DETERMINING THE EFFECTIVE SYSTEM
DAMPING OF HIGHWAY BRIDGES
By
Maria Q. Feng, Professor
and
Sung Chil Lee, Post- doctoral Researcher
Department of Civil & Environmental Engineering
University of California, Irvine
CA- UCI- 2009- 001
June 2009
Final Report Submitted to
the California Department of Transportation
under Contract No: RTA59A0495
DETERMINING THE EFFECTIVE SYSTEM
DAMPING OF HIGHWAY BRIDGES
Final Report Submitted to the Caltrans under
Contract No: RTA59A0495
By
Maria Q. Feng, Professor
and
Sung Chil Lee, Post- doctoral Researcher
Department of Civil & Environmental Engineering
University of California, Irvine
CA- UCI- 2009- 001
June 2009
ii
STATE OF CALIFORNIA ⋅ DEPARTMENT OF TRASPORTATION
TECHNICAL REPORT DOCUMENTAION PAGE
TR0003 ( REV. 9/ 99)
1. REPORT NUMBER
CA- UCI- 2009- 001
2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER
4. TITLE AND SUBTITLE
DETERMINING THE EFFECTIVE SYSTEM DAMPING OF
HIGHWAY BRIDGES
5. REPORT DATE
June, 2009
6. PERFORMING ORGANIZATION CODE
UC Irvine
7. AUTHOR
Maria Q. Feng, and Sungchil Lee
8. PERFORMING ORGANIZATION REPORT
NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Civil and Environmental Engineering E4120 Engineering Gateway
University of California, Irvine
Irvine, CA 92697- 2175
10. WORK UNIT NUMBER
11. CONTRACT OR GRANT NUMBER
RTA59A0495
12. SPONSORING AGENCY AND ADDRESS
California Department of Transportation ( Caltrans)
Sacramento, CA
13. TYPE OF REPORT AND PERIOD
COVERED
Final Report
14. SPONSORING AGENT CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT
This project investigates four methods for modeling modal damping ratios of short- span and isolated
concrete bridges subjected to strong ground motion, which can be used for bridge seismic analysis
and design based on the response spectrum method. The seismic demand computation of highway
bridges relies mainly on the design spectrum method, which requires effective modal damping.
However, high damping components, such as embankments of short- span bridges under strong
ground motion and isolation bearings make bridges non- proportionally damped systems for which
modal damping cannot be calculated using the conventional modal analysis. In this project four
methods are investigated for estimating the effective system modal damping, including complex
modal analysis ( CMA), neglecting off- diagonal elements in damping matrix method ( NODE),
composite damping rule ( CDR), and optimization in time domain and frequency domain ( OPT) and
applied to a short- span bridge and an isolated bridge.
The results show that among the four damping estimating methods, the NODE method is the most
efficient and the conventional assumption of 5% modal damping ratio is too conservative for short-span
bridges when energy dissipation is significant at the bridge boundaries. From the analysis of
isolated bridge case, the effective system damping is very close to the damping ratio of isolation
bearing.
17. KEYWORDS
Effective Damping, Concrete Bridge,
Response Spectrum Method
18. DISTRIBUTION STATEMENT
No restrictions.
19. SECURITY CLASSIFICATION ( of this report)
Unclassified
20. NUMBER OF PAGES
288
21. COST OF REPORT CHARGED
ii i
DISCLAIMER: The contents of this report reflect the views of the
authors who are responsible for the facts and the
accuracy of the data presented herein. The contents
do not necessarily reflect the official views or
policies of the State of California or the Federal
Highway Administration. This report does not
constitute a standard, specification or regulation.
The United States Government does not endorse
products or manufacturers. Trade and
manufacturers’ names appear in this report only
because they are considered essential to the object
of the document.
iv
SUMMARY
The overall objective of this project is to study the fundamental issue of damping in
bridge structural systems involving significantly different damping components and to
develop a more rational method to determine the approximation of seismic demand of
isolated bridges and short bridge. Within the framework of the current response
spectrum method, on which the design of highway bridges primarily relies, four damping
estimation methods including the complex modal analysis method, neglecting off-diagonal
elements method, optimization method, and composite damping rule method,
are explored to compute the equivalent modal damping ratio of short- span bridges and
isolated bridges.
From the application to a real short- span bridge utilizing earthquake data recorded at the
bridge site, the effective system damping ratio of the bridge was determined to be as
large as 25% under strong ground motions, which is much higher than the conventional
damping ratio used for design of such bridges. Meanwhile, the simulation with the 5%
damping ratio produced nearly two times the demand of the measured data, which
implies that the 5% value used in practice may be too low for the design of short- span
bridges considering the strong ground motions which should be sustained.
