cable-stayed bridge crossing the sea between Yobuko

and Kabe Island. It has a main span of 250 meters, two side spans of 121 meters,

a width of 10.9 meters, and a pier height of 103.6 meters. The total mass of the

bridge is approximately 28,802 Mg.

3.3 Finite Element Model

For the computer simulations, the bridge is modeled using three-dimensional

finite elements. Truss elements are employed to represent the cables and beam

elements to represent the deck and towers. The damping matrix of the bridge is

considered to be orthogonal, with a damping ratio of 1 per cent in all its modes.

15

The simulations are carried out with the commercial computer program Cosmos/M, an

all-purpose finite element program for personal computers [3]. The model is built

with 112 elements and 85 nodes. It considers six degrees of freedom per node, with

a total of 456 degrees of freedom. The towers are assumed fixed at the foundation

level, while the ends of the deck are assumed constrained against vertical motion

and rotation about a longitudinal axis, but free to displace horizontally along the

longitudinal direction and rotate about a vertical and a transversal axis. Spring,

mass, and damper elements, available in the program, are used to model the

components of the damped resonant appendage. Cosmos/M recognizes the non-classical

damping nature of the bridge when the appendage is added to it.

3.4 Bridge Dynamic Properties

The bridge dynamic characteristics are determined to identify its significant

modes of vibration and to obtain the modal parameters that are needed to select the

parameters of an efficient resonant appendage. The natural frequencies and mode

shapes for the first five modes of the bridge model, calculated with Cosmos/M, are

shown in Figures 3.3 through 3.7.

3.5 Parameters and Location of Resonant Appendage

From the inspection of the mode shapes presented in the previous section, it

can be seen that the bridge undergoes a significant longitudinal motion in its

fourth modes shape, i.e., in its mode with a natural frequency of 0.31252 Hz.

Since, depending on the characteristics of the ground motion experienced, this

longitudinal motion may be sometimes excessive, the use a single-degree-of-freedom

resonant appendage may be a desirable alternative to reduce the response of the

structure in this mode. In the work herein reported, it is thus assumed that the

resonant appendage will be utilized to damp the fourth mode of the bridge and,

hence, such a longitudinal motion.

16

In accordance to the guidelines introduced in Section 2.4, a resonant

appendage may be effective in reducing the response of the bridge in its fourth mode

if the appendage is tuned to the natural frequency in that mode, its damping ratio

is high, it satisfies Eq. 2.25, and is located at the point of maximum displacement

in such a mode. Consequently, if 'k represents the mode shape amplitude along the

direction of the excitation of the point in the bridge to which the appendage will I J,

be attached times the corresponding participation factor, wb its natural frequency,

I

and M* the corresponding generalized mass, all in the bridge mode under consider-

ation, the appendage mass, ma, stiffness, ka, and damping constant, Ca, for a given

appendage damping ratio, ea, can be determined by means of the following formulas:

Ica - chi i 2

ma = (3.1) ('k)2

ka = Wb 2 ma (3.2)

Ca = 2 ea COb ma (3.3)

Similarly, from the inspection of the aforementioned fourth mode shape, it is

apparent that the best location for the appendage is any point on the bridge deck,

as the deck is the bridge component that undergoes the largest displacement along

the bridge's longitudinal direction in such a mode.

To test for the sensitivity of the reduction in response to the damping ratio

of the appendage, the investigation is carried out with four different values for

this parameter: 10, 15, 20, and 30 per cent of critical. Thus, since for the fourth

mode of the bridge wb = 1.9636 rad/sec, Dk = 0.66402, and M* = 8,522,315 Kg , the

parameters of the appendages considered in this study are those listed in Table 3.2.

It is noted from the values in Table 3.2 that the mass of the appendage

17

increases as its damping ratio also increases. This is a consideration that should

be weighed in the design process when selecting the damping ratio for the appendage.

Since the total weight of the bridge is 28,802 Mg, it may be noted also that the

weight of the appendage represents about 0.67, 1.51, 2.68, and 6.04 per cent of the

bridge total weight for the cases of 10, 15, 20, and 30 per cent damping,

respectively.

3.6 Dynamic Properties of Bridge-Appendage System

As predicted by the theory presented in Chapter 2, the addition of a resonant

appendage to the bridge splits in two its mode of vibration whose frequency is equal

to the natural frequency of the appendage. One of these modes will have a frequency

that is slightly lower than the original frequency and the other one that is

slightly higher. If this shift in frequencies is significant, this effect alone may

cause a significant change in the response of the bridge, particularly when the

frequency of the bridge mode in question lies close to one of the dominant

frequencies of the ground motion that excites the structure. It thus of interest

to determine the mode shapes and natural frequencies of the bridge after the

appendage is attached to it and assess the influence of such a frequency shift in

the reduction, or lack of reduction, in the bridge's response.

Some of the natural frequencies and mode shapes of the bridge-appendage system

formed by the bridge under analysis and one of the appendages whose parameters are

given in Table 3.2 are shown in Figures 3.8 through 3.15. These mode shapes and

natural frequencies correspond to the two new modes that result from tuning the

appendage to the fourth mode of the bridge. The other mode shapes and natural

frequencies are virtually identical to the natural frequencies and mode shapes of

the bridge without the appendage.

4

18

3.7 Earthquake Ground Notions

To assess the effectiveness of the selected resonant appendages under

different excitations, the bridge is analyzed under three different ground

acceleration time histories. These are:

First ten seconds of the N-S accelerogram recorded during the May 18,

1940, El Centro earthquake

E-W accelerogram recorded at Foster City during the October 17, 1989,

Loma Prieta earthquake

E-W accelerogram recorded at UC Santa Cruz during the October 17, 1989,

Loma Prieta earthquake

These time histories and their respective displacement and acceleration

response spectra are shown in Figures 3.16 through 3.21.

3.8 Results

The bridge is analyzed first by itself and then with one of the appendages

described above attached to the left end of the deck. For each of the excitations

considered, the time histories of the longitudinal displacements of the left end of

the deck and the top of the left tower are obtained first without any appendage and

then with one of the appendages whose parameters are listed in Table 3.2. Because

of the limitations of the computer program used, the analysis is performed with no

phase lag between the base motions at the bridge supports.

The results of the analysis are presented in Figures 3.22 through 3.31 for El

Centro, in Figures 3.32 through 3.41 for Foster City, and in Figures 3.42 through

3.51 for U.C. Santa Cruz. To aid in their interpretation, the maximum displacements

obtained with the appendage in each case are compared with the corresponding maximum

values when no appendage is considered. Such maximum displacements and comparison

19

are given •in Tables 3.3 and 3.4. Similarly, to assess the order of magnitude of

the relative motion of the appendage with respect to its support, the variation with

time of this relative motion is presented in Figures 3.52 through 3.54 for the

appendage with 20 per cent damping. It is observed from these plots that the peak

relative motion equals 0.28, 0.58, and 0.10 meters when the bridge is subjected,

respectively, to El Centro, Foster City, and UC Santa Cruz earthquakes.

*

RM

Chapter 4

EXPERIMENTAL STUDY

4.1 Introduction

A small-scale cable-stayed bridge and an appendage consisting of a small mass,

a small spring, and a small viscous damper are built and tested on a pair of shaking

tables under various earthquake excitations to verify the effectiveness of the

proposed resonant appendages. This chapter describes the structural model and the

appendage used in the test, the equipment and excitations employed, and the results

obtained therefrom.

4.2 Model Description

4.2.1 Structural Model

The bridge model is 12-feet long, made of aluminum, and designed as a

simple span bridge with its cables supported by two towers 29 inches in height. The

cables are arranged over two parallel planes, using a harp configuration. Its

geometry and dimensions are given in Figures 4.1 and 4.2, and the characteristics

of its components in Table 4.1. To add mass to the bridge, one-pound steel weights

are attached with hot glue along the bottom of the deck and along the sides of the

towers as shown in Figure 4.1 . The typical prestressing tensions listed in Table

4.2 are applied to the cables to keep the deck more or less straight under its own

and the added weights. These tensions are applied by means of the guitar string

gears that are installed to connect the cables to the bridge deck. They are

determined in an approximate fashion by measuring in each cable the deflection

caused by a weight of known value connected to the midpoint of the cable. The

abutments rest on ball bearings and attached to the towers by means of relatively

rigid braces to make them undergo the same base motion as the towers. Similarly,

the ends of the bridge deck rest on low-friction teflon plates and are attached to

21

the abutments by means of the clamps shown in Figure 4.3. This type of support is

designed to allow the horizontal motion and rotation of the deck, but prevent, at

the same time, the uplifting of the deck ends caused by the vertical motion of the

deck. In this way, the model simulates the support conditions of actual cable-

stayed bridges. A lubricant is used to minimize the friction between the deck and

the abutment, and the abutment and the supporting ball bearings.

