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FINAL REPORT TO THE CALIFORNIA DEPARTMENT OF TRANSPORTATION SOCIO ECONOMIC EFFECT OF SEISMIC RETROFIT IMPLEMENTED ON BRIDGES IN LOS ANGELES HIGHWAY NETWORK RTA 59A0304 By Masanobu Shinozuka1, Professor and Youwei Zhou1, Graduate Research Assistant Sang Hoon Kim1, Post Doctoral Researcher Yuko Murachi1, Visiting Researcher Swagata Banerjee1, Graduate Research Assistant Sunbin Cho2, Research Engineer Howard Chung2, Research Engineer 1 Department of Civil and Environmental Engineering University of California, Irvine 2 ImageCat, Inc. October 2005 STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION TECHNICAL REPORT DOCUMENTATION PAGE TR0003 ( REV. 10/ 98) 1. REPORT NUMBER F/ CA/ SD 2005/ 03 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER 5. REPORT DATE October 2005 4. TITLE AND SUBTITLE SOCIO ECONOMIC EFFECT OF SEISMIC RETROFIT IMPLEMENTED ON BRIDGES IN THE LOS ANGELES HIGHWAY NETWORK 6. PERFORMING ORGANIZATION CODE 7. AUTHOR( S) Masanobu Shinozuka, Youwei Zhou, Sanghoon Kim, Yuko Murachi, Swagata Banerjee, Sunbin Cho and Howard Chung 8. PERFORMING ORGANIZATION REPORT NO. 10. WORK UNIT NUMBER 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Civil and Environmental Engineering The Henry Samueli School of Engineering University of California, Irvine Irvine, California 92697 11. CONTRACT OR GRANT NUMBER RTA 59A0304 13. TYPE OF REPORT AND PERIOD COVERED Final Report 12. SPONSORING AGENCY AND ADDRESS California Department of Transportation Division of Research and Innovation, MS 83 1227 O Street Sacramento CA 95814 14. SPONSORING AGENCY CODE 15. SUPPLEMENTAL NOTES 16. ABSTRACT This research studied socio economic effect of the seismic retrofit implemented on bridges in Los Angeles Area Freeway Network. Firstly, advanced FE ( Finite Element) modeling and nonlinear time history analysis are carried out to evaluate the seismic performance in the form of fragility curve, of representative bridges before and after retrofit. This analysis resulted in the determination of retrofit effect in such a way that we can quantify, through the change in fragility parameters, the improvement of bridge seismic performance after retrofit. Secondly, an integrated traffic assignment model is introduced to consider change in the post earthquake OD characteristics due to building damage, and is utilized to evaluate the post earthquake network performance of the damaged freeway network in terms of daily travel delay ( compared with the travel time associated with the freeway network not damaged) and attendant opportunity cost. Furthermore, the process of system restoration is simulated to estimate the total social cost based on bridge functionality restoration ( repair / replacement) process. The benefit from the retrofit is defined as the combined social and bridge restoration cost avoided by comparing the total social and bridge restoration cost before and after bridge retrofit. The benefit resulting from combined social and bridge restoration cost avoided together with the bridge retrofit cost are used for a cost benefit analysis. The result shows that the retrofit is cost effective if both social and bridge restoration cost avoided are considered, and the bridge restoration cost avoided can only contribute a small portion of the initial bridge retrofit cost. 17. KEY WORDS Bridge Fragility, Retrofit, Seismic Risk Analysis, Traffic Assignment, Monte Carlo Simulation, Loss Estimation, Cost Benefit Analysis 18. DISTRIBUTION STATEMENT No restrictions. This document is available to the public through the National Technical Information Service, Springfield, VA 22161 19. SECURITY CLASSIFICATION ( of this report) Unclassified 20. NUMBER OF PAGES 314 21. PRICE Reproduction of completed page authorized DISCLAIMER: The Opinions, findings, and conclusions expressed in this publication are those of the authors and not necessarily those of the STATE OF CALIFORNIA ABSTRACT This research studied socio economic effect of the seismic retrofit implemented on bridges in Los Angeles Area Freeway Network. Firstly, advanced FE ( Finite Element) modeling and nonlinear time history analysis are carried out to evaluate the seismic performance in the form of fragility curve, of representative bridges before and after retrofit. This analysis resulted in the determination of retrofit effect in such a way that we can quantify, through the change in fragility parameters, the improvement of bridge seismic performance after retrofit. Secondly, an integrated traffic assignment model is introduced to consider change in the post earthquake OD characteristics due to building damage, and is utilized to evaluate the post earthquake network performance of the damaged freeway network in terms of daily travel delay ( compared with the travel time associated with the freeway network not damaged) and attendant opportunity cost. Furthermore, the process of system restoration is simulated to estimate the total social cost based on bridge functionality restoration ( repair / replacement) process. The benefit from the retrofit is defined as the combined social and bridge restoration cost avoided by comparing the total social and bridge restoration cost before and after bridge retrofit. The benefit resulting from combined social and bridge restoration cost avoided together with the bridge retrofit cost are used for a cost benefit analysis. The result shows that the retrofit is cost effective if both social and bridge restoration cost avoided are considered, and the bridge restoration cost avoided can only contribute a small portion of the initial bridge retrofit cost. i ACKNOWLEDGEMENT The research presented in this report was sponsored by the California Department of Transportation ( Caltrans) with Li Hong Sheng as contract monitor. The authors are indebted to Caltrans for its support of this project and to Li Hong Sheng for his most valuable advice. ii TABLE OF CONTENTS Abstract……………………………………………………………………………..……. i Acknowledgements……………………………………………………………………… ii Table of Contents……………………………………………………………………..… iii List of Figures…………………………………………………………………………... vi List of Tables…………………………………………………………………………… xii Chapter 1 Introduction…………………………………………………………………... 1 Chapter 2 Development of Analytical Fragility Curves……………………….………... 7 2.1 Introduction……………………………………………………………… …. 7 2.2 Column Retrofit with Steel Jacketing……………………………………….. 8 2.2.1 Background………………………………………………………... 8 2.2.2 Steel Jacketing…………………………………………………….. 9 2.2.3 Stress Strain Relationship for Confined Concrete……………….... 9 2.3 Bridge Model………………………………………………………….……. 11 2.3.1 Bridge Description………………………………………………... 11 2.3.2 Bridge Modeling………………………………………………….. 12 2.4 Development of Moment Curvature Relationship………………………….. 15 2.4.1 Moment Curvature Relationship in Longitudinal Direction…….... 15 2.5 Bridge Response Analysis…………………………………………………... 19 2.5.1 Input Ground Motions…………………………………………….. 19 2.5.2 Response of Structures………………………………………….…. 22 2.6 Fragility Analysis of Bridges………………………………………………... 26 2.6.1 Fragility Analysis……………………………………………….…. 26 2.6.2 Damage States……………………………………………………... 27 2.7 Pounding, Soil, Jacketing, and Restrainer Effects on Fragility Curves……... 28 2.7.1 Pounding at Expansion Joint…………………………………….… 28 2.7.2 Numerical Simulation for Pounding………………………………. 31 2.7.3 Pounding Effects on Fragility Curves……………………………... 33 2.7.4 Pounding and Soil Effects on Fragility Curves……………………. 38 2.7.5 Jacketing and Restrainer Effects on Fragility Curves……………... 43 iii 2.8 Fragility Enhancement after Column Retrofit………………………………. 48 2.8.1 Fragility Curves after Retrofit by Column Jacketing for Longitudinal Direction……………………………………………. 48 2.8.1.1 Enhancement for Circular Column……………………… 55 2.8.1.2 Enhancement for Oblong Shape Column……………….. 56 2.8.1.3 Enhancement for Rectangular Column………………….. 56 2.8.1.4 Enhancement for All Types of Columns………………... 57 2.8.2 Enhancement after Calibrating the Analytical Fragility Curves....... 58 2.8.3 Fragility Curves after Retrofit by Column Jacketing for Transverse Direction………………………………………………. 61 Chapter 3 Development of Empirical Fragility Curves…………………………….…… 67 3.1 Empirical Bridge Damage Data……………………………………………... 67 3.2 Bridge Classification………………………………………………………… 70 3.3 Parameter Estimation………………………………………………………... 71 3.4 Enhancement of Empirical Fragility Curves………………………………… 98 Chapter 4 Seismic Hazard Modeling for Spatially Distributed Highway System……... 101 4.1 Highway Network: Spatially Distributed System………………………….. 101 4.2 Deterministic Seismic Hazard…………………………………………….... 102 4.3 Probabilistic Seismic Hazard………………………………………………. 103 Chapter 5 Methodology for System Performance Evaluation of Highway Network….. 107 5.1 Overview………………………………………………………………...…. 107 5.2 Site Ground Motion……………………………………………………..… 108 5.3 Network Modeling………………………………………………………… 109 5.4 Bridge Damage State Simulation………………………………………….. 110 5.5 Assignment of Link Damage State and Residual Capacity…………..…… 111 5.6 Traffic Demand: Origin Destination Data………………………………… 113 5.6.1 1996 SCAG Origin Destination Data……………………………….. 113 5.6.2 Origin Destination Data Condensation…………………………….... 117 5.6.3 Origin Destination Data Change After Earthquake…………………. 118 5.7 The Integrated Model……………………………………………...………. 121 5.8 Drivers’ Delay………………………………………………………...…… 123 iv 5.9 Opportunity Cost………………………………………………………….. 124 Chapter 6 Direct Economic Loss: Bridge Repair Cost………………………………… 127 6.1 Number of Seismically Damaged Bridges ………………………………… 127 6.2 Bridge Repair Cost Estimation in an Earthquake………………………….. 134 6.3 Expected Annual Repair Cost of a Site Specific Bridge……………..……. 137 6.3.1 Annual Probability of Damage………………………………...… 137 6.3.2 Expected Annual Repair Cost before Retrofit…………………… 138 6.3.3 Expected Annual Repair Cost after Retrofit………………...…… 138 6.3.4 System Annual Bridges Repair Cost…………………….……….. 139 Chapter 7 Social Cost Estimation……………………………………………………… 143 7.1 Daily Social Cost………………………………………………………...… 143 7.1.1 Daily Social Cost under no Retrofit Condition………………...… 143 7.1.2 Retrofit Effect on Daily Social Cost………………………….….. 147 7.2 System Restoration..………………………………………………..……… 154 7.2.1 System Restoration Based on Bridge Repair Process………….... 154 7.2.2 System Restoration Based on Bridge Functionality Restoration... 157 7.2.3 OD Recovery……………………………………………..……… 158 7.2.4 System Restoration Curve and Total Social Cost……………..… 159 7.3 Economic Loss Estimation Related to System Social Cost…………...…… 167 Chapter 8 Cost effectiveness Analysis……………………………………………….... 171 8.1 Introduction………………………………………………………………… 171 8.2 Retrofit Cost ………..………………………...……………………...…….. 171 8.3 Benefit from Retrofit………..…………………………………………...…. 168 8.4 Cost Effectiveness Evaluation…………………………………………...… 179 Chapter 9 Conclusions ………………………………………………………………… 189 Appendix A Moment Rotation Curves of Bridge Columns………..…………….…..... 193 Appendix B Integrated Traffic Assignment Model.………………………..………….. 257 Appendix C HighwaySRA Manual………………………..………………………..… 293 Reference……………………………………………………………………...……….. 311 v LIST OF FIGURES Fig. 2.1 Stress Strain Model for Concrete in Compression 10 Fig. 2.2 Elevation of Sample Bridges 13 Fig. 2.3 Nonlinearities in Bridge Model 15 Fig. 2.4 Column 2 of Bridge 1 before retrofit 16 Fig. 2.5 Column 2 of Bridge 1 after retrofit 18 Fig. 2.6 Acceleration Time Histories Generated for Los Angeles 21 Fig. 2.7 Responses at Column End of Bridge 1 23 Fig. 2.8 Displacement at Expansion Joints of Bridge 1 25 Fig. 2.9 Gap Element 31 Fig. 2.10 Ground Motion Time History for LA01 32 Fig. 2.11 Pounding Force at Expansion Joint 32 Fig. 2.12 Structural Responses without Pounding 32 Fig. 2.13 Structural Responses with Pounding 32 Fig. 2.14 Pounding Effect on Fragility Curves of Bridge 2 34 Fig. 2.15 Pounding Effect on Fragility Curves of Bridge 3 35 Fig. 2.16 Pounding Effect on Fragility Curves of Bridge 4 36 Fig. 2.17 Pounding Effect on Fragility Curves of Bridge 5 37 Fig. 2.18 Pounding and Soil Effects on Fragility Curves of Bridge 2 39 Fig. 2.19 Pounding and Soil Effects on Fragility Curves of Bridge 3 40 Fig. 2.20 Pounding and Soil Effects on Fragility Curves of Bridge 4 41 Fig. 2.21 Pounding and Soil Effects on Fragility Curves of Bridge 5 42 Fig. 2.22 Jacketing and Restrainer Effects on Fragility Curves of Bridge 2 44 Fig. 2.23 Jacketing and Restrainer Effects on Fragility Curves of Bridge 3 45 Fig. 2.24 Jacketing and Restrainer Effects on Fragility Curves of Bridge 4 46 Fig. 2.25 Jacketing and Restrainer Effects on Fragility Curves of Bridge 5 47 Fig. 2.26 Retrofit Effect on Fragility Curves of Bridge 1 ( Longitudinal) 50 Fig. 2.27 Retrofit Effect on Fragility Curves of Bridge 2 ( Longitudinal) 51 Fig. 2.28 Retrofit Effect on Fragility Curves of Bridge 3 ( Longitudinal) 52 Fig. 2.29 Retrofit Effect on Fragility Curves of Bridge 4 ( Longitudinal) 53 vi Fig. 2.30 Retrofit Effect on Fragility Curves of Bridge 5 ( Longitudinal) 54 Fig. 2.31 Enhancement Curve for Circular Columns with Steel Jacketing 55 Fig. 2.32 Enhancement Curve for Oblong Columns with Steel Jacketing 56 Fig. 2.33 Enhancement Curve for Rectangular Columns with Steel Jacketing 57 Fig. 2.34 Enhancement Curve for Five Sample Bridges with Steel Jacketing 58 Fig. 2.35 Empirical Fragility Curves and Calibrated Analytical Fragility Curves of Bridge 2 60 Fig. 2.36 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 62 Fig. 2.37 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 63 Fig. 2.38 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 64 Fig. 2.39 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 65 Fig. 2.40 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 66 Fig. 3.1 1994 Northridge Earthquake: PGA Distribution 68 Fig. 3.2 1994 Northridge Earthquake: PGV Distribution 68 Fig. 3.3 Fragility Curve on the basis of PGA ( Composite) 77 Fig. 3.4 Fragility Curve on the basis of PGV ( Composite) 77 Fig. 3.5 Fragility Curve in PGA ( Single Span) 78 Fig. 3.6 Fragility Curve in PGA ( Multiple Span) 78 Fig. 3.7 Fragility Curve in PGA ( Skew 00 200) 79 Fig. 3.8 Fragility Curve in PGA ( Skew 200 600) 79 Fig. 3.9 Fragility Curve in PGA ( Skew > 600) 80 Fig. 3.10 Fragility Curve in PGA ( Soil A) 80 Fig. 3.11 Fragility Curve in PGA ( Soil B) 81 Fig. 3.12 Fragility Curve in PGA ( Soil C) 81 Fig. 3.13 Fragility Curve in PGA ( Single Span / Skew 00 200) 82 Fig. 3.14 Fragility Curve in PGA ( Single Span/ Skew 200 600) 82 Fig. 3.15 Fragility Curve in PGA ( Single Span/ Skew > 600) 83 Fig. 3.16 Fragility Curve in PGA ( Multiple Span / Skew 00 200) 83 Fig. 3.17 Fragility Curve in PGA ( Multiple Span/ Skew 200 600) 84 Fig. 3.18 Fragility Curve in PGA ( Multiple Span/ Skew > 600) 84 Fig. 3.19 Fragility Curve in PGA ( Skew 00 200/ Soil A) 85 vii Fig. 3.20 Fragility Curve in PGA ( Skew 00 200/ Soil B) 85 Fig. 3.21 Fragility Curve in PGA ( Skew 00 200/ Soil C) 86 Fig. 3.22 Fragility Curve in PGA ( Skew 200 600/ Soil A) 86 Fig. 3.23 Fragility Curve in PGA ( Skew 200 600/ Soil B) 87 Fig. 3.24 Fragility Curve in PGA ( Skew 200 600/ Soil C) 87 Fig. 3.25 Fragility Curve in PGA ( Skew > 600/ Soil A) 88 Fig. 3.26 Fragility Curve in PGA ( Skew > 600/ Soil C) 88 Fig. 3.27 Fragility Curve in PGA ( Single Span / Soil A) 89 Fig. 3.28 Fragility Curve in PGA ( Single Span / Soil B) 89 Fig. 3.29 Fragility Curve in PGA ( Single Span / Soil C) 90 Fig. 3.30 Fragility Curve in PGA ( Multiple Span / Soil A) 90 Fig. 3.31 Fragility Curve in PGA ( Multiple Span / Soil B) 91 Fig. 3.32 Fragility Curve in PGA ( Multiple Span / Soil C) 91 Fig. 3.33 Fragility Curve in PGA ( Single Span / Skew 00 200 / Soil A) 92 Fig. 3.34 Fragility Curve in PGA ( Single Span / Skew 00 200 / Soil B) 92 Fig. 3.35 Fragility Curve in PGA ( Single Span / Skew 00 200 / Soil C) 93 Fig. 3.36 Fragility Curve in PGA ( Single Span / Skew 200 600 / Soil C) 93 Fig. 3.37 Fragility Curve in PGA ( Single Span / Skew > 600 / Soil C) 94 Fig. 3.38 Fragility Curve in PGA ( Multiple Span / Skew 00 200 / Soil A) 94 Fig. 3.39 Fragility Curve in PGA ( Multiple Span / Skew 00 200 / Soil C) 95 Fig. 3.40 Fragility Curve in PGA ( Multiple Span / Skew 200 600 / Soil A) 95 Fig. 3.41 Fragility Curve in PGA ( Multiple Span / Skew 200 600 / Soil B) 96 Fig. 3.42 Fragility Curve in PGA ( Multiple Span / Skew 200 600 / Soil C) 96 Fig. 3.43 Fragility Curve in PGA ( Multiple Span / Skew > 600 / Soil A) 97 Fig. 3.44 Fragility Curve in PGA ( Multiple Span / Skew > 600 / Soil C) 97 Fig. 3.45 Enhanced Fragility Curve ( Minor) 99 Fig. 3.46 Enhanced Fragility Curve ( Moderate) 99 Fig. 3.47 Enhanced Fragility Curve ( Major) 100 Fig. 3.48 Enhanced Fragility Curve ( Collapse) 100 Fig. 5.1 Flow Chart for System Performance Evaluation 107 Fig. 5.2 Highway Network: Link, Node and Bridge Component 109 viii Fig. 5.3 Network Model: Los Angeles and Orange County 109 Fig. 5.4 Detour after Northridge Earthquake ( January 20th 1994) 113 Fig. 5.5 1996 Southern California Origin Destination Data 115 Fig. 5.6 OD Data Condensation: Thiessen Polygon 117 Fig. 5.7 Integrated Trip Reduction And Network Models 120 Fig. 6.1 Bridge Damage in Elysian Park 7.1 ( Without Retrofit) 128 Fig. 6.2 Link Damage in Elysian Park 7.1 ( Without Retrofit) 129 Fig. 6.3 Bridge Damage in Elysian Park 7.1 ( 23% Retrofit) 129 Fig. 6.4 Link Damage in Elysian Park 7.1 ( 23% Retrofit) 130 Fig. 6.5 Bridge Damage in Elysian Park 7.1 ( 100% Retrofit) 130 Fig. 6.6 Link Damage in Elysian Park 7.1 ( 100% Retrofit) 131 Fig. 7.1 System Risk Curve in terms of Daily Drivers’ Delay Under different Link Residual Capacity Assumptions ( without retrofit) 145 Fig. 7.2 System Risk Curve in terms of Daily Opportunity Cost Under different Link Residual Capacity Assumptions ( without retrofit) 146 Fig. 7.3 System Risk Curve in terms of Daily Social Cost Under different Link Residual Capacity Assumptions ( without retrofit) 146 Fig. 7.4 Effect of Retrofit on System Risk Curve ( Assumption 1) 150 Fig. 7.5 Effect of Retrofit on System Risk Curve ( Assumption 2) 151 Fig. 7.6 Effect of Retrofit on System Risk Curve ( Assumption 3) 154 Fig. 7.7 Probability Distribution of Functions used to Model Repair Processes 156 Fig. 7.8 Restoration Curves for Highway Bridges ( after ATC 13, 1985) 158 Fig. 7.9 System Recovery Curves After Elysian Park M7.1 ( No Retrofit ) 159 Fig. 7.10 System Recovery Curves After Elysian Park M7.1 ( 23% Retrofit ) 160 Fig. 7.11 System Recovery Curves After Elysian Park M7.1 ( 100% Retrofit ) 160 Fig. 7.12 Retrofit Effect on System Recovery Curve( Elysian Park 7.1, Assumption 1) 162 Fig. 7.13 Retrofit Effect on System Recovery Curve( Elysian Park 7.1, Assumption 2) 162 Fig. 7.14 Retrofit Effect on System Recovery Curve( Elysian Park 7.1, Assumption 3) 163 ix Fig. 7.15 Fig. 7.15 Retrofit Effect on System Recovery Curve ( Elysian Park 7.1, HAZUS) 163 Fig. A. 1 Moment Curvature Analysis of Bridge 1( Longitudinal) 195 Fig. A. 2 Moment Curvature Analysis of Bridge 2( Longitudinal) 196 Fig. A. 3 Moment Curvature Analysis of Bridge 3( Longitudinal) 200 Fig. A. 4 Moment Curvature Analysis of Bridge 4( Longitudinal) 201 Fig. A. 5 Moment Curvature Analysis of Bridge 5( Longitudinal) 220 Fig. A. 6 Moment Curvature Analysis of Bridge 3( Transverse) 225 Fig. A. 7 Moment Curvature Analysis of Bridge 4( Transverse) 234 Fig. A. 8 Moment Curvature Analysis of Bridge 5( Transverse) 245 Fig. B. 1 Framework of Trip Reduction Estimation 258 Fig. B. 2 Integrated Analysis of Trip Reduction and Network Models 262 Fig. B. 3 Personal Trip Reductions caused by Regional Building Damage 263 Fig. B. 4 Comparison of EPEDAT Fragility and the Study Estimated Fragility for Selected Building Occupancy Types 272 Fig. B. 5 Trip Reduction Rate 279 Fig. B. 6 Estimation Procedure of Truck Trip Reduction 279 Fig. B. 7 Truck Trip Reduction Ratio 286 Fig. B. 8 Conceptual OD Recovery Model 292 Fig. C. 1 Main Interface 295 Fig. C. 2 Menu “ Map” 296 Fig. C. 3 Map of Study Region 297 Fig. C. 4 Map of Faults 297 Fig. C. 5 Map of Freeway Network 298 Fig. C. 6 Map of Bridges 298 Fig. C. 7 Menu “ Inventory” 299 Fig. C. 8 Inventory: Bridges 299 Fig. C. 9 Inventory: Network Links 300 Fig. C. 10 Menu “ Hazard” 300 Fig. C. 11 Predefined Events 301 Fig. C. 12 Importing PGA and MMI Shape Map 301 x Fig. C. 13 User Defined Event 302 Fig. C. 14 Current Event Information 302 Fig. C. 15 Analysis: Setting 303 Fig. C. 16 Bridge Fragility Setting 303 Fig. C. 17 Residual Link Performance Setting 304 Fig. C. 18 Economic Loss Estimation Parameter Setting 304 Fig. C. 19 Risk Analysis Option 305 Fig. C. 20 Menu “ Results” 306 Fig. C. 21 PGA distribution 306 Fig. C. 22 Display Bridge Damage States 307 Fig. C. 23 Display Link Damage States 308 Fig. C. 24 System Performance 309 Fig. C. 25 Economic Loss 310 xi LIST OF TABLES Table 2.1 Description of Five ( 5) Sample Bridges 12 Table 2.2 Description of Los Angeles Ground Motions 20 Table 2.3 Description of Damaged States 28 Table 2.4 Peak Ductility Demand of First Left Column of Sample Bridges 28 Table 2.5 Number of Damaged Bridges: Pounding Effect in Bridge 2 34 Table 2.6 Number of Damaged Bridges: Pounding Effect in Bridge 3 35 Table 2.7 Number of Damaged Bridges: Pounding Effect in Bridge 4 36 Table 2.8 Number of Damaged Bridges: Pounding Effect in Bridge 5 37 Table 2.9 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 2 39 Table 2.10 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 3 40 Table 2.11 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 4 41 Table 2.12 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 5 42 Table 2.13 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 44 Table 2.14 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 45 Table 2.15 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 46 Table 2.16 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 47 Table 2.17 Number of Damaged Bridges: Retrofit Effect in Bridge 1 ( longitudinal) 50 Table 2.18 Number of Damaged Bridges: Retrofit Effect in Bridge 2 ( longitudinal) 51 Table 2.19 Number of Damaged Bridges: Retrofit Effect in Bridge 3 ( longitudinal) 52 Table 2.20 Number of Damaged Bridges: Retrofit Effect in Bridge 4 ( longitudinal) 53 xii Table 2.21 Number of Damaged Bridges: Retrofit Effect in Bridge 5 ( longitudinal) 54 Table 2.22 Simulated Ductility Capacities of Sample Bridges 59 Table 2.23 Fragility Curves based on Adjusted Damage States Definitions 61 Table 2.24 Enhancement Ratios Comparison 61 Table 2.25 Number of Damaged Bridges: Retrofit Effect in Bridge 1( Transverse) 62 Table 2.26 Number of Damaged Bridges: Retrofit Effect in Bridge 2 ( Transverse) 63 Table 2.27 Number of Damaged Bridges: Retrofit Effect in Bridge 3 ( Transverse) 64 Table 2.28 Number of Damaged Bridges: Retrofit Effect in Bridge 4 ( Transverse) 65 Table 2.29 Number of Damaged Bridges: Retrofit Effect in Bridge 5 ( Transverse) 66 Table 3.1 Summary of Bridge Damage Status in 1994 Northridge Earthquake 67 Table 3.2 Bridge Seismic Damage Table in the 1994 Northridge Earthquake 69 Table 3.3 Empirical Fragility Curve: First Level ( Composite) 71 Table 3.4 Empirical Fragility Curve: Second Level 71 Table 3.5 Empirical Fragility Curve: Third Level 72 Table 3.6 Empirical Fragility Curve: Fourth Level 74 Table 4.1 Probabilistic Scenario Earthquake Set 105 Table 5.1 Assumption for Residual Link Capacity 113 Table 5.2 Trip ratios of each directional trip for 4 time span 115 Table 5.3 3 hours average of am peak and midday applied peak ratio and car occupancy rates 115 Table 6.1 Damaged Bridges in Elysian Park M7.1( Total 3133 bridges) 131 Table 6.2 Comparison: Number of Damaged Bridges 132 Table 6.3 Reduction in Number of Damaged Bridges 133 Table 6.4 Damage Ratios for Highway Bridge Components ( from HAZUS 99) 136 Table 6.5 Expected Bridge Repair Cost ( in $ thousand) 136 xiii Table 6.6 Annual Probability of Sustaining Damage for a Bridge before and after Retrofit 140 Table 6.7 Annual Bridge Repair Cost in the System 141 Table 7.1 Daily Travel Delay and Opportunity Cost ( without retrofit) 144 Table 7.2 Daily Travel Delay and Opportunity Cost ( 23% retrofit) 147 Table 7.3 Daily Travel Delay and Opportunity Cost ( 100% Retrofit) 148 Table 7.4 Restoration Function for Highway Bridges ( after ATC 13, 1985) 158 Table 7.5 Total Travel Delay and Opportunity Cost ( no retrofit) 164 Table 7.6 Total Travel Delay and Opportunity Cost ( 23% retrofit) 165 Table 7.7 Total Travel Delay and Opportunity Cost ( 100% retrofit) 166 Table 7.8 Economic Loss due to System Dysfunction ( No Retrofit ) 168 Table 7.9 Economic Loss due to System Dysfunction ( 23% Retrofit) 169 Table 7.10 Economic Loss due to System Dysfunction ( 100% Retrofit) 170 Table 8.1 Annual Avoided Social Loss ( Assumption 1) 174 Table 8.2 Annual Avoided Social Loss ( Assumption 2) 175 Table 8.3 Annual Avoided Social Loss ( Assumption 3) 176 Table 8.4 Annual Avoided Social Loss ( HAZUS Model) 177 Table 8.5 Discount Factor 179 Table 8.6 Cost Effectiveness Analysis ( Discount Rate 3%) 182 Table 8.7 Cost Effectiveness Analysis ( Discount Rate 5%) 184 Table 8.8 Cost Effectiveness Analysis ( Discount Rate 7%) 186 Table 8.9 Cost Benefit Analysis Summary 188 Table B. 1 Building fragility by structure types 265 Table B. 2 Southern California Building Stock by Structure Types 267 Table B. 3 Southern California Building Stock by Occupancy Types 268 Table B. 4 Summary of Southern California Buildings by structure and occupancy Types 269 Table B. 5 Vulnerability of Building Occupancy 271 Table B. 6 Activity Population by Building Occupancy Types 275 Table B. 7 Trip Types and Building Occupancy types 276 Table B. 8 Person Trip Reduction Rates 277 xiv Table B. 9 Truck Generation Rate 281 Table B. 10 Building Usage and Truck trip Generating Industries 282 Table B. 11 Estimated PCE by Industries 283 Table B. 12 Calibration of Decay Function Parameter 290 xv Chapter 1 Introduction Past experience showed too often that earthquake damage to highway components ( e. g., bridges, roadways, tunnels, retaining walls, etc.) can severely disrupt traffic flows and thus negatively impacting on the economy of the region as well as post earthquake emergency response and recovery. Furthermore, the extent of these impacts will depend not only on the nature and magnitude of the seismic damage sustained by the individual components, but also on the mode of functional impairment of the highway system as a network resulting from physical damage of its components. In order to estimate the effects of the earthquake on the performance of the transportation network, an analytical framework must be developed to integrate bridge and other structural performance model and transportation network model in the context of seismic risk assessment. Highway transportation networks are complex with many engineered components placed in equally complex hazardous environments, natural or manmade. Among the engineered components, bridges represent potentially the most vulnerable components under earthquake conditions as demonstrated as vividly in the San Fernando, Loma Prieta, Northridge and Kobe Earthquakes. Recognizing this, the Caltrans’ seismic retrofit program has been underway since the 1971 San Fernando Earthquake, and accelerated since the 1989 Loma Prieta event. At this time ( June, 2005), 23% of Caltrans freeway bridges in Los Angeles and Orange Counties have been retrofitted by the steel and composite jacketing of the columns as well as rebuilding and upgrading of the restraining devises at expansion joints for which the seismic retrofit was deemed necessary. It is therefore most timely at this time to assess not only the engineering significance of such retrofit but also the socio economic benefit arising therefrom. 