The four damping estimating methods are also applied to an isolated bridge. By
approximating non- linear isolation bearings with equivalent viscoelastic elements, an
equivalent linear analysis is carried out. The estimation of the seismic demand based on
the response spectrum method using the effective system damping computed by the four
methods is verified by comparing the response with that from the non- linear time history
analysis. Equivalent damping ratio of isolation bearing varies from 10% to 28% under
v
ground motions. For isolated bridges, majority of the energy dissipation takes place in
isolation bearings, but contribution from the bridge structural damping should also be
considered. A simplified way of determining the effective system damping of the
isolated bridge is suggested as the summation of equivalent damping ratio of isolation
bearing and the half of the damping ratio of bridge structure. Also, from the relation
between the effective system damping ratio and ground motion characteristics, a simple
approximation to predict the effective system damping of isolated bridges is suggested.
v i
Table of Contents
TECHNICAL REPORT PAGE……………………………………………………........... ii
DISCLAMER…………………………………….…………………………………........ iii
SUMMARY……………………………………………………………….…….............. iv
TABLE OF CONTENTS…………………………………………..…………................. vi
LIST OF FIGURES……………………………………………………………………... xii
LIST OF TABLES……………………………………….…………………………….. xvii
ACKNOWLEDGEMENT………………………………………..…………................... xx
1. INTRODUCTION ......................................................................................................... 1
1.1 Background ............................................................................................................ 1
1.2 Effective Modal Damping ..................................................................................... 4
1.3 Objectives and Scope ............................................................................................. 6
2. LITERATURE REVIEW ............................................................................................... 8
2.1 Energy Dissipation and EMDR of Short- span Bridge ........................................... 8
2.2 Energy Dissipation and EMDR of Short- span Bridge ......................................... 12
v ii
2.3 Energy Dissipation and EMDR of Short- span Bridge ......................................... 16
3. EFFECTIVE SYSTEM DAMPING ESTIMATION METHOD .................................. 19
3.1 Complex Modal Analysis ( CMA) Method .......................................................... 19
3.1.1 Normal Modal Analysis ................................................................................ 19
3.1.2 Complex Modal Analysis and EMDR Estimation ........................................ 21
3.1.3 Procedure of CMA Method .......................................................................... 23
3.2 Neglecting Off- Diagonal Elements ( NODE) Method ........................................ 24
3.2.1 Basic Principles ........................................................................................... 24
3.2.2 Error Criteria of NODE Method .................................................................. 26
3.2.3 Procedure of NODE Method ....................................................................... 26
3.3 Optimization ( OPT) Method ............................................................................... 28
3.3.1 Basic Principle ............................................................................................. 28
3.3.2 Time Domain ............................................................................................... 29
3.3.3 Frequency Domain ....................................................................................... 30
3.3.4 Procedure of OPT Method ........................................................................... 33
3.4 Composite Damping Rule ( CDR) Method ......................................................... 37
3.4.1 Basic Principle ............................................................................................. 37
3.4.2 Procedure of CDR Method .......................................................................... 41
3.5 Summary ............................................................................................................. 42
v iii
4. APPLICATION TO SHORT- SPAN BRIDGE ............................................................ 44
4.1 Analysis Procedure ............................................................................................. 44
4.2 Example Bridge and Earthquake Recordings ..................................................... 48
4.2.1 Description of Painter St. Overpass ............................................................. 48
4.2.2 Recorded Earthquakes and Dynamic Responses ......................................... 48
4.3 Seismic Response From NP- Model .................................................................... 52
4.3.1 Modeling of Concrete Structure ................................................................... 52
4.3.2 Estimation of Boundary Condition and Response from NP- Model ............ 53
4.3.3 Natural Frequency and Mode Shape ............................................................ 56
4.4 EMDR Estimation ............................................................................................... 59
4.4.1 Complex Modal Analysis ( CMA) Method .................................................. 59
4.4.2 Neglecting Off- Diagonal Elements ( NODE) Method ................................. 64
4.4.3 Composite Damping Rule ( CDR) Method .................................................. 67
4.4.4 Optimization ( OPT) Method ........................................................................ 70
4.5 Comparison with Current Design Method .......................................................... 77
4.6 EMDR and Ground Motion Characteristics ........................................................ 80
4.6.1 Ground Motion Parameters .......................................................................... 80
4.6.2 Relationship of EMDR with Ground Motion Parameters ........................... 80