The total weight of the bridge model, without the abutments and the braces

that join these to the towers, is approximately 41.3 lb.

4.2.2 Model Analytical Modal Properties

Before the experimental testing and the actual construction of the mass-

spring-damper system to be used as a resonant appendage, the bridge model is

represented with finite elements and its modal properties determined numerically

using the computer program Cosmos/M. These modal properties are needed to (a)

determine the bridge parameters that are required to apply the design recommenda-

tions of Section 2.4; and (b) decide on the best location and orientation for the

appendage.

The finite element model is implemented using truss elements to represent the

cables, three-dimensional plate elements to represent the deck, and three-

dimensional beam elements to represent the towers. The towers are assumed fixed at

the foundation leiel, while the ends of the deck are assumed constrained against

vertical motion and rotation about a longitudinal axis, but free to displace

horizontally along the longitudinal direction and rotate about a vertical and a

transversal axis. A total of 390 elements and 411 nodes are considered with, at six

degrees of freedom per node, a total of 2436 degrees of freedom. As with the Yobuko

bridge, spring, mass, and damper elements are used to model the components of the

22

damped resonant appendage.

The first four natural frequencies and mode shapes obtained with this finite

element model of the bridge are shown in Figures 4.4 through 4.7. For the first

mode, the mode shape amplitude of the deck's left end, multiplied by the correspond-

ing participation factor, and the generalized mass are respectively equal to 1.4543

and 0.2969 lb-sec2/in.


The appendage for the bridge model is designed for a target damping ratio

increment of 25 per cent; that is, for H. - bI = 0.25. On the basis of the

results for the Yobuko bridge, this value is selected so that, on the one hand, it

is high enough to secure a significant damping augmentation, but, on the other hand,

it is low enough to keep the appendage mass within practical limits. Similarly, it

is observed from the mode shapes presented in the foregoing section that the mode

with the most significant longitudinal motion is the first mode and that the deck

is the bridge component that displaces the most along the longitudinal direction.

Moreover, it is observed that in this mode all the points along the deck virtually

undergo the same horizontal displacement. Therefore, the appendage is designed to

damp the bridge first mode and minimize thus such a longitudinal motion. Likewise,

it is decided to place the appendage at any point along the deck, as it makes no

significant difference which point is selected. Because of space limitations, the

appendage is attached to a point near the deck's center.

In accordance with the appendage characteristics selected above, the

assumption of a damping ratio of 7 per cent for the first mode of the bridge model,

the first-mode values reported in the previous section, and Eqs. 3.1 through 3.3,

the parameters for the appendage that theoretically may damp the response of the

23

bridge in its first mode are thus those listed in Table 4.3.

The design of the appendage is based upon the popular mathematical model of

a single-degree -of -freedom system; that is, it consists of a weight attached to a

spring and a damper in parallel. Its dimensions are sketched in Figure 4.8 and its

configuration shown in Figure 4.9. Its weight is provided by a piece of a steel bar,

and its spring by a conventional off-the-shelf one. The damper used is a commercial

air cylinder with a valve that permits the variation of its damping characteristics.

A photograph of this commercial air cylinder is presented in Figure 4.10.

4.2.4 Analytical Modal Properties of Bridge-Appendage System

To assess, as in the case of the Yobuko bridge, the significance of the

frequency shift induced by the attachment to the bridge model of an appendage with

the parameters listed in Table 4.3, the bridge model natural frequencies and mode

shapes are also cetermined for the case in which such an appendage is attached to

the left end of the bridge deck. The obtained first five such natural frequencies

and mode shapes are shown in Figures 4.11 through 4.15.

4.3 Equipment

4.3.1 Introductory Remarks

The following sections briefly describe the equipment used in the shaking

table experiment. The primary equipment used are the shaking tables themselves,

accelerometers, a HP3562A dynamic signal analyzer, and a HP computer with a MODAL

3.0 SE system installed. The configurations for each set-up are detailed in Section

4.5.

4. 3 . 2 Accelerometers

The accelerometers used to measure the accelerations in the bridge model are

I

24

PCB Piezotronics Structcel motion sensors, Model 330A, connected to a HP3562A signal

analyzer. These accelerometers, employing a mass loaded, differential structure

with integral microelectronics as their sensing element, convert accelerations into

voltage signals compatible with readout and analyzing equipment. Together with

their mounting sockets, they weigh 3 grams each. They have a nominal sensitivity

of 200 mV/g and operate over a frequency range of 1 to 1000 Hz. They are capable

of measuring accelerations up to 10 g, with a resolution of 0.001 g. In a

I I

calibration test run with one of the shaking tables, an average conversion factor

of 4749 mV/g is found when these accelerometers are used in conjunction with the

HP3562A signal analyzer.

4.3.3 HP3562A Dynamic Signal Analyzer

The Hewlett-Packard HP3562A is a dynamic signal analyzer which interprets the

response of an instrumented structure. The HP3562A, a dual-channel Fast-Fourier

Transform (FFT) analyzer, acquires a certain number of data readings over a

designated length of time, then averages them and generates the frequency response

curve by FFT computations. It is also possible to generate the response in the time

domain by using an option known as Time Capture. When this option is used, the

HP3562A collects real-time data recorded from the input channel and analyzes the

captured data.

4.3.4 Shaking Tables

The shaking tables used have a horizontal surface of 9 in X 12 in and are

driven by Electro-Seis electromagnetic shakers from APS Dynamics, Inc. One of the

shakers is a model 113 and the other a model 400. Both have a capacity to support

up to 50 lb of vertical load and a maximum stroke of 6.25 in. A personal computer

implemented with a digital to analog converter board is used to drive each of the

two electromagnetic shakers according to a specified earthquake ground acceleration

25

record. The input to each of the shakers is the same, but a phase lag between the

two can be specified.

4.3.5 Modal 3.0 SE System

The Modal 3.0 SE system is a modal analysis program designed to operate in

conjunction with a multichannel FFT analyzer, such as the HP3562A. The program is

installed on a Hewlett-Packard desktop computer. The program identifies the modal

properties of a structure by use of a frequency response method. After the

frequency response curves are obtained by the HP3562A, they are transferred to the

Modal 3.0 SE program through a floppy disk. The program then processes the measured

data by curve fitting; that is, by matching an analytical function to a band of

measured data points. The fitting is done in a least squared error sense.

4.4 Experimental Dynamic Properties

4.4.1 Introductory Remarks

Because the design of the appendage is based on the dynamic properties of the

bridge model that are determined by means of a finite element analysis, it is

necessary to determine experimentally these dynamic properties to verify that the

assumptions made in the design of the appendage are indeed adequate, and to ensure,

thus, that the selected appendage will be effective in reducing the response of the

bridge model. The following sections describe the dynamic properties of the bridge

and appendage models that are obtained experimentally.

4.4.2 Bridge Model

To determine its dynamic characteristics, the bridge model is tested under

random excitations generated by the HP3562A dynamic signal analyzer and its modal

properties identified by means of the Modal 3.0 SE system described above. The

natural frequencies, damping ratios, and mode shapes thus obtained for the first

26

three modes of the model are shown in Figures 4.16 through 4.18. Note that the mode

shapes in these figures represent the motion of half the bridge deck and along its

longitudinal direction.


I I

Because of its simple design, the appendage model behaves basically as a

single-degree-of-freedom system. Thus, its dynamic properties are obtained by

I *

determining first the values of its mass and spring and damping constants, and by

using then the well-known formulas of Structural Dynamics for single-degree -of -

freedom systems. The values for the weight, spring constant, and damping constant

of the system used as appendage are listed in Table 4.4. The value of the spring

constant is determined directly from the spring manufacturer's specifications. The

value of the damping constant is obtained by measuring, after a predetermined

setting of the regulating valve, the velocity with which the cylinder shaft is

pulled down by a weight of known value attached to it. Note that the values in

Table 4.4 closely correspond to the design ones given in Table 4.3. Note too that

since the total weight of the bridge model is 41.3 lb, the weight of the appendage

represents about 8 per cent of the total weight of the bridge.

According to the weight, spring constant, and damping constant listed in Table

4.4, the appendage model's undamped natural frequency and damping ratio are thus

equal to 6.26 Hz and 32 per cent, respectively. Observe that this natural frequency

of 6.26 Hz matches very closely the fundamental natural frequency of the bridge

model, which, according to Figure 4.16, is equal to 6.23 Hz.