1 The purpose of this research therefore is to assess the socio economic impact of seismic retrofit implemented on the Caltrans’ bridges on the freeway network in the Los Angeles and Orange Counties. The research concentrates on the evaluation of the socio economic benefit resulting from the retrofit performed on the Caltrans’ bridges primarily by means of column jacketing with steel. The three major tasks of this research are ( 1) development of fragility curves of the bridge, ( 2) assessment of the seismic performance of the freeway and ( 3) related socio economic analysis. In order to perform a seismic risk analysis of a highway network, it is imperative to identify seismic vulnerability of bridges associated with various states of damage. As a widely practiced approach, the vulnerability information is expressed in the form of fragility curve to account for a multitude of uncertain sources involved ( Shinozuka et al, 2003a). In Chapter 2, a manageable number of representative bridges are selected for the fragility analysis. Finite Element Model for each of the representative bridges, without or with retrofit ( column jacketing with steel) is developed and used to perform nonlinear dynamic time history analysis. Based on the result of this dynamic analysis, a family of fragility curves associated with various states of damage are estimated with a statistical procedure. The seismic performance improvement of the retrofitted bridges is evident in that the median value of fragility curve of these bridges is significantly increased. The median value is one of the two fragility parameters with the other being the log standard deviation. The enhancement ratios for median values of analytical fragility curves are then applied to empirical fragility curves based on bridge damage data obtained from the 1994 Northridge Earthquake to consider the effect of the bridge retrofit ( Chapter 3). The 2 enhancement ratio is defined as ( median value for retrofitted bridges) / ( median value for bridges not retrofitted). After the introduction of major features of seismic risk analysis for spatially distributed system, both deterministic and probabilistic seismic modeling methods are described in Chapter 4. Particularly, a set of 47 probabilistic scenario earthquakes is provided for the probabilistic seismic risk analysis for the highway transportation network in Los Angles and Orange Counties. In chapter 5, a methodology is developed to evaluate the seismic performance of highway transportation network in terms of related social cost. Based on fragility curves developed above and the site ground motion originating from scenarios, the damage states of bridges are simulated, which determine the reduced link traffic capacity. A comprehensive traffic assignment analysis, which features realistic consideration of trip reduction and recovery after a damaging earthquake, is then performed in the degraded highway network with variable OD input. The daily social cost, including the traffic delay time and opportunity cost, is used to measure the post event performance of the damaged highway network. The enhancement of the network performance is then studied by comparing the social cost in using fragility curves of bridges with and without retrofit in the network performance simulation under the same scenario earthquake. Chapter 6 describes the method for estimation of bridge restoration ( repair/ replacement) cost. For the given scenarios, the expected bridge repair cost is calculated for each of the 3 cases of bridge retrofit status: No retrofit, 23% retrofit ( current status) and 100% retrofit, assuming that no freeway bridges ( in Los Angeles and Orange County), 23% of them ( actual % at the time of writing this report) and 100% of them have been retrofitted. To estimate the total social cost resulting from an earthquake, the network 3 restoration curves are developed in Chapter 7. Using a probabilistic time dependent bridge repair model, the new set of bridge damage states are determined based on Monte Carlo simulation at any given time point after an earthquake. The traffic assignment analysis is performed again to obtain the corresponding daily social cost for the partially restored network. The integration of the daily social cost over the restoration period gives the total social cost in time for a particular earthquake event. The economic loss due to the time cost is estimated by considering the local unit time value. Whether a retrofit strategy is cost effective is evaluated by a cost benefit analysis introduced in Chapter 8. The restoration cost for the damaged bridges, the retrofit cost and economic loss due to social cost are estimated. The difference between the economic loss without and with retrofit represents the cost avoided. The economic benefit is then measured by the cost avoided minus the cost of retrofit. The economic analysis is performed for each of the probabilistic scenario earthquakes and expected annual benefit of the retrofit measure obtained by considering the annual probabilities of these scenarios. The results show that the bridge restoration cost avoided alone cannot compensate for the retrofit cost. However, when the social cost avoided is considered, the cost effectiveness ratios in both retrofit cases are much larger than 1, indicating very high benefit for the public obtained from the Caltrans bridge retrofit measures. Chapter 9 summarizes the conclusions obtained from this research. At the end of the report, three documents are appended. Appendix A provides the cross sections and moment rotation relationship of 5 sample bridges’ columns before and after retrofit. Appendix B describes the background of the traffic assignment model integrating the OD change due to earthquake damage. In Appendix C, A GIS based 4 Program for Highway Seismic Risk Analysis ( HighwaySRA) developed at UCI is introduced and its usage and functionality are demonstrated in a manual which is part of the Appendix C. 5 6 Chapter 2 Development of Analytical Fragility Curve for Bridges 2.1 Introduction Several recent destructive earthquakes, particularly the 1989 Loma Prieta and 1994 Northridge earthquakes in California, and the 1995 Hanshin Awaji ( Kobe) earthquake in Japan, caused significant damage to a large number of highway structures that were seismically deficient ( Basoz and Kiremidjian 1998, Buckle 1994). The investigation of these negative consequences gave rise to serious discussions about seismic design philosophy and extensive research activity on the retrofit of existing bridges as well as the seismic design of new bridges. In this respect, this study presents an approach for the seismic assessment of older bridges retrofitted by steel jacketing of the columns having substandard seismic characteristics and by restrainers at expansion joints to prevent bridge decks from unseating. The main objective of the study is focused to evaluate the effects of column retrofit with steel jacketing on the ductility capacity of bridge columns. The Caltrans’ seismic retrofit program was underway prior to the 1994 Northridge earthquake and was accelerated after the 1989 Loma Prieta event. This resulted in implementation of steel and composite jacketing of the columns, and of installing and upgrading of the restraining devices at expansion joints for many bridges for which the seismic retrofit was deemed necessary. Therefore, it is most timely to assess the engineering significance and benefit from such retrofit. 7 This study first develops moment curvature curves of bridge columns and then performs nonlinear dynamic time history analyses producing fragility curves for five ( 5) sample bridges before and after retrofitting their columns with steel jacketing. The effect of retrofit is demonstrated by means of the ratio of the median value of the fragility curve for retrofitted column to that of the column before retrofit. This ratio is referred to as fragility enhancement. The fragility enhancement is found to be more significant for more severe state of damage. It is then assumed that the same fragility enhancement is applicable to the empirical fragility curves developed from the Northridge damage data ( Chapter 3). The fragility curves for four ( 4) of sample bridges are also developed before and after retrofitting its expansion joints with restrainers. This physical improvement of the seismic vulnerability due to steel jacketing becomes evident in terms of enhanced fragility curves shifting those associated with the bridges before retrofit to the right when plotted as functions of PGA ( Peak Ground Acceleration). Thus, this study makes it possible to evaluate the improvement of the highway network performance resulting from such retrofit by providing basic information for fragility enhancement. 2.2 Column Retrofit with Steel Jacketing 2.2.1 Background Concrete columns of earlier design often lack flexural strength, flexural ductility and shear strength. One of the main causes for these structural inadequacies is lap splices in critical regions and/ or premature termination of longitudinal reinforcement. A number of column retrofit techniques, such as steel jacketing, wire pre stressing and composite material jacketing, have been developed and tested. Although advanced composite 8 materials and other methods have been recently studied, the steel jacketing has been widely applied to bridge retrofit as the most common retrofit technique. Chai et al. ( 1991) observed that confinement of the concrete columns can be improved if transverse reinforcement layers are placed relatively close together along the longitudinal axis by restraining the lateral expansion of the concrete. It makes it possible for the compression zone to sustain higher compression stresses and much higher compression strains before failure occurs. Obviously, however, this is for original design and construction, but not applicable to existing bridges, to enhance the performance of columns by adding transverse reinforcement layers. In this respect, this study focuses on the steel jacketing technique for retrofitting existing bridge columns to improve their seismic performance. 2.2.2 Steel Jacketing An experiment was performed by Chai et al. ( 1991) to investigate the retrofit of circular columns with steel jacketing. In this experiment, for circular columns, two half shells of steel plate rolled to a radius slightly larger than that of the column are positioned over the area to be retrofitted and are site welded up the vertical seams to provide a continuous tube with a small annular gap around the column. This gap is grouted with pure cement. It is typical that the jacket is cut to provide a space of about 50 mm ( 2 in) between the jacket and any supporting member. It is for the jacket to avoid the possibility to act as compressing reinforcement by bearing against the supporting member at large drift angles. It is noted that the jacket is effective only in passive confinement and the level of confinement depends on the hoop strength and stiffness of the steel jacket. 9 10 The thickness of steel jacket is calculated from the following equation ( Priestley et al., 1996). ' 0.18( 0.004) cm cc j yj sm Df t f ε ε − = ( 2.1) where cm ε is the strain at maximum stress in concrete, sm ε the strain at maximum stress in steel jacket, D the diameter of circular column, ' cc f the compressive strength of confined concrete and yj f the yield stress of steel jacket. 2.2.3 Compression Stress Strain Relationships for Confined Concrete The effect of confinement is to increase the compression strength and ultimate strain of concrete as illustrated in Fig 2.1 ( after Priestley et al., 1996). Many different stress strain relationships have been developed for confined concrete. Most of these are applicable under certain specific conditions. A recent model applicable to all cross sectional shapes and at all levels of confinement is used for the analysis defined by the key equations that also appears in Priestley et al. ( 1996). Fig 2.1 Stress Strain Model for Concrete in Compression ε t εco 2εco εsp εcc εcu f'cc f'c First hoop Confined concrete fracture Assumed for cover concrete Unconfined concrete f't Ec Esec Compressive strain, ε c Compressive stress, fc 2.3 Bridge Model Not all but a manageable number of bridges, representing typical bridges in California and covering many types of bridge structures, have been selected for the fragility analysis. 2.3.1 Bridge Description Five ( 5) sample bridges used for example analysis are listed in Table 2.1 and shown in Fig 2.2. Bridge 1 has the overall length of 34 m ( 112 ft) with three spans. The superstructure consists of a longitudinally reinforced concrete deck slab 10 m ( 32.8 ft ) wide and it is supported by two sets of columns ( and by an abutment at each end). Each set has three columns of circular cross section with 0.8 m ( 31.5 in) diameter. Bridge 2 has an overall length of 242 m ( 794 ft) with five spans and an expansion joint in the center span. This bridge is supported by four columns of equal height of 21 m ( 69 ft) between the abutments at the ends. Each column has a circular cross section with 2.4 m diameter. The deck has a 3 cell concrete box type girder section 13 m ( 42.6 ft) wide and 2 m ( 6.6 ft) deep. Bridge 3 has an overall length of 226 m ( 741 ft) with five spans, consisting of three frames separated by two expansion joints. The columns have varying lengths with longer ones in the center span and shorter ones near the abutments. The superstructure consists of a RC box girder to the left of the left expansion joint and to the right of the right expansion joint, and a prestressed box girder in the central span. The deck has a 6 cell box girder section 20 m ( 65.6 ft) wide and 2.6 m ( 8.5 ft) deep, and the column section is octagonal. 11 Bridge 4 has an overall length of 483 m ( 1584 ft) with ten spans and four expansion joints. This bridge is supported by nine columns having different heights. Each column has a rectangular cross section which is 1.2 m ( 3.9 ft) by 3.7 m ( 12.1 ft) in dimension. The deck has a 5 cell concrete box type girder section 17 m ( 56 ft) wide and 2 m ( 6.6 ft) deep. Bridge 5 has an overall length of 500 m ( 1640 ft) with twelve spans and an expansion joint. This bridge is supported by eleven columns of equal height of 12.8 m ( 42.0 ft) between the abutments at the ends. Each column section is oblong in shape. The deck has a 4 cell concrete box type girder section 15 m ( 49.2 ft) wide and 2 m ( 6.6 ft) deep. Table 2.1 Description of Five ( 5) Sample Bridges Bridges Overall Length meter ( foot) Number of Spans Number of Hinges Column Height meter ( foot) 1 34( 112) 3 0 4.7 ( 15.4) 2 242( 794) 5 1 21.0 ( 68.9) 3 226( 741) 5 2 9.5  24.7( 31.2 81.0) 4 483 ( 1584) 10 4 9.5  34.4 ( 31.2 112.83) 5 500 ( 1640) 12 1 12.8 ( 42.0) 13.5 m10.5 m 10.0 m4.7 m 4.7 m34.0 m ( a) Bridge 1 12 53.38 m 41.18 m 41.18 m 53.38 m53.38 m21.0 m 242.0 mExpansionJoint10.7 m ( b) Bridge 2 52.0 m 39.0 m 27.5 m 63.5 m44.0 m 9.5 m 226.0 m21.3 m24.7 m17.5 m 10.4 m10.4 m ( c) Bridge 3 483 m 30.2 m 50 m 46 m 63 m 43 m 55 m 54 m 33 m 33 m 54 m 52 m 31.9 m 16.8 m9.5 m 13.7 m 17.7 m 17.2 m 28.4 m 34.4 m ( d) Bridge 4 500.0 m12.8 m10@ 43.58m= 435.8m32.1 m32.1 m ( e) Bridge 5 Fig 2.2 Elevation of Sample Bridges 2.3.2 Bridge Modeling The bridges are modeled to exhibit the nonlinear behavior of the columns. A column is modeled as an elastic zone with a pair of plastic zones at each end of the column. Each plastic zone is then modeled to consist of a nonlinear rotational spring and 13 a rigid element depicted in Fig 2.3. The plastic hinge formed in the bridge column is assumed to have bilinear hysteretic characteristics. Furthermore, pounding effect at the expansion joint of the bridges is reflected in the structural response analysis, so that the fragility information of the structure becomes more realistic. In this respect, the expansion joint is constrained in the relative vertical movement, while freely allowing horizontal opening movement and rotation. The closure at the joint, however, is restricted by a gap element when the relative motion of adjacent decks exhausts the initial gap width of 2.54 cm ( 1 in) leading to deck pounding. A hoop element sustaining tension only is used for the bridge retrofitted by restrainers at expansion joints and the opening is restricted by the element when the relative motion exhausts the initial slack of 1.27 cm ( 0.5 in). Springs are also attached to the bases of the columns to account for soil effects, while two abutments are modeled as roller supports. To reflect the cracked state of a concrete bridge column for the seismic response analysis, an effective moment of inertia is employed, making the period of the bridge longer. 14 15 kHook Linear Column Potential Plastic Hinge Relative Displacement Force ks αks θy Rotation Μ y M Μ y = 36539 kN m α= 0.01675 θy = 3.7x10 3 rad Potential Plastic Hinge 1.27 cm Expansion Joint Deck 2.54 cm kGap Fig 2.3 Nonlinearities in Bridge Model 2.4 Development of Moment Curvature Relationship The column ductility program developed by Kushiyama ( 2002) ( the code is attached in Appendix A) is used to model the moment curvature relationship of plastic hinges for columns. The critical parameter used to describe the nonlinear structural response in this study is the ductility demand. The ductility demand is defined as / y θ θ , where θ is the rotation of a bridge column in its plastic hinge and y θ is the corresponding rotation at the yield point. Nonlinear response characteristics associated with the bridges are based on moment curvature curve analysis taking axial loads as well as confinement effects into account. The moment curvature relationship used in this study for the nonlinear spring is bilinear without any stiffness degradation. Its parameters are established according to the equations in Priestley et al. ( 1996). 2.4.1 Moment Curvature Curves for Longitudinal Direction of Bridges In Fig 2.4 and 2.5, Section of the column, stress strain relationship, distribution of axial force, P M interaction diagram, moment curvature curve and moment rotation curve for column 2 of Bridge 1 before and after retrofit are plotted. The cross sections and the moment rotation curves of all the other columns of Bridge 1 5 are provided in Appendix A. One of results, for example, shows that the moment curvature curve after retrofit gives a much better performance than that before retrofit by 4 times based on curvature at the ultimate compressive strain and by 1.6 times at the ultimate moment. 16  15  10  5 0 5 10 15  15  10  5 0 5 10 15 Dimension( in) D im e n s io n ( in ) 0 0.005 0.01 0.015 0.02 0 1000 2000 3000 4000 5000 6000 Confined Concrete Compressive Strain C o m p re s s iv e S tre s s ( p s i ) Cover Concrete ( a) Section of Column ( b) Stress Strain Relationship 0 0.005 0.01 0.015 0.02 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 6 Strain A x ia l fo rc e ( lb ) Confined Concrete Cover Concrete Steel Bar Total 0 0.5 1 1.5 2 2.5 x 10 4 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Mn ( kips in) P n ( k i p s ) ( c) Distribution of Axial Force ( d) P M Interaction Diagram 0 0.5 1 1.5 2 x 10  3 0 200 400 600 800 1000 1200 1400 1600 Curvature ( 1/ in) M o m e n t ( k i p s  ft ) 0 0.01 0.02 0.03 0.04 0.05 0.06 0 200 400 600 800 1000 1200 1400 1600 Rotation ( radian) M o m e n t ( k i p s  ft ) M y = 1442kips ft M u = 1516kips ft θ y = 0.006871rad θ u = 0.05732rad K eff = 2.099e+ 005kips ft α = 0.006918 ( e) Moment Curvature Curve ( f) Moment Rotation Curve Fig 2.4 Column 2 of Bridge 1 Before Retrofit Dimension ( in) Compression Stress ( psi) Axial Force ( pf) Pn ( kips) Moment ( kip ft) Moment ( kip ft) 18  15  10  5 0 5 10 15  15  10  5 0 5 10 15 Dimension( in) D im e n s io n ( in ) 0 0.01 0.02 0.03 0.04 0 1000 2000 3000 4000 5000 6000 7000 Confined Concrete Compressive Strain C o m p re s s iv e S tre s s ( p s i ) Cover Concrete ( a) Section of Column ( b) Stress Strain Relationship 0 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 x 10 6 Strain A x ia l fo rc e ( lb ) Confined Concrete Cover Concrete Steel Bar Total 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 0 1000 2000 3000 4000 5000 6000 7000 Mn ( kips in) P n ( k i p s ) ( c) Distribution of Axial Force ( d) P M Interaction Diagram 0 1 2 3 4 x 10  3 0 500 1000 1500 2000 2500 Curvature ( 1/ in) M o m e n t ( k i p s  ft ) 0 0.05 0.1 0.15 0.2 0 500 1000 1500 2000 2500 Rotation ( radian) M o m e n t ( k i p s  ft ) M y = 1749kips ft M u = 2249kips ft θ y = 0.006428rad θ u = 0.1202rad K eff = 2.722e+ 005kips ft α = 0.01614 ( e) Moment Curvature Curve ( f) Moment Rotation Curve Fig 2.5 Column 2 of Bridge 1 After Retrofit Dimension ( in) Compression Stress ( psi) Axial Force ( pf) Pn ( kips) Moment ( kip ft) Moment ( kip ft) 2.5 Bridge Response Analysis The SAP2000/ Nonlinear finite element computer code ( Computer and Structures, 2002) is utilized for the extensive two dimensional response analysis of the bridge under sixty ( 60) Los Angeles earthquake time histories ( http:// nisee. berkeley. edu/ data/ strong_ motion/ sacsteel/ ground_ motions. html) listed in Table 2.2, to develop the fragility curves before and after column retrofit with steel jackets. 2.5.1 Input Ground Motions These acceleration time histories were derived from historical records with some linear adjustments and consist of three ( 3) groups ( each consisting of 20 time histories) having probabilities of exceedance of 10% in 50 years, 2% in 50 years and 50% in 50 years, respectively. A typical acceleration time history in each group is plotted in the same scale to compare the magnitude of the acceleration in Fig 2.6. 19 Table 2.2 Description of Los Angeles Ground Motions 10% Exceedence in 50 yr 2% Exceedence in 50 yr 50% Exceedence in 50 yr SAC Name DT ( sec) Duration ( sec) PGA ( cm/ sec2) SAC Name DT ( sec) Duration ( sec) PGA ( cm/ sec2) SAC Name DT ( sec) Duration ( sec) PGA ( cm/ sec2) LA01 0.02 39.38 452.03 LA21 0.02 59.98 1258.00 LA41 0.01 39.38 578.34 LA02 0.02 39.38 662.88 LA22 0.02 59.98 902.75 LA42 0.01 39.38 326.81 LA03 0.01 39.38 386.04 LA23 0.01 24.99 409.95 LA43 0.01 39.08 140.67 LA04 0.01 39.38 478.65 LA24 0.01 24.99 463.76 LA44 0.01 39.08 109.45 LA05 0.01 39.38 295.69 LA25 0.005 14.945 851.62 LA45 0.02 78.60 141.49 LA06 0.01 39.38 230.08 LA26 0.005 14.945 925.29 LA46 0.02 78.60 156.02 LA07 0.02 79.98 412.98 LA27 0.02 59.98 908.70 LA47 0.02 79.98 331.22 LA08 0.02 79.98 417.49 LA28 0.02 59.98 1304.10 LA48 0.02 79.98 301.74 LA09 0.02 79.98 509.70 LA29 0.02 49.98 793.45 LA49 0.02 59.98 312.41 LA10 0.02 79.98 353.35 LA30 0.02 49.98 972.58 LA50 0.02 59.98 535.88 LA11 0.02 39.38 652.49 LA31 0.01 29.99 1271.20 LA51 0.02 43.92 765.65 LA12 0.02 39.38 950.93 LA32 0.01 29.99 1163.50 LA52 0.02 43.92 619.36 LA13 0.02 59.98 664.93 LA33 0.01 29.99 767.26 LA53 0.02 26.14 680.01 LA14 0.02 59.98 644.49 LA34 0.01 29.99 667.59 LA54 0.02 26.14 775.05 LA15 0.005 14.945 523.30 LA35 0.01 29.99 973.16 LA55 0.02 59.98 507.58 LA16 0.005 14.945 568.58 LA36 0.01 29.99 1079.30 LA56 0.02 59.98 371.66 LA17 0.02 59.98 558.43 LA37 0.02 59.98 697.84 LA57 0.02 79.46 248.14 LA18 0.02 59.98 801.44 LA38 0.02 59.98 761.31 LA58 0.02 79.46 226.54 LA19 0.02 59.98 999.43 LA39 0.02 59.98 490.58 LA59 0.02 39.98 753.70 LA20 0.02 59.98 967.61 LA40 0.02 59.98 613.28 LA60 0.02 39.98 469.07 20 02040Time ( s) 60  1500 1000 500050010001500Acceleration ( cm/ s2) PGA= 662.88 cm/ s2 ( a) 10% Probability of Exceedence in 50 Years 02040Time( s) 60  1500 1000 500050010001500Acceleration ( cm/ s2) PGA= 1,304.10 cm/ s2 ( b) 2% Probability of Exceedence in 50 Years 01020304Time( s) 0  1500 1000 500050010001500Acceleration ( cm/ s2) PGA= 109.45 cm/ s2 ( c) 50% Probability of Exceedence in 50 Years Fig 2.6 Acceleration Time Histories Generated for Los Angeles 21 2.5.2 Responses of Structures Typical responses at column bottom end of Bridge 1 are plotted in Fig 2.7 with the acceleration time history in Fig 2.6a as input. It is reasonable to expect that the rotation after retrofit is generally smaller than before, while the accelerations do not necessarily behave that way and can be quite different each other. It is noted that some higher fluctuations in acceleration response appear after retrofit because the column becomes stiffer than before. 0102030405Time ( s) 0  0.400.4Acceleration at column end ( m/ s2) before retrofit ( a) Acceleration before retrofit 010203040Time ( s) 50  0.400.4Acceleration at column end ( m/ s2) after retrofit ( b) Acceleration after retrofit 22 0102030405Time ( s) 0  0.008 0.00400.0040.008Rotation at column end ( rad) before retrofit ( c) Rotation before retrofit 010203040Time ( s) 50  0.008 0.00400.0040.008Rotation at column end ( rad) after retrofit ( d) Rotation after retrofit Fig 2.7 Responses at Column End of Bridge 1 23 Typical responses at expansion joints of Bridge 1 are also plotted in Fig 2.8 to show the differences of the structural behaviors for the cases without and with considering gap and hook elements. 010203040Time ( sec)  40 2002040Displacement at Expansion Joints ( cm) Left JointRight Joint 010203040Time ( sec)  10010203040Relative Displacement ( cm) Right Left ( a) Without Gap and Hook Elements 24 010203040Time ( sec)  40 2002040Displacement at Expansion Joints ( cm) Left JointRight Joint 010203040Time ( sec)  3 2 1012Displacement at Expansion Joints ( cm) Right Left ( b) with Gap and Hook Elements Fig 2.8 Displacement at Expansion Joints of Bridge 1 25 26 2.6. Fragility Analysis of Bridges 2.6.1 Fragility Parameter Estimation It is assumed that the fragility curves can be expressed in the form of two parameter lognormal distribution functions, and the estimation of the two parameters ( median and log standard deviation) is performed with the aid of the maximum likelihood method. A common log standard deviation, which forces the fragility curves not to intersect, can also be estimated. The following likelihood formulation described by Shinozuka et al. ( 2000) is introduced for the purpose of this method. Although this method can be used for any number of damage states, it is assumed here for the ease of demonstration of analytical procedure that there are five states of damage including the state of ( almost) no damage. A family of four ( 4) fragility curves exists in this case where events E1, E2, E3, E4 and E5, respectively, indicate the state of ( almost) no, ( at least) slight, ( at least) moderate, ( at least) extensive damage and complete collapse. Pik = P( ai, Ek) in turn indicates the probability that a bridge selected randomly from the sample will be in the damage state Ek when subjected to ground motion intensity expressed by PGA = ai. All fragility curves are then represented ln( /) ( ; ,) i j j j jj j a c F a c ς ς = Φ ( 2.2) where Φ(⋅) is the standard normal distribution function, cj and j ς are the median and log standard deviation of the fragility curves for the damage state of “( at least) slight”, “( at least) moderate”, “( at least) major” and “ complete” identified by j = 1, 2, 3 and 4. From this definition of fragility curves, and under the assumption that the log standard deviation is equal to ς common to all the fragility curves, one obtains; 27 Pi1= P( ai, E1)= 1F1( ai ; c1 , ς) − ( 2.3) 2 2 1122 ( , )(;,)(;,) i i ii P PaEFacFac = = ς ς − ( 2.4) 3 3 2233 ( , )(;,)(;,) i i ii P PaEFacFac = = ς ς − ( 2.5) 4 4 3344 ( , )(;,)(;,) i i ii P PaEFacFac = = ς ς − ( 2.6) 5 5 44 ( , )(;,) i i i P PaEFac = = ς ( 2.7) The likelihood function can then be introduced as 5 1 2 34 1 1 ( , ,,,)(;) ik n x k i k i k L c cccPaE ς = = = ΠΠ ( 2.8) Where 1 ik x = ( 2.9) if the damage state Ek occurs in the bridge subjected to a = ai, and 0 ik x = ( 2.10) otherwise. Then the maximum likelihood estimates c0j for cj and 0 ς for ς are obtained by solving the following equations, 1 2341234 ln(,,,,) ln(,,,,) 0 j L c cccLcccc c ς ς ς ∂ ∂ = = ∂ ∂ ( 1,2,3,4) j = ( 2.11) by implementing a straightforward optimization algorithm. 