4.7 Summary ............................................................................................................. 84
ix
5. APPLICATION TO ISOLATED BRIDGE ................................................................. 86
5.1 Analysis Procedure ............................................................................................. 87
5.2 Equivalent Linearization of Isolation Bearing .................................................... 90
5.2.1 Bi- linear Model of Isolation Bearing ............................................................ 90
5.2.2 Equivalent Linearization of Isolation Bearing ............................................. 91
5.3 Example Bridge and Ground Motions ................................................................. 94
5.3.1 Description of Example Bridge .................................................................... 94
5.3.2 Modal Analysis Results ................................................................................ 98
5.3.3 Ground Motions .......................................................................................... 100
5.4 Seismic Response from Bi- linear Model .......................................................... 101
5.5 Seismic Response from NP- Model ................................................................... 104
5.5.1 Results of Equivalent Linearization of Isolation Bearing .......................... 104
5.5.2 Seismic Response from NP- Model ............................................................ 114
5.6 EMDR Estimation ................................................................................... 119
5.6.1 Complex Modal Analysis ( CMA) Method ................................................ 119
5.6.2 Neglecting off- Diagonal Elements ( NODE) Method ................................ 123
5.6.3 Composite Damping Rule ( CDR) Method ................................................ 127
5.6.4 Optimization ( OPT) Method ...................................................................... 130
5.6.5 Comparison of EMDR ............................................................................... 133
x
5.7 Seismic Response from Modal Combination ................................................... 138
5.8 Comparison with Current Design Method ........................................................ 144
5.9 Effects of Ground Motion Characteristics ....................................................... 149
5.9.1 Effects on Ductility Ratio .......................................................................... 149
5.9.2 Effects on Equivalent Linearization of Isolation Bearing ......................... 152
5.9.3 Effects on EMDR ....................................................................................... 155
5.10 Summary ......................................................................................................... 158
6. CONCLUSIONS AND RECOMMENDED FUTURE WORKS .............................. 160
6.1 Conclusions ....................................................................................................... 160
6.2 Recommended Procedures ................................................................................ 164
REFERENCES ............................................................................................................... 169
APPENDIX A : EFFECTS OF GROUND MOTION PARAMETERS ON SHORT-SPAN
BRIDGE .............................................................................................................. 174
A. 1 Free Field Ground Motions .............................................................................. 174
A. 2 EMDR and Ground Motion Parameters .......................................................... 174
APPENDIX B : EQUIVALENT LINEAR SYSTEM OF ISOLATION BEARINGS AND
SEISMIC ANALYSIS RESULTS .................................................................................. 180
x i
B. 1 Equivalent Linear System ............................................................................... 180
B. 2 Seismic Analysis Results from NP- Model ...................................................... 180
APPENDIX C : EFFECTS OF GROUND MOTION PARAMETERS ON ISOLATED
BRIDGE .......................................................................................................................... 187
C. 1 Effects on Ductility Ratio................................................................................. 187
C. 2 Effects on Equivalent Linearization of Isolation Bearings .............................. 191
C. 3 Effects on EMDR ............................................................................................. 195
APPENDIX D : MECHANICALLY STABILIZED EARTH ( MSE) WALLS .............. 207
D. 1 Mechanically Stabilized Walls ......................................................................... 200
D. 1.1 MSE Wall Design ...................................................................................... 201
D. 1.2 Initial Design Steps ................................................................................... 203
D. 2 MSE Wall Example Problem 1 ......................................................................... 238
APPENDIX E : ABUTMENT DESIGN EXAMPLE ..................................................... 254
E. 1 Given Conditions ............................................................................................... 254
E. 2 Permanent Loads ( DC & EV) ........................................................................... 256
E. 3 Earthquake Load ( AE) ....................................................................................... 257
E. 4 Live Load Surcharge ( LS) ................................................................................. 258
E. 5 Design Piles ....................................................................................................... 259
x ii
E. 6 Check Shear in Footing ..................................................................................... 262
E. 7 Design Footing Reinforcement ......................................................................... 266
E. 7.1 Top Transverse Reinforcement Design for Strength Limit State ............... 266