4.5 Experimental Set-Up

For the test, the bridge model is mounted on the shaking tables as depicted

in Figure 4.1 and as illustrated, without and with the appendage, in Figures 4.19

27

and 4.20, respectively. The equipment is arranged according to the schematic

configuration in Figure 4.21 for the determination of its dynamic properties, and

according to the one shown in Figure 4.22 for its response analysis. In both cases,

the bridge base plates are tightly fastened to the shaking tables. Accelerometers

are attached with hot glue along the longitudinal direction of the bridge. One is

attached at one of the base plates; the other to either the center of the bridge 40

deck or the top of one of the towers. The shaking tables are then activated by

starting a program in the personal computer with the digital to analog converter

board. This program activates the two shaking tables with the desired acceleration

time history and the desired phase lag between the two shaking tables.

4.6 Base Acceleration Time Histories

The three acceleration time histories defined in Section 3.7 are also used .for

the experimental test. However, scale factors are used to scale down the intensity

of the accelerations and minimize thus the chances of damaging the model. These

scale factors are different for each of the excitations considered and are varied

for some of the tests. They are defined in the sections devoted to the presentation

of results.

4.7 Experimental Results

4.7.1 Introduction

The bridge model is tested on the shaking tables by itself first and then with

the appendage previously described attached to middle of the deck as shown in Figure

4.23. The accelerations at a point on the deck and the top of the tower are 4

measured each time. The relative motion between the deck and the mass of the

appendage is also measured. In addition, several other cases are considered to

assess the influence in the effectiveness of the added appendage of:

28

a phase lag between the motions at the two supports;

a shock-type base motion;

an appendage frequency slightly out of tune;

tuning according to Den Hartog's formula;

no relative motion between appendage and bridge deck.

For all cases, except the last three, the El Centro record is used with a

I

I scale factor of 0.5, the Foster City one with a scale factor of 0.7, and the UC

Santa Cruz with a scale factor of 0.4. For the last three, such scale factors are,

respectively, 0.6, 0.8, and 0.5.

The obtained results are presented in the sections below separately for each

of the above cases.

4.7.2 Test with no Phase Lag

For this test, the two shaking tables are set to move with exactly the same

base motion and no time lag between them. The time histories recorded during this

test are presented in Figures 4.24 through 4.29 for the longitudinal accelerations

at the left end of the bridge model deck, Figures 4.30 through 4.35 for the

longitudinal accelerations at the top of the left tower, and Figures 4.36 through

4.41 for the vertical accelerations at the deck's center. The acceleration time

histories of the appendage mass and the point on the deck that supports this mass

are also recorded to investigate the relative motion of the appendage mass. These

time histories are shown in Figures 4.42 through 4.47. A compilation of the peak

accelerations observed without and with the appendage and the corresponding

reduction factors are given in Tables 4.5 and 4.6.

29

4.7.3 Test with Phase Lag

To simulate the out-of-phase motion that is common for the supports of long-

span structures such as cable-stayed bridges, the bridge model is also tested with

the shaking tables set to generate the same base motion, but with an arbitrary time

lag of 0.4 seconds between the two of them. In this way, one of the towers and the

abutment braced to it move out of phase with respect the other tower and the other

abutment, and, as result, the bridge is subjected to a non-uniform base excitation.

Figures 4.48 through 4.53 show the time-histories obtained from this test. Table I

4.7 lists the corresponding maximum values and reduction factors.

4.7.4 Test with Shock-Type Base Motion

The idea behind the use of a high damping appendage to reduce the seismic

response of a structure is that, by attaching the appendage to the structure, the

damping in the structure is augmented. Thus, the mechanism involved in the

reduction of response is simple one of energy dissipation induced by the additional

energy dissipating device. In other words, a reduction in response is attained

because the damping ratio of the structure is increased. It is well known, however,

that damping is not very effective in reducing the response of a structure to loads

of high intensity and short duration; that is, shock-type loads. It is of interest,

therefore, to test the bridge model under such type of loading and assess the

effectiveness of a damped appendage in such a case. For this purpose, the bridge

model is subjected to a cut version of the El Centro record that exhibits its peak

early at the beginning of the record. This record and the corresponding response

spectra are shown in Figures 4.54 and 4.55. The acceleration response of the bridge

deck end, without and with the appendage, is presented in Figures 4.56 and 4.57.

It is noted from these figures that despite the large peak at the beginning of the

excitation, a reduction factor of 0.23 is attained with the addition of the

appendage.

30

4.7.5 Test with Slightly Out-of-Tune Appendage

A question that often comes to mind when dealing with the use of a resonant

appendage is the effectiveness of the appendage when it is not perfectly tuned to

the structure. The question is a natural one since in a practical situation a

perfect tuning cannot be attained or guaranteed to remain at the same level during

IL

the entire lifetime of the structure. Although this is a problem that deserves a

thorough future investigation, the bridge model is tested with the appendage

slightly out of tune to shed some light into it. Figures 4.58 through 4.63 depict

the results obtained, under the three ground motions being considered, without an

appendage and when the appendage is slightly out of tune. For this test, the

appendage mass is reduced by 165 grams, whereby its natural frequency is increased

to a value of 6.63 Hz. This signifies a detuning of about 6 per cent. Table 4.8

lists the maximum accelerations without and with the appendage, and the correspond-

ing reduction factors.

4.7.6 Test with Tuning According to Den Hartog's Formula

In some of his early work, Den Hartog shows [4] that the optimum tuning of a

damped vibration absorber to a single-degree-of freedom system subjected to a

sinusoidal forcing function is attained when that tuning is done according to the

following formula:

f = 1 / (1 + IL)

(4.1)

In this formula,

natural frequency of absorber f = frequency ratio = ----------------------------------- (4.2)

natural frequency of main system

and

31

absorber mass p = mass ratio =

(43) mass of main system

Thus, although it should be clear from the theory presented in Chapter 2 that

the resonant appendages being investigated differ conceptually and in several other

aspects from the vibration absorbers proposed by Den Hartog, there are nevertheless

some similarities between the two of them, such that some may wonder weather or not

a greater reduction in response may be obtained when the proposed appendages are

tuned according to the above formula.

In an attempt to answer this question, the tests with the bridge model are

extended to include a case in which the appendage is tuned to the bridge according

to Den Hartog's formula. For this purpose, a damped appendage is designed and

constructed using Den Hartog's formula, and the bridge model tested again with and

without such an appendage. The parameters of the appendage, given in Table 4.9, are

determined on the basis of the spring constant of a commercial spring, the

generalized mass and natural frequency of the bridge model in its first mode, and

the tuning formula described above. Since tuning according to Den Hartog's method

does not involve damping, the damping constant for the system is selected

arbitrarily as the one that corresponds to a commercial air cylinder of the type

shown in Figure 4.10, but with no valve. The results from this test, with the

appendage, are presented in Figures 4.64 through 4.66. Those without the appendage

are the same as those presented in Figures 4.58, 4.60, and 4.62. The peak values

and reduction factors are presented in Table 4.10.

4.7.7 Test with appendage glued to bridge deck

To assess the effect on the response of the bridge caused by the weight of the

32

appendage alone, the bridge model is also tested with the appendage glued to its

deck. By gluing the appendage to the bridge deck, the relative motion between the

appendage mass and its point of attachment is prevented and thus the appendage

damper and spring are not activated. This test is of interest to assess what

portion of the reduction in the bridge response is attained by the change in the

mass of the bridge and what by the dynamic effect of the appendage and the added

energy dissipation mechanism. The results of this test are given in Figures 4.67

through 4.69. The comparison between the peak values of the bridge response without

an appendage and with the glued appendage is shown in Table 4.11.

33

Chapter 5

SW(MARY AND CONCLUSIONS

5.1 Summary

A theoretical formulation has been presented to show that, if certain

conditions are satisfied, the addition of an appendage with a relatively small mass

and a high damping ratio may be an effective way to increase the inherent damping

in structures and reduce, thus, their response to earthquake excitations. Based on

this formulation, it is then postulated that the addition of a small appendage with

a high damping ratio, a natural frequency equal to the dominant mode of the

structure, and parameters that satisfy a given relationship, may be used to reduce

the seismic response of cable-stayed bridges. In addition, numerical and

experimental studies are conducted to verify the postulate, establish some

preliminary guidelines for the selection of the parameters and location of the

appendage, and gain some insight into the feasibility of their application to actual

practice. In the numerical study, an actual cable-stayed bridge is modeled with

finite elements and analyzed with and without the proposed appendages under three

different earthquake excitations. Appendages with damping ratios of 10, 15, 20, and

30 per cent are considered. In the experimental test, a 12-foot long cable-stayed

bridge and an appendage consisting of a small mass, a small spring, and a small

viscous damper are built and tested, without and with the appendage, on a pair of

shaking tables set to reproduce three recorded earthquake ground motions. The

damping ratio of the appendage in this test is of 32 per cent.

