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), cross-section
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