2.6.2 Definition of Damage States A set of five ( 5) different damage states recommended by Dutta and Mander ( 1999) are introduced in Table 2.3 which displays the description of these five damage states and the corresponding drift limits for a typical column. For each limit state, the drift limit can be transformed to peak ductility demand of the columns for the purpose of this study. Table 2.4 lists the values of these ductility demands for five ( 5) sample bridges. Table 2.3 Description of Damaged States Damage state Description Drift limits Almost no First yield 0.005 Slight Cracking, spalling 0.007 Moderate Loss of anchorage 0.015 Extensive Incipient column collapse 0.025 Complete Column collapse 0.050 Table 2.4 Peak Ductility Demand of First Left Column of Sample Bridges Bridge 1 Bridge 2 Bridge 3 Bridge 4 Bridge 5 Damage state before retrofit after retrofit before retrofit After retrofit before retrofit after retrofit before retrofit after retrofit before retrofit after retrofit Almost no 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 Slight 1.3 1.8 1.5 1.8 1.2 2.1 1.5 2.5 1.7 2.5 Moderate 2.6 4.9 3.5 5.2 2.2 6.4 3.5 8.2 4.3 8.3 Extensive 4.3 8.9 6.0 9.3 3.5 11.7 6.1 15.5 7.5 15.7 Complete 8.3 18.7 12.3 19.7 6.5 25.2 12.4 33.6 15.7 34.0 2.7 Pounding and Soil Effects on Fragility Curves The section presents the fragility curves taking the effect of pounding at expansion joints on concrete bridge response to earthquake ground motions into consideration. The primary objective of this section is to develop fragility curves of the sample bridges and quantify the effect of pounding at expansion joints of the bridges. The effect of pounding at expansion joints on the seismic response is systematically examined and the resulting fragility curves are compared with those for the cases without pounding. 2.7.1 Pounding at Expansion Joint 28 Pounding at expansion joints ( hinges) might have been another source of extensive damage during past earthquakes. In fact, the collapse of the 483 m ( 1610 ft) long bridge at the Interstate 5 and State Road 14 Interchange located approximately 12 km ( 7.5 mile) from the epicenter during the 1994 Northridge earthquake is an example suggesting that the effect of pounding at expansion joint might have caused the significant failure investigating damage states ( Buckle 1994). A preliminary investigation was performed by Shinozuka et al. ( 2002b) on impact phenomena as well as effects of seismically induced pounding at expansion joints of typical California bridges, through which it was found that pounding has significant effects on the acceleration and velocity responses, but little effects on the displacement responses. Although pounding effect is found to have negligible effect on the ductility demand, a need is felt to quantify the effect of pounding at the expansion joints by developing fragility curves of highway bridges, particularly for multi span long bridges with expansion joints. In order to investigate the effect of pounding of bridges, four ( 4) sample bridge models are considered for the nonlinear time history analysis. As described earlier in the Section 2.3, two ( 2) of them have mid overall lengths, but one hinge with same column height and two hinges with different column height. The other two have long overall lengths, but one hinge with same column height and four hinges with different column height. It is typical for a California highway bridge with more than four spans to have expansion joints located nearly at inflection points ( i. e., 1/ 4 to 1/ 5th of spans). The bridge superstructure consists of reinforced or prestressed concrete box girders. For 29 30 example, the material and cross sectional properties of Bridge 2 as follows: Young's modulus= 27.793 Gpa ( 4.03×106 ksi), mass density= 2.401 Mg/ m3 ( 62.428 kip/ ft3), crosssection area and moment of inertia are respectively 6.701 m2 ( 72.13 ft2) and 4.625 m4 ( 535.86 ft4) for box girders, while they are 4.670 m2 ( 50.27 ft2 ) and 0.620 m4 ( 71.83 ft4) for columns. Perhaps one of the most difficult to analyze nonlinear behaviors that occur in bridge systems idealized to include gap elements is the closing of a gap between different segments of the bridge. The usual gap element shown as Fig 2.9 has the following physical properties: 1) The element cannot develop a force until the opening d0 is closed; and 2) the element can only develop a compression force. Note that the numerical convergence of the response analysis particularly at the gap element can be very slow if a large elastic stiffness k is used. In order to minimize the difficulty associated with this problem, the stiffness k should not be over 1,000 times the stiffness of the elements adjacent to the gap according to the authors’ experience. This kind of dynamic contact problem involving two adjacent structural segments usually does not have a simple, unique solution. In fact, it is impractical to use continuum mechanics analysis in the vicinity of the contact area for local stress and strain evaluation and at the same time to pursue structural dynamic analysis to evaluate the bridge response as a system including, for example, ductility demand at the column ends. A viable alternative appears to be the deployment of the finite element analysis with gap elements having the stiffness value k selected from sensitivity analysis of gap element stiffness ( Shinozuka et al., 2003c). kd0ji Fig 2.9 Gap Element 2.7.2 Numerical Simulation for Pounding Numerical simulation were performed for the four ( 4) sample bridges under sixty ( 60) Los Angeles earthquakes for the cases without pounding and with pounding by considering gap element at expansion joints. The computer code SAP2000/ Nonlinear was utilized in order to calculate the state of damage of the structure under ground acceleration time histories. The structural responses with pounding were compared to those without pounding, in order to highlight how pounding affects the structural response behaviors. Numerical simulations were carried out under LA01 earthquake as shown Fig 2.10. Pounding force time history was also presented as shown Fig 2.11. Time histories of acceleration and displacement at the expansion joint, and rotation of the column end are plotted as shown Fig 2.12 and Fig 2.13 for the cases without and with pounding, respectively. From these results, it is observed that ( 1) the pounding takes place twenty three ( 23) times during the duration of the earthquake, ( 2) the acceleration is affected much more by pounding than displacement and rotation are; ( 3) the peak value of the rotation at column end can be reduced by pounding. It is indicated that the pounding are not usually capable of causing large deformation to bridge structures while it may cause significantly 31 high axial compressive stress locally leading to a possible local damage at the contact area at the expansion joint. 0102030405Time ( sec) 0  0.500.5Acceleration ( g) 0102030405Time ( sec) 0  16000 12000 8000 40000Force ( ton) Fig 2.10 Ground Motion Time History for LA01 Fig 2.11 Pounding Force at Expansion Joint 0102030405Time ( sec) 0  80 4004080Acceleration ( cm/ sec2) 0102030405Time ( sec) 0  8000 6000 4000 200002000Acceleration ( cm/ sec2) ( a) Acceleration at Expansion Joint ( a) Acceleration at Expansion Joint 0102030405Time ( sec) 0  2 1012Displacement ( cm) 0102030405Time( sec) 0  2 1012Displacement ( cm) ( b) Displacement at Expansion Joint ( b) Displacement at Expansion Joint 0102030405Time ( sec) 0 0.004 0.00200.0020.004Rotation ( rad) 0102030405Time ( sec) 0  0.004 0.00200.0020.004Rotation ( rad) ( c) Rotation at Column Bottom ( c) Rotation at Column Bottom Fig 2.12 Structural Responses without Pounding Fig 2.13 Structural Responses with Pounding 32 2.7.3 Pounding Effects on Fragility Curves The fragility curves for the four ( 4) sample bridges associated with the states of damage mentioned in the previous section were plotted as a function of peak ground acceleration in Figs 2.14 2.17, while the number of damaged bridges is listed in Tables 2.5 2.8, respectively. Each Fig has two curves for the cases without pounding and with pounding to compare how much the curves are shifted to left or right ( more or less fragile). It is noted here that the log standard deviation in each of Figs 2.14 2.17 was obtained by taking the whole events involving the cases without and with pounding using Equation 2.11 for these fragility curves. This is for the reason that the pair of fragility curves in each Fig is not theoretically expected to intersect each other. The fragility curves in pairs produced mixed results in such a way that the pounding effect is even beneficial for some damage states, while it appears detrimental for some cases. In particular, if the number of bridges at a certain state of damage counted, it can be clearly seen that the pounding does not increase the number of damaged bridges ( or the ductility factor) in general. It is noted that bridge characteristics, such as overall length, number of spans, number of expansion joints and height of columns, might not a major factor to change the trend of the fragility curves by increasing or decreasing ductility demand. High response amplifications due to pounding might result only if the colliding bridge segments separated by an expansion joint are significantly different in natural period, however this condition does not usually exist in the bridge structure. 33 34 Table 2.5 Number of Damaged Bridges: Pounding Effect in Bridge 2 sample size= 60 Damage States without Pounding with Pounding No 8 9 Almost No 8 9 Slight 8 9 Moderate 8 4 Extensive 14 16 Complete 14 13 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.18, ζ 0= 0.86) with Pounding ( c0= 0.23, ζ 0= 0.86) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.29, ζ 0= 0.86) with Pounding ( c0= 0.32, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.46, ζ 0= 0.86) with Pounding ( c0= 0.53, ζ 0= 0.86) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.61, ζ 0= 0.86) with Pounding ( c0= 0.60, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 1.01, ζ 0= 0.86) with Pounding ( c0= 1.01, ζ 0= 0.86) ( d) Extensive Damage ( e) Complete Collapse Fig 2.14 Pounding Effect on Fragility Curves of Bridge 2 35 Table 2.6 Number of Damaged Bridges: Pounding Effect in Bridge Model 3 sample size= 60 Damage States without Pounding with Pounding No 1 1 Almost No 2 3 Slight 13 13 Moderate 4 2 Extensive 14 20 Complete 26 21 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.11, ζ 0= 0.86) with Pounding ( c0= 0.11, ζ 0= 0.86) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.15, ζ 0= 0.86) with Pounding ( c0= 0.18, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.35, ζ 0= 0.86) with Pounding ( c0= 0.39, ζ 0= 0.86) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.43, ζ 0= 0.86) with Pounding ( c0= 0.40, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.66, ζ 0= 0.86) with Pounding ( c0= 0.76, ζ 0= 0.86) ( d) Extensive Damage ( e) Complete Collapse Fig 2.15 Pounding Effect on Fragility Curves of Bridge 3 36 Table 2.7 Number of Damaged Bridges: Pounding Effect in Bridge 4 sample size= 60 Damage States without Pounding with Pounding No 1 1 Almost No 2 2 Slight 6 7 Moderate 7 6 Extensive 9 7 Complete 35 37 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.03, ζ 0= 0.95) with Pounding ( c0= 0.03, ζ 0= 0.95) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.07, ζ 0= 0.95) with Pounding ( c0= 0.13, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.20, ζ 0= 0.95) with Pounding ( c0= 0.27, ζ 0= 0.95) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.32, ζ 0= 0.95) with Pounding ( c0= 0.31, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.49, ζ 0= 0.95) with Pounding ( c0= 0.45, ζ 0= 0.95) ( d) Extensive Damage ( e) Complete Collapse Fig 2.16 Pounding Effect on Fragility Curves of Bridge 4 37 Table 2.8 Number of Damaged Bridges: Pounding Effect in Bridge 5 sample size= 60 Damage States without Pounding with Pounding No 8 7 Almost No 1 3 Slight 18 17 Moderate 9 9 Extensive 14 14 Complete 10 10 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.28, ζ 0= 0.70) with Pounding ( c0= 0.26, ζ 0= 0.70) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.28, ζ 0= 0.70) with Pounding ( c0= 0.31, ζ 0= 0.70) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.54, ζ 0= 0.70) with Pounding ( c0= 0.54, ζ 0= 0.70) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.69, ζ 0= 0.70) with Pounding ( c0= 0.69, ζ 0= 0.70) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 1.01, ζ 0= 0.70) with Pounding ( c0= 1.01, ζ 0= 0.70) ( d) Extensive Damage ( e) Complete Collapse Fig 2.17 Pounding Effect on Fragility Curves of Bridge 5 2.7.4 Pounding and Soil Effects on Fragility Curves The fragility curves for the four ( 4) sample bridges associated with the states of damage mentioned in the previous section were plotted as a function of peak ground acceleration in Figs 2.18, 2.19, 2.20 and 2.21, while the number of damaged bridges is listed in Tables 2.9, 2.10, 2.11 and 2.12, respectively. Each Fig has four ( 4) curves for the following four ( 4) cases: CASE 1: without pounding effects and without soil effects CASE 2: with pounding effects and without soil effects CASE 3: without pounding effects and with soil effects CASE 4: with pounding effects and with soil effects In order to compare how much the curves are shifted to left or right ( more or less fragile) due to the effects of pounding and/ or soil, the four ( 4) curves were put into one Figure. It is noted that the log standard deviation in each of Figs 2.18, 2.19, 2.20 and 2.21 was obtained by taking the whole events involving the four ( 4) cases using equation 2.11 for these fragility curves. This is for the reason that the pair of fragility curves in each Fig is not theoretically expected to intersect each other. The fragility curves produced mixed results in such a way that the pounding and/ or soil effects are even beneficial for some damage states, while it appears detrimental for some cases. In particular, if the number of bridges at a certain state of damage counted, there is a definite effect but it is hard to say any trend. 38 39 Table 2.9 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 2 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 8 10 8 8 Almost No 9 5 12 9 Slight 8 11 7 8 Moderate 10 9 7 12 Extensive 15 14 16 13 Complete 10 11 10 10 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.28, ζ 0= 0.95) CASE 2 ( c0= 0.29, ζ 0= 0.95) CASE 3 ( c0= 0.29, ζ 0= 0.95) CASE 4 ( c0= 0.31, ζ 0= 0.95) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.52, ζ 0= 0.95) CASE 2 ( c0= 0.43, ζ 0= 0.95) CASE 3 ( c0= 0.61, ζ 0= 0.95) CASE 4 ( c0= 0.54, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.70, ζ 0= 0.95) CASE 2 ( c0= 0.74, ζ 0= 0.95) CASE 3 ( c0= 0.77, ζ 0= 0.95) CASE 4 ( c0= 0.70, ζ 0= 0.95) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.02, ζ 0= 0.95) CASE 2 ( c0= 1.02, ζ 0= 0.95) CASE 3 ( c0= 0.98, ζ 0= 0.95) CASE 4 ( c0= 1.08, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.91, ζ 0= 0.95) CASE 2 ( c0= 1.91, ζ 0= 0.95) CASE 3 ( c0= 2.13, ζ 0= 0.95) CASE 4 ( c0= 2.13, ζ 0= 0.95) ( d) Extensive Damage ( e) Complete Collapse Fig 2.18 Pounding and Soil Effects on Fragility Curves of Bridge 2 40 Table 2.10 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 3 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 2 2 2 10 Almost No 2 2 10 5 Slight 10 10 7 5 Moderate 8 5 5 3 Extensive 12 13 12 11 Complete 26 28 24 26 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.12, ζ 0= 0.77) CASE 2 ( c0= 0.12, ζ 0= 0.77) CASE 3 ( c0= 0.12, ζ 0= 0.77) CASE 4 ( c0= 0.22, ζ 0= 0.77) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.17, ζ 0= 0.77) CASE 2 ( c0= 0.19, ζ 0= 0.77) CASE 3 ( c0= 0.24, ζ 0= 0.77) CASE 4 ( c0= 0.31, ζ 0= 0.77) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.34, ζ 0= 0.77) CASE 2 ( c0= 0.33, ζ 0= 0.77) CASE 3 ( c0= 0.40, ζ 0= 0.77) CASE 4 ( c0= 0.40, ζ 0= 0.77) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.46, ζ 0= 0.77) CASE 2 ( c0= 0.40, ζ 0= 0.77) CASE 3 ( c0= 0.48, ζ 0= 0.77) CASE 4 ( c0= 0.45, ζ 0= 0.77) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.66, ζ 0= 0.77) CASE 2 ( c0= 0.62, ζ 0= 0.77) CASE 3 ( c0= 0.69, ζ 0= 0.77) CASE 4 ( c0= 0.66, ζ 0= 0.77) ( d) Extensive Damage ( e) Complete Collapse Fig 2.19 Pounding and Soil Effects on Fragility Curves of Bridge 3 41 Table 2.11 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 4 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 5 3 9 7 Almost No 5 5 6 5 Slight 15 11 13 10 Moderate 11 7 9 8 Extensive 15 18 14 16 Complete 9 16 9 14 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.18, ζ 0= 1.01) CASE 2 ( c0= 0.08, ζ 0= 1.01) CASE 3 ( c0= 0.21, ζ 0= 1.01) CASE 4 ( c0= 0.05, ζ 0= 1.01) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.25, ζ 0= 1.01) CASE 2 ( c0= 0.19, ζ 0= 1.01) CASE 3 ( c0= 0.30, ζ 0= 1.01) CASE 4 ( c0= 0.24, ζ 0= 1.01) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.49, ζ 0= 1.01) CASE 2 ( c0= 0.33, ζ 0= 1.01) CASE 3 ( c0= 0.54, ζ 0= 1.01) CASE 4 ( c0= 0.42, ζ 0= 1.01) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.69, ζ 0= 1.01) CASE 2 ( c0= 0.50, ζ 0= 1.01) CASE 3 ( c0= 0.71, ζ 0= 1.01) CASE 4 ( c0= 0.57, ζ 0= 1.01) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.09, ζ 0= 1.01) CASE 2 ( c0= 0.88, ζ 0= 1.01) CASE 3 ( c0= 1.16, ζ 0= 1.01) CASE 4 ( c0= 0.98, ζ 0= 1.01) ( d) Extensive Damage ( e) Complete Collapse Fig 2.20 Pounding and Soil Effects on Fragility Curves of Bridge 4 42 Table 2.12Number of Damaged Bridges: Pounding and Soil Effects in Bridge 5 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 9 9 11 10 Almost No 7 6 4 4 Slight 14 15 12 13 Moderate 11 11 14 14 Extensive 13 13 12 12 Complete 6 6 7 7 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.44, ζ 0= 0.78) CASE 2 ( c0= 0.44, ζ 0= 0.78) CASE 3 ( c0= 0.35, ζ 0= 0.78) CASE 4 ( c0= 0.32, ζ 0= 0.78) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.55, ζ 0= 0.78) CASE 2 ( c0= 0.54, ζ 0= 0.78) CASE 3 ( c0= 0.50, ζ 0= 0.78) CASE 4 ( c0= 0.47, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.88, ζ 0= 0.78) CASE 2 ( c0= 0.88, ζ 0= 0.78) CASE 3 ( c0= 0.80, ζ 0= 0.78) CASE 4 ( c0= 0.80, ζ 0= 0.78) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.18, ζ 0= 0.78) CASE 2 ( c0= 1.18, ζ 0= 0.78) CASE 3 ( c0= 1.18, ζ 0= 0.78) CASE 4 ( c0= 1.18, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.83, ζ 0= 0.78) CASE 2 ( c0= 1.83, ζ 0= 0.78) CASE 3 ( c0= 1.84, ζ 0= 0.78) CASE 4 ( c0= 1.84, ζ 0= 0.78) ( d) Extensive Damage ( e) Complete Collapse Fig 2.21 Pounding and Soil Effects on Fragility Curves of Bridge 5 2.7.5 Jacketing and Restrainer Effects on Fragility Curves The fragility curves for the four ( 4) sample bridges associated with the states of damage mentioned in the previous section were plotted as a function of peak ground acceleration in Figs 2.23, 2.24, 2.25 and 2.26, while the number of damaged bridges is listed in Tables 2.14, 2.15, 2.16 and 2.17, respectively. Each Fig has four ( 4) curves for the following four ( 4) cases: CASE 1: without jacketing and without restrainer CASE 2: with jacketing and without restrainer CASE 3: without jacketing and with restrainer CASE 4: with jacketing and with restrainer In order to compare how much the curves are shifted to left or right ( more or less fragile) due to the effects of jacketing and/ or restrainer, the four ( 4) curves were put into one Fig. It is noted that the log standard deviation in each of Figs 2.23, 2.24, 2.25 and 2.26 was obtained by taking the whole events involving the four ( 4) cases using equation 2.11 for these fragility curves. This is for the reason that the pair of fragility curves in each figure is not theoretically expected to intersect each other. The damage state of a bridge is defined in terms of the maximum value of the peak ductility demands sustained by all the column ends. In this context, comparison between fragility curves in Figs 2.23 2.26 indicates that the bridge is less susceptible for damage to the ground motion after column retrofit than before, while the effect of restrainers at expansion joints is found to be negligible or even adversely affects on the column responses. 43 44 Table 2.13 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 10 14 10 15 Almost No 4 8 6 7 Slight 11 18 11 18 Moderate 11 11 9 13 Extensive 13 8 15 6 Complete 11 1 9 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.29, ζ 0= 1.10) CASE 2 ( c0= 0.39, ζ 0= 1.10) CASE 3 ( c0= 0.29, ζ 0= 1.10) CASE 4 ( c0= 0.46, ζ 0= 1.10) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.39, ζ 0= 1.10) CASE 2 ( c0= 0.64, ζ 0= 1.10) CASE 3 ( c0= 0.49, ζ 0= 1.10) CASE 4 ( c0= 0.64, ζ 0= 1.10) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.70, ζ 0= 1.10) CASE 2 ( c0= 1.19, ζ 0= 1.10) CASE 3 ( c0= 0.74, ζ 0= 1.10) CASE 4 ( c0= 1.19, ζ 0= 1.10) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.05, ζ 0= 1.10) CASE 2 ( c0= 1.97, ζ 0= 1.10) CASE 3 ( c0= 1.05, ζ 0= 1.10) CASE 4 ( c0= 2.50, ζ 0= 1.10) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.91, ζ 0= 1.10) CASE 2 ( c0= 6.12, ζ 0= 1.10) CASE 3 ( c0= 2.78, ζ 0= 1.10) CASE 4 ( c0= 6.12, ζ 0= 1.10) ( d) Extensive Damage ( e) Complete Collapse Fig 2.22 Jacketing and Restrainer Effects on Fragility Curves of Bridge 2 45 Table 2.14 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 3 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 1 3 2 5 Almost No 3 13 3 4 Slight 9 16 4 15 Moderate 6 11 9 14 Extensive 13 13 6 14 Complete 28 4 36 8 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.11, ζ 0= 0.95) CASE 2 ( c0= 0.16, ζ 0= 0.95) CASE 3 ( c0= 0.10, ζ 0= 0.95) CASE 4 ( c0= 0.15, ζ 0= 0.95) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.19, ζ 0= 0.95) CASE 2 ( c0= 0.36, ζ 0= 0.95) CASE 3 ( c0= 0.15, ζ 0= 0.95) CASE 4 ( c0= 0.27, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.33, ζ 0= 0.95) CASE 2 ( c0= 0.62, ζ 0= 0.95) CASE 3 ( c0= 0.27, ζ 0= 0.95) CASE 4 ( c0= 0.48, ζ 0= 0.95) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.40, ζ 0= 0.95) CASE 2 ( c0= 0.86, ζ 0= 0.95) CASE 3 ( c0= 0.39, ζ 0= 0.95) CASE 4 ( c0= 0.73, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.62, ζ 0= 0.95) CASE 2 ( c0= 2.31, ζ 0= 0.95) CASE 3 ( c0= 0.48, ζ 0= 0.95) CASE 4 ( c0= 1.15, ζ 0= 0.95) ( d) Extensive Damage ( e) Complete Collapse Fig 2.23 Jacketing and Restrainer Effects on Fragility Curves of Bridge 3 46 Table 2.15 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 4 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 3 8 3 8 Almost No 5 13 4 10 Slight 11 15 14 14 Moderate 7 12 5 9 Extensive 18 9 12 15 Complete 16 3 22 4 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.08, ζ 0= 1.06) CASE 2 ( c0= 0.18, ζ 0= 1.06) CASE 3 ( c0= 0.13, ζ 0= 1.06) CASE 4 ( c0= 0.20, ζ 0= 1.06) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.19, ζ 0= 1.06) CASE 2 ( c0= 0.39, ζ 0= 1.06) CASE 3 ( c0= 0.21, ζ 0= 1.06) CASE 4 ( c0= 0.38, ζ 0= 1.06) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.33, ζ 0= 1.06) CASE 2 ( c0= 0.69, ζ 0= 1.06) CASE 3 ( c0= 0.39, ζ 0= 1.06) CASE 4 ( c0= 0.62, ζ 0= 1.06) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.50, ζ 0= 1.06) CASE 2 ( c0= 1.02, ζ 0= 1.06) CASE 3 ( c0= 0.50, ζ 0= 1.06) CASE 4 ( c0= 0.81, ζ 0= 1.06) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.87, ζ 0= 1.06) CASE 2 ( c0= 1.87, ζ 0= 1.06) CASE 3 ( c0= 0.72, ζ 0= 1.06) CASE 4 ( c0= 2.31, ζ 0= 1.06) ( d) Extensive Damage ( e) Complete Collapse Fig 2.24 Jacketing and Restrainer Effects on Fragility Curves of Bridge 4 47 Table 2.16 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 5 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 4 6 3 5 Almost No 5 9 6 11 Slight 12 20 12 19 Moderate 10 11 11 13 Extensive 14 14 15 12 Complete 15 0 13 0 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.33, ζ 0= 0.88) CASE 2 ( c0= 0.44, ζ 0= 0.88) CASE 3 ( c0= 0.25, ζ 0= 0.88) CASE 4 ( c0= 0.37, ζ 0= 0.88) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.58, ζ 0= 0.88) CASE 2 ( c0= 0.72, ζ 0= 0.88) CASE 3 ( c0= 0.58, ζ 0= 0.88) CASE 4 ( c0= 0.74, ζ 0= 0.88) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.90, ζ 0= 0.88) CASE 2 ( c0= 1.35, ζ 0= 0.88) CASE 3 ( c0= 0.90, ζ 0= 0.88) CASE 4 ( c0= 1.35, ζ 0= 0.88) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.20, ζ 0= 0.88) CASE 2 ( c0= 1.77, ζ 0= 0.88) CASE 3 ( c0= 1.25, ζ 0= 0.88) CASE 4 ( c0= 1.84, ζ 0= 0.88) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.74, ζ 0= 0.88) CASE 2 ( c0= 2.68, ζ 0= 0.88) CASE 3 ( c0= 1.82, ζ 0= 0.88) CASE 4 ( c0= 2.68, ζ 0= 0.88) ( d) Extensive Damage ( e) Complete Collapse Fig 2.25 Jacketing and Restrainer Effects on Fragility Curves of Bridge 5 2.8 Fragility Enhancement After Column Retrofit 2.8.1 Fragility Curves After Retrofit for Longitudinal Direction The fragility curves for five ( 5) sample bridges associated with those damage states are plotted in Figs 2.26, 2.27, 2.28, 2.29 and 2.30, while the number of damaged bridges is listed in Tables 2.17, 2.18, 2.19, 2.20 and 2.21, respectively, for the cases before retrofit and after retrofit as a function of peak ground acceleration. It is noted here that the log standard deviation for the pair of fragility curves in each of Figs is obtained by considering both two cases ( before and after retrofit) together and calculating the optimal values from equation 2.11 for these fragility curves. This is for the reason that the bridge with jacketed columns is expected to be less vulnerable to ground motion than the bridge with the columns not jacketed and therefore we expect that the pair of these fragility curves should not theoretically intersect. The damage state of a bridge in this case is defined in terms of the maximum value of the peak ductility demands sustained by all the column ends. In this context, comparison between the two curves in each of Figs 2.26 2.30 indicates that the bridge is less susceptible to damage from the ground motion after retrofit than before. The simulated fragility curves in this case demonstrate that, for all levels of damage states, the median fragility values after retrofit are larger than the corresponding values before retrofit. This implies the following: if the number of Type 1 bridges suffering from a certain state of damage is counted, on average, the damage is smaller when the bridge is subjected to these sixty ( 60) earthquakes after retrofit than before retrofit. The number is listed in Tables 2.17 2.21 for before and after retrofit to Bridge 1~ 5. The result in Tables 2.17 2.21 is consistent with the observation that the fragility enhancement is found to be 48 more significant for more severe state of damage in general. This is not unexpected because the ductility demands for more severe states of damage increase after retrofit by much larger multiples than those that occurred before retrofit. 