5.2 Conclusions

The investigation has verified the postulated theory and shown that damped

resonant appendages may be indeed suitable as damping augmenting devices in bridge

structures and effective in reducing their seismic response. In the numerical

34

study, it is found that, for the analyzed bridge, appendages with a weight equal to

0.67, 1.5, 2.7, and 6.0 per cent of the total weight of the bridge and damping

ratios of 10, 15, 20, and 30 per cent can respectively reduce the peak longitudinal

response of the bridge deck by 88, 88, 88, and 87 per cent, and that of the top of

the bridge towers by 86, 85, 85, and 84 per cent. Similarly, the experimental study

I

shows that an appendage with a weight equal to 8 per cent of the weight of the

bridge and a damping ratio of 32 per cent is capable of reducing the peak

longitudinal response of the bridge deck by 41 per cent and that of the top of the

bridge towers by 12 per cent.

The investigation has also verified the design guidelines derived from the

theory presented in Chapter 2 and utilized in the design of the appendages

considered in this study. It may be established, therefore, that these design

guidelines are adequate for the selection of the parameters and location that make

a resonant appendage effective in reducing the seismic response of cable-stayed

bridges.

5.3 Feasibility Assessment

The study shows that there are three major drawbacks in the use of damped

resonant appendages as a means to reduce the earthquake response of cable-stayed

bridges. The first one is the size of the appendage mass that is needed to attain

a substantial reduction in a bridge's response. It is observed that the mass needed

for an effective appendage increases if the selected damping ratio for the appendage

is increased. The second one is the uncertainty in the tuning of the appendage to

the desired bridge frequency. The effectiveness of a resonant appendage diminishes

when it is not perfectly tuned to the structure. The third is that there exists a

dependence of the reduction in response attained by means of such an appendage on

the characteristics of the ground motion exciting the bridge. This reduction in

35

response is large for resonant ground motions and is progressively less as the

dominant frequency of the ground motion gets farther apart from the natural

frequency of the bridge to which the appendage is tuned.

At first sight, these three drawbacks may appear to be a hindrance to the

practical application of the proposed technique; they can mislead one to believe

that a large mass is needed to attain a large reduction factor, that the system

might not work well because in a practical application it is difficult to predict

with certainty the natural frequencies of a bridge, and that the technique might be

effective only for some but not all possible excitations. It is felt, however, that

these limitations can be overcome in the design process. For instance, for a given

bridge a designer can find ways to minimize the mass needed for the construction of

the appendage by using the mass of parts and appurtenances of the bridge.

Similarly, in weighing the use of an appendage as a possible design solution, the

designer can consider the uncertainty in the natural frequencies of the bridge and

a reduced effectiveness in the performance of the appendage. Moreover, in the

design of a resonant appendage the designer should borne in mind that what an

attached appendage does to a structure is simply to augment its damping characteris-

tics. As such, since beyond a certain limit additional damping will not signifi-

cantly reduce a structure's response any further, a high damping ratio, and hence

a large appendage mass, will not be necessary in most cases. That implies too that

the appendage will be effective under those ground motions that in the absence of

damping would induce a large structural response and, hence, under the critical

ground motions that govern the design of the structure.

In view of the above and the results from the numerical and experimental

studies, it is concluded that damped resonant appendages may be a convenient

alternative against a conventional design. However, it is not possible to assert

V

with the information obtained from this investigation whether or not the use of a

resonant appendage can offer a more economical solution than a conventional design.

Only a cost analysis with a real bridge and a real design can shed some light into

this question.

5.4 Recommendations for Future Research I'

The results of the study suggest that damped resonant appendages may be an

I

effective method to reduce the response of cable-stayed bridges to earthquake ground

motions. Nonetheless, extensive further research is needed before they can be

implemented into practice. Among others, additional studies are required to

investigate: (a) their effectiveness in medium to large scale models, under three-

dimensional ground motions, at various locations along the deck and towers of the

bridge, and in bridges with significant higher and closely-spaced modes; (b) the

importance of uncertainties in bridge and appendage parameters; (c) the behavior of

the bridge-appendage system under excitations that load the bridge beyond its

elastic range; and (d) the effectiveness of multiple single-degree -of -freedom and

single multi-degree-of-freedom appendages. Likewise, additional work is needed to

develop: (a) reliable, functional, and economical prototype appendages; (b)

practical configurations for their installation; and (c) guidelines for the seismic

design of bridges with damped resonant appendages.

I

37

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C. H. Thornton, D. A. Cuoco, and E. E. Velivasakis, "Taming Structural Vibrations," Civil Engineering, ASCE, Nov. 1990

S. Unjoh, A. M. Abdel-Ghaffar, and S. F. Masri, "A study on the effectiveness of structural control for cable-stayed bridges," Proc. 2nd. Workshop on Bridge Engineering: Research in Progress, Reno, Nevada, Oct. 1990, pp. 51-54.

R. Villaverde and N.M. Newmark, Seismic Response of Light Attachment to Buildings, SRS No. 469, University of Illinois at Urbana-Champaign, Feb. 1980.

R. Villaverde, "Reduction in seismic response with heavily-damped vibration absorbers," Earthquake Engineering and Structural Dynamics, Vol. 13, 1985, pp. 33-42.

G. B. Warburton, "Effect of absorbers on the response of structures," Proc. 9th European Conference on Earthquake Engineering., Moscow, 1990, Vol. 8, pp. 189-198.

A. C. Webster & M. P. Levy, "A case of the shakes," Civil Engineering, ASCE, Vol. 62, Feb. 1992, pp. 58-60.

P. H. Wirsching and J. T. P. Yao, "Safety design concepts for seismic struc-tures," Comput. Struc., Vol. 3, 1973, pp. 809-826.

P. H. Wirsching and G. W. Campbell, "Minimal structural response under random excitation using the vibration absorber," Earthq. Eng. Struc. Dyn., Vol. 2, 1974, pp. 303-312.

ACKNOWLEDGNENTS

The authors greatly benefitted from the information exchange with and

suggestions offered by Messrs. T. Takeda, Y. Murayama, K. Kanda, T. Ichinomiya, and

Y. Okimi of Kajima Technical Research Institute during and after their visit to UCI.

The authors also wish to express their sincere gratitude to R. Kazanjy, manager of

UCI's Structures Laboratory for his professional help and insight. Thanks are due

to Lucinda Auciello too for her assistance in the development of the project.

The investigation was generously funded by Kajima Technical Research Institute

through a research grant given to CUREe, California Universities for Research on

Earthquake Engineering. The support offered by CUREe and Kajima are here gratefully

acknowledged.

01

40

Table 3.1. Properties of structural elements in simplified model of Yobuko Bridge

Element E (t/m2)

A I I, I Unit (m2) () (rn') (m') weight

(t/m3)

Girder 3.5 X 106 7.0 9.5 4.0 70.0 3.0

Each 3.5 X 106 7.5 3.0 10.0 9.0 2.5 tower

Upper 3.5 X 106 5.0 4.0 3.6 1.5 2.5 cross beam

Lower 3.5 X 106 5.0 4.0 3.6 1.5 2.5 cross beam

Pier 3.0 X 106 70.0 1500 350 1000 2.5

Cable 1 1.86 X 107 0.017 - - - 10.0

Cable 2 1.86 X 10 0.017 - - - 10.0

Cable 3 1.86 X 107 0.013 - - - 10.0

Cable 4 1.86 X 107 0.006

Table 3.2. Parameters of appendages in simulation study

Parameter Damping Ratio (%)

10 15 20 j 30

Mass (Mg) 192.905 434.035 771.618 1739.553

Stiffness (kN/m) 743.807 1673.565 2975.227 6707.415

Damping constant (kN-sec/m)

83.334 272.731 636.371 2113.662

41

Table 3.3. Maximum longitudinal displacements of bridge deck end without and with appendage

Excita- Disp. 10 % 15 % 20 % 30 % tion with

no Damping

Appendage Damping

Appendage Damping

Appendage Damping

Appendage appendage

Disp. R.F. Disp. R.F. Disp. R.F. Disp. R.F. (m) (m) (m) (m) (m)