E. 7.2 Bottom Transverse Reinforcement Design for Strength Limit State ......... 270
E. 7.3 Longitudinal Reinforcement Design for Strength Limit State ................... 273
E. 8 Flexural Design of the Stem .............................................................................. 275
E. 9 Splice Length ..................................................................................................... 281
E. 10 Flexural Design of the Backwall ( parapet) ..................................................... 282
x iii
List of Figures
Figure 3.2.1 Non- proportional damping of short- span bridge ......................................... 25
Figure 3.3.1 Flow chart of optimization method .............................................................. 35
Figure 3.3.2 Frequency response function for different EMDR ....................................... 36
Figure 4.1.1 Analysis procedure ...................................................................................... 47
Figure 4.2.1 Description of PSO and sensor locations .................................................... 49
Figure 4.2.2 Cape Mendocino/ Petrolina Earthquake in 1992 ........................................... 50
Figure 4.2.3 Acceleration response at embankment of PSO ............................................ 51
Figure 4.2.4 Acceleration response at deck of PSO ......................................................... 51
Figure 4.3.1 Finite element model of PSO ....................................................................... 53
Figure 4.3.2 Optimization algorithm for estimating boundary condition ......................... 55
Figure 4.3.3 Comparison of response time history .......................................................... 56
Figure 4.3.4 Comparison of power spectral density ........................................................ 56
Figure 4.3.5 Mode shape of PSO ..................................................................................... 58
Figure 4.4.1 Element degree- of- freedom ......................................................................... 60
Figure 4.4.2 Response time history from CMA method .................................................. 63
Figure 4.4.3 Response time history from NODE method ................................................ 66
Figure 4.4.4 Response time from CDR method ............................................................... 69
Figure 4.4.5 FRF of NP- Model and P- Model after optimization .................................... 74
Figure 4.4.6 Response time history from OPT method in time domain .......................... 75
Figure 4.4.7 Response time history from OPT method in frequency domain ................. 75
Figure 4.5.1 Relative error of response spectrum method with measured response ....... 79
x iv
Figure 4.6.1 Relationship between ground motion intensity and EMDR ........................ 83
Figure 5.1.1 Analysis procedure diagram ......................................................................... 89
Figure 5.2.1 Bi- linear hysteretic force- displacement model of isolator ........................... 91
Figure 5.3.1 Isolated bridge model ................................................................................... 95
Figure 5.3.2 Effective stiffness of isolation bearing ( α = 0.154) ..................................... 97
Figure 5.3.3 Effective damping ratio of isolation bearing ( α = 0.154) ............................ 97
Figure 5.3.4 Mode shape of isolated bridge case ............................................................. 99
Figure 5.4.1 Definition of deck and pier top displacement ............................................ 103
Figure 5.4.2 Ratio of maximum to minimum response of deck and pier top ................. 103
Figure 5.5.1 Effective stiffness of isolation bearing P- 3 ................................................ 106
Figure 5.5.2 Effective stiffness of isolation bearing P- 3 with ductility ratio ................ 106
Figure 5.5.3 Dissipated energy of equivalent linear system and bilinear model ............ 111
Figure 5.5.4 Damping ratio of isolator P- 3 ..................................................................... 112
Figure 5.5.5 Damping coefficient of isolator P- 3 ........................................................... 112
Figure 5.5.6 Damping ratio vs. ductility ratio of isolator P- 3 ......................................... 113
Figure 5.5.7 Damping coefficient vs. ductility ratio of isolation bearing P- 3 ................ 113
Figure 5.5.8 Relative error of AASHTO method ........................................................... 116
Figure 5.5.9 Relative error of Caltrans 94 method ......................................................... 116
Figure 5.5.10 Relative error of Caltrans 96 method ....................................................... 117
Figure 5.5.11 Relative error with average ductility ratio by AASHTO method ............ 117
Figure 5.5.12 Relative error with average ductility ratio by Caltrans 94 method .......... 118
Figure 5.5.13 Relative error with average ductility ratio by Caltrans 96 method .......... 118
Figure 5.6.1 EMDR from CMA method ......................................................................... 126
Figure 5.6.2 EMDR from NODE method ....................................................................... 126
Figure 5.6.3 Relative mode shape amplitude of isolated bridge deck ............................ 129
x v
Figure 5.6.4 EMDR from composite damping rule method ........................................... 129
Figure 5.6.5 EMDR from time domain optimization method ........................................ 132
Figure 5.6.6 EMDR from AASHTO method .................................................................. 134
Figure 5.6.7 EMDR from Caltrans 94 method ............................................................... 135
Figure 5.6.8 EMDR from Caltrans 96 method ............................................................... 135
Figure 5.6.9 EMDR with ductility ratio from AASHTO method ................................... 136
Figure 5.6.10 EMDR with ductility ratio from Caltrans 94 method ............................... 136