49 50 Table 2.17 Number of Damaged Bridges: Retrofit Effect in Bridge 1 sample size= 60 Damage States before Retrofit after Retrofit No 4 7 Almost No 5 9 Slight 10 16 Moderate 7 13 Extensive 17 13 Complete 17 2 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.36, ζ 0= 0.84) after retrofit ( c0= 0.47, ζ 0= 0.84) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.45, ζ 0= 0.84) after retrofit ( c0= 0.75, ζ 0= 0.84) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.80, ζ 0= 0.84) after retrofit ( c0= 1.25, ζ 0= 0.84) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.04, ζ 0= 0.84) after retrofit ( c0= 1.73, ζ 0= 0.84) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.66, ζ 0= 0.84) after retrofit ( c0= 5.05, ζ 0= 0.84) ( d) Extensive Damage ( e) Complete Collapse Fig 2.26 Retrofit Effect on Fragility Curves of Bridge 1 ( Longitudinal) 51 Table 2.18 Number of Damaged Bridges: Retrofit Effect in Bridge 2 sample size= 60 Damage States before Retrofit after Retrofit No 9 10 Almost No 4 9 Slight 10 19 Moderate 7 12 Extensive 16 6 Complete 14 4 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.38, ζ 0= 0.96) after retrofit ( c0= 0.39, ζ 0= 0.96) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.44, ζ 0= 0.96) after retrofit ( c0= 0.52, ζ 0= 0.96) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.65, ζ 0= 0.96) after retrofit ( c0= 1.11, ζ 0= 0.96) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.86, ζ 0= 0.96) after retrofit ( c0= 2.13, ζ 0= 0.96) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.47, ζ 0= 0.96) after retrofit ( c0= 3.00, ζ 0= 0.96) ( d) Extensive Damage ( e) Complete Collapse Fig 2.27 Retrofit Effect on Fragility Curves of Bridge 2 ( Longitudinal) 52 Table 2.19 Number of Damaged Bridges: Retrofit Effect in Bridge 3 sample size= 60 Damage States before Retrofit after Retrofit No 2 3 Almost No 2 13 Slight 9 16 Moderate 6 11 Extensive 13 13 Complete 28 4 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.12, ζ 0= 0.97) after retrofit ( c0= 0.15, ζ 0= 0.97) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.19, ζ 0= 0.97) after retrofit ( c0= 0.36, ζ 0= 0.97) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.33, ζ 0= 0.97) after retrofit ( c0= 0.62, ζ 0= 0.97) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.40, ζ 0= 0.97) after retrofit ( c0= 0.86, ζ 0= 0.97) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.62, ζ 0= 0.97) after retrofit ( c0= 2.31, ζ 0= 0.97) ( d) Extensive Damage ( e) Complete Collapse Fig 2.28 Retrofit Effect on Fragility Curves of Bridge 3 ( Longitudinal) 53 Table 2.20 Number of Damaged Bridges: Retrofit Effect in Bridge 4 sample size= 60 Damage States before Retrofit after Retrofit No 2 4 Almost No 5 17 Slight 12 14 Moderate 7 13 Extensive 18 9 Complete 16 3 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.09, ζ 0= 1.22) after retrofit ( c0= 0.18, ζ 0= 1.22) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.13, ζ 0= 1.22) after retrofit ( c0= 0.39, ζ 0= 1.22) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.35, ζ 0= 1.22) after retrofit ( c0= 0.67, ζ 0= 1.22) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.50, ζ 0= 1.22) after retrofit ( c0= 1.02, ζ 0= 1.22) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.88, ζ 0= 1.22) after retrofit ( c0= 1.87, ζ 0= 1.22) ( d) Extensive Damage ( e) Complete Collapse Fig 2.29 Retrofit Effect on Fragility Curves of Bridge 4 ( Longitudinal) 54 Table 2.21 Number of Damaged Bridges: Retrofit Effect in Bridge 5 sample size= 60 Damage States before Retrofit after Retrofit No 9 9 Almost No 7 12 Slight 14 24 Moderate 11 12 Extensive 13 2 Complete 6 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.44, ζ 0= 0.83) after retrofit ( c0= 0.44, ζ 0= 0.83) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.55, ζ 0= 0.83) after retrofit ( c0= 0.67, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.88, ζ 0= 0.83) after retrofit ( c0= 1.30, ζ 0= 0.83) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.18, ζ 0= 0.83) after retrofit ( c0= 1.99, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.83, ζ 0= 0.83) after retrofit ( c0= 2.08, ζ 0= 0.83) ( d) Extensive Damage ( e) Complete Collapse Fig 2.30 Retrofit Effect on Fragility Curves of Bridge 5 ( Longitudinal) The result shows, for example, that the effect of column retrofit on the seismic performance is excellent in explaining that the bridges are up to three times less fragile for Bridge 1 ( complete damage) and two for Bridge 2 ( complete damage) after retrofit compared to the case before retrofit in terms of the median values. 2.8.1.1 Enhancement after Retrofit for Circular Column Considering Bridge 1 and 2 which have circular columns and corresponding sets of fragility curves before and after retrofit, the average fragility enhancement over these two ( 2) bridges at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” is plotted through curve fitting as shown in Fig 2.31. This curve shows 20%, 34%, 58%, 98% and 167% improvement for each damage state described on the x axis in Fig 2.31. Almost_ NoSlightModerateExtensiveCollapseDamage States04080120160200Increase in Percentage (%) y= 11.8 e 0.53xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.31 Enhancement Curve for Circular Columns with Steel Jacketing 55 2.8.1.2 Enhancement after Retrofit for Oblong Shape Column For Bridge 3 and 5 with oblong columns, the fragility enhancement is developed in Fig 2.28 and 2.30. Considering these two ( 2) sample bridges with oblong columns and corresponding sets of fragility curves before and after retrofit, the average fragility enhancement over these two ( 2) sample bridges at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” is plotted through curve fitting as shown in Fig 2.38. This curve shows 20%, 34%, 58%, 99% and 170% improvement for each damage state described on the x axis in Fig 32. Almost_ NoSlightModerateExtensiveCollapseDamage States04080120160Increase in Percentage (%) y= 11.4 e 0.54xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.32 Enhancement Curve for Oblong Columns with Steel Jacketing 2.8.1.3 Enhancement after Retrofit for Rectangular Column For Bridge 4 with rectangular columns, the fragility enhancement is developed in Fig 2.30. 56 Considering the sample bridge with rectangular columns and corresponding sets of fragility curves before and after retrofit at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” is plotted through curve fitting as shown in Fig 2.33. It is noted that the effect of retrofit is not good for Bridge 4 because the geometric shape after retrofit [ Fig C4 ( b1)~( b9)] is not efficient for steel jacketing to produce confinement effect. Almost_ NoSlightModerateExtensiveCollapseDamageStates04080120160200Increase in Percentage (%) y= 132 e  0.04xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.33 Enhancement Curve for Rectangular Columns with Steel Jacketing 2.8.1.4 Enhancement after Retrofit for All Types of Column Considering all the sample bridges and corresponding sets of fragility curves before and after retrofit at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” 57 is plotted through curve fitting as shown in Fig 2.34. This curve shows 40%, 55%, 75%, 104% and 143% improvement for each damage state described on the x axis in Fig 2.34 Almost_ NoSlightModerateExtensiveCollapseDamageStates04080120160200Increase in Percentage (%) y= 28.8 e 0.32xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.34 Enhancement Curve for Five Sample Bridges with Steel Jacketing 2.8.2 Enhancement after Calibrating the Analytical Fragility Curves As described in the earlier part, analytical fragility curves are obtained using the damage state definitions given by Dutta and Mander ( Table 2.4). To compare these analytically obtained fragility curves with past earthquake bridge damage data, empirical fragility curves for a third level subset ( considering ‘ multiple span’ and ‘ soil type C’) have been developed ( Shinozuka et al. 2003a) ( see Chapter 3). Results indicate that the analytical curves are more probable to exceed a damage state than empirical ones ( Shinozuka and Banerjee, 2004). They have defined the damage states of bridges for slight, moderate and extensive damage levels in terms of threshold ductility capacities, 58 for what, the analytical fragility curves will be consistent with empirical curves. These new definitions of threshold ductility capacities have extended to develop the fragility curves after retrofit. Fig 2.35 shows the empirical fragility curves and simulated fragility curves for three already stated damage states of Bridge 2. Obtained threshold ductility capacities at each damage states for bridge 2, 4, and 5 before and after retrofit are tabulated in Table 2.22 . Table 2.22 Simulated Ductility Capacities of Sample Bridges Bridge 2 Bridge 4 Bridge 5 Damage state before retrofit before retrofit after retrofit after retrofit before retrofit after retrofit Slight 4.5 5.4 6.9 11.5 4.5 6.62 Moderate 6.5 9.66 7.31 17.13 8.4 16.21 Extensive 16.8 26.04 14.5 36.84 12.8 26.8 Slight Damage 00.20.40.60.8100.20.40.60.81Empirical CurveAnalytical CurveProbability of Exceeding Damage State 59 60 00.20.40.60.8100.20.40.60.81Empirical CurveAnalytica Curve Probability of Exceeding Damage State PGA ( g) Moderate Damage 00.20.40.60.8100.20.40.60.81Empirical CurveAnalytical Curve Probability of Exceeding Damage State PGA ( g) Extensive Damage Fig 2.35 Empirical Fragility Curves and Calibrated Analytical Fragility Curves of Bridge 2 Based on the new definitions of damage states, the fragility curves of bridge 2( Circular Column), 4 ( Rectangular Column) and 5 ( Oblong Column) before and after retrofit are estimated again. Table 2.23 give the fragility parameters, and the enhancement ratios based on the 2 set of definitions of damage states are provided in Table 2.24. 61 Table 2.23 Fragility Curves based on Adjusted Damage States Definitions Bridge 2 Bridge 4 Bridge 5 Damage State 0 c ( g) ' 0 c ( g) 0 ζ 0 c ( g) ' 0 c ( g) 0 ζ 0 c ( g) ' 0 c ( g) 0 ζ At least minor 0.59 0.76 0.56 0.56 0.83 0.52 0.51 0.74 0.42 At least moderate 0.71 1.48 0.67 0.62 1.08 0.64 0.66 1.33 0.44 At least extensive 1.13 6.12 1.27 1.08 2.53 0.71 / / / Table 2.24 Enhancement Ratios Comparison Bridge 2 Bridge 4 Bridge 5 Damage State Mander’s Calibrated Mander’s Calibrated Mander’s Calibrated At least minor 18% 28% 200% 48% 22% 46% At least moderate 71% 109% 91% 74% 48% 102% At least extensive 148% 440% 104% 134% 69% / 2.8.3 Fragility Curves for Transverse Direction The fragility curves for five ( 5) sample bridges associated with those damage states are plotted in Figs 2.36, 2.37, 2.38, 2.39 and 2.40, while the number of damaged bridges is listed in tables 2.25, 2.26, 2.27, 2.28 and 2.29, respectively, for the cases before retrofit and after retrofit as a function of peak ground acceleration. 62 Table 2.25 Number of Damaged Bridges: Retrofit Effect in Bridge 1 sample size= 60 Damage States before Retrofit after Retrofit No 5 8 Almost No 3 8 Slight 10 17 Moderate 9 13 Extensive 18 13 Collapse 15 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.35, ζ 0= 0.83) after retrofit ( c0= 0.55, ζ 0= 0.83) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.55, ζ 0= 0.83) after retrofit ( c0= 0.75, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.77, ζ 0= 0.83) after retrofit ( c0= 1.28, ζ 0= 0.83) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.06, ζ 0= 0.83) after retrofit ( c0= 1.75, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.79, ζ 0= 0.83) after retrofit ( c0= 6.12, ζ 0= 0.83) ( d) Extensive Damage ( e) Complete Collapse Fig 2.36 Retrofit Effect on Fragility Curves of Bridge 1 ( Transverse) 63 Table 2.27 Number of Damaged Bridges: Retrofit Effect in Bridge 2 sample size= 60 Damage States before Retrofit after Retrofit No 4 4 Almost No 4 16 Slight 16 18 Moderate 6 11 Extensive 18 9 Collapse 12 2 0 0.20.40.60.81 PGA ( g) 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.03, ζ 0= 1.80) after retrofit ( c0= 0.08, ζ 0= 1.80) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.07, ζ 0= 1.80) after retrofit ( c0= 0.40, ζ 0= 1.80) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.53, ζ 0= 1.80) after retrofit ( c0= 0.92, ζ 0= 1.80) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.68, ζ 0= 1.80) after retrofit ( c0= 1.43, ζ 0= 1.80) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.31, ζ 0= 1.80) after retrofit ( c0= 2.03, ζ 0= 1.80) ( d) Extensive Damage ( e) Complete Collapse Fig 2.37 Retrofit Effect on Fragility Curves of Bridge 2 ( Transverse) 64 Table 2.28 Number of Damaged Bridges: Retrofit Effect in Bridge 3 sample size= 60 Damage States before Retrofit after Retrofit No 5 8 Almost No 5 8 Slight 4 22 Moderate 8 14 Extensive 14 7 Collapse 24 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.33, ζ 0= 0.78) after retrofit ( c0= 0.39, ζ 0= 0.78) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.46, ζ 0= 0.78) after retrofit ( c0= 0.55, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.52, ζ 0= 0.78) after retrofit ( c0= 1.10, ζ 0= 0.78) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.71, ζ 0= 0.78) after retrofit ( c0= 1.62, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.04, ζ 0= 0.78) after retrofit ( c0= 2.08, ζ 0= 0.78) ( d) Extensive Damage ( e) Complete Collapse Fig 2.38 Retrofit Effect on Fragility Curves of Bridge 3 ( Transverse) 65 Table 2.29 Number of Damaged Bridges: Retrofit Effect in Bridge 4 sample size= 60 Damage States before Retrofit after Retrofit No 1 3 Almost No 3 11 Slight 12 17 Moderate 12 12 Extensive 15 14 Collapse 17 2 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.17, ζ 0= 0.99) after retrofit ( c0= 0.27, ζ 0= 0.99) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.22, ζ 0= 0.99) after retrofit ( c0= 0.56, ζ 0= 0.99) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.55, ζ 0= 0.99) after retrofit ( c0= 0.93, ζ 0= 0.99) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.83, ζ 0= 0.99) after retrofit ( c0= 1.30, ζ 0= 0.99) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.24, ζ 0= 0.99) after retrofit ( c0= 4.64, ζ 0= 0.99) ( d) Extensive Damage ( e) Complete Collapse Fig 2.39 Retrofit Effect on Fragility Curves of Bridge 4 ( Transverse) 66 Table 2.30 Number of Damaged Bridges: Retrofit Effect in Bridge 5 sample size= 60 Damage States before Retrofit after Retrofit No 7 9 Almost No 2 6 Slight 10 16 Moderate 10 13 Extensive 14 13 Collapse 17 3 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.38, ζ 0= 0.68) after retrofit ( c0= 0.42, ζ 0= 0.68) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.42, ζ 0= 0.68) after retrofit ( c0= 0.58, ζ 0= 0.68) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.63, ζ 0= 0.68) after retrofit ( c0= 0.90, ζ 0= 0.68) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.86, ζ 0= 0.68) after retrofit ( c0= 1.27, ζ 0= 0.68) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.24, ζ 0= 0.68) after retrofit ( c0= 1.99, ζ 0= 0.68) ( d) Extensive Damage ( e) Complete Collapse Fig 2.40 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) Chapter 3 Development of Empirical Fragility Curves for Bridges 3.1 Empirical Bridge Damage Data The 1994 Northridge Earthquake caused tremendous damages to the human building environment. However, the damage investigation after the event provided valuable data basis for developing empirical fragility curves. After the event, 2209 highway bridges around Los Angeles Area were investigated and the damage of each bridge was classified as one of the five states: No, Minor, Moderate, Major or Collapse. Table 3.1 provides the summary of the bridge damage condition. The site ground motion of each bridge structure can be derived from any ground motion spatial distribution ( contour) map. Figs. 3.1 and 3.2 show PGA and PGV distribution in the 1994 Northridge Earthquake, which are acquired from the TriNet Shakemap ( http:// www. trinet. org/ trinet. html). Table 3.2 lists part of the bridge damage table including bridge site ground motion determined from these two maps. Table 3.1 Summary of Bridge Damage Status in the 1994 Northridge Earthquake Damage State No Damage Minor Damage Moderate Damage Major Damage Collapse Damage Total Number 1978 84 94 47 6 2209 67 Fig 3.1 1994 Northridge Earthquake: PGA Distribution Fig 3.2 1994 Northridge Earthquake: PGV Distribution PGA( g) PGV( cm/ s) 68 Table 3.2 Seismic Damages of Bridges in the 1994 Northridge Earthquake ID BRIDGE_ NO Damage States ShakeMap PGA ( g) ShakeMap PGV ( cm/ s) LAT LONG 1 53 1301 MOD* 0.2 16 34.0227  118.2500 2 53 1471 0.12 12 33.7500  118.2687 3 53 2618 0.08 6 33.7667  118.2353 4 53 2216G MAJ* 0.76 114 34.2667  118.4697 5 53 1907G MOD 0.24 20 34.1383  118.2333 6 53 0595 0.28 16 34.0353  118.2187 7 53 1851 MOD 0.28 28 33.9863  118.4000 8 53 2549H 0.12 10 33.8687  118.2843 9 53 1637F MOD 0.4 42 34.0257  118.4237 10 53 1790H MOD 0.24 20 34.1520  118.2747 11 53 1717H MIN* 0.28 14 34.0353  118.1677 12 53 1627G MAJ 0.4 50 34.0257  118.4343 13 53 2673 0.28 18 34.0520  118.2227 14 53 1424 MOD 0.24 16 34.0757  118.2217 15 53 2142F 0.12 10 33.8697  118.1863 16 53 0707F 0.2 14 34.0393  118.2697 17 53 2700G 0.12 10 33.9080  118.1010 18 53 1714G MOD 0.28 14 34.0353  118.1677 19 53 0845 0.2 22 33.9353  118.3903 20 53 2731 0.08 10 33.8373  118.2040 21 52 0331R 0.28 22 34.2859  118.8650 22 53 2143F 0.12 10 33.8697  118.1843 23 53 2318G 0.16 14 34.1500  118.1530 24 53 2327F MAJ 0.6 72 34.2667  118.4383 25 53 2329G MAJ 0.6 72 34.2667  118.4383 26 53 2102G MAJ 0.4 46 34.2863  118.4030 27 53 0405 0.28 18 34.0520  118.2227 28 52 0118 MOD 0.24 20 34.3917  118.9150 29 53 1960F COL* 0.6 76 34.3350  118.5083 30 53 1238G 0.2 18 33.9167  118.3667 31 53 2104F MOD 0.4 44 34.2853  118.4020 32 52 0413 0.2 18 34.2011  118.9758 33 53 2627 0.08 10 33.7843  118.2217 34 53 1964F COL 0.6 76 34.3353  118.5056 35 53 1962F MOD 0.64 76 34.3343  118.5040 36 53 2200S MOD 0.48 48 34.4010  118.4540 37 53 1790 MIN 0.24 18 34.1510  118.2717 …. * MIN: Minor Damage MOD: Moderate Damage MAJ: Major Damage COL: Collapse 69 70 3.2 Bridge Classification In this research, the bridges are classified into different subsets according to the following three distinct attributes; ( A) It is either single span ( S) or multiple span ( M), ( B) it is built on either hard soil ( SA), medium soil ( SB) or soft soil ( SC) in the definition of UBC94, and ( C) it has a skew angle 1 θ ( less than 20o), 2 θ ( between 20o and 60o) or 3 θ ( larger than 60o). To begin with, one might consider the first level hypothesis that the entire sample is taken from a statistically homogeneous population of bridges. The second level subsets are created by dividing the sample either ( A) into two groups of bridges, one with single spans and the other with multiple spans, ( B) into three groups, the first with soil condition SA, the second with SB and the third with SC, or ( C) into three groups depending on the skew angles 1 θ , 2 θ and 3 θ . The third and fourth level sub groupings were also considered for the development of corresponding fragility curves under PGA and PGV as ground motion intensity index ( Shinozuka et al, 2003a). 3.3 Parameter Estimation It is assumed that the curves can be expressed in the form of two parameter lognormal distribution functions, and the estimation of the two parameters ( median and log standard deviation) is performed with the aid of the maximum likelihood method. For this purpose, PGA and PGV values are used to represent the intensity of the seismic ground motion. The likelihood method for fragility parameter estimation was described in Chapter 2. The median values and log standard deviations of all levels of attribute combinations are listed in Table 3.3 3.6. Note that, if an element of a matrix in these 71 tables shows N/ A, it indicates that no sub sample was found for the particular combination of bridge attributes the element signifies. The family of fragility curves corresponding to the first level is plotted in Fig3.3 and 3.4. The curve with a “ minor” designation represents, at each PGA or PGV value a , the probability that “ at least a minor” state of damage will be sustained by a bridge ( arbitrarily chosen from the sample of bridges) when it is subjected to PGA or PGV a . The same meaning applies to other curves with their respective damage state designations. All the other fragility curves in PGA are plotted in Figs 3.5 3.44 Table 3.3 First Level ( Composite) Fragility Curve Table 3.4 Second Level Fragility Curve ( a) Number of Span PGA ( g) PGV ( cm/ s) Span Damage State c ς c ς Min 0.89 0.66129 0.98 Mod 1.15 0.66188 0.98 Maj 1.76 0.66357 0.98 Single Col N/ A 0.66N/ A0.98 Min 0.56 0.6663 0.92 Mod 0.70 0.6687 0.92 Maj 1.09 0.66163 0.92 Multiple Col 2.16 0.66428 0.92 ( b) Skew Angle PGA ( g) PGV ( cm/ s) Skew Damage State c ς c ς Min 0.82 0.76108 1.07 Mod 1.10 0.76164 1.07 Maj 1.86 0.76343 1.07 00 200 Col 3.49 0.76833 1.07 PGA ( g) PGV ( cm/ s) Damage State c ς c ς Min 0.64 0.7076 0.98 Mod 0.80 0.70106 0.98 Maj 1.25 0.70200 0.98 Col 2.55 0.70555 0.98 72 Min 0.60 0.7170 0.98 Mod 0.72 0.7190 0.98 Maj 1.15 0.71173 0.98 200 600 Col 3.18 0.71769 0.98 Min 0.42 0.5242 0.75 Mod 0.52 0.5256 0.75 Maj 0.74 0.5296 0.75 > 600 Col 1.26 0.52212 0.75 ( c) Soil Type PGA ( g) PGV ( cm/ s) Soil Damage State c ς c ς Min 0.87 0.75110 1.03 Mod 1.10 0.75151 1.03 Maj 1.51 0.75234 1.03 A Col N/ A 0.75N/ A1.03 Min 0.64 0.7165 0.81 Mod 0.84 0.7191 0.81 Maj 1.24 0.71145 0.81 B Col N/ A 0.71N/ A0.81 Min 0.61 0.6974 0.98 Mod 0.76 0.69102 0.98 Maj 1.22 0.69199 0.98 C Col 2.35 0.69523 0.98 Table 3.5 Third Level Fragility Curve ( a) Span/ Skew PGA ( g) PGV ( cm/ s) Span Skew Damage State c ς c ς Minor 1.37 0.82 276 1.28 Moderate 2.04 0.82 502 1.28 Major 3.56 0.82 1179 1.28 00 200 Collapse N/ A N/ A N/ A N/ A Minor 0.63 0.43 82 0.7 Moderate 0.70 0.43 98 0.7 Major 0.96 0.43 164 0.7 200 600 Collapse N/ A N/ A N/ A N/ A Minor 0.62 0.13 86 0.10 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A Single > 600 Collapse N/ A N/ A N/ A N/ A Minor 0.68 0.71 82 0.98 Moderate 0.91 0.71 122 0.98 Major 1.52 0.71 251 0.98 Multiple 00 200 Collapse 2.76 0.71 574 0.98 73 Minor 0.56 0.74 63 0.99 Moderate 0.69 0.74 84 0.99 Major 1.11 0.74 162 0.99 200 600 Collapse 3.14 0.74 716 0.99 Minor 0.38 0.38 37 0.58 Moderate 0.42 0.38 43 0.58 Major 0.56 0.38 68 0.58 > 600 Collapse 0.67 0.38 92 0.58 ( b) Span/ Soil PGA ( g) PGV ( cm/ s) Span Soil Damage State c ς c ς Minor 0.90 0.40 116 0.50 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor 0.68 0.50 68 0.50 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.74 0.57 106 0.90 Moderate 0.91 0.57 144 0.90 Major 1.37 0.57 274 0.90 Single C Collapse N/ A N/ A N/ A N/ A Minor 0.64 0.64 66 0.81 Moderate 0.77 0.64 83 0.81 Major 1.05 0.64 125 0.81 A Collapse N/ A N/ A N/ A N/ A Minor 0.47 0.45 44 0.53 Moderate 0.56 0.45 57 0.53 Major 0.76 0.45 86 0.53 B Collapse N/ A N/ A N/ A N/ A Minor 0.56 0.67 65 0.96 Moderate 0.7 0.67 89 0.96 Major 1.11 0.67 173 0.96 Multiple C Collapse 2.11 0.67 435 0.96 ( c) Skew/ Soil PGA ( g) PGV ( cm/ s) Skew Soil Damage State c ς c ς Minor 0.70 0.50 61 0.50 Moderate 0.98 0.50 90 0.50 Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor 0.80 0.50 75 0.5 00 200 B Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A Collapse N/ A N/ A N/ A N/ A Minor 0.74 0.72 98 1.04 Moderate 0.97 0.72 144 1.04 Major 1.61 0.72 299 1.04 C Collapse 2.99 0.72 728 1.04 Minor 0.73 0.48 79 0.50 Moderate 0.73 0.48 79 0.50 Major 0.83 0.48 88 0.50 A Collapse N/ A N/ A N/ A N/ A Minor 0.49 0.38 48 0.48 Moderate 0.57 0.38 68 0.48 Major 0.57 0.38 68 0.48 B Collapse N/ A N/ A N/ A N/ A Minor 0.57 0.72 66 0.57 Moderate 0.69 0.72 86 0.69 Major 1.19 0.72 187 1.19 200 600 C Collapse 3.07 0.72 759 3.07 Minor 0.26 0.11 21 0.10 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.48 0.48 57 0.74 Moderate 0.59 0.48 76 0.74 Major 0.74 0.48 107 0.74 > 600 C Collapse 0.87 0.48 137 0.74 74 75 Table 3.6 Fourth Level Fragility Curve ( Span/ Skew/ Soil) PGA ( g) PGV ( cm/ s) Span Skew Soil Damage State c ς c ς Minor 0.63 0.22 81 0.40 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor 0.63 0.50 63 0.5 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.98 0.57 239 1.16 Moderate 1.19 0.57 340 1.16 Major 1.85 0.57 780 1.16 00 200 C Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.53 0.39 64 0.64 Moderate 0.60 0.39 78 0.64 Major 0.84 0.39 134 0.64 200 600 C Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Min
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Title  Socioeconomic effect of seismic retrofit implemented on bridges in Los Angeles highway network 
Subject  TG315.S63 2008; BridgesCaliforniaLos Angeles Metropolitan AreaMaintenance and repairCosts.; BridgesEarthquake effectsCaliforniaLos Angeles Metropolitan Area.; Earthquake resistant designCaliforniaLos Angeles Metropolitan AreaCosts.; Externalities (Economics) 
Description  "Performing organization report no.: F/CA/SD2005/03"Technical report documentation page.; "December 2008."; Facsimile reprint. Originally published: Irvine, Calif. : University of California Irvine, Dept. of Civil and Environmental Engineering, 2005.; Final report.; Performed by University of California, Irvine, Dept. of Civil and Environmental Engineering for California Dept. of Transportation, Engineering Services Center and California Dept. of Transportation, Division of Research and Innovation under contract no. 
Publisher  California Dept. of Transportation; Available through the National Technical Information Service 
Contributors  Shinozuka, Masanobu.; California. Dept. of Transportation. Engineering Services Center.; University of California, Irvine. Dept. of Civil and Environmental Engineering. 
Type  Text 
Language  eng 
Relation  http://worldcat.org/oclc/619408096/viewonline 
DateIssued  2008] 
FormatExtent  xv, 314 p. : col. ill., charts (some col.), col. maps ; 28 cm. 