El 0.3333 0.2545 0.76 0.2361 0.71 0.2163 0.65 0.1759 0.53 Centro

Foster 0.4450 0.3768 0.85 0.3248 0.73 0.3128 0.70 0.3070 0.69 City

U.C. 0.4617 0.0536 0.12 0.0548 0.12 0.0559 0.12 0.0605 0.13 Santa

11 Cruz Note: R.F. = Reduction Factor

Table 3.4. Maximum longitudinal displacements of tower top without and with appendage

Excita- Disp. 10 % 15 % 20 % 30 % tion with Damping Damping Damping Damping

no Appendage Appendage Appendage Appendage appendage

Disp. R.F. Disp R.F. Disp. R.F. Disp. R.F. (m)

El 0.4548 0.3067 0.67 0.2902 0.64 0.2712 0.59 0.2299 0.50 Centro

Foster 0.4534 0.4001 0.88 0.3608 0.80 0.3582 0.79 0.3523 0.78 City

U.C. 0.4533 0.0647 0.14 0.0658 0.15 0.0677 0.15 0.0718 0.16 Santa Cruz Note: R.F. = Reduction Factor

42

Table 4.1. Characteristics of bridge model components

Component Material Shape Dimensions (in)

Cables Stainless Circular 1/64 in diam. steel

Deck Aluminum Rectangular 144 X 3 X 1/8

Towers Aluminum Rectangular 29 X 1-1/2 X 3/4 X tube 1/16

Cross beams Aluminum Rectangular 3-1/2 X 3/4 X 1/2 X tube 1/16

Abutments Aluminum Rectangular 2-1/2 X 2-1/2 X 1/4 tube

Braces Aluminum Rectangular 1-1/2 X 3/4 X 1/8 tube

Base plates Aluminum Rectangular 7-1/2 X 7-1/2 X 1/4

Bearing Plastic Rectangular 3/4 X 1 X 1-1/2 plates

Ball Stainless Spherical 13/32 in diam. bearings steel

Table 4.2. Prestressing tensions in cables of bridge model

Cable plane

Location number from tower top

Tension (ib)

North 1 0.56

North 2 0.25

North 3 0.13

North 4 0.04

South 1 0.72

South 2 0.44

out 3 0.18

South 4 0.12

43

Table 4.3. Design parameters of appendage model

Parameter Design value

Weight 3.232 lb

Spring constant 13.010 lb/in

Damping constant 0.216 lb-sec/in

Table 4.4. Parameters of appendage experimental model

Parameter Selected value

Weight 3.38 lb



Table 4.5. Maximum longitudinal accelerations of bridge deck end and left tower top without and with appendage

Excitation Acceleration of deck left end (mV)

Acceleration of left tower top (mV)

No appendage

With appendage

R.F. No appendage

With appendage

R.F.

El Centro 787.2 463.8 0.59 1047.6 981.7 0.94

Foster City 613.4 417.8 0.68 606.2 531.0 0.88

U.C. Santa Criiz

965.7 622.9 0.65 753.6 732.7 0.97

Note: R.F. = Reduction Factor

44

Table 4.6. Maximum vertical accelerations of bridge deck center without and with appendage

Excitation Acceleration (mY) ____

No appendage

With appendage

R.F.

El Centro 1090.2 997.1 0.91

Foster City 729.1 539.9 0.74

U.C. Santa Cruz

1139.4 791.0 0.69


Table 4.7. Maximum longitudinal accelerations of bridge end without and with appendage in test with phase lag

Excitation Acceleration (mV) _____

No appendage

With appendage

R.F.

El Centro 573.2

Foster City 733.4 398.3 0.54

U.C. Santa Cruz

1055.2 509.7 0.48

Note: R.F. = Reduction Faci

45

Table 4.8. Maximum longitudinal accelerations of bridge deck center without appendage and with appendage slightly out of tune

Excitation Acceleration (mV)

No appendage

With appendage

R.F.

El Centro 1272.3 1259.2 0.99

Foster City 1042.8 914.6 0.88

U.C. Santa Cruz11

1091.2 839.9 0.77


Table 4.9. Parameters of appendage model tuned with Den Hartog's formula

Parameter Selected value

Weight 5.96 lb



46

Table 4.10. Maximum longitudinal accelerations of bridge deck center without and with appendage in test with tuning according to Den Hartog's fonnula

Excitation Acceleration (mV)

No appendage

With appendage

R.F.

El Centro 1272.3 0.

Foster City 1042.8 783.7 0.75

U.C. Santa Cruz

1091.2 868.5 0.80

Note: R.F. = Reduction actor

Table 4.11. Maximum longitudinal accelerations of bridge deck center without appendage and with appendage glued to bridge deck

Excitation Acceleration (mV) _

No appendage

With appendage

R.F.

El Centro 1272.3 1381.6 1.09

Foster City 1042.8 1102.9 1.06

U.C. Santa Cruz

1091.2 1149.0 1.05

Note: R.F. = Reduction Factöi

47

Dimensions in

IER E

Figure 3.1. Elevation of simplified bridge model in simulation study

5.0 m

15.0 m

20.0 m 60.0 m

7.5 m

10.0 m

37.5 m

Figure 3.2. Simplified model of pier and tower of bridge in simulation study

49

ODEI I Vobuko Bridge REQ I 0.077SO Hz

01 0

Figure 3.3. Natural frequency and mode shape in first mode of Yobuko bridge

flODE: 2 Vobuko Bridqe PREQ I 0.286iS Hz

Figure 3.4. Natural frequency and mode shape in second mode of Yobuko bridge

IlODIfl 3 FREI1 I 0.2986I Hz

obuko Bridqe

L.

Figure 3.5. Natural frequency and mode shape in third mode of Yobuko bridge

"ODES q Vobuko Bridq FREQ : 0.31252 Hz

Figure 3.6. Natural frequency and mode shape in fourth mode of Yobuko bridge

"ODEI B Yobuko FRQ : 0.18127 Hz

Bridqe

N

L.

Figure 3.7. Natural frequency and mode shape in fifth mode of Yobuko bridge

U' U,

fIOBE! 3 FREQ I 0.2971S Hz

VObuko Bridge ui-u-i io dnping

Figure 3.8. Natural frequency and mode shape in third mode of Yobuko bridge with 10 per cent damping appendage

01.

I1ODE B FREO 0.2ei8 Hz

L. 'fobuko Bridge uikh i dnpinq

Figure 3.9. Natural frequency and mode shape in fifth mode of Yobuko bridge with 10 per cent damping appendage

Lfl

flODEI 3 FRE1 I 0.299GS Hz

Vobuku Bridge ui-th iS dnpinq


"ODE I S FREQ : 0.33605 Hz

L. 'fobuko Bridqe ui-th 15/ dz*npinq


(11 '.0

I1ODE: 2 FREQ 0.28228 Hz

-- 1------- -----

Vobuku Bridge ui-th 20 danping


C' 0

f1ODE I B FR1 I 0.3q3B9 Hz

Vobuko Bridge ui-th 20 dznpinq


ODE: 2 RER : 0.26799 Hz

'fobuko Bridge ui-th 3/ danpinq

Figure 3.14. Natural frequency and mode shape in second mode of Yobuko bridge with 30 per cent damping appendage

I1ODEZ S FREO : 0.35930 Hz

L. 'fobuko Bridge iikh 30% danpinq


,.I

Time (sec)

Figure 3.16. First ten seconds of N-S ground acceleration record of May 18, 1940, El Centro earthquake

300

0 0

___

-300

1989 Foster City]

() 10 20 q fl 41) F0 :ii

Time (Sec) Figure 3.17. E-W ground acceleration record at Foster City of October 17, 1989, Loma Prieta earthquake

500

Ii

-30(

5 10 15 20

Time (sec) 25 30 35 40

0) U'

Figure 3.18. E-W ground acceleration record at UC Santa Cruz of October 17, 1989, Loma Prieta earthquake

RESPONSE SPECTRA El Cenfro N—S, May 18, 1 940

0.4

0.35

0.3 _

0.25

0 A \1 0.2

U "A E015

V W.A 0.1 0

(1)

0.05

0 0.10 1.00 10.00 100.00

Frequency (Hz)

RESPONSE SPECTRA El Cenfro N—S, May 18, 1940

3 ---

2.5 ----- 0)

C

.9 2 ----- - 0 L a)

-- -- C) U

0 v IV - -

U a)

0 --- -.=-- 0.01 0.10 1.00 10.00 100.00

Frequency (Hz)

j.I

Figure 3.19. Response spectra for 0, 1, 5, 10 and 20 per cent damping of 1940 El Centro accelerogram

I ,

I

•• •

IIIIllhI•tIIIIIHIIIIIHIUIIIIIHI I . •IIIflIIIUUI1IAIIIUIIIHIII•IIIIIIII uiiiiiii•iniiiuiiooiuiiiuiiii

•111111111fl1llh1111Hh11111flh11 iuiniii•rnu uiin•uiuiiiiivuinim I •uniohiuuiiinu••uiniiiusuiiiiii • O

iliffllIIh!IIINHIUIIIIIllI •IIIIIIII•IIIIHki!!UIIIIIUIIIIHI

0I I I II Oh hill

RESPONSE SPECTRA Foster City E—W, October 17, 1989

3.5

0)

2.5

uuuiii•uuiiii•ii•uiu ••uuiuii 111111 IIIIIIIIIIIEIIIINIIIIIUIIIII IIIIIIIEUIIIIflHi)ii'ii!iHiIIUIIIIIIH IIIIIllhIRIIIiP1'iiOiiiIIIIIIII

INVA :!UU!lIII

0.10 1.00 10.00 100.00 Frequency (Hz)

Figure 3.20. Response spectra for 0, 1, 5, 10 and 20 per cent damping of 1989 Foster City accelerogram

67

RESPONSE SPECTRA UC Santa Cruz, October 17, 1989

1

0.01 0.1 1 10 100 Frequency (Hz)

Figure 3.21. Response spectra for 0, 1, 5, 10 and 20 per cent damping of 1989 UC Santa Cruz accelerogram

68

0.4

OW

0.2

E 0.1 4-

w E 0 C)

0. (I) 0

-0.2

-0.3

-0.4 L

0 1 2 3 4 5

Time ( sec )

6 7 8 9 10

Figure 3.22. Displacement response of deck end of Yobuko bridge with no appendage under the 1940 El Centro earthquake

0.4

0.3

0.2

Deck end 1940 El Centro Appendage with

10% damping

-4 0

-0.2

-0.3

-0.4

0

1 2 3 4 5 6 7 8 9

10

Time ( sec )

Figure 3.23. Displacement response of deck end of Yobuko bridge with 10 per cent damping appendage under the 1940 El Centro earthquake

0.4

0.3

0.2

E 0.1 4- C

E 0 0 C, 0

a

-0.2

-0.3

-0.4

0 1 2 3 4 5

Time ( sec )

6 7 8 9 10


0.4

0.3

Deck end 1940 El Centro

0.2 Appendage with

20% damping

E 0.1

r.J

-0.2

-0.3

-0.4

0

1 2 3 4 5 6 7 8 9

10

Time ( sec )


0.4

0.3

0.2

E 0.1

a) E 0

0

-0.2

-0.3

-n 4

0 I 2 3 4 5 6 7 8 9 10

Time ( sec )


0.5

0.4

0.3

_- 0.2 E V.-. 0.1

E 0 C) a CC a- C,,

-0.2

-0.3

-0.4

-0.5

Tower top 1940 El Centro No appendage

0 1 2 3 4 5 6 7 8 9 10

Time ( sec )

Figure 3.27. Displacement response of tower top of Yobuko bridge with no appendage under the 1940 El Centro earthquake

0.5

0.4

0.3

_-. 0.2 E - 0.1

E 0 CD C)

CL

MA

m

-0.2

-0.3

-0.4

-0.5

Tower top 1940 El Centro Appendage with

10% damping

0 1 2 3 4 5 6 7 8 9 10

Time ( sec )

Figure 3.28. Displacement response of tower top of Yobuko bridge with 10 per cent damping appendage under the 1940 El Centro earthquake

I

0.5

0.4

0.3

- 0.2 E . 0.1 C 0 E 0 0 U Co 0.1

-0.2

-0.3

-0.4

-0.5


15% damping

0 1 2 3 4 5 6 7 8 9 10

Time ( sec )

Figure 329. Displacement response of tower top of Yobuko bridge with 15 per cent damping appendage under the 1940 El Centro earthquake

0.5

0.4

0.3

- 0.2 E - 0.1 .- 0 E 0

0.1

Mn CI

(3 0 0.

-0.2

-0.3

-0.4

-0.5


20% dampIng

0 1 2 3 4 5 6 7 8 9 10

Time ( sec )


S

0.5

0.4

0.3

- 0.2 E — 0.1 C 0 E 0

0.1

U 0 0.

-0.2

-0.3

-0.4

-0.5

Tower top - 1940 El Centro

Appendage with - 30% damping

0 1 2 3 4 5 6 7 8 9 10

Time ( sec )


0.5

0.4

0.3

-. 0.2 E - 0.1 4.. C w E 0 w C)

a.

-0.2

-0.3

-0.4

-0.5

-4 kD

Deck end J1989 Foster City

I No Appendage

0 6 12 18 24 30 36 42 48 54 60

Time ( sec )

Figure 3.32. Displacement response of deck end of Yobuko bridge with no appendage under the 1989 Foster City earthquake

0.5

0.4

0.3

- 0.2 E - 0.1 C a, E 0

a, 0.

-0.2

-0.3

-0.4

-0.5

- Deck end 1989 Foster City

- Appendage with 10% damping

0 6 12 18 24 30 36 42 48 54 60

Time ( sec )

Figure 3.33. Displacement response of deck end of Yobuko bridge with 10 per cent damping appendage under the 1989 Foster City earthquake

0.5

0.4

0.3

- 0.2 E —. 0.1 C C, E 0 C, U

0. U,

-0.2

-0.3

-0.4

-0.5

Deck end 1989 Foster City Appendage with

15% damping

0 6 12 18 24 30 36 42 48 54 60

Time ( sec )


0.5

0.4

0.3

_. 0.2 E '- 0.1 4-

0 E 0 0 U 0 a. (I,

-0.2

-0.3

-0.4

-0.5


20% damping

0 6 12 18 24 30 36 42 48 54 60

Time ( sec )


0.5

0.4

0.3

- 0.2

E 0.1

4-

w E 0 0) 0


30% dampIng

-0.2

-0.3

-0.5

0 6 12 18 24 30 36 42 48 54 60

Time ( sec )


0.5

0.4

0.3

- 0.2 E -. 0.1 4-

E 0 w 0

0.

-0.2

-0.3

-0.4

-0.5

to

Tower top 1989 Foster City

No Appendage

0 6 12 18 24 30 36 42 48 54 60

Time ( sec )

Figure 3.37. Displacement response of tower top of Yobuko bridge with no appendage under the 1989 Foster City earthquake

0.5

0.4

0.3

- 0.2 E — 0.1 C 0 E 0 w 0 0 0.1 0. Cl,

-0.2

-0.3

-0.4

-0.5

- Deck end 1989 U. C. Santa Cruz

- Appendage with 30% damping

0 4 8 12 16 20 24 28 32 36 40

Time ( sec )

Figure 3.46. Displacement response of deck end of Yobuko bridge with 30 per cent damping appendage under the 1989 U.C. Santa Cruz earthquake

0.5

0.4

Tower top 1989 U. C. Santa Cruz

0.3

No appendage

._. 0.2 E

0.1

E 0

a. 0.1

-0.2

-0.3

-0.4

-0.5

0 4 8 12 16 20 24 28 32 36 40

TIme(sec)

Figure 3.47. Displacement response of tower top of Yobuko bridge with no appendage under the 1989 U.C. Santa Cruz earthquake

*

LA

0.4

0.3

-. 0.2 E

0.1

-0.3

-0.4

-0.5


Appendage with 10% dampIng

0 4.02 8.02 12 16 20 24 28 32 36

Time ( sec )

Figure 3.48. Displacement response of tower top of Yobuko bridge with 10 per cent damping appendage under the 1989 U.C. Santa Cruz earthquake

0.5

0.4

0.3

- 0.2 E — 0.1 C

E 0 a) C.)

a. 0.1

-0.2

-0.3

-0.4

-0.5


Appendage with 15% damping

0 4 8 12 16 20 24 28 32 36 40

Time ( sec )


4

0.5

0.4

0.3

. 0.2 E — 0.1 C 0 E 0 C, U CC

-0.2

-0.3

-0.4

-0.5

Tower end 1989 U. C. Santa Cruz

Appendage with 20% damping

0 4 8 12 16 20 24 28 32 36 40

Time(sec)


0.5

0.4

0.3

-0.3

-0.4

-0.5

0 4 8 12 16 20 24 28 32 36 40

Time ( sec )


- 4 a

0.3

Relative displacement 0.2

Appendage vs. Deck 1940 El Centro 20% damping

-0.2

-0.3

0 2 4 6 8 10

Time ( sec )

Figure 3.52. Relative displacement of 20 per cent damping appendage in Yobuko bridge under the 1940 El Centro earthquake

0.4

Relative displacement Appendage vs. Deck

1989 Foster City

E 0. 2

20% damping

MIM

-0.4

0 6 12 18 24 30 36 42 48 54 60

Time ( sec )

Figure 3.53. Relative displacement of 20 per cent damping appendage in Yobuko bridge under the 1989 Foster City earthquake

"ODES 3 PREt : 1.1S7 Hz

/

Experimen11 Model

Figure 4.6. Analytical third natural frequency and mode shape mode of experimental model

0OD

MODE: FRE€ : 15.73q Hz

*

L Experimen-1a1 Model

Figure 4.7. Analytical fourth natural frequency and mode shape mode of experimental model

3 7 0 9 10 1.