Transcript  FINAL REPORT TO THE CALIFORNIA DEPARTMENT OF TRANSPORTATION SOCIO ECONOMIC EFFECT OF SEISMIC RETROFIT IMPLEMENTED ON BRIDGES IN LOS ANGELES HIGHWAY NETWORK RTA 59A0304 By Masanobu Shinozuka1, Professor and Youwei Zhou1, Graduate Research Assistant Sang Hoon Kim1, Post Doctoral Researcher Yuko Murachi1, Visiting Researcher Swagata Banerjee1, Graduate Research Assistant Sunbin Cho2, Research Engineer Howard Chung2, Research Engineer 1 Department of Civil and Environmental Engineering University of California, Irvine 2 ImageCat, Inc. October 2005 STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION TECHNICAL REPORT DOCUMENTATION PAGE TR0003 ( REV. 10/ 98) 1. REPORT NUMBER F/ CA/ SD 2005/ 03 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER 5. REPORT DATE October 2005 4. TITLE AND SUBTITLE SOCIO ECONOMIC EFFECT OF SEISMIC RETROFIT IMPLEMENTED ON BRIDGES IN THE LOS ANGELES HIGHWAY NETWORK 6. PERFORMING ORGANIZATION CODE 7. AUTHOR( S) Masanobu Shinozuka, Youwei Zhou, Sanghoon Kim, Yuko Murachi, Swagata Banerjee, Sunbin Cho and Howard Chung 8. PERFORMING ORGANIZATION REPORT NO. 10. WORK UNIT NUMBER 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Civil and Environmental Engineering The Henry Samueli School of Engineering University of California, Irvine Irvine, California 92697 11. CONTRACT OR GRANT NUMBER RTA 59A0304 13. TYPE OF REPORT AND PERIOD COVERED Final Report 12. SPONSORING AGENCY AND ADDRESS California Department of Transportation Division of Research and Innovation, MS 83 1227 O Street Sacramento CA 95814 14. SPONSORING AGENCY CODE 15. SUPPLEMENTAL NOTES 16. ABSTRACT This research studied socio economic effect of the seismic retrofit implemented on bridges in Los Angeles Area Freeway Network. Firstly, advanced FE ( Finite Element) modeling and nonlinear time history analysis are carried out to evaluate the seismic performance in the form of fragility curve, of representative bridges before and after retrofit. This analysis resulted in the determination of retrofit effect in such a way that we can quantify, through the change in fragility parameters, the improvement of bridge seismic performance after retrofit. Secondly, an integrated traffic assignment model is introduced to consider change in the post earthquake OD characteristics due to building damage, and is utilized to evaluate the post earthquake network performance of the damaged freeway network in terms of daily travel delay ( compared with the travel time associated with the freeway network not damaged) and attendant opportunity cost. Furthermore, the process of system restoration is simulated to estimate the total social cost based on bridge functionality restoration ( repair / replacement) process. The benefit from the retrofit is defined as the combined social and bridge restoration cost avoided by comparing the total social and bridge restoration cost before and after bridge retrofit. The benefit resulting from combined social and bridge restoration cost avoided together with the bridge retrofit cost are used for a cost benefit analysis. The result shows that the retrofit is cost effective if both social and bridge restoration cost avoided are considered, and the bridge restoration cost avoided can only contribute a small portion of the initial bridge retrofit cost. 17. KEY WORDS Bridge Fragility, Retrofit, Seismic Risk Analysis, Traffic Assignment, Monte Carlo Simulation, Loss Estimation, Cost Benefit Analysis 18. DISTRIBUTION STATEMENT No restrictions. This document is available to the public through the National Technical Information Service, Springfield, VA 22161 19. SECURITY CLASSIFICATION ( of this report) Unclassified 20. NUMBER OF PAGES 314 21. PRICE Reproduction of completed page authorized DISCLAIMER: The Opinions, findings, and conclusions expressed in this publication are those of the authors and not necessarily those of the STATE OF CALIFORNIA ABSTRACT This research studied socio economic effect of the seismic retrofit implemented on bridges in Los Angeles Area Freeway Network. Firstly, advanced FE ( Finite Element) modeling and nonlinear time history analysis are carried out to evaluate the seismic performance in the form of fragility curve, of representative bridges before and after retrofit. This analysis resulted in the determination of retrofit effect in such a way that we can quantify, through the change in fragility parameters, the improvement of bridge seismic performance after retrofit. Secondly, an integrated traffic assignment model is introduced to consider change in the post earthquake OD characteristics due to building damage, and is utilized to evaluate the post earthquake network performance of the damaged freeway network in terms of daily travel delay ( compared with the travel time associated with the freeway network not damaged) and attendant opportunity cost. Furthermore, the process of system restoration is simulated to estimate the total social cost based on bridge functionality restoration ( repair / replacement) process. The benefit from the retrofit is defined as the combined social and bridge restoration cost avoided by comparing the total social and bridge restoration cost before and after bridge retrofit. The benefit resulting from combined social and bridge restoration cost avoided together with the bridge retrofit cost are used for a cost benefit analysis. The result shows that the retrofit is cost effective if both social and bridge restoration cost avoided are considered, and the bridge restoration cost avoided can only contribute a small portion of the initial bridge retrofit cost. i ACKNOWLEDGEMENT The research presented in this report was sponsored by the California Department of Transportation ( Caltrans) with Li Hong Sheng as contract monitor. The authors are indebted to Caltrans for its support of this project and to Li Hong Sheng for his most valuable advice. ii TABLE OF CONTENTS Abstract……………………………………………………………………………..……. i Acknowledgements……………………………………………………………………… ii Table of Contents……………………………………………………………………..… iii List of Figures…………………………………………………………………………... vi List of Tables…………………………………………………………………………… xii Chapter 1 Introduction…………………………………………………………………... 1 Chapter 2 Development of Analytical Fragility Curves……………………….………... 7 2.1 Introduction……………………………………………………………… …. 7 2.2 Column Retrofit with Steel Jacketing……………………………………….. 8 2.2.1 Background………………………………………………………... 8 2.2.2 Steel Jacketing…………………………………………………….. 9 2.2.3 Stress Strain Relationship for Confined Concrete……………….... 9 2.3 Bridge Model………………………………………………………….……. 11 2.3.1 Bridge Description………………………………………………... 11 2.3.2 Bridge Modeling………………………………………………….. 12 2.4 Development of Moment Curvature Relationship………………………….. 15 2.4.1 Moment Curvature Relationship in Longitudinal Direction…….... 15 2.5 Bridge Response Analysis…………………………………………………... 19 2.5.1 Input Ground Motions…………………………………………….. 19 2.5.2 Response of Structures………………………………………….…. 22 2.6 Fragility Analysis of Bridges………………………………………………... 26 2.6.1 Fragility Analysis……………………………………………….…. 26 2.6.2 Damage States……………………………………………………... 27 2.7 Pounding, Soil, Jacketing, and Restrainer Effects on Fragility Curves……... 28 2.7.1 Pounding at Expansion Joint…………………………………….… 28 2.7.2 Numerical Simulation for Pounding………………………………. 31 2.7.3 Pounding Effects on Fragility Curves……………………………... 33 2.7.4 Pounding and Soil Effects on Fragility Curves……………………. 38 2.7.5 Jacketing and Restrainer Effects on Fragility Curves……………... 43 iii 2.8 Fragility Enhancement after Column Retrofit………………………………. 48 2.8.1 Fragility Curves after Retrofit by Column Jacketing for Longitudinal Direction……………………………………………. 48 2.8.1.1 Enhancement for Circular Column……………………… 55 2.8.1.2 Enhancement for Oblong Shape Column……………….. 56 2.8.1.3 Enhancement for Rectangular Column………………….. 56 2.8.1.4 Enhancement for All Types of Columns………………... 57 2.8.2 Enhancement after Calibrating the Analytical Fragility Curves....... 58 2.8.3 Fragility Curves after Retrofit by Column Jacketing for Transverse Direction………………………………………………. 61 Chapter 3 Development of Empirical Fragility Curves…………………………….…… 67 3.1 Empirical Bridge Damage Data……………………………………………... 67 3.2 Bridge Classification………………………………………………………… 70 3.3 Parameter Estimation………………………………………………………... 71 3.4 Enhancement of Empirical Fragility Curves………………………………… 98 Chapter 4 Seismic Hazard Modeling for Spatially Distributed Highway System……... 101 4.1 Highway Network: Spatially Distributed System………………………….. 101 4.2 Deterministic Seismic Hazard…………………………………………….... 102 4.3 Probabilistic Seismic Hazard………………………………………………. 103 Chapter 5 Methodology for System Performance Evaluation of Highway Network….. 107 5.1 Overview………………………………………………………………...…. 107 5.2 Site Ground Motion……………………………………………………..… 108 5.3 Network Modeling………………………………………………………… 109 5.4 Bridge Damage State Simulation………………………………………….. 110 5.5 Assignment of Link Damage State and Residual Capacity…………..…… 111 5.6 Traffic Demand: Origin Destination Data………………………………… 113 5.6.1 1996 SCAG Origin Destination Data……………………………….. 113 5.6.2 Origin Destination Data Condensation…………………………….... 117 5.6.3 Origin Destination Data Change After Earthquake…………………. 118 5.7 The Integrated Model……………………………………………...………. 121 5.8 Drivers’ Delay………………………………………………………...…… 123 iv 5.9 Opportunity Cost………………………………………………………….. 124 Chapter 6 Direct Economic Loss: Bridge Repair Cost………………………………… 127 6.1 Number of Seismically Damaged Bridges ………………………………… 127 6.2 Bridge Repair Cost Estimation in an Earthquake………………………….. 134 6.3 Expected Annual Repair Cost of a Site Specific Bridge……………..……. 137 6.3.1 Annual Probability of Damage………………………………...… 137 6.3.2 Expected Annual Repair Cost before Retrofit…………………… 138 6.3.3 Expected Annual Repair Cost after Retrofit………………...…… 138 6.3.4 System Annual Bridges Repair Cost…………………….……….. 139 Chapter 7 Social Cost Estimation……………………………………………………… 143 7.1 Daily Social Cost………………………………………………………...… 143 7.1.1 Daily Social Cost under no Retrofit Condition………………...… 143 7.1.2 Retrofit Effect on Daily Social Cost………………………….….. 147 7.2 System Restoration..………………………………………………..……… 154 7.2.1 System Restoration Based on Bridge Repair Process………….... 154 7.2.2 System Restoration Based on Bridge Functionality Restoration... 157 7.2.3 OD Recovery……………………………………………..……… 158 7.2.4 System Restoration Curve and Total Social Cost……………..… 159 7.3 Economic Loss Estimation Related to System Social Cost…………...…… 167 Chapter 8 Cost effectiveness Analysis……………………………………………….... 171 8.1 Introduction………………………………………………………………… 171 8.2 Retrofit Cost ………..………………………...……………………...…….. 171 8.3 Benefit from Retrofit………..…………………………………………...…. 168 8.4 Cost Effectiveness Evaluation…………………………………………...… 179 Chapter 9 Conclusions ………………………………………………………………… 189 Appendix A Moment Rotation Curves of Bridge Columns………..…………….…..... 193 Appendix B Integrated Traffic Assignment Model.………………………..………….. 257 Appendix C HighwaySRA Manual………………………..………………………..… 293 Reference……………………………………………………………………...……….. 311 v LIST OF FIGURES Fig. 2.1 Stress Strain Model for Concrete in Compression 10 Fig. 2.2 Elevation of Sample Bridges 13 Fig. 2.3 Nonlinearities in Bridge Model 15 Fig. 2.4 Column 2 of Bridge 1 before retrofit 16 Fig. 2.5 Column 2 of Bridge 1 after retrofit 18 Fig. 2.6 Acceleration Time Histories Generated for Los Angeles 21 Fig. 2.7 Responses at Column End of Bridge 1 23 Fig. 2.8 Displacement at Expansion Joints of Bridge 1 25 Fig. 2.9 Gap Element 31 Fig. 2.10 Ground Motion Time History for LA01 32 Fig. 2.11 Pounding Force at Expansion Joint 32 Fig. 2.12 Structural Responses without Pounding 32 Fig. 2.13 Structural Responses with Pounding 32 Fig. 2.14 Pounding Effect on Fragility Curves of Bridge 2 34 Fig. 2.15 Pounding Effect on Fragility Curves of Bridge 3 35 Fig. 2.16 Pounding Effect on Fragility Curves of Bridge 4 36 Fig. 2.17 Pounding Effect on Fragility Curves of Bridge 5 37 Fig. 2.18 Pounding and Soil Effects on Fragility Curves of Bridge 2 39 Fig. 2.19 Pounding and Soil Effects on Fragility Curves of Bridge 3 40 Fig. 2.20 Pounding and Soil Effects on Fragility Curves of Bridge 4 41 Fig. 2.21 Pounding and Soil Effects on Fragility Curves of Bridge 5 42 Fig. 2.22 Jacketing and Restrainer Effects on Fragility Curves of Bridge 2 44 Fig. 2.23 Jacketing and Restrainer Effects on Fragility Curves of Bridge 3 45 Fig. 2.24 Jacketing and Restrainer Effects on Fragility Curves of Bridge 4 46 Fig. 2.25 Jacketing and Restrainer Effects on Fragility Curves of Bridge 5 47 Fig. 2.26 Retrofit Effect on Fragility Curves of Bridge 1 ( Longitudinal) 50 Fig. 2.27 Retrofit Effect on Fragility Curves of Bridge 2 ( Longitudinal) 51 Fig. 2.28 Retrofit Effect on Fragility Curves of Bridge 3 ( Longitudinal) 52 Fig. 2.29 Retrofit Effect on Fragility Curves of Bridge 4 ( Longitudinal) 53 vi Fig. 2.30 Retrofit Effect on Fragility Curves of Bridge 5 ( Longitudinal) 54 Fig. 2.31 Enhancement Curve for Circular Columns with Steel Jacketing 55 Fig. 2.32 Enhancement Curve for Oblong Columns with Steel Jacketing 56 Fig. 2.33 Enhancement Curve for Rectangular Columns with Steel Jacketing 57 Fig. 2.34 Enhancement Curve for Five Sample Bridges with Steel Jacketing 58 Fig. 2.35 Empirical Fragility Curves and Calibrated Analytical Fragility Curves of Bridge 2 60 Fig. 2.36 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 62 Fig. 2.37 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 63 Fig. 2.38 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 64 Fig. 2.39 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 65 Fig. 2.40 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) 66 Fig. 3.1 1994 Northridge Earthquake: PGA Distribution 68 Fig. 3.2 1994 Northridge Earthquake: PGV Distribution 68 Fig. 3.3 Fragility Curve on the basis of PGA ( Composite) 77 Fig. 3.4 Fragility Curve on the basis of PGV ( Composite) 77 Fig. 3.5 Fragility Curve in PGA ( Single Span) 78 Fig. 3.6 Fragility Curve in PGA ( Multiple Span) 78 Fig. 3.7 Fragility Curve in PGA ( Skew 00 200) 79 Fig. 3.8 Fragility Curve in PGA ( Skew 200 600) 79 Fig. 3.9 Fragility Curve in PGA ( Skew > 600) 80 Fig. 3.10 Fragility Curve in PGA ( Soil A) 80 Fig. 3.11 Fragility Curve in PGA ( Soil B) 81 Fig. 3.12 Fragility Curve in PGA ( Soil C) 81 Fig. 3.13 Fragility Curve in PGA ( Single Span / Skew 00 200) 82 Fig. 3.14 Fragility Curve in PGA ( Single Span/ Skew 200 600) 82 Fig. 3.15 Fragility Curve in PGA ( Single Span/ Skew > 600) 83 Fig. 3.16 Fragility Curve in PGA ( Multiple Span / Skew 00 200) 83 Fig. 3.17 Fragility Curve in PGA ( Multiple Span/ Skew 200 600) 84 Fig. 3.18 Fragility Curve in PGA ( Multiple Span/ Skew > 600) 84 Fig. 3.19 Fragility Curve in PGA ( Skew 00 200/ Soil A) 85 vii Fig. 3.20 Fragility Curve in PGA ( Skew 00 200/ Soil B) 85 Fig. 3.21 Fragility Curve in PGA ( Skew 00 200/ Soil C) 86 Fig. 3.22 Fragility Curve in PGA ( Skew 200 600/ Soil A) 86 Fig. 3.23 Fragility Curve in PGA ( Skew 200 600/ Soil B) 87 Fig. 3.24 Fragility Curve in PGA ( Skew 200 600/ Soil C) 87 Fig. 3.25 Fragility Curve in PGA ( Skew > 600/ Soil A) 88 Fig. 3.26 Fragility Curve in PGA ( Skew > 600/ Soil C) 88 Fig. 3.27 Fragility Curve in PGA ( Single Span / Soil A) 89 Fig. 3.28 Fragility Curve in PGA ( Single Span / Soil B) 89 Fig. 3.29 Fragility Curve in PGA ( Single Span / Soil C) 90 Fig. 3.30 Fragility Curve in PGA ( Multiple Span / Soil A) 90 Fig. 3.31 Fragility Curve in PGA ( Multiple Span / Soil B) 91 Fig. 3.32 Fragility Curve in PGA ( Multiple Span / Soil C) 91 Fig. 3.33 Fragility Curve in PGA ( Single Span / Skew 00 200 / Soil A) 92 Fig. 3.34 Fragility Curve in PGA ( Single Span / Skew 00 200 / Soil B) 92 Fig. 3.35 Fragility Curve in PGA ( Single Span / Skew 00 200 / Soil C) 93 Fig. 3.36 Fragility Curve in PGA ( Single Span / Skew 200 600 / Soil C) 93 Fig. 3.37 Fragility Curve in PGA ( Single Span / Skew > 600 / Soil C) 94 Fig. 3.38 Fragility Curve in PGA ( Multiple Span / Skew 00 200 / Soil A) 94 Fig. 3.39 Fragility Curve in PGA ( Multiple Span / Skew 00 200 / Soil C) 95 Fig. 3.40 Fragility Curve in PGA ( Multiple Span / Skew 200 600 / Soil A) 95 Fig. 3.41 Fragility Curve in PGA ( Multiple Span / Skew 200 600 / Soil B) 96 Fig. 3.42 Fragility Curve in PGA ( Multiple Span / Skew 200 600 / Soil C) 96 Fig. 3.43 Fragility Curve in PGA ( Multiple Span / Skew > 600 / Soil A) 97 Fig. 3.44 Fragility Curve in PGA ( Multiple Span / Skew > 600 / Soil C) 97 Fig. 3.45 Enhanced Fragility Curve ( Minor) 99 Fig. 3.46 Enhanced Fragility Curve ( Moderate) 99 Fig. 3.47 Enhanced Fragility Curve ( Major) 100 Fig. 3.48 Enhanced Fragility Curve ( Collapse) 100 Fig. 5.1 Flow Chart for System Performance Evaluation 107 Fig. 5.2 Highway Network: Link, Node and Bridge Component 109 viii Fig. 5.3 Network Model: Los Angeles and Orange County 109 Fig. 5.4 Detour after Northridge Earthquake ( January 20th 1994) 113 Fig. 5.5 1996 Southern California Origin Destination Data 115 Fig. 5.6 OD Data Condensation: Thiessen Polygon 117 Fig. 5.7 Integrated Trip Reduction And Network Models 120 Fig. 6.1 Bridge Damage in Elysian Park 7.1 ( Without Retrofit) 128 Fig. 6.2 Link Damage in Elysian Park 7.1 ( Without Retrofit) 129 Fig. 6.3 Bridge Damage in Elysian Park 7.1 ( 23% Retrofit) 129 Fig. 6.4 Link Damage in Elysian Park 7.1 ( 23% Retrofit) 130 Fig. 6.5 Bridge Damage in Elysian Park 7.1 ( 100% Retrofit) 130 Fig. 6.6 Link Damage in Elysian Park 7.1 ( 100% Retrofit) 131 Fig. 7.1 System Risk Curve in terms of Daily Drivers’ Delay Under different Link Residual Capacity Assumptions ( without retrofit) 145 Fig. 7.2 System Risk Curve in terms of Daily Opportunity Cost Under different Link Residual Capacity Assumptions ( without retrofit) 146 Fig. 7.3 System Risk Curve in terms of Daily Social Cost Under different Link Residual Capacity Assumptions ( without retrofit) 146 Fig. 7.4 Effect of Retrofit on System Risk Curve ( Assumption 1) 150 Fig. 7.5 Effect of Retrofit on System Risk Curve ( Assumption 2) 151 Fig. 7.6 Effect of Retrofit on System Risk Curve ( Assumption 3) 154 Fig. 7.7 Probability Distribution of Functions used to Model Repair Processes 156 Fig. 7.8 Restoration Curves for Highway Bridges ( after ATC 13, 1985) 158 Fig. 7.9 System Recovery Curves After Elysian Park M7.1 ( No Retrofit ) 159 Fig. 7.10 System Recovery Curves After Elysian Park M7.1 ( 23% Retrofit ) 160 Fig. 7.11 System Recovery Curves After Elysian Park M7.1 ( 100% Retrofit ) 160 Fig. 7.12 Retrofit Effect on System Recovery Curve( Elysian Park 7.1, Assumption 1) 162 Fig. 7.13 Retrofit Effect on System Recovery Curve( Elysian Park 7.1, Assumption 2) 162 Fig. 7.14 Retrofit Effect on System Recovery Curve( Elysian Park 7.1, Assumption 3) 163 ix Fig. 7.15 Fig. 7.15 Retrofit Effect on System Recovery Curve ( Elysian Park 7.1, HAZUS) 163 Fig. A. 1 Moment Curvature Analysis of Bridge 1( Longitudinal) 195 Fig. A. 2 Moment Curvature Analysis of Bridge 2( Longitudinal) 196 Fig. A. 3 Moment Curvature Analysis of Bridge 3( Longitudinal) 200 Fig. A. 4 Moment Curvature Analysis of Bridge 4( Longitudinal) 201 Fig. A. 5 Moment Curvature Analysis of Bridge 5( Longitudinal) 220 Fig. A. 6 Moment Curvature Analysis of Bridge 3( Transverse) 225 Fig. A. 7 Moment Curvature Analysis of Bridge 4( Transverse) 234 Fig. A. 8 Moment Curvature Analysis of Bridge 5( Transverse) 245 Fig. B. 1 Framework of Trip Reduction Estimation 258 Fig. B. 2 Integrated Analysis of Trip Reduction and Network Models 262 Fig. B. 3 Personal Trip Reductions caused by Regional Building Damage 263 Fig. B. 4 Comparison of EPEDAT Fragility and the Study Estimated Fragility for Selected Building Occupancy Types 272 Fig. B. 5 Trip Reduction Rate 279 Fig. B. 6 Estimation Procedure of Truck Trip Reduction 279 Fig. B. 7 Truck Trip Reduction Ratio 286 Fig. B. 8 Conceptual OD Recovery Model 292 Fig. C. 1 Main Interface 295 Fig. C. 2 Menu “ Map” 296 Fig. C. 3 Map of Study Region 297 Fig. C. 4 Map of Faults 297 Fig. C. 5 Map of Freeway Network 298 Fig. C. 6 Map of Bridges 298 Fig. C. 7 Menu “ Inventory” 299 Fig. C. 8 Inventory: Bridges 299 Fig. C. 9 Inventory: Network Links 300 Fig. C. 10 Menu “ Hazard” 300 Fig. C. 11 Predefined Events 301 Fig. C. 12 Importing PGA and MMI Shape Map 301 x Fig. C. 13 User Defined Event 302 Fig. C. 14 Current Event Information 302 Fig. C. 15 Analysis: Setting 303 Fig. C. 16 Bridge Fragility Setting 303 Fig. C. 17 Residual Link Performance Setting 304 Fig. C. 18 Economic Loss Estimation Parameter Setting 304 Fig. C. 19 Risk Analysis Option 305 Fig. C. 20 Menu “ Results” 306 Fig. C. 21 PGA distribution 306 Fig. C. 22 Display Bridge Damage States 307 Fig. C. 23 Display Link Damage States 308 Fig. C. 24 System Performance 309 Fig. C. 25 Economic Loss 310 xi LIST OF TABLES Table 2.1 Description of Five ( 5) Sample Bridges 12 Table 2.2 Description of Los Angeles Ground Motions 20 Table 2.3 Description of Damaged States 28 Table 2.4 Peak Ductility Demand of First Left Column of Sample Bridges 28 Table 2.5 Number of Damaged Bridges: Pounding Effect in Bridge 2 34 Table 2.6 Number of Damaged Bridges: Pounding Effect in Bridge 3 35 Table 2.7 Number of Damaged Bridges: Pounding Effect in Bridge 4 36 Table 2.8 Number of Damaged Bridges: Pounding Effect in Bridge 5 37 Table 2.9 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 2 39 Table 2.10 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 3 40 Table 2.11 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 4 41 Table 2.12 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 5 42 Table 2.13 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 44 Table 2.14 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 45 Table 2.15 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 46 Table 2.16 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 47 Table 2.17 Number of Damaged Bridges: Retrofit Effect in Bridge 1 ( longitudinal) 50 Table 2.18 Number of Damaged Bridges: Retrofit Effect in Bridge 2 ( longitudinal) 51 Table 2.19 Number of Damaged Bridges: Retrofit Effect in Bridge 3 ( longitudinal) 52 Table 2.20 Number of Damaged Bridges: Retrofit Effect in Bridge 4 ( longitudinal) 53 xii Table 2.21 Number of Damaged Bridges: Retrofit Effect in Bridge 5 ( longitudinal) 54 Table 2.22 Simulated Ductility Capacities of Sample Bridges 59 Table 2.23 Fragility Curves based on Adjusted Damage States Definitions 61 Table 2.24 Enhancement Ratios Comparison 61 Table 2.25 Number of Damaged Bridges: Retrofit Effect in Bridge 1( Transverse) 62 Table 2.26 Number of Damaged Bridges: Retrofit Effect in Bridge 2 ( Transverse) 63 Table 2.27 Number of Damaged Bridges: Retrofit Effect in Bridge 3 ( Transverse) 64 Table 2.28 Number of Damaged Bridges: Retrofit Effect in Bridge 4 ( Transverse) 65 Table 2.29 Number of Damaged Bridges: Retrofit Effect in Bridge 5 ( Transverse) 66 Table 3.1 Summary of Bridge Damage Status in 1994 Northridge Earthquake 67 Table 3.2 Bridge Seismic Damage Table in the 1994 Northridge Earthquake 69 Table 3.3 Empirical Fragility Curve: First Level ( Composite) 71 Table 3.4 Empirical Fragility Curve: Second Level 71 Table 3.5 Empirical Fragility Curve: Third Level 72 Table 3.6 Empirical Fragility Curve: Fourth Level 74 Table 4.1 Probabilistic Scenario Earthquake Set 105 Table 5.1 Assumption for Residual Link Capacity 113 Table 5.2 Trip ratios of each directional trip for 4 time span 115 Table 5.3 3 hours average of am peak and midday applied peak ratio and car occupancy rates 115 Table 6.1 Damaged Bridges in Elysian Park M7.1( Total 3133 bridges) 131 Table 6.2 Comparison: Number of Damaged Bridges 132 Table 6.3 Reduction in Number of Damaged Bridges 133 Table 6.4 Damage Ratios for Highway Bridge Components ( from HAZUS 99) 136 Table 6.5 Expected Bridge Repair Cost ( in $ thousand) 136 xiii Table 6.6 Annual Probability of Sustaining Damage for a Bridge before and after Retrofit 140 Table 6.7 Annual Bridge Repair Cost in the System 141 Table 7.1 Daily Travel Delay and Opportunity Cost ( without retrofit) 144 Table 7.2 Daily Travel Delay and Opportunity Cost ( 23% retrofit) 147 Table 7.3 Daily Travel Delay and Opportunity Cost ( 100% Retrofit) 148 Table 7.4 Restoration Function for Highway Bridges ( after ATC 13, 1985) 158 Table 7.5 Total Travel Delay and Opportunity Cost ( no retrofit) 164 Table 7.6 Total Travel Delay and Opportunity Cost ( 23% retrofit) 165 Table 7.7 Total Travel Delay and Opportunity Cost ( 100% retrofit) 166 Table 7.8 Economic Loss due to System Dysfunction ( No Retrofit ) 168 Table 7.9 Economic Loss due to System Dysfunction ( 23% Retrofit) 169 Table 7.10 Economic Loss due to System Dysfunction ( 100% Retrofit) 170 Table 8.1 Annual Avoided Social Loss ( Assumption 1) 174 Table 8.2 Annual Avoided Social Loss ( Assumption 2) 175 Table 8.3 Annual Avoided Social Loss ( Assumption 3) 176 Table 8.4 Annual Avoided Social Loss ( HAZUS Model) 177 Table 8.5 Discount Factor 179 Table 8.6 Cost Effectiveness Analysis ( Discount Rate 3%) 182 Table 8.7 Cost Effectiveness Analysis ( Discount Rate 5%) 184 Table 8.8 Cost Effectiveness Analysis ( Discount Rate 7%) 186 Table 8.9 Cost Benefit Analysis Summary 188 Table B. 1 Building fragility by structure types 265 Table B. 2 Southern California Building Stock by Structure Types 267 Table B. 3 Southern California Building Stock by Occupancy Types 268 Table B. 4 Summary of Southern California Buildings by structure and occupancy Types 269 Table B. 5 Vulnerability of Building Occupancy 271 Table B. 6 Activity Population by Building Occupancy Types 275 Table B. 7 Trip Types and Building Occupancy types 276 Table B. 8 Person Trip Reduction Rates 277 xiv Table B. 9 Truck Generation Rate 281 Table B. 10 Building Usage and Truck trip Generating Industries 282 Table B. 11 Estimated PCE by Industries 283 Table B. 12 Calibration of Decay Function Parameter 290 xv Chapter 1 Introduction Past experience showed too often that earthquake damage to highway components ( e. g., bridges, roadways, tunnels, retaining walls, etc.) can severely disrupt traffic flows and thus negatively impacting on the economy of the region as well as post earthquake emergency response and recovery. Furthermore, the extent of these impacts will depend not only on the nature and magnitude of the seismic damage sustained by the individual components, but also on the mode of functional impairment of the highway system as a network resulting from physical damage of its components. In order to estimate the effects of the earthquake on the performance of the transportation network, an analytical framework must be developed to integrate bridge and other structural performance model and transportation network model in the context of seismic risk assessment. Highway transportation networks are complex with many engineered components placed in equally complex hazardous environments, natural or manmade. Among the engineered components, bridges represent potentially the most vulnerable components under earthquake conditions as demonstrated as vividly in the San Fernando, Loma Prieta, Northridge and Kobe Earthquakes. Recognizing this, the Caltrans’ seismic retrofit program has been underway since the 1971 San Fernando Earthquake, and accelerated since the 1989 Loma Prieta event. At this time ( June, 2005), 23% of Caltrans freeway bridges in Los Angeles and Orange Counties have been retrofitted by the steel and composite jacketing of the columns as well as rebuilding and upgrading of the restraining devises at expansion joints for which the seismic retrofit was deemed necessary. It is therefore most timely at this time to assess not only the engineering significance of such retrofit but also the socio economic benefit arising therefrom. 1 The purpose of this research therefore is to assess the socio economic impact of seismic retrofit implemented on the Caltrans’ bridges on the freeway network in the Los Angeles and Orange Counties. The research concentrates on the evaluation of the socio economic benefit resulting from the retrofit performed on the Caltrans’ bridges primarily by means of column jacketing with steel. The three major tasks of this research are ( 1) development of fragility curves of the bridge, ( 2) assessment of the seismic performance of the freeway and ( 3) related socio economic analysis. In order to perform a seismic risk analysis of a highway network, it is imperative to identify seismic vulnerability of bridges associated with various states of damage. As a widely practiced approach, the vulnerability information is expressed in the form of fragility curve to account for a multitude of uncertain sources involved ( Shinozuka et al, 2003a). In Chapter 2, a manageable number of representative bridges are selected for the fragility analysis. Finite Element Model for each of the representative bridges, without or with retrofit ( column jacketing with steel) is developed and used to perform nonlinear dynamic time history analysis. Based on the result of this dynamic analysis, a family of fragility curves associated with various states of damage are estimated with a statistical procedure. The seismic performance improvement of the retrofitted bridges is evident in that the median value of fragility curve of these bridges is significantly increased. The median value is one of the two fragility parameters with the other being the log standard deviation. The enhancement ratios for median values of analytical fragility curves are then applied to empirical fragility curves based on bridge damage data obtained from the 1994 Northridge Earthquake to consider the effect of the bridge retrofit ( Chapter 3). The 2 enhancement ratio is defined as ( median value for retrofitted bridges) / ( median value for bridges not retrofitted). After the introduction of major features of seismic risk analysis for spatially distributed system, both deterministic and probabilistic seismic modeling methods are described in Chapter 4. Particularly, a set of 47 probabilistic scenario earthquakes is provided for the probabilistic seismic risk analysis for the highway transportation network in Los Angles and Orange Counties. In chapter 5, a methodology is developed to evaluate the seismic performance of highway transportation network in terms of related social cost. Based on fragility curves developed above and the site ground motion originating from scenarios, the damage states of bridges are simulated, which determine the reduced link traffic capacity. A comprehensive traffic assignment analysis, which features realistic consideration of trip reduction and recovery after a damaging earthquake, is then performed in the degraded highway network with variable OD input. The daily social cost, including the traffic delay time and opportunity cost, is used to measure the post event performance of the damaged highway network. The enhancement of the network performance is then studied by comparing the social cost in using fragility curves of bridges with and without retrofit in the network performance simulation under the same scenario earthquake. Chapter 6 describes the method for estimation of bridge restoration ( repair/ replacement) cost. For the given scenarios, the expected bridge repair cost is calculated for each of the 3 cases of bridge retrofit status: No retrofit, 23% retrofit ( current status) and 100% retrofit, assuming that no freeway bridges ( in Los Angeles and Orange County), 23% of them ( actual % at the time of writing this report) and 100% of them have been retrofitted. To estimate the total social cost resulting from an earthquake, the network 3 restoration curves are developed in Chapter 7. Using a probabilistic time dependent bridge repair model, the new set of bridge damage states are determined based on Monte Carlo simulation at any given time point after an earthquake. The traffic assignment analysis is performed again to obtain the corresponding daily social cost for the partially restored network. The integration of the daily social cost over the restoration period gives the total social cost in time for a particular earthquake event. The economic loss due to the time cost is estimated by considering the local unit time value. Whether a retrofit strategy is cost effective is evaluated by a cost benefit analysis introduced in Chapter 8. The restoration cost for the damaged bridges, the retrofit cost and economic loss due to social cost are estimated. The difference between the economic loss without and with retrofit represents the cost avoided. The economic benefit is then measured by the cost avoided minus the cost of retrofit. The economic analysis is performed for each of the probabilistic scenario earthquakes and expected annual benefit of the retrofit measure obtained by considering the annual probabilities of these scenarios. The results show that the bridge restoration cost avoided alone cannot compensate for the retrofit cost. However, when the social cost avoided is considered, the cost effectiveness ratios in both retrofit cases are much larger than 1, indicating very high benefit for the public obtained from the Caltrans bridge retrofit measures. Chapter 9 summarizes the conclusions obtained from this research. At the end of the report, three documents are appended. Appendix A provides the cross sections and moment rotation relationship of 5 sample bridges’ columns before and after retrofit. Appendix B describes the background of the traffic assignment model integrating the OD change due to earthquake damage. In Appendix C, A GIS based 4 Program for Highway Seismic Risk Analysis ( HighwaySRA) developed at UCI is introduced and its usage and functionality are demonstrated in a manual which is part of the Appendix C. 5 6 Chapter 2 Development of Analytical Fragility Curve for Bridges 2.1 Introduction Several recent destructive earthquakes, particularly the 1989 Loma Prieta and 1994 Northridge earthquakes in California, and the 1995 Hanshin Awaji ( Kobe) earthquake in Japan, caused significant damage to a large number of highway structures that were seismically deficient ( Basoz and Kiremidjian 1998, Buckle 1994). The investigation of these negative consequences gave rise to serious discussions about seismic design philosophy and extensive research activity on the retrofit of existing bridges as well as the seismic design of new bridges. In this respect, this study presents an approach for the seismic assessment of older bridges retrofitted by steel jacketing of the columns having substandard seismic characteristics and by restrainers at expansion joints to prevent bridge decks from unseating. The main objective of the study is focused to evaluate the effects of column retrofit with steel jacketing on the ductility capacity of bridge columns. The Caltrans’ seismic retrofit program was underway prior to the 1994 Northridge earthquake and was accelerated after the 1989 Loma Prieta event. This resulted in implementation of steel and composite jacketing of the columns, and of installing and upgrading of the restraining devices at expansion joints for many bridges for which the seismic retrofit was deemed necessary. Therefore, it is most timely to assess the engineering significance and benefit from such retrofit. 7 This study first develops moment curvature curves of bridge columns and then performs nonlinear dynamic time history analyses producing fragility curves for five ( 5) sample bridges before and after retrofitting their columns with steel jacketing. The effect of retrofit is demonstrated by means of the ratio of the median value of the fragility curve for retrofitted column to that of the column before retrofit. This ratio is referred to as fragility enhancement. The fragility enhancement is found to be more significant for more severe state of damage. It is then assumed that the same fragility enhancement is applicable to the empirical fragility curves developed from the Northridge damage data ( Chapter 3). The fragility curves for four ( 4) of sample bridges are also developed before and after retrofitting its expansion joints with restrainers. This physical improvement of the seismic vulnerability due to steel jacketing becomes evident in terms of enhanced fragility curves shifting those associated with the bridges before retrofit to the right when plotted as functions of PGA ( Peak Ground Acceleration). Thus, this study makes it possible to evaluate the improvement of the highway network performance resulting from such retrofit by providing basic information for fragility enhancement. 2.2 Column Retrofit with Steel Jacketing 2.2.1 Background Concrete columns of earlier design often lack flexural strength, flexural ductility and shear strength. One of the main causes for these structural inadequacies is lap splices in critical regions and/ or premature termination of longitudinal reinforcement. A number of column retrofit techniques, such as steel jacketing, wire pre stressing and composite material jacketing, have been developed and tested. Although advanced composite 8 materials and other methods have been recently studied, the steel jacketing has been widely applied to bridge retrofit as the most common retrofit technique. Chai et al. ( 1991) observed that confinement of the concrete columns can be improved if transverse reinforcement layers are placed relatively close together along the longitudinal axis by restraining the lateral expansion of the concrete. It makes it possible for the compression zone to sustain higher compression stresses and much higher compression strains before failure occurs. Obviously, however, this is for original design and construction, but not applicable to existing bridges, to enhance the performance of columns by adding transverse reinforcement layers. In this respect, this study focuses on the steel jacketing technique for retrofitting existing bridge columns to improve their seismic performance. 2.2.2 Steel Jacketing An experiment was performed by Chai et al. ( 1991) to investigate the retrofit of circular columns with steel jacketing. In this experiment, for circular columns, two half shells of steel plate rolled to a radius slightly larger than that of the column are positioned over the area to be retrofitted and are site welded up the vertical seams to provide a continuous tube with a small annular gap around the column. This gap is grouted with pure cement. It is typical that the jacket is cut to provide a space of about 50 mm ( 2 in) between the jacket and any supporting member. It is for the jacket to avoid the possibility to act as compressing reinforcement by bearing against the supporting member at large drift angles. It is noted that the jacket is effective only in passive confinement and the level of confinement depends on the hoop strength and stiffness of the steel jacket. 9 10 The thickness of steel jacket is calculated from the following equation ( Priestley et al., 1996). ' 0.18( 0.004) cm cc j yj sm Df t f ε ε − = ( 2.1) where cm ε is the strain at maximum stress in concrete, sm ε the strain at maximum stress in steel jacket, D the diameter of circular column, ' cc f the compressive strength of confined concrete and yj f the yield stress of steel jacket. 2.2.3 Compression Stress Strain Relationships for Confined Concrete The effect of confinement is to increase the compression strength and ultimate strain of concrete as illustrated in Fig 2.1 ( after Priestley et al., 1996). Many different stress strain relationships have been developed for confined concrete. Most of these are applicable under certain specific conditions. A recent model applicable to all cross sectional shapes and at all levels of confinement is used for the analysis defined by the key equations that also appears in Priestley et al. ( 1996). Fig 2.1 Stress Strain Model for Concrete in Compression ε t εco 2εco εsp εcc εcu f'cc f'c First hoop Confined concrete fracture Assumed for cover concrete Unconfined concrete f't Ec Esec Compressive strain, ε c Compressive stress, fc 2.3 Bridge Model Not all but a manageable number of bridges, representing typical bridges in California and covering many types of bridge structures, have been selected for the fragility analysis. 2.3.1 Bridge Description Five ( 5) sample bridges used for example analysis are listed in Table 2.1 and shown in Fig 2.2. Bridge 1 has the overall length of 34 m ( 112 ft) with three spans. The superstructure consists of a longitudinally reinforced concrete deck slab 10 m ( 32.8 ft ) wide and it is supported by two sets of columns ( and by an abutment at each end). Each set has three columns of circular cross section with 0.8 m ( 31.5 in) diameter. Bridge 2 has an overall length of 242 m ( 794 ft) with five spans and an expansion joint in the center span. This bridge is supported by four columns of equal height of 21 m ( 69 ft) between the abutments at the ends. Each column has a circular cross section with 2.4 m diameter. The deck has a 3 cell concrete box type girder section 13 m ( 42.6 ft) wide and 2 m ( 6.6 ft) deep. Bridge 3 has an overall length of 226 m ( 741 ft) with five spans, consisting of three frames separated by two expansion joints. The columns have varying lengths with longer ones in the center span and shorter ones near the abutments. The superstructure consists of a RC box girder to the left of the left expansion joint and to the right of the right expansion joint, and a prestressed box girder in the central span. The deck has a 6 cell box girder section 20 m ( 65.6 ft) wide and 2.6 m ( 8.5 ft) deep, and the column section is octagonal. 11 Bridge 4 has an overall length of 483 m ( 1584 ft) with ten spans and four expansion joints. This bridge is supported by nine columns having different heights. Each column has a rectangular cross section which is 1.2 m ( 3.9 ft) by 3.7 m ( 12.1 ft) in dimension. The deck has a 5 cell concrete box type girder section 17 m ( 56 ft) wide and 2 m ( 6.6 ft) deep. Bridge 5 has an overall length of 500 m ( 1640 ft) with twelve spans and an expansion joint. This bridge is supported by eleven columns of equal height of 12.8 m ( 42.0 ft) between the abutments at the ends. Each column section is oblong in shape. The deck has a 4 cell concrete box type girder section 15 m ( 49.2 ft) wide and 2 m ( 6.6 ft) deep. Table 2.1 Description of Five ( 5) Sample Bridges Bridges Overall Length meter ( foot) Number of Spans Number of Hinges Column Height meter ( foot) 1 34( 112) 3 0 4.7 ( 15.4) 2 242( 794) 5 1 21.0 ( 68.9) 3 226( 741) 5 2 9.5  24.7( 31.2 81.0) 4 483 ( 1584) 10 4 9.5  34.4 ( 31.2 112.83) 5 500 ( 1640) 12 1 12.8 ( 42.0) 13.5 m10.5 m 10.0 m4.7 m 4.7 m34.0 m ( a) Bridge 1 12 53.38 m 41.18 m 41.18 m 53.38 m53.38 m21.0 m 242.0 mExpansionJoint10.7 m ( b) Bridge 2 52.0 m 39.0 m 27.5 m 63.5 m44.0 m 9.5 m 226.0 m21.3 m24.7 m17.5 m 10.4 m10.4 m ( c) Bridge 3 483 m 30.2 m 50 m 46 m 63 m 43 m 55 m 54 m 33 m 33 m 54 m 52 m 31.9 m 16.8 m9.5 m 13.7 m 17.7 m 17.2 m 28.4 m 34.4 m ( d) Bridge 4 500.0 m12.8 m10@ 43.58m= 435.8m32.1 m32.1 m ( e) Bridge 5 Fig 2.2 Elevation of Sample Bridges 2.3.2 Bridge Modeling The bridges are modeled to exhibit the nonlinear behavior of the columns. A column is modeled as an elastic zone with a pair of plastic zones at each end of the column. Each plastic zone is then modeled to consist of a nonlinear rotational spring and 13 a rigid element depicted in Fig 2.3. The plastic hinge formed in the bridge column is assumed to have bilinear hysteretic characteristics. Furthermore, pounding effect at the expansion joint of the bridges is reflected in the structural response analysis, so that the fragility information of the structure becomes more realistic. In this respect, the expansion joint is constrained in the relative vertical movement, while freely allowing horizontal opening movement and rotation. The closure at the joint, however, is restricted by a gap element when the relative motion of adjacent decks exhausts the initial gap width of 2.54 cm ( 1 in) leading to deck pounding. A hoop element sustaining tension only is used for the bridge retrofitted by restrainers at expansion joints and the opening is restricted by the element when the relative motion exhausts the initial slack of 1.27 cm ( 0.5 in). Springs are also attached to the bases of the columns to account for soil effects, while two abutments are modeled as roller supports. To reflect the cracked state of a concrete bridge column for the seismic response analysis, an effective moment of inertia is employed, making the period of the bridge longer. 14 15 kHook Linear Column Potential Plastic Hinge Relative Displacement Force ks αks θy Rotation Μ y M Μ y = 36539 kN m α= 0.01675 θy = 3.7x10 3 rad Potential Plastic Hinge 1.27 cm Expansion Joint Deck 2.54 cm kGap Fig 2.3 Nonlinearities in Bridge Model 2.4 Development of Moment Curvature Relationship The column ductility program developed by Kushiyama ( 2002) ( the code is attached in Appendix A) is used to model the moment curvature relationship of plastic hinges for columns. The critical parameter used to describe the nonlinear structural response in this study is the ductility demand. The ductility demand is defined as / y θ θ , where θ is the rotation of a bridge column in its plastic hinge and y θ is the corresponding rotation at the yield point. Nonlinear response characteristics associated with the bridges are based on moment curvature curve analysis taking axial loads as well as confinement effects into account. The moment curvature relationship used in this study for the nonlinear spring is bilinear without any stiffness degradation. Its parameters are established according to the equations in Priestley et al. ( 1996). 2.4.1 Moment Curvature Curves for Longitudinal Direction of Bridges In Fig 2.4 and 2.5, Section of the column, stress strain relationship, distribution of axial force, P M interaction diagram, moment curvature curve and moment rotation curve for column 2 of Bridge 1 before and after retrofit are plotted. The cross sections and the moment rotation curves of all the other columns of Bridge 1 5 are provided in Appendix A. One of results, for example, shows that the moment curvature curve after retrofit gives a much better performance than that before retrofit by 4 times based on curvature at the ultimate compressive strain and by 1.6 times at the ultimate moment. 16  15  10  5 0 5 10 15  15  10  5 0 5 10 15 Dimension( in) D im e n s io n ( in ) 0 0.005 0.01 0.015 0.02 0 1000 2000 3000 4000 5000 6000 Confined Concrete Compressive Strain C o m p re s s iv e S tre s s ( p s i ) Cover Concrete ( a) Section of Column ( b) Stress Strain Relationship 0 0.005 0.01 0.015 0.02 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 6 Strain A x ia l fo rc e ( lb ) Confined Concrete Cover Concrete Steel Bar Total 0 0.5 1 1.5 2 2.5 x 10 4 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Mn ( kips in) P n ( k i p s ) ( c) Distribution of Axial Force ( d) P M Interaction Diagram 0 0.5 1 1.5 2 x 10  3 0 200 400 600 800 1000 1200 1400 1600 Curvature ( 1/ in) M o m e n t ( k i p s  ft ) 0 0.01 0.02 0.03 0.04 0.05 0.06 0 200 400 600 800 1000 1200 1400 1600 Rotation ( radian) M o m e n t ( k i p s  ft ) M y = 1442kips ft M u = 1516kips ft θ y = 0.006871rad θ u = 0.05732rad K eff = 2.099e+ 005kips ft α = 0.006918 ( e) Moment Curvature Curve ( f) Moment Rotation Curve Fig 2.4 Column 2 of Bridge 1 Before Retrofit Dimension ( in) Compression Stress ( psi) Axial Force ( pf) Pn ( kips) Moment ( kip ft) Moment ( kip ft) 18  15  10  5 0 5 10 15  15  10  5 0 5 10 15 Dimension( in) D im e n s io n ( in ) 0 0.01 0.02 0.03 0.04 0 1000 2000 3000 4000 5000 6000 7000 Confined Concrete Compressive Strain C o m p re s s iv e S tre s s ( p s i ) Cover Concrete ( a) Section of Column ( b) Stress Strain Relationship 0 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 x 10 6 Strain A x ia l fo rc e ( lb ) Confined Concrete Cover Concrete Steel Bar Total 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 0 1000 2000 3000 4000 5000 6000 7000 Mn ( kips in) P n ( k i p s ) ( c) Distribution of Axial Force ( d) P M Interaction Diagram 0 1 2 3 4 x 10  3 0 500 1000 1500 2000 2500 Curvature ( 1/ in) M o m e n t ( k i p s  ft ) 0 0.05 0.1 0.15 0.2 0 500 1000 1500 2000 2500 Rotation ( radian) M o m e n t ( k i p s  ft ) M y = 1749kips ft M u = 2249kips ft θ y = 0.006428rad θ u = 0.1202rad K eff = 2.722e+ 005kips ft α = 0.01614 ( e) Moment Curvature Curve ( f) Moment Rotation Curve Fig 2.5 Column 2 of Bridge 1 After Retrofit Dimension ( in) Compression Stress ( psi) Axial Force ( pf) Pn ( kips) Moment ( kip ft) Moment ( kip ft) 2.5 Bridge Response Analysis The SAP2000/ Nonlinear finite element computer code ( Computer and Structures, 2002) is utilized for the extensive two dimensional response analysis of the bridge under sixty ( 60) Los Angeles earthquake time histories ( http:// nisee. berkeley. edu/ data/ strong_ motion/ sacsteel/ ground_ motions. html) listed in Table 2.2, to develop the fragility curves before and after column retrofit with steel jackets. 2.5.1 Input Ground Motions These acceleration time histories were derived from historical records with some linear adjustments and consist of three ( 3) groups ( each consisting of 20 time histories) having probabilities of exceedance of 10% in 50 years, 2% in 50 years and 50% in 50 years, respectively. A typical acceleration time history in each group is plotted in the same scale to compare the magnitude of the acceleration in Fig 2.6. 19 Table 2.2 Description of Los Angeles Ground Motions 10% Exceedence in 50 yr 2% Exceedence in 50 yr 50% Exceedence in 50 yr SAC Name DT ( sec) Duration ( sec) PGA ( cm/ sec2) SAC Name DT ( sec) Duration ( sec) PGA ( cm/ sec2) SAC Name DT ( sec) Duration ( sec) PGA ( cm/ sec2) LA01 0.02 39.38 452.03 LA21 0.02 59.98 1258.00 LA41 0.01 39.38 578.34 LA02 0.02 39.38 662.88 LA22 0.02 59.98 902.75 LA42 0.01 39.38 326.81 LA03 0.01 39.38 386.04 LA23 0.01 24.99 409.95 LA43 0.01 39.08 140.67 LA04 0.01 39.38 478.65 LA24 0.01 24.99 463.76 LA44 0.01 39.08 109.45 LA05 0.01 39.38 295.69 LA25 0.005 14.945 851.62 LA45 0.02 78.60 141.49 LA06 0.01 39.38 230.08 LA26 0.005 14.945 925.29 LA46 0.02 78.60 156.02 LA07 0.02 79.98 412.98 LA27 0.02 59.98 908.70 LA47 0.02 79.98 331.22 LA08 0.02 79.98 417.49 LA28 0.02 59.98 1304.10 LA48 0.02 79.98 301.74 LA09 0.02 79.98 509.70 LA29 0.02 49.98 793.45 LA49 0.02 59.98 312.41 LA10 0.02 79.98 353.35 LA30 0.02 49.98 972.58 LA50 0.02 59.98 535.88 LA11 0.02 39.38 652.49 LA31 0.01 29.99 1271.20 LA51 0.02 43.92 765.65 LA12 0.02 39.38 950.93 LA32 0.01 29.99 1163.50 LA52 0.02 43.92 619.36 LA13 0.02 59.98 664.93 LA33 0.01 29.99 767.26 LA53 0.02 26.14 680.01 LA14 0.02 59.98 644.49 LA34 0.01 29.99 667.59 LA54 0.02 26.14 775.05 LA15 0.005 14.945 523.30 LA35 0.01 29.99 973.16 LA55 0.02 59.98 507.58 LA16 0.005 14.945 568.58 LA36 0.01 29.99 1079.30 LA56 0.02 59.98 371.66 LA17 0.02 59.98 558.43 LA37 0.02 59.98 697.84 LA57 0.02 79.46 248.14 LA18 0.02 59.98 801.44 LA38 0.02 59.98 761.31 LA58 0.02 79.46 226.54 LA19 0.02 59.98 999.43 LA39 0.02 59.98 490.58 LA59 0.02 39.98 753.70 LA20 0.02 59.98 967.61 LA40 0.02 59.98 613.28 LA60 0.02 39.98 469.07 20 02040Time ( s) 60  1500 1000 500050010001500Acceleration ( cm/ s2) PGA= 662.88 cm/ s2 ( a) 10% Probability of Exceedence in 50 Years 02040Time( s) 60  1500 1000 500050010001500Acceleration ( cm/ s2) PGA= 1,304.10 cm/ s2 ( b) 2% Probability of Exceedence in 50 Years 01020304Time( s) 0  1500 1000 500050010001500Acceleration ( cm/ s2) PGA= 109.45 cm/ s2 ( c) 50% Probability of Exceedence in 50 Years Fig 2.6 Acceleration Time Histories Generated for Los Angeles 21 2.5.2 Responses of Structures Typical responses at column bottom end of Bridge 1 are plotted in Fig 2.7 with the acceleration time history in Fig 2.6a as input. It is reasonable to expect that the rotation after retrofit is generally smaller than before, while the accelerations do not necessarily behave that way and can be quite different each other. It is noted that some higher fluctuations in acceleration response appear after retrofit because the column becomes stiffer than before. 0102030405Time ( s) 0  0.400.4Acceleration at column end ( m/ s2) before retrofit ( a) Acceleration before retrofit 010203040Time ( s) 50  0.400.4Acceleration at column end ( m/ s2) after retrofit ( b) Acceleration after retrofit 22 0102030405Time ( s) 0  0.008 0.00400.0040.008Rotation at column end ( rad) before retrofit ( c) Rotation before retrofit 010203040Time ( s) 50  0.008 0.00400.0040.008Rotation at column end ( rad) after retrofit ( d) Rotation after retrofit Fig 2.7 Responses at Column End of Bridge 1 23 Typical responses at expansion joints of Bridge 1 are also plotted in Fig 2.8 to show the differences of the structural behaviors for the cases without and with considering gap and hook elements. 