-1 - - -

.,,-- -

: It

-

p.

Figure 4.10. Photograph of damper in appendage model

111

11on: I FRE : BfG7 Hz

Exprinen±1 liodel with 2B damping

Figure 4.11. Analytical first natural frequency and mode shape mode of experimental model with resonant appendage

I1ODI 2 FREI1 : B.'f87 Hz

S i

I

Eper-inen-tzd rioclel with 2 lamping

Figure 4.12. Analytical second natural frequency and mode shape mode of experimental model with resonant appendage

flOBE: 3 FRE I 6.9I1 Hz

Experiinent1 •liodeI iiitli 2E Ilamping

Figure 4.13. Analytical third natural frequency and mode shape mode of experimental model with resonant appendage

F1OBE: ' FREIL& I iO.±S Hz

Experinan±1 Iloclel with 2B clamping

Figure 4.14. Analytical fourth natural frequency and mode shape mode of experimental model with resonant appendage

I1ODE: FRE*L : 16.79'I Hz

Experiinen±1 liodel with 2E clanping

Figure 4.15. Analytical fifth natural frequency and mode shape mode of experimental model with resonant appendage

Mode: 1 Freq: 6.23 Hz Damp: 7.67

Figure 4.16. Natural frequency, damping ratio, and mode shape in first mode of experimental model

Mode: 2 Freq: 9.94 Hz Damp: - 4.88 >

z

Figure 4.17. Natural frequency, damping ratio, and mode shape in second mode of experimental model

'.0

Mode: 3 Freq: 14.53Hz Damp: 4.25 >'

x

Figure 4.18. Natural frequency, damping ratio, and mode shape in third mode of experimental model

Figure 4.19. Experimental bridge model on shaking tables

120

___

- ..

Am ....

I

Accelerometer

N

Figure 4.21. Schematic equipment arrangement for determination of dynamic properties

122

F Signal]Analyzer

Accelerometer

N

Figure 4.22. Schematic equipment arrangement for response analysis

123

X=4.9434 Sec Ya=-787 . 2mV

CAF 1.26

31.5 m

/Div

FReal

LV

—1..26

Pxd Y 0.0 Sec 40.0

Figure 4.24. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.5 times El Centro earthquake

U

a

X47852 Sec Ya463 786mV

CAP 1. 26

3 15 m

/D±v

Real

V

—I 26

Pxd Y 0.0 Sec 40.0 Figure 4.25. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.5 times El

Centro earthquake

X=16.291 Sec Ya==-613 . 39mV CAP TIM UF 1.26t

Real

V

—1 26

Fxd Y 0.0 Sec 40.0

Figure 4.26. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.7 tImes c' Foster City earthquake

X=15.229 Sec Ya=47 . B33mV CAP TIM BUF :1.25

35 m

/Dv

Real ___________________ Iry

CO

V

Fxd Y0.0 Sec 40.0

Figure 4.27. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.7 times Foster City earthquake

CAP 1.26

Real

V

*

X==14.912 Sec Ya=-965 68mV

Fxd Y 0.0 Sec 40.0

Figure 4.28. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.4 times UC Santa Cruz earthquake

LI 0

X=13.039 Sec Ya=622 . 929mV

CAP TIM E,UF 1.26

315 m

/Di.v

e a 1

- .26

Pxd Y0.0 Sec 40.0

Figure 4.29. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.4 times UC Santa Cruz earthquake

a a a

X=5.0391 Sec Ya=1.04763 V

CAP TIM BUF 1.26E

315 m

/Div

Real

V

—1.26

Fxd Y 0.0 Sec 4U.0

Figure 4.30. Longitudinal acceleration response of tower top of bridge model without appendage under 0.5 times El Centro earthquake

X=5.1621 Sec Ya==981 . 679mV

CAP TIM BUF 1.261 0v2

3 1 E rr

/Div

Real

—1..

Fxd Y 0.0 Sec 4..

Figure 4.31. Longitudinal acceleration response of tower top of bridge model with appendage under 0.5 times El Centro earthquake

X=11.916, Sec Ya=-606 . 2lmV CAP TIMUF

uv 1.26fl

315 m

/Div

Real

LVA

.26

Fxd Y 0.0 Sec 40.0

Figure 4.32. Longitudinal acceleration response of tower top of bridge model without appendage under 0.7 times Foster City earthquake

XL5.689 Sec Ya=-530 . 97mV CAP TIM UP L.26l 0v2

31 m

/Div

Real

V

—L .26

Pxd Y Sec 4..

Figure 4.33. Longitudinal acceleration response of tower top of bridge model with appendage under 0.7 times Foster City earthquake

4

X=13.207 Sec Ya=-753 . 61mV

CAP TIM B.UF 0v2 1.26

315 m

/D±v

Real

-1.26

Fxd Y 0.0 Sec 40.0

Figure 4.34. Longitudinal acceleration response of tower top of bridge model without appendage under 0.4 times UC Santa Cruz earthquake

L) Ui

X=15.264 Sec Ya=732. 662mV

CAP TIM BUF 1.261

0v2

Real

V

-1.26

Fxd Y

Sec 4..

Figure 4.35. Longitudinal acceleration response of tower top of bridge model with appendage under 0.4 tImes UC Santa Cruz earthquake

a

X==4.9082 Sec Ya=-1.0902 V

CAP TIM ELUF 1.26

315 m

/D±v

Real

V

-1.26

Fxd Y 0.0 Sec 40.0

Figure 4.36. Vertical acceleration response of left deck center of bridge model without appendage under 0.5 tImes El Centro earthquake

X=5.6816 Sec Ya=-997. 12mV

CAFE 1.26

315 m

/Div

Real

V

I - 26

Fxd Y

TIM E,LJF Dvi 1 1

U

MINEW

Sec

Figure 4.37. Vertical acceleration response of left deck center of bridge model with appendage under 0.5 times El Centro earthquake

Real

V

CAP TIM B.UF 1.26

3 15 m

/Dav

IT 11 , 1

X=15.156 Sec Ya=-729. 14mV

P><d YO.O Sec 40.0

Figure 4.38. Vertical acceleration response of left deck center of bridge model without appendage under 0.7 times Foster City earthquake

X=15.738 Sec Ya=-539. 9mV EAF TIM BUF 1.26

1

315 m

/Div

A e a 1

v

Fxd YO.O Sec 40.0

Figure 4.39. Vertical acceleration response of left deck center of bridge model with appendage under 0.7 tImes Foster City earthquake

A

><=L.385 Sec Ya=—.1394 V

CAP TIM EJUF .261 T

31..5 m

/Div

Real

V

—L.26

Fxd Y 0.0 Sec 40.0

Figure 4.40. Vertical acceleration response of left deck center of bridge model without appendage under 0.4 times UC Santa Cruz earthquake

X=3.414 Sec Ya=-791. OmV CAP TIM UF 1.261

315 m

/Div

Rea1

V

—1 . 26

Fxd Y 0.0 Sec 40.0

Figure 4.41. Vertical acceleration response of left deck center of bridge model with appendage under 0.4 times UC Santa Cru.z earthquake

0.5

0.4

0.3

U) e.g

0 > 0.1

C 0

0

0.2

-0.3

-0.4

-0.5

5.048 7.539 10.35 12.99 15.58 18.68 Time ( sec )

Figure 4.42. Longitudinal acceleration response of appendage mass under 0.5 times El Centro earthquake

Bridge deck 1940 El Centro

- 0.2 U)