010203040Time ( sec)  40 2002040Displacement at Expansion Joints ( cm) Left JointRight Joint 010203040Time ( sec)  10010203040Relative Displacement ( cm) Right Left ( a) Without Gap and Hook Elements 24 010203040Time ( sec)  40 2002040Displacement at Expansion Joints ( cm) Left JointRight Joint 010203040Time ( sec)  3 2 1012Displacement at Expansion Joints ( cm) Right Left ( b) with Gap and Hook Elements Fig 2.8 Displacement at Expansion Joints of Bridge 1 25 26 2.6. Fragility Analysis of Bridges 2.6.1 Fragility Parameter Estimation It is assumed that the fragility curves can be expressed in the form of two parameter lognormal distribution functions, and the estimation of the two parameters ( median and log standard deviation) is performed with the aid of the maximum likelihood method. A common log standard deviation, which forces the fragility curves not to intersect, can also be estimated. The following likelihood formulation described by Shinozuka et al. ( 2000) is introduced for the purpose of this method. Although this method can be used for any number of damage states, it is assumed here for the ease of demonstration of analytical procedure that there are five states of damage including the state of ( almost) no damage. A family of four ( 4) fragility curves exists in this case where events E1, E2, E3, E4 and E5, respectively, indicate the state of ( almost) no, ( at least) slight, ( at least) moderate, ( at least) extensive damage and complete collapse. Pik = P( ai, Ek) in turn indicates the probability that a bridge selected randomly from the sample will be in the damage state Ek when subjected to ground motion intensity expressed by PGA = ai. All fragility curves are then represented ln( /) ( ; ,) i j j j jj j a c F a c ς ς = Φ ( 2.2) where Φ(⋅) is the standard normal distribution function, cj and j ς are the median and log standard deviation of the fragility curves for the damage state of “( at least) slight”, “( at least) moderate”, “( at least) major” and “ complete” identified by j = 1, 2, 3 and 4. From this definition of fragility curves, and under the assumption that the log standard deviation is equal to ς common to all the fragility curves, one obtains; 27 Pi1= P( ai, E1)= 1F1( ai ; c1 , ς) − ( 2.3) 2 2 1122 ( , )(;,)(;,) i i ii P PaEFacFac = = ς ς − ( 2.4) 3 3 2233 ( , )(;,)(;,) i i ii P PaEFacFac = = ς ς − ( 2.5) 4 4 3344 ( , )(;,)(;,) i i ii P PaEFacFac = = ς ς − ( 2.6) 5 5 44 ( , )(;,) i i i P PaEFac = = ς ( 2.7) The likelihood function can then be introduced as 5 1 2 34 1 1 ( , ,,,)(;) ik n x k i k i k L c cccPaE ς = = = ΠΠ ( 2.8) Where 1 ik x = ( 2.9) if the damage state Ek occurs in the bridge subjected to a = ai, and 0 ik x = ( 2.10) otherwise. Then the maximum likelihood estimates c0j for cj and 0 ς for ς are obtained by solving the following equations, 1 2341234 ln(,,,,) ln(,,,,) 0 j L c cccLcccc c ς ς ς ∂ ∂ = = ∂ ∂ ( 1,2,3,4) j = ( 2.11) by implementing a straightforward optimization algorithm. 2.6.2 Definition of Damage States A set of five ( 5) different damage states recommended by Dutta and Mander ( 1999) are introduced in Table 2.3 which displays the description of these five damage states and the corresponding drift limits for a typical column. For each limit state, the drift limit can be transformed to peak ductility demand of the columns for the purpose of this study. Table 2.4 lists the values of these ductility demands for five ( 5) sample bridges. Table 2.3 Description of Damaged States Damage state Description Drift limits Almost no First yield 0.005 Slight Cracking, spalling 0.007 Moderate Loss of anchorage 0.015 Extensive Incipient column collapse 0.025 Complete Column collapse 0.050 Table 2.4 Peak Ductility Demand of First Left Column of Sample Bridges Bridge 1 Bridge 2 Bridge 3 Bridge 4 Bridge 5 Damage state before retrofit after retrofit before retrofit After retrofit before retrofit after retrofit before retrofit after retrofit before retrofit after retrofit Almost no 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 Slight 1.3 1.8 1.5 1.8 1.2 2.1 1.5 2.5 1.7 2.5 Moderate 2.6 4.9 3.5 5.2 2.2 6.4 3.5 8.2 4.3 8.3 Extensive 4.3 8.9 6.0 9.3 3.5 11.7 6.1 15.5 7.5 15.7 Complete 8.3 18.7 12.3 19.7 6.5 25.2 12.4 33.6 15.7 34.0 2.7 Pounding and Soil Effects on Fragility Curves The section presents the fragility curves taking the effect of pounding at expansion joints on concrete bridge response to earthquake ground motions into consideration. The primary objective of this section is to develop fragility curves of the sample bridges and quantify the effect of pounding at expansion joints of the bridges. The effect of pounding at expansion joints on the seismic response is systematically examined and the resulting fragility curves are compared with those for the cases without pounding. 2.7.1 Pounding at Expansion Joint 28 Pounding at expansion joints ( hinges) might have been another source of extensive damage during past earthquakes. In fact, the collapse of the 483 m ( 1610 ft) long bridge at the Interstate 5 and State Road 14 Interchange located approximately 12 km ( 7.5 mile) from the epicenter during the 1994 Northridge earthquake is an example suggesting that the effect of pounding at expansion joint might have caused the significant failure investigating damage states ( Buckle 1994). A preliminary investigation was performed by Shinozuka et al. ( 2002b) on impact phenomena as well as effects of seismically induced pounding at expansion joints of typical California bridges, through which it was found that pounding has significant effects on the acceleration and velocity responses, but little effects on the displacement responses. Although pounding effect is found to have negligible effect on the ductility demand, a need is felt to quantify the effect of pounding at the expansion joints by developing fragility curves of highway bridges, particularly for multi span long bridges with expansion joints. In order to investigate the effect of pounding of bridges, four ( 4) sample bridge models are considered for the nonlinear time history analysis. As described earlier in the Section 2.3, two ( 2) of them have mid overall lengths, but one hinge with same column height and two hinges with different column height. The other two have long overall lengths, but one hinge with same column height and four hinges with different column height. It is typical for a California highway bridge with more than four spans to have expansion joints located nearly at inflection points ( i. e., 1/ 4 to 1/ 5th of spans). The bridge superstructure consists of reinforced or prestressed concrete box girders. For 29 30 example, the material and cross sectional properties of Bridge 2 as follows: Young's modulus= 27.793 Gpa ( 4.03×106 ksi), mass density= 2.401 Mg/ m3 ( 62.428 kip/ ft3), crosssection area and moment of inertia are respectively 6.701 m2 ( 72.13 ft2) and 4.625 m4 ( 535.86 ft4) for box girders, while they are 4.670 m2 ( 50.27 ft2 ) and 0.620 m4 ( 71.83 ft4) for columns. Perhaps one of the most difficult to analyze nonlinear behaviors that occur in bridge systems idealized to include gap elements is the closing of a gap between different segments of the bridge. The usual gap element shown as Fig 2.9 has the following physical properties: 1) The element cannot develop a force until the opening d0 is closed; and 2) the element can only develop a compression force. Note that the numerical convergence of the response analysis particularly at the gap element can be very slow if a large elastic stiffness k is used. In order to minimize the difficulty associated with this problem, the stiffness k should not be over 1,000 times the stiffness of the elements adjacent to the gap according to the authors’ experience. This kind of dynamic contact problem involving two adjacent structural segments usually does not have a simple, unique solution. In fact, it is impractical to use continuum mechanics analysis in the vicinity of the contact area for local stress and strain evaluation and at the same time to pursue structural dynamic analysis to evaluate the bridge response as a system including, for example, ductility demand at the column ends. A viable alternative appears to be the deployment of the finite element analysis with gap elements having the stiffness value k selected from sensitivity analysis of gap element stiffness ( Shinozuka et al., 2003c). kd0ji Fig 2.9 Gap Element 2.7.2 Numerical Simulation for Pounding Numerical simulation were performed for the four ( 4) sample bridges under sixty ( 60) Los Angeles earthquakes for the cases without pounding and with pounding by considering gap element at expansion joints. The computer code SAP2000/ Nonlinear was utilized in order to calculate the state of damage of the structure under ground acceleration time histories. The structural responses with pounding were compared to those without pounding, in order to highlight how pounding affects the structural response behaviors. Numerical simulations were carried out under LA01 earthquake as shown Fig 2.10. Pounding force time history was also presented as shown Fig 2.11. Time histories of acceleration and displacement at the expansion joint, and rotation of the column end are plotted as shown Fig 2.12 and Fig 2.13 for the cases without and with pounding, respectively. From these results, it is observed that ( 1) the pounding takes place twenty three ( 23) times during the duration of the earthquake, ( 2) the acceleration is affected much more by pounding than displacement and rotation are; ( 3) the peak value of the rotation at column end can be reduced by pounding. It is indicated that the pounding are not usually capable of causing large deformation to bridge structures while it may cause significantly 31 high axial compressive stress locally leading to a possible local damage at the contact area at the expansion joint. 0102030405Time ( sec) 0  0.500.5Acceleration ( g) 0102030405Time ( sec) 0  16000 12000 8000 40000Force ( ton) Fig 2.10 Ground Motion Time History for LA01 Fig 2.11 Pounding Force at Expansion Joint 0102030405Time ( sec) 0  80 4004080Acceleration ( cm/ sec2) 0102030405Time ( sec) 0  8000 6000 4000 200002000Acceleration ( cm/ sec2) ( a) Acceleration at Expansion Joint ( a) Acceleration at Expansion Joint 0102030405Time ( sec) 0  2 1012Displacement ( cm) 0102030405Time( sec) 0  2 1012Displacement ( cm) ( b) Displacement at Expansion Joint ( b) Displacement at Expansion Joint 0102030405Time ( sec) 0 0.004 0.00200.0020.004Rotation ( rad) 0102030405Time ( sec) 0  0.004 0.00200.0020.004Rotation ( rad) ( c) Rotation at Column Bottom ( c) Rotation at Column Bottom Fig 2.12 Structural Responses without Pounding Fig 2.13 Structural Responses with Pounding 32 2.7.3 Pounding Effects on Fragility Curves The fragility curves for the four ( 4) sample bridges associated with the states of damage mentioned in the previous section were plotted as a function of peak ground acceleration in Figs 2.14 2.17, while the number of damaged bridges is listed in Tables 2.5 2.8, respectively. Each Fig has two curves for the cases without pounding and with pounding to compare how much the curves are shifted to left or right ( more or less fragile). It is noted here that the log standard deviation in each of Figs 2.14 2.17 was obtained by taking the whole events involving the cases without and with pounding using Equation 2.11 for these fragility curves. This is for the reason that the pair of fragility curves in each Fig is not theoretically expected to intersect each other. The fragility curves in pairs produced mixed results in such a way that the pounding effect is even beneficial for some damage states, while it appears detrimental for some cases. In particular, if the number of bridges at a certain state of damage counted, it can be clearly seen that the pounding does not increase the number of damaged bridges ( or the ductility factor) in general. It is noted that bridge characteristics, such as overall length, number of spans, number of expansion joints and height of columns, might not a major factor to change the trend of the fragility curves by increasing or decreasing ductility demand. High response amplifications due to pounding might result only if the colliding bridge segments separated by an expansion joint are significantly different in natural period, however this condition does not usually exist in the bridge structure. 33 34 Table 2.5 Number of Damaged Bridges: Pounding Effect in Bridge 2 sample size= 60 Damage States without Pounding with Pounding No 8 9 Almost No 8 9 Slight 8 9 Moderate 8 4 Extensive 14 16 Complete 14 13 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.18, ζ 0= 0.86) with Pounding ( c0= 0.23, ζ 0= 0.86) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.29, ζ 0= 0.86) with Pounding ( c0= 0.32, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.46, ζ 0= 0.86) with Pounding ( c0= 0.53, ζ 0= 0.86) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.61, ζ 0= 0.86) with Pounding ( c0= 0.60, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 1.01, ζ 0= 0.86) with Pounding ( c0= 1.01, ζ 0= 0.86) ( d) Extensive Damage ( e) Complete Collapse Fig 2.14 Pounding Effect on Fragility Curves of Bridge 2 35 Table 2.6 Number of Damaged Bridges: Pounding Effect in Bridge Model 3 sample size= 60 Damage States without Pounding with Pounding No 1 1 Almost No 2 3 Slight 13 13 Moderate 4 2 Extensive 14 20 Complete 26 21 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.11, ζ 0= 0.86) with Pounding ( c0= 0.11, ζ 0= 0.86) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.15, ζ 0= 0.86) with Pounding ( c0= 0.18, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.35, ζ 0= 0.86) with Pounding ( c0= 0.39, ζ 0= 0.86) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.43, ζ 0= 0.86) with Pounding ( c0= 0.40, ζ 0= 0.86) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.66, ζ 0= 0.86) with Pounding ( c0= 0.76, ζ 0= 0.86) ( d) Extensive Damage ( e) Complete Collapse Fig 2.15 Pounding Effect on Fragility Curves of Bridge 3 36 Table 2.7 Number of Damaged Bridges: Pounding Effect in Bridge 4 sample size= 60 Damage States without Pounding with Pounding No 1 1 Almost No 2 2 Slight 6 7 Moderate 7 6 Extensive 9 7 Complete 35 37 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.03, ζ 0= 0.95) with Pounding ( c0= 0.03, ζ 0= 0.95) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.07, ζ 0= 0.95) with Pounding ( c0= 0.13, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.20, ζ 0= 0.95) with Pounding ( c0= 0.27, ζ 0= 0.95) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.32, ζ 0= 0.95) with Pounding ( c0= 0.31, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.49, ζ 0= 0.95) with Pounding ( c0= 0.45, ζ 0= 0.95) ( d) Extensive Damage ( e) Complete Collapse Fig 2.16 Pounding Effect on Fragility Curves of Bridge 4 37 Table 2.8 Number of Damaged Bridges: Pounding Effect in Bridge 5 sample size= 60 Damage States without Pounding with Pounding No 8 7 Almost No 1 3 Slight 18 17 Moderate 9 9 Extensive 14 14 Complete 10 10 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.28, ζ 0= 0.70) with Pounding ( c0= 0.26, ζ 0= 0.70) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.28, ζ 0= 0.70) with Pounding ( c0= 0.31, ζ 0= 0.70) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.54, ζ 0= 0.70) with Pounding ( c0= 0.54, ζ 0= 0.70) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 0.69, ζ 0= 0.70) with Pounding ( c0= 0.69, ζ 0= 0.70) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State without Pounding ( c0= 1.01, ζ 0= 0.70) with Pounding ( c0= 1.01, ζ 0= 0.70) ( d) Extensive Damage ( e) Complete Collapse Fig 2.17 Pounding Effect on Fragility Curves of Bridge 5 2.7.4 Pounding and Soil Effects on Fragility Curves The fragility curves for the four ( 4) sample bridges associated with the states of damage mentioned in the previous section were plotted as a function of peak ground acceleration in Figs 2.18, 2.19, 2.20 and 2.21, while the number of damaged bridges is listed in Tables 2.9, 2.10, 2.11 and 2.12, respectively. Each Fig has four ( 4) curves for the following four ( 4) cases: CASE 1: without pounding effects and without soil effects CASE 2: with pounding effects and without soil effects CASE 3: without pounding effects and with soil effects CASE 4: with pounding effects and with soil effects In order to compare how much the curves are shifted to left or right ( more or less fragile) due to the effects of pounding and/ or soil, the four ( 4) curves were put into one Figure. It is noted that the log standard deviation in each of Figs 2.18, 2.19, 2.20 and 2.21 was obtained by taking the whole events involving the four ( 4) cases using equation 2.11 for these fragility curves. This is for the reason that the pair of fragility curves in each Fig is not theoretically expected to intersect each other. The fragility curves produced mixed results in such a way that the pounding and/ or soil effects are even beneficial for some damage states, while it appears detrimental for some cases. In particular, if the number of bridges at a certain state of damage counted, there is a definite effect but it is hard to say any trend. 38 39 Table 2.9 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 2 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 8 10 8 8 Almost No 9 5 12 9 Slight 8 11 7 8 Moderate 10 9 7 12 Extensive 15 14 16 13 Complete 10 11 10 10 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.28, ζ 0= 0.95) CASE 2 ( c0= 0.29, ζ 0= 0.95) CASE 3 ( c0= 0.29, ζ 0= 0.95) CASE 4 ( c0= 0.31, ζ 0= 0.95) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.52, ζ 0= 0.95) CASE 2 ( c0= 0.43, ζ 0= 0.95) CASE 3 ( c0= 0.61, ζ 0= 0.95) CASE 4 ( c0= 0.54, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.70, ζ 0= 0.95) CASE 2 ( c0= 0.74, ζ 0= 0.95) CASE 3 ( c0= 0.77, ζ 0= 0.95) CASE 4 ( c0= 0.70, ζ 0= 0.95) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.02, ζ 0= 0.95) CASE 2 ( c0= 1.02, ζ 0= 0.95) CASE 3 ( c0= 0.98, ζ 0= 0.95) CASE 4 ( c0= 1.08, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.91, ζ 0= 0.95) CASE 2 ( c0= 1.91, ζ 0= 0.95) CASE 3 ( c0= 2.13, ζ 0= 0.95) CASE 4 ( c0= 2.13, ζ 0= 0.95) ( d) Extensive Damage ( e) Complete Collapse Fig 2.18 Pounding and Soil Effects on Fragility Curves of Bridge 2 40 Table 2.10 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 3 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 2 2 2 10 Almost No 2 2 10 5 Slight 10 10 7 5 Moderate 8 5 5 3 Extensive 12 13 12 11 Complete 26 28 24 26 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.12, ζ 0= 0.77) CASE 2 ( c0= 0.12, ζ 0= 0.77) CASE 3 ( c0= 0.12, ζ 0= 0.77) CASE 4 ( c0= 0.22, ζ 0= 0.77) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.17, ζ 0= 0.77) CASE 2 ( c0= 0.19, ζ 0= 0.77) CASE 3 ( c0= 0.24, ζ 0= 0.77) CASE 4 ( c0= 0.31, ζ 0= 0.77) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.34, ζ 0= 0.77) CASE 2 ( c0= 0.33, ζ 0= 0.77) CASE 3 ( c0= 0.40, ζ 0= 0.77) CASE 4 ( c0= 0.40, ζ 0= 0.77) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.46, ζ 0= 0.77) CASE 2 ( c0= 0.40, ζ 0= 0.77) CASE 3 ( c0= 0.48, ζ 0= 0.77) CASE 4 ( c0= 0.45, ζ 0= 0.77) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.66, ζ 0= 0.77) CASE 2 ( c0= 0.62, ζ 0= 0.77) CASE 3 ( c0= 0.69, ζ 0= 0.77) CASE 4 ( c0= 0.66, ζ 0= 0.77) ( d) Extensive Damage ( e) Complete Collapse Fig 2.19 Pounding and Soil Effects on Fragility Curves of Bridge 3 41 Table 2.11 Number of Damaged Bridges: Pounding and Soil Effects in Bridge 4 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 5 3 9 7 Almost No 5 5 6 5 Slight 15 11 13 10 Moderate 11 7 9 8 Extensive 15 18 14 16 Complete 9 16 9 14 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.18, ζ 0= 1.01) CASE 2 ( c0= 0.08, ζ 0= 1.01) CASE 3 ( c0= 0.21, ζ 0= 1.01) CASE 4 ( c0= 0.05, ζ 0= 1.01) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.25, ζ 0= 1.01) CASE 2 ( c0= 0.19, ζ 0= 1.01) CASE 3 ( c0= 0.30, ζ 0= 1.01) CASE 4 ( c0= 0.24, ζ 0= 1.01) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.49, ζ 0= 1.01) CASE 2 ( c0= 0.33, ζ 0= 1.01) CASE 3 ( c0= 0.54, ζ 0= 1.01) CASE 4 ( c0= 0.42, ζ 0= 1.01) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.69, ζ 0= 1.01) CASE 2 ( c0= 0.50, ζ 0= 1.01) CASE 3 ( c0= 0.71, ζ 0= 1.01) CASE 4 ( c0= 0.57, ζ 0= 1.01) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.09, ζ 0= 1.01) CASE 2 ( c0= 0.88, ζ 0= 1.01) CASE 3 ( c0= 1.16, ζ 0= 1.01) CASE 4 ( c0= 0.98, ζ 0= 1.01) ( d) Extensive Damage ( e) Complete Collapse Fig 2.20 Pounding and Soil Effects on Fragility Curves of Bridge 4 42 Table 2.12Number of Damaged Bridges: Pounding and Soil Effects in Bridge 5 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 9 9 11 10 Almost No 7 6 4 4 Slight 14 15 12 13 Moderate 11 11 14 14 Extensive 13 13 12 12 Complete 6 6 7 7 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.44, ζ 0= 0.78) CASE 2 ( c0= 0.44, ζ 0= 0.78) CASE 3 ( c0= 0.35, ζ 0= 0.78) CASE 4 ( c0= 0.32, ζ 0= 0.78) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.55, ζ 0= 0.78) CASE 2 ( c0= 0.54, ζ 0= 0.78) CASE 3 ( c0= 0.50, ζ 0= 0.78) CASE 4 ( c0= 0.47, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.88, ζ 0= 0.78) CASE 2 ( c0= 0.88, ζ 0= 0.78) CASE 3 ( c0= 0.80, ζ 0= 0.78) CASE 4 ( c0= 0.80, ζ 0= 0.78) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.18, ζ 0= 0.78) CASE 2 ( c0= 1.18, ζ 0= 0.78) CASE 3 ( c0= 1.18, ζ 0= 0.78) CASE 4 ( c0= 1.18, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.83, ζ 0= 0.78) CASE 2 ( c0= 1.83, ζ 0= 0.78) CASE 3 ( c0= 1.84, ζ 0= 0.78) CASE 4 ( c0= 1.84, ζ 0= 0.78) ( d) Extensive Damage ( e) Complete Collapse Fig 2.21 Pounding and Soil Effects on Fragility Curves of Bridge 5 2.7.5 Jacketing and Restrainer Effects on Fragility Curves The fragility curves for the four ( 4) sample bridges associated with the states of damage mentioned in the previous section were plotted as a function of peak ground acceleration in Figs 2.23, 2.24, 2.25 and 2.26, while the number of damaged bridges is listed in Tables 2.14, 2.15, 2.16 and 2.17, respectively. Each Fig has four ( 4) curves for the following four ( 4) cases: CASE 1: without jacketing and without restrainer CASE 2: with jacketing and without restrainer CASE 3: without jacketing and with restrainer CASE 4: with jacketing and with restrainer In order to compare how much the curves are shifted to left or right ( more or less fragile) due to the effects of jacketing and/ or restrainer, the four ( 4) curves were put into one Fig. It is noted that the log standard deviation in each of Figs 2.23, 2.24, 2.25 and 2.26 was obtained by taking the whole events involving the four ( 4) cases using equation 2.11 for these fragility curves. This is for the reason that the pair of fragility curves in each figure is not theoretically expected to intersect each other. The damage state of a bridge is defined in terms of the maximum value of the peak ductility demands sustained by all the column ends. In this context, comparison between fragility curves in Figs 2.23 2.26 indicates that the bridge is less susceptible for damage to the ground motion after column retrofit than before, while the effect of restrainers at expansion joints is found to be negligible or even adversely affects on the column responses. 43 44 Table 2.13 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 2 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 10 14 10 15 Almost No 4 8 6 7 Slight 11 18 11 18 Moderate 11 11 9 13 Extensive 13 8 15 6 Complete 11 1 9 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.29, ζ 0= 1.10) CASE 2 ( c0= 0.39, ζ 0= 1.10) CASE 3 ( c0= 0.29, ζ 0= 1.10) CASE 4 ( c0= 0.46, ζ 0= 1.10) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.39, ζ 0= 1.10) CASE 2 ( c0= 0.64, ζ 0= 1.10) CASE 3 ( c0= 0.49, ζ 0= 1.10) CASE 4 ( c0= 0.64, ζ 0= 1.10) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.70, ζ 0= 1.10) CASE 2 ( c0= 1.19, ζ 0= 1.10) CASE 3 ( c0= 0.74, ζ 0= 1.10) CASE 4 ( c0= 1.19, ζ 0= 1.10) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.05, ζ 0= 1.10) CASE 2 ( c0= 1.97, ζ 0= 1.10) CASE 3 ( c0= 1.05, ζ 0= 1.10) CASE 4 ( c0= 2.50, ζ 0= 1.10) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.91, ζ 0= 1.10) CASE 2 ( c0= 6.12, ζ 0= 1.10) CASE 3 ( c0= 2.78, ζ 0= 1.10) CASE 4 ( c0= 6.12, ζ 0= 1.10) ( d) Extensive Damage ( e) Complete Collapse Fig 2.22 Jacketing and Restrainer Effects on Fragility Curves of Bridge 2 45 Table 2.14 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 3 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 1 3 2 5 Almost No 3 13 3 4 Slight 9 16 4 15 Moderate 6 11 9 14 Extensive 13 13 6 14 Complete 28 4 36 8 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.11, ζ 0= 0.95) CASE 2 ( c0= 0.16, ζ 0= 0.95) CASE 3 ( c0= 0.10, ζ 0= 0.95) CASE 4 ( c0= 0.15, ζ 0= 0.95) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.19, ζ 0= 0.95) CASE 2 ( c0= 0.36, ζ 0= 0.95) CASE 3 ( c0= 0.15, ζ 0= 0.95) CASE 4 ( c0= 0.27, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.33, ζ 0= 0.95) CASE 2 ( c0= 0.62, ζ 0= 0.95) CASE 3 ( c0= 0.27, ζ 0= 0.95) CASE 4 ( c0= 0.48, ζ 0= 0.95) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.40, ζ 0= 0.95) CASE 2 ( c0= 0.86, ζ 0= 0.95) CASE 3 ( c0= 0.39, ζ 0= 0.95) CASE 4 ( c0= 0.73, ζ 0= 0.95) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.62, ζ 0= 0.95) CASE 2 ( c0= 2.31, ζ 0= 0.95) CASE 3 ( c0= 0.48, ζ 0= 0.95) CASE 4 ( c0= 1.15, ζ 0= 0.95) ( d) Extensive Damage ( e) Complete Collapse Fig 2.23 Jacketing and Restrainer Effects on Fragility Curves of Bridge 3 46 Table 2.15 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 4 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 3 8 3 8 Almost No 5 13 4 10 Slight 11 15 14 14 Moderate 7 12 5 9 Extensive 18 9 12 15 Complete 16 3 22 4 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.08, ζ 0= 1.06) CASE 2 ( c0= 0.18, ζ 0= 1.06) CASE 3 ( c0= 0.13, ζ 0= 1.06) CASE 4 ( c0= 0.20, ζ 0= 1.06) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.19, ζ 0= 1.06) CASE 2 ( c0= 0.39, ζ 0= 1.06) CASE 3 ( c0= 0.21, ζ 0= 1.06) CASE 4 ( c0= 0.38, ζ 0= 1.06) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.33, ζ 0= 1.06) CASE 2 ( c0= 0.69, ζ 0= 1.06) CASE 3 ( c0= 0.39, ζ 0= 1.06) CASE 4 ( c0= 0.62, ζ 0= 1.06) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.50, ζ 0= 1.06) CASE 2 ( c0= 1.02, ζ 0= 1.06) CASE 3 ( c0= 0.50, ζ 0= 1.06) CASE 4 ( c0= 0.81, ζ 0= 1.06) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.87, ζ 0= 1.06) CASE 2 ( c0= 1.87, ζ 0= 1.06) CASE 3 ( c0= 0.72, ζ 0= 1.06) CASE 4 ( c0= 2.31, ζ 0= 1.06) ( d) Extensive Damage ( e) Complete Collapse Fig 2.24 Jacketing and Restrainer Effects on Fragility Curves of Bridge 4 47 Table 2.16 Number of Damaged Bridges: Jacketing and Restrainer Effects on Bridge 5 sample size= 60 Damage States Case1 Case2 Case3 Case4 No 4 6 3 5 Almost No 5 9 6 11 Slight 12 20 12 19 Moderate 10 11 11 13 Extensive 14 14 15 12 Complete 15 0 13 0 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.33, ζ 0= 0.88) CASE 2 ( c0= 0.44, ζ 0= 0.88) CASE 3 ( c0= 0.25, ζ 0= 0.88) CASE 4 ( c0= 0.37, ζ 0= 0.88) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.58, ζ 0= 0.88) CASE 2 ( c0= 0.72, ζ 0= 0.88) CASE 3 ( c0= 0.58, ζ 0= 0.88) CASE 4 ( c0= 0.74, ζ 0= 0.88) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 0.90, ζ 0= 0.88) CASE 2 ( c0= 1.35, ζ 0= 0.88) CASE 3 ( c0= 0.90, ζ 0= 0.88) CASE 4 ( c0= 1.35, ζ 0= 0.88) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.20, ζ 0= 0.88) CASE 2 ( c0= 1.77, ζ 0= 0.88) CASE 3 ( c0= 1.25, ζ 0= 0.88) CASE 4 ( c0= 1.84, ζ 0= 0.88) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State CASE 1 ( c0= 1.74, ζ 0= 0.88) CASE 2 ( c0= 2.68, ζ 0= 0.88) CASE 3 ( c0= 1.82, ζ 0= 0.88) CASE 4 ( c0= 2.68, ζ 0= 0.88) ( d) Extensive Damage ( e) Complete Collapse Fig 2.25 Jacketing and Restrainer Effects on Fragility Curves of Bridge 5 2.8 Fragility Enhancement After Column Retrofit 2.8.1 Fragility Curves After Retrofit for Longitudinal Direction The fragility curves for five ( 5) sample bridges associated with those damage states are plotted in Figs 2.26, 2.27, 2.28, 2.29 and 2.30, while the number of damaged bridges is listed in Tables 2.17, 2.18, 2.19, 2.20 and 2.21, respectively, for the cases before retrofit and after retrofit as a function of peak ground acceleration. It is noted here that the log standard deviation for the pair of fragility curves in each of Figs is obtained by considering both two cases ( before and after retrofit) together and calculating the optimal values from equation 2.11 for these fragility curves. This is for the reason that the bridge with jacketed columns is expected to be less vulnerable to ground motion than the bridge with the columns not jacketed and therefore we expect that the pair of these fragility curves should not theoretically intersect. The damage state of a bridge in this case is defined in terms of the maximum value of the peak ductility demands sustained by all the column ends. In this context, comparison between the two curves in each of Figs 2.26 2.30 indicates that the bridge is less susceptible to damage from the ground motion after retrofit than before. The simulated fragility curves in this case demonstrate that, for all levels of damage states, the median fragility values after retrofit are larger than the corresponding values before retrofit. This implies the following: if the number of Type 1 bridges suffering from a certain state of damage is counted, on average, the damage is smaller when the bridge is subjected to these sixty ( 60) earthquakes after retrofit than before retrofit. The number is listed in Tables 2.17 2.21 for before and after retrofit to Bridge 1~ 5. The result in Tables 2.17 2.21 is consistent with the observation that the fragility enhancement is found to be 48 more significant for more severe state of damage in general. This is not unexpected because the ductility demands for more severe states of damage increase after retrofit by much larger multiples than those that occurred before retrofit. 