0 > 0.1

a) a) 0 < -0.2

-0.3

0.4

-0.5

1.565 5.048 7.539 10.35 12.99 15.88 18.68

Time ( sec )

0.5

0.4

0.3

Figure 4.43. Longitudinal acceleration response of appendage support under 0.5 times El Centro earthquake

0.5

0.4

Appendage mass 1989 Foster City

U,

0.3

il U) 4-

0 >0.1

0 4- Co

w -0.1 C) C) < -0.2

-0.3

-0.4

-0.5

2.049 8.705 14.36 19.68 24.97 30.54

Time ( sec )

36.22 42.29

Figure 4.44. Longitudinal acceleration response of appendage mass under 0.7 times Foster City earthquake

0.5

0.4 Bridge deck

0.3 1989 Foster City

IS

-0.3

-0.5

2.049 8.714 14.43 19.69 24.98

Time ( sec )

30.57 36.25 42.32

Figure 4.45. Longitudinal acceleration response of appendage support under 0.7 times Foster City earthquake

0

Appendage mass 1989 U. C. Santa Cruz

0.3

0 > 0.1

CD U U < -0.2

-0.3

MMI

1.087 7.917 13.95 19.35 25.6 31.26 36.76 42.42

Time ( sec )

0.5

0.4

Figure 4.46. Longitudinal acceleration response of appendage mass under 0.4 times UC Santa Cruz earthquake

Bridge deck 1989 U. C. Santa Cruz

0.3

IM

0) C.) a 4 -0.2

-0.3

-0.4

-0.5

1.087 7.917 13.98 19.38 25.63 31.41 36.9 42.51

Time ( sec )

0.5

0.4

Figure 4.47. Longitudinal acceleration response of appendage support under 0.4 times UC Santa Cruz earthquake

Figure 4.48. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.5 times El Centro earthquake with phase lag of 0.4 sec

X=5.3359 Sec Ya=573. 1.8mV

CAP TIM B.UF 1..26

315 m

/Di.v

:eal

-L .26

Pxd Y0.0 Sec 40.0

Figure 4.49. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.5 times El Centro earthquake with phase lag of 0.4 sec

X=16.516 Sec Ya=-733 35mV CAP TIM UF 1.261

315 m

/Di.v

Real

V

—1.26

Fxd Y 0.0 Sec 40.0

Figure 4.50. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.7 times Foster City earthquake with phase lag of 0.4 sec

X=18.896 Sec Ya=398 344mV CAP TIM EUF 1.26

315 m

/D± v

:eal

- I . 26

Fxd Y0.0 Sec 40.0

Figure 4.51. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.7 times Foster City earthquake with phase lag of 0.4 sec

*

X=13.449 Sec Ya=-1.0662 V

CAP TIM QUF 1.261

315 m

/Div

Real

V

-1.26

Fxd Y 0.0 Sec 40.0

Figure 4.52. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.4 times UC Santa Cruz earthquake with phase lag of 0.4 sec

U, L)

X=13.172 Sec Ya=509. 6BBmV

CAP TIM E.UF 1.26

Real

V

—1.26

Fxd Y Sec 4..

Figure 4.53. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.4 times UC Santa Cruz earthquake with phase lag of 0.4 sec

a

2 c'J 4c C-) ci) CO

E 0

0 01 01

a) ci) -2 0 0

3OO

00- 1

0 11V

V

V V

001

Time (sec)

Figure 4.54. Cut version of N-S 1940 El Centro ground acceleration record

RESPONSE SPECTRA Cut Version N—S 1940 El Centro

0.4 - - - - -

0.35

E A\

CD

0.2 - __ 0.1

0.05 _

O ---

0.1 1 10 100 Frequency (Hz)

. •

- S •A• -

11111111 11111111 _ 11111111 _ 11111111 • 11111111 11111111 lilLilil 11111111

11111111 IIIIHIIIII1iAiIIIIIIII • 11111111 IIIIIIHiki!IIi!IIIIIII a 11111111 IIIIIHiIL!!ItEk!iiIIIIII

, 4Il.u.uI li~ro 000ijjiiIIHijiuiioi I I I II

- I -

Figure 4.55. Response spectra for 0, 1, 5, 10, and 20 per cent damping of ground acceleration record in Figure 4.54

156

X=5.2871 Sec Ya=1.27233 V CAP TIM UF 2.OE

499 m

/Div

Real

V

—2.0

Fxd Y 0.0 Sec 40.0

Figure 4.58. Longitudinal acceleration response of deck center of bridge model without appendage under 0.6 times El Centro earthquake j i

X=2.6984 Eec Ya=-1.21.47 V CAP TIM BUF 1.261 0v2

3LE rr

/Div

Real

V

___c Fxd Y 0.0 ec 4..

Figure 4.56. Longitudinal acceleration response of left deck end of bridge model without appendage under 0.5 times the cut version of El Centro earthquake

X=3.7812 Sec Ya=-277 . 77mV

CAP TIM EUP 1.26

315 m

/Di.v

Real

V

-1.26

Fxd Y 0.0 Sec 40.0

Figure 4.57. Longitudinal acceleration response of left deck end of bridge model with appendage under 0.5 times the cut version of El Centro earthquake

X=5.1523 Sec Ya=-1.2592 V

CAP TINv1SUF 2.0

499 m

/Div

Real

V

U

—2.0

Pxd Y Sec 4..

Figure 4.59. Longitudinal acceleration response of deck center of bridge model with out-of-tune appendage under 0.6 times El Centro earthquake

><=15.338 Sec Ya=1.04279 V

CAP 2.0

499 m

/Div

Real

.YA

TIM 1JP

1*

—2.0

Fxd Y 0.0 Sec 40.0

Figure 4.60. Longitudinal acceleration response of deck center of bridge model without appendage under 0.8 times Foster City earthquake

X=18.916 Sec Ya=914 . 603mV CAP TIM BUF 2.0

499 m

/Di.v n

: e a 1 4 —

—2.0

Pxd Y0.0 - Sec 40.0

Figure 4.61. Longitudinal acceleration response of deck center of bridge model with out-of-tune appendage under 0.8 times Foster City earthquake

X=14.639 Sec Ya=1.091L5 V CAP TIM qUF 2.01

499 m

/Di.v

Real

V

2.0

Fxd Y 0.0 Sec 40.0

Figure 4.62. Longitudinal acceleration response of deck center of bridge model without appendage under 0.5 tImes UC Santa Cruz earthquake

X=1.E.674 Sec Ya=-839 . 9mV CAP TIM EUP 2.0

499 m

/Div

Real

V Cbj

—2.0

Fxd Y [SS

Sec 4..

Figure 4.63. Longitudinal acceleration response of deck center of bridge model with out-of-tune appendage under 0.5 times UC Santa Cruz earthquake

Figure 4.64. Longitudinal acceleration response of deck center of bridge model with appendage tuned according to Den Harlog theory under 0.6 times El Centro earthquake

>K=15.763 5ec Ya=783. 7O2mV CAP TIM BUF 1...59(

39-7, ri-

/Div

a-' a-'

V

Fxd YO.O 5ac

Figure 4.65. Longitudinal acceleration response of deck center of bridge model with appendage tuned according to Den Hartog theory under 0.8 times Foster City earthquake

X=16.475 Sec Ya=-868 54mV CAP TIM BUF 1591

397 m

/Di.v

Real.

V

- . 59

Fxd Y 0.0 Sec 40.0

Figure 4.66. Longitudinal acceleration response of deck center of bridge model with appendage tuned according to Den Hartog theory under 0.5 times UC Santa Cruz earthquake

X=6.127 Sec Ya=1.3859 V

CAP TIM EJUF 2.01 Cvi

a.' 00

49 m

/Dv

Real

V

' Fxd YO.O Sec

Figure 4.67. Longitudinal acceleration response of deck center of bridge model with appendage glued to bridge deck under 0.5 times El Centro earthquake

X=15.14B Sec Ya=.10286 V

CAP TIM QUF 2.01 T

499 m

/Di.v

Real kD

—2.0

Fxd Y 0.0 Sec 40.0

Figure 4.68. Longitudinal acceleration response of deck center of bridge model with appendage glued to bridge deck under 0.7 times Foster City earthquake

X=L5.592 Sec Ya=1.14902 V CAP TIM BIJF 2.0

499 m

n /Dav

:eal

—20

Fxd Y0.0 Sec 40.0

Figure 4.69. Longitudinal acceleration response of deck center of bridge model with appendage glued to bridge deck under 0.4 times UC Santa Cruz earthquake

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