49 50 Table 2.17 Number of Damaged Bridges: Retrofit Effect in Bridge 1 sample size= 60 Damage States before Retrofit after Retrofit No 4 7 Almost No 5 9 Slight 10 16 Moderate 7 13 Extensive 17 13 Complete 17 2 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.36, ζ 0= 0.84) after retrofit ( c0= 0.47, ζ 0= 0.84) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.45, ζ 0= 0.84) after retrofit ( c0= 0.75, ζ 0= 0.84) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.80, ζ 0= 0.84) after retrofit ( c0= 1.25, ζ 0= 0.84) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.04, ζ 0= 0.84) after retrofit ( c0= 1.73, ζ 0= 0.84) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.66, ζ 0= 0.84) after retrofit ( c0= 5.05, ζ 0= 0.84) ( d) Extensive Damage ( e) Complete Collapse Fig 2.26 Retrofit Effect on Fragility Curves of Bridge 1 ( Longitudinal) 51 Table 2.18 Number of Damaged Bridges: Retrofit Effect in Bridge 2 sample size= 60 Damage States before Retrofit after Retrofit No 9 10 Almost No 4 9 Slight 10 19 Moderate 7 12 Extensive 16 6 Complete 14 4 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.38, ζ 0= 0.96) after retrofit ( c0= 0.39, ζ 0= 0.96) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.44, ζ 0= 0.96) after retrofit ( c0= 0.52, ζ 0= 0.96) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.65, ζ 0= 0.96) after retrofit ( c0= 1.11, ζ 0= 0.96) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.86, ζ 0= 0.96) after retrofit ( c0= 2.13, ζ 0= 0.96) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.47, ζ 0= 0.96) after retrofit ( c0= 3.00, ζ 0= 0.96) ( d) Extensive Damage ( e) Complete Collapse Fig 2.27 Retrofit Effect on Fragility Curves of Bridge 2 ( Longitudinal) 52 Table 2.19 Number of Damaged Bridges: Retrofit Effect in Bridge 3 sample size= 60 Damage States before Retrofit after Retrofit No 2 3 Almost No 2 13 Slight 9 16 Moderate 6 11 Extensive 13 13 Complete 28 4 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.12, ζ 0= 0.97) after retrofit ( c0= 0.15, ζ 0= 0.97) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.19, ζ 0= 0.97) after retrofit ( c0= 0.36, ζ 0= 0.97) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.33, ζ 0= 0.97) after retrofit ( c0= 0.62, ζ 0= 0.97) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.40, ζ 0= 0.97) after retrofit ( c0= 0.86, ζ 0= 0.97) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.62, ζ 0= 0.97) after retrofit ( c0= 2.31, ζ 0= 0.97) ( d) Extensive Damage ( e) Complete Collapse Fig 2.28 Retrofit Effect on Fragility Curves of Bridge 3 ( Longitudinal) 53 Table 2.20 Number of Damaged Bridges: Retrofit Effect in Bridge 4 sample size= 60 Damage States before Retrofit after Retrofit No 2 4 Almost No 5 17 Slight 12 14 Moderate 7 13 Extensive 18 9 Complete 16 3 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.09, ζ 0= 1.22) after retrofit ( c0= 0.18, ζ 0= 1.22) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.13, ζ 0= 1.22) after retrofit ( c0= 0.39, ζ 0= 1.22) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.35, ζ 0= 1.22) after retrofit ( c0= 0.67, ζ 0= 1.22) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.50, ζ 0= 1.22) after retrofit ( c0= 1.02, ζ 0= 1.22) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.88, ζ 0= 1.22) after retrofit ( c0= 1.87, ζ 0= 1.22) ( d) Extensive Damage ( e) Complete Collapse Fig 2.29 Retrofit Effect on Fragility Curves of Bridge 4 ( Longitudinal) 54 Table 2.21 Number of Damaged Bridges: Retrofit Effect in Bridge 5 sample size= 60 Damage States before Retrofit after Retrofit No 9 9 Almost No 7 12 Slight 14 24 Moderate 11 12 Extensive 13 2 Complete 6 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.44, ζ 0= 0.83) after retrofit ( c0= 0.44, ζ 0= 0.83) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.55, ζ 0= 0.83) after retrofit ( c0= 0.67, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.88, ζ 0= 0.83) after retrofit ( c0= 1.30, ζ 0= 0.83) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.18, ζ 0= 0.83) after retrofit ( c0= 1.99, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.83, ζ 0= 0.83) after retrofit ( c0= 2.08, ζ 0= 0.83) ( d) Extensive Damage ( e) Complete Collapse Fig 2.30 Retrofit Effect on Fragility Curves of Bridge 5 ( Longitudinal) The result shows, for example, that the effect of column retrofit on the seismic performance is excellent in explaining that the bridges are up to three times less fragile for Bridge 1 ( complete damage) and two for Bridge 2 ( complete damage) after retrofit compared to the case before retrofit in terms of the median values. 2.8.1.1 Enhancement after Retrofit for Circular Column Considering Bridge 1 and 2 which have circular columns and corresponding sets of fragility curves before and after retrofit, the average fragility enhancement over these two ( 2) bridges at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” is plotted through curve fitting as shown in Fig 2.31. This curve shows 20%, 34%, 58%, 98% and 167% improvement for each damage state described on the x axis in Fig 2.31. Almost_ NoSlightModerateExtensiveCollapseDamage States04080120160200Increase in Percentage (%) y= 11.8 e 0.53xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.31 Enhancement Curve for Circular Columns with Steel Jacketing 55 2.8.1.2 Enhancement after Retrofit for Oblong Shape Column For Bridge 3 and 5 with oblong columns, the fragility enhancement is developed in Fig 2.28 and 2.30. Considering these two ( 2) sample bridges with oblong columns and corresponding sets of fragility curves before and after retrofit, the average fragility enhancement over these two ( 2) sample bridges at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” is plotted through curve fitting as shown in Fig 2.38. This curve shows 20%, 34%, 58%, 99% and 170% improvement for each damage state described on the x axis in Fig 32. Almost_ NoSlightModerateExtensiveCollapseDamage States04080120160Increase in Percentage (%) y= 11.4 e 0.54xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.32 Enhancement Curve for Oblong Columns with Steel Jacketing 2.8.1.3 Enhancement after Retrofit for Rectangular Column For Bridge 4 with rectangular columns, the fragility enhancement is developed in Fig 2.30. 56 Considering the sample bridge with rectangular columns and corresponding sets of fragility curves before and after retrofit at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” is plotted through curve fitting as shown in Fig 2.33. It is noted that the effect of retrofit is not good for Bridge 4 because the geometric shape after retrofit [ Fig C4 ( b1)~( b9)] is not efficient for steel jacketing to produce confinement effect. Almost_ NoSlightModerateExtensiveCollapseDamageStates04080120160200Increase in Percentage (%) y= 132 e  0.04xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.33 Enhancement Curve for Rectangular Columns with Steel Jacketing 2.8.1.4 Enhancement after Retrofit for All Types of Column Considering all the sample bridges and corresponding sets of fragility curves before and after retrofit at each state of damage is computed and plotted as a function of the state of damage. An analytical function is interpolated and the “ enhancement curve” 57 is plotted through curve fitting as shown in Fig 2.34. This curve shows 40%, 55%, 75%, 104% and 143% improvement for each damage state described on the x axis in Fig 2.34 Almost_ NoSlightModerateExtensiveCollapseDamageStates04080120160200Increase in Percentage (%) y= 28.8 e 0.32xx= 1 for Almost_ No 2 for Slight 3 for Moderate 4 for Extensive 5 for Collapse Fig 2.34 Enhancement Curve for Five Sample Bridges with Steel Jacketing 2.8.2 Enhancement after Calibrating the Analytical Fragility Curves As described in the earlier part, analytical fragility curves are obtained using the damage state definitions given by Dutta and Mander ( Table 2.4). To compare these analytically obtained fragility curves with past earthquake bridge damage data, empirical fragility curves for a third level subset ( considering ‘ multiple span’ and ‘ soil type C’) have been developed ( Shinozuka et al. 2003a) ( see Chapter 3). Results indicate that the analytical curves are more probable to exceed a damage state than empirical ones ( Shinozuka and Banerjee, 2004). They have defined the damage states of bridges for slight, moderate and extensive damage levels in terms of threshold ductility capacities, 58 for what, the analytical fragility curves will be consistent with empirical curves. These new definitions of threshold ductility capacities have extended to develop the fragility curves after retrofit. Fig 2.35 shows the empirical fragility curves and simulated fragility curves for three already stated damage states of Bridge 2. Obtained threshold ductility capacities at each damage states for bridge 2, 4, and 5 before and after retrofit are tabulated in Table 2.22 . Table 2.22 Simulated Ductility Capacities of Sample Bridges Bridge 2 Bridge 4 Bridge 5 Damage state before retrofit before retrofit after retrofit after retrofit before retrofit after retrofit Slight 4.5 5.4 6.9 11.5 4.5 6.62 Moderate 6.5 9.66 7.31 17.13 8.4 16.21 Extensive 16.8 26.04 14.5 36.84 12.8 26.8 Slight Damage 00.20.40.60.8100.20.40.60.81Empirical CurveAnalytical CurveProbability of Exceeding Damage State 59 60 00.20.40.60.8100.20.40.60.81Empirical CurveAnalytica Curve Probability of Exceeding Damage State PGA ( g) Moderate Damage 00.20.40.60.8100.20.40.60.81Empirical CurveAnalytical Curve Probability of Exceeding Damage State PGA ( g) Extensive Damage Fig 2.35 Empirical Fragility Curves and Calibrated Analytical Fragility Curves of Bridge 2 Based on the new definitions of damage states, the fragility curves of bridge 2( Circular Column), 4 ( Rectangular Column) and 5 ( Oblong Column) before and after retrofit are estimated again. Table 2.23 give the fragility parameters, and the enhancement ratios based on the 2 set of definitions of damage states are provided in Table 2.24. 61 Table 2.23 Fragility Curves based on Adjusted Damage States Definitions Bridge 2 Bridge 4 Bridge 5 Damage State 0 c ( g) ' 0 c ( g) 0 ζ 0 c ( g) ' 0 c ( g) 0 ζ 0 c ( g) ' 0 c ( g) 0 ζ At least minor 0.59 0.76 0.56 0.56 0.83 0.52 0.51 0.74 0.42 At least moderate 0.71 1.48 0.67 0.62 1.08 0.64 0.66 1.33 0.44 At least extensive 1.13 6.12 1.27 1.08 2.53 0.71 / / / Table 2.24 Enhancement Ratios Comparison Bridge 2 Bridge 4 Bridge 5 Damage State Mander’s Calibrated Mander’s Calibrated Mander’s Calibrated At least minor 18% 28% 200% 48% 22% 46% At least moderate 71% 109% 91% 74% 48% 102% At least extensive 148% 440% 104% 134% 69% / 2.8.3 Fragility Curves for Transverse Direction The fragility curves for five ( 5) sample bridges associated with those damage states are plotted in Figs 2.36, 2.37, 2.38, 2.39 and 2.40, while the number of damaged bridges is listed in tables 2.25, 2.26, 2.27, 2.28 and 2.29, respectively, for the cases before retrofit and after retrofit as a function of peak ground acceleration. 62 Table 2.25 Number of Damaged Bridges: Retrofit Effect in Bridge 1 sample size= 60 Damage States before Retrofit after Retrofit No 5 8 Almost No 3 8 Slight 10 17 Moderate 9 13 Extensive 18 13 Collapse 15 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.35, ζ 0= 0.83) after retrofit ( c0= 0.55, ζ 0= 0.83) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.55, ζ 0= 0.83) after retrofit ( c0= 0.75, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.77, ζ 0= 0.83) after retrofit ( c0= 1.28, ζ 0= 0.83) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.06, ζ 0= 0.83) after retrofit ( c0= 1.75, ζ 0= 0.83) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.79, ζ 0= 0.83) after retrofit ( c0= 6.12, ζ 0= 0.83) ( d) Extensive Damage ( e) Complete Collapse Fig 2.36 Retrofit Effect on Fragility Curves of Bridge 1 ( Transverse) 63 Table 2.27 Number of Damaged Bridges: Retrofit Effect in Bridge 2 sample size= 60 Damage States before Retrofit after Retrofit No 4 4 Almost No 4 16 Slight 16 18 Moderate 6 11 Extensive 18 9 Collapse 12 2 0 0.20.40.60.81 PGA ( g) 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.03, ζ 0= 1.80) after retrofit ( c0= 0.08, ζ 0= 1.80) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.07, ζ 0= 1.80) after retrofit ( c0= 0.40, ζ 0= 1.80) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.53, ζ 0= 1.80) after retrofit ( c0= 0.92, ζ 0= 1.80) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.68, ζ 0= 1.80) after retrofit ( c0= 1.43, ζ 0= 1.80) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.31, ζ 0= 1.80) after retrofit ( c0= 2.03, ζ 0= 1.80) ( d) Extensive Damage ( e) Complete Collapse Fig 2.37 Retrofit Effect on Fragility Curves of Bridge 2 ( Transverse) 64 Table 2.28 Number of Damaged Bridges: Retrofit Effect in Bridge 3 sample size= 60 Damage States before Retrofit after Retrofit No 5 8 Almost No 5 8 Slight 4 22 Moderate 8 14 Extensive 14 7 Collapse 24 1 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.33, ζ 0= 0.78) after retrofit ( c0= 0.39, ζ 0= 0.78) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.46, ζ 0= 0.78) after retrofit ( c0= 0.55, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.52, ζ 0= 0.78) after retrofit ( c0= 1.10, ζ 0= 0.78) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.71, ζ 0= 0.78) after retrofit ( c0= 1.62, ζ 0= 0.78) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.04, ζ 0= 0.78) after retrofit ( c0= 2.08, ζ 0= 0.78) ( d) Extensive Damage ( e) Complete Collapse Fig 2.38 Retrofit Effect on Fragility Curves of Bridge 3 ( Transverse) 65 Table 2.29 Number of Damaged Bridges: Retrofit Effect in Bridge 4 sample size= 60 Damage States before Retrofit after Retrofit No 1 3 Almost No 3 11 Slight 12 17 Moderate 12 12 Extensive 15 14 Collapse 17 2 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.17, ζ 0= 0.99) after retrofit ( c0= 0.27, ζ 0= 0.99) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.22, ζ 0= 0.99) after retrofit ( c0= 0.56, ζ 0= 0.99) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.55, ζ 0= 0.99) after retrofit ( c0= 0.93, ζ 0= 0.99) ( b) Slight Damage © Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.83, ζ 0= 0.99) after retrofit ( c0= 1.30, ζ 0= 0.99) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.24, ζ 0= 0.99) after retrofit ( c0= 4.64, ζ 0= 0.99) ( d) Extensive Damage ( e) Complete Collapse Fig 2.39 Retrofit Effect on Fragility Curves of Bridge 4 ( Transverse) 66 Table 2.30 Number of Damaged Bridges: Retrofit Effect in Bridge 5 sample size= 60 Damage States before Retrofit after Retrofit No 7 9 Almost No 2 6 Slight 10 16 Moderate 10 13 Extensive 14 13 Collapse 17 3 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.38, ζ 0= 0.68) after retrofit ( c0= 0.42, ζ 0= 0.68) ( a) Almost No Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.42, ζ 0= 0.68) after retrofit ( c0= 0.58, ζ 0= 0.68) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.63, ζ 0= 0.68) after retrofit ( c0= 0.90, ζ 0= 0.68) ( b) Slight Damage ( c) Moderate Damage 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 0.86, ζ 0= 0.68) after retrofit ( c0= 1.27, ζ 0= 0.68) 0 0.20.40.60.81 PGA ( g) 0 0.2 0.4 0.6 0.8 1 Probability of Exceeding a Damage State before retrofit ( c0= 1.24, ζ 0= 0.68) after retrofit ( c0= 1.99, ζ 0= 0.68) ( d) Extensive Damage ( e) Complete Collapse Fig 2.40 Retrofit Effect on Fragility Curves of Bridge 5 ( Transverse) Chapter 3 Development of Empirical Fragility Curves for Bridges 3.1 Empirical Bridge Damage Data The 1994 Northridge Earthquake caused tremendous damages to the human building environment. However, the damage investigation after the event provided valuable data basis for developing empirical fragility curves. After the event, 2209 highway bridges around Los Angeles Area were investigated and the damage of each bridge was classified as one of the five states: No, Minor, Moderate, Major or Collapse. Table 3.1 provides the summary of the bridge damage condition. The site ground motion of each bridge structure can be derived from any ground motion spatial distribution ( contour) map. Figs. 3.1 and 3.2 show PGA and PGV distribution in the 1994 Northridge Earthquake, which are acquired from the TriNet Shakemap ( http:// www. trinet. org/ trinet. html). Table 3.2 lists part of the bridge damage table including bridge site ground motion determined from these two maps. Table 3.1 Summary of Bridge Damage Status in the 1994 Northridge Earthquake Damage State No Damage Minor Damage Moderate Damage Major Damage Collapse Damage Total Number 1978 84 94 47 6 2209 67 Fig 3.1 1994 Northridge Earthquake: PGA Distribution Fig 3.2 1994 Northridge Earthquake: PGV Distribution PGA( g) PGV( cm/ s) 68 Table 3.2 Seismic Damages of Bridges in the 1994 Northridge Earthquake ID BRIDGE_ NO Damage States ShakeMap PGA ( g) ShakeMap PGV ( cm/ s) LAT LONG 1 53 1301 MOD* 0.2 16 34.0227  118.2500 2 53 1471 0.12 12 33.7500  118.2687 3 53 2618 0.08 6 33.7667  118.2353 4 53 2216G MAJ* 0.76 114 34.2667  118.4697 5 53 1907G MOD 0.24 20 34.1383  118.2333 6 53 0595 0.28 16 34.0353  118.2187 7 53 1851 MOD 0.28 28 33.9863  118.4000 8 53 2549H 0.12 10 33.8687  118.2843 9 53 1637F MOD 0.4 42 34.0257  118.4237 10 53 1790H MOD 0.24 20 34.1520  118.2747 11 53 1717H MIN* 0.28 14 34.0353  118.1677 12 53 1627G MAJ 0.4 50 34.0257  118.4343 13 53 2673 0.28 18 34.0520  118.2227 14 53 1424 MOD 0.24 16 34.0757  118.2217 15 53 2142F 0.12 10 33.8697  118.1863 16 53 0707F 0.2 14 34.0393  118.2697 17 53 2700G 0.12 10 33.9080  118.1010 18 53 1714G MOD 0.28 14 34.0353  118.1677 19 53 0845 0.2 22 33.9353  118.3903 20 53 2731 0.08 10 33.8373  118.2040 21 52 0331R 0.28 22 34.2859  118.8650 22 53 2143F 0.12 10 33.8697  118.1843 23 53 2318G 0.16 14 34.1500  118.1530 24 53 2327F MAJ 0.6 72 34.2667  118.4383 25 53 2329G MAJ 0.6 72 34.2667  118.4383 26 53 2102G MAJ 0.4 46 34.2863  118.4030 27 53 0405 0.28 18 34.0520  118.2227 28 52 0118 MOD 0.24 20 34.3917  118.9150 29 53 1960F COL* 0.6 76 34.3350  118.5083 30 53 1238G 0.2 18 33.9167  118.3667 31 53 2104F MOD 0.4 44 34.2853  118.4020 32 52 0413 0.2 18 34.2011  118.9758 33 53 2627 0.08 10 33.7843  118.2217 34 53 1964F COL 0.6 76 34.3353  118.5056 35 53 1962F MOD 0.64 76 34.3343  118.5040 36 53 2200S MOD 0.48 48 34.4010  118.4540 37 53 1790 MIN 0.24 18 34.1510  118.2717 …. * MIN: Minor Damage MOD: Moderate Damage MAJ: Major Damage COL: Collapse 69 70 3.2 Bridge Classification In this research, the bridges are classified into different subsets according to the following three distinct attributes; ( A) It is either single span ( S) or multiple span ( M), ( B) it is built on either hard soil ( SA), medium soil ( SB) or soft soil ( SC) in the definition of UBC94, and ( C) it has a skew angle 1 θ ( less than 20o), 2 θ ( between 20o and 60o) or 3 θ ( larger than 60o). To begin with, one might consider the first level hypothesis that the entire sample is taken from a statistically homogeneous population of bridges. The second level subsets are created by dividing the sample either ( A) into two groups of bridges, one with single spans and the other with multiple spans, ( B) into three groups, the first with soil condition SA, the second with SB and the third with SC, or ( C) into three groups depending on the skew angles 1 θ , 2 θ and 3 θ . The third and fourth level sub groupings were also considered for the development of corresponding fragility curves under PGA and PGV as ground motion intensity index ( Shinozuka et al, 2003a). 3.3 Parameter Estimation It is assumed that the curves can be expressed in the form of two parameter lognormal distribution functions, and the estimation of the two parameters ( median and log standard deviation) is performed with the aid of the maximum likelihood method. For this purpose, PGA and PGV values are used to represent the intensity of the seismic ground motion. The likelihood method for fragility parameter estimation was described in Chapter 2. The median values and log standard deviations of all levels of attribute combinations are listed in Table 3.3 3.6. Note that, if an element of a matrix in these 71 tables shows N/ A, it indicates that no sub sample was found for the particular combination of bridge attributes the element signifies. The family of fragility curves corresponding to the first level is plotted in Fig3.3 and 3.4. The curve with a “ minor” designation represents, at each PGA or PGV value a , the probability that “ at least a minor” state of damage will be sustained by a bridge ( arbitrarily chosen from the sample of bridges) when it is subjected to PGA or PGV a . The same meaning applies to other curves with their respective damage state designations. All the other fragility curves in PGA are plotted in Figs 3.5 3.44 Table 3.3 First Level ( Composite) Fragility Curve Table 3.4 Second Level Fragility Curve ( a) Number of Span PGA ( g) PGV ( cm/ s) Span Damage State c ς c ς Min 0.89 0.66129 0.98 Mod 1.15 0.66188 0.98 Maj 1.76 0.66357 0.98 Single Col N/ A 0.66N/ A0.98 Min 0.56 0.6663 0.92 Mod 0.70 0.6687 0.92 Maj 1.09 0.66163 0.92 Multiple Col 2.16 0.66428 0.92 ( b) Skew Angle PGA ( g) PGV ( cm/ s) Skew Damage State c ς c ς Min 0.82 0.76108 1.07 Mod 1.10 0.76164 1.07 Maj 1.86 0.76343 1.07 00 200 Col 3.49 0.76833 1.07 PGA ( g) PGV ( cm/ s) Damage State c ς c ς Min 0.64 0.7076 0.98 Mod 0.80 0.70106 0.98 Maj 1.25 0.70200 0.98 Col 2.55 0.70555 0.98 72 Min 0.60 0.7170 0.98 Mod 0.72 0.7190 0.98 Maj 1.15 0.71173 0.98 200 600 Col 3.18 0.71769 0.98 Min 0.42 0.5242 0.75 Mod 0.52 0.5256 0.75 Maj 0.74 0.5296 0.75 > 600 Col 1.26 0.52212 0.75 ( c) Soil Type PGA ( g) PGV ( cm/ s) Soil Damage State c ς c ς Min 0.87 0.75110 1.03 Mod 1.10 0.75151 1.03 Maj 1.51 0.75234 1.03 A Col N/ A 0.75N/ A1.03 Min 0.64 0.7165 0.81 Mod 0.84 0.7191 0.81 Maj 1.24 0.71145 0.81 B Col N/ A 0.71N/ A0.81 Min 0.61 0.6974 0.98 Mod 0.76 0.69102 0.98 Maj 1.22 0.69199 0.98 C Col 2.35 0.69523 0.98 Table 3.5 Third Level Fragility Curve ( a) Span/ Skew PGA ( g) PGV ( cm/ s) Span Skew Damage State c ς c ς Minor 1.37 0.82 276 1.28 Moderate 2.04 0.82 502 1.28 Major 3.56 0.82 1179 1.28 00 200 Collapse N/ A N/ A N/ A N/ A Minor 0.63 0.43 82 0.7 Moderate 0.70 0.43 98 0.7 Major 0.96 0.43 164 0.7 200 600 Collapse N/ A N/ A N/ A N/ A Minor 0.62 0.13 86 0.10 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A Single > 600 Collapse N/ A N/ A N/ A N/ A Minor 0.68 0.71 82 0.98 Moderate 0.91 0.71 122 0.98 Major 1.52 0.71 251 0.98 Multiple 00 200 Collapse 2.76 0.71 574 0.98 73 Minor 0.56 0.74 63 0.99 Moderate 0.69 0.74 84 0.99 Major 1.11 0.74 162 0.99 200 600 Collapse 3.14 0.74 716 0.99 Minor 0.38 0.38 37 0.58 Moderate 0.42 0.38 43 0.58 Major 0.56 0.38 68 0.58 > 600 Collapse 0.67 0.38 92 0.58 ( b) Span/ Soil PGA ( g) PGV ( cm/ s) Span Soil Damage State c ς c ς Minor 0.90 0.40 116 0.50 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor 0.68 0.50 68 0.50 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.74 0.57 106 0.90 Moderate 0.91 0.57 144 0.90 Major 1.37 0.57 274 0.90 Single C Collapse N/ A N/ A N/ A N/ A Minor 0.64 0.64 66 0.81 Moderate 0.77 0.64 83 0.81 Major 1.05 0.64 125 0.81 A Collapse N/ A N/ A N/ A N/ A Minor 0.47 0.45 44 0.53 Moderate 0.56 0.45 57 0.53 Major 0.76 0.45 86 0.53 B Collapse N/ A N/ A N/ A N/ A Minor 0.56 0.67 65 0.96 Moderate 0.7 0.67 89 0.96 Major 1.11 0.67 173 0.96 Multiple C Collapse 2.11 0.67 435 0.96 ( c) Skew/ Soil PGA ( g) PGV ( cm/ s) Skew Soil Damage State c ς c ς Minor 0.70 0.50 61 0.50 Moderate 0.98 0.50 90 0.50 Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor 0.80 0.50 75 0.5 00 200 B Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A Collapse N/ A N/ A N/ A N/ A Minor 0.74 0.72 98 1.04 Moderate 0.97 0.72 144 1.04 Major 1.61 0.72 299 1.04 C Collapse 2.99 0.72 728 1.04 Minor 0.73 0.48 79 0.50 Moderate 0.73 0.48 79 0.50 Major 0.83 0.48 88 0.50 A Collapse N/ A N/ A N/ A N/ A Minor 0.49 0.38 48 0.48 Moderate 0.57 0.38 68 0.48 Major 0.57 0.38 68 0.48 B Collapse N/ A N/ A N/ A N/ A Minor 0.57 0.72 66 0.57 Moderate 0.69 0.72 86 0.69 Major 1.19 0.72 187 1.19 200 600 C Collapse 3.07 0.72 759 3.07 Minor 0.26 0.11 21 0.10 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.48 0.48 57 0.74 Moderate 0.59 0.48 76 0.74 Major 0.74 0.48 107 0.74 > 600 C Collapse 0.87 0.48 137 0.74 74 75 Table 3.6 Fourth Level Fragility Curve ( Span/ Skew/ Soil) PGA ( g) PGV ( cm/ s) Span Skew Soil Damage State c ς c ς Minor 0.63 0.22 81 0.40 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor 0.63 0.50 63 0.5 Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.98 0.57 239 1.16 Moderate 1.19 0.57 340 1.16 Major 1.85 0.57 780 1.16 00 200 C Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A B Collapse N/ A N/ A N/ A N/ A Minor 0.53 0.39 64 0.64 Moderate 0.60 0.39 78 0.64 Major 0.84 0.39 134 0.64 200 600 C Collapse N/ A N/ A N/ A N/ A Minor N/ A N/ A N/ A N/ A Moderate N/ A N/ A N/ A N/ A Major N/ A N/ A N/ A N/ A A Collapse N/ A N/ A N/ A N/ A Min 



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