Figure 5.6.11 EMDR with ductility ratio from Caltrans 96 method ............................... 137
Figure 5.7.1 Relative error from AASHTO method ....................................................... 141
Figure 5.7.2 Relative error from Caltrans 94 method ..................................................... 142
Figure 5.7.3 Relative error of Caltrans 96 method ......................................................... 143
Figure 5.8.1 Damping ratio from AASHTO method ( CMA) ......................................... 146
Figure 5.8.2 Damping ratio from Caltrans 94 method ( CMA) ...................................... 146
Figure 5.8.3 Damping ratio from Caltrans 96 method ( CMA) ...................................... 147
Figure 5.8.4 Comparison of RMSE ( Ductility ratio < 15) ............................................... 147
Figure 5.8.5 Comparison of RMSE ( Ductility ratio > 15) .............................................. 148
Figure 5.9.1 Ductility ratio and response spectrum intensity and energy dissipation index
............................................................................................................................... .. 151
Figure 5.9.2 Ductility ratio and peak ground acceleration ............................................ 151
Figure 5.9.3 Effective stiffness and response spectrum intensity ................................... 153
Figure 5.9.4 Effective stiffness and energy dissipation index ........................................ 153
Figure 5.9.5 Effective damping ratio and response spectrum intensity .......................... 154
Figure 5.9.6 Effective damping ratio and energy dissipation index ............................... 154
Figure 5.9.7 EMDR with ground motion parameters ( AASHTO) ................................. 156
Figure 5.9.8 EMDR with ground motion parameters ( Caltrans 94) ............................... 156
x vi
Figure 5.9.9 EMDR with ground motion parameters ( Caltrans 96) ............................... 157
Figure 6.2.1 Recommended procedure for EMDR ......................................................... 168
Figure A. 1.1 Cape Mendocino Earthquake in 1986 ....................................................... 175
Figure A. 1.2 Cape Mendocino Earthquake in 1986 ( Aftershock) ................................. 175
Figure A. 1.3 Cape Mendocino Earthquake in 1987 ...................................................... 176
Figure A. 1.4 Cape Mendocino/ Petrolina Earthquake in 1992 ....................................... 176
Figure A. 1.5 Cape Mendocino/ Petrolina Earthquake in 1992 ( Aftershock 1) ............... 177
Figure A. 1.6 Cape Mendocino/ Petrolina Earthquake in 1992 ( Aftershock 2) .............. 177
Figure A. 2.1 PGA and EMDR ........................................................................................ 178
Figure A. 2.2 Time duration and EMDR ......................................................................... 178
Figure A. 2.3 Ground motion intensity and EMDR ( 1) .................................................. 179
Figure A. 2.4 Ground motion intensity and EMDR ( 2) .................................................. 179
Figure C. 1.1 Ductility ratio and time duration parameters ............................................. 188
Figure C. 1.2 Ductility ratio and intensity parameters ..................................................... 188
Figure C. 1.3 Ductility ratio and damage parameters ...................................................... 189
Figure C. 1.4 Ductility ratio and spectrum intensity parameters ..................................... 189
Figure C. 1.5 Ductility ratio and peak ground acceleration ............................................. 190
Figure C. 1.6 Ductility ratio and response spectrum intensity and energy dissipation index
............................................................................................................................... .. 190
Figure C. 2.1 Effects of PGA on equivalent linearization .............................................. 192
Figure C. 2.2 Effects of RSI on equivalent linearization ................................................. 193
Figure C. 2.3 Effects of EDI on equivalent linearization ................................................ 194
Figure C. 3.1 EMDR with PGA ....................................................................................... 195
Figure C. 3.2 EMDR with root mean square acceleration ............................................... 196
Figure C. 3.3 EMDR with average intensity ................................................................... 196
x vii
Figure C. 3.4 EMDR with bracketed duration ................................................................. 197
Figure C. 3.5 EMDR with acceleration spectrum intensity ............................................. 197
Figure C. 3.6 EMDR with effective peak acceleration .................................................... 198
Figure C. 3.7 EMDR with effective peak velocity .......................................................... 198
Figure C. 3.8 EMDR with cumulative intensity .............................................................. 199
Figure C. 3.9 EMDR with Cumulative absolute velocity ................................................ 199
Figure D. 1.1 Potential external failure mechanisms for MSE walls. ............................ 202
Figure D. 1.2 Pressure diagram for MSE walls .............................................................. 205
Figure D. 1.3 Distribution of stress from concentrated vertical load ............................. 206
Figure D. 1.4 Distribution of stress from concentrated horizontal loads for external and
internal stability calculations ................................................................................... 207
Figure D. 1.5 Pressure diagram for MSE walls with sloping backslope ........................ 208
Figure D. 1.6 Pressure diagram for MSE walls with broken backslope ......................... 209
Figure D. 1.7 Calculation of eccentricity for sloping backslope condition .................... 211
Figure D. 1.8 Potential failure surface for internal stability design of MSE wall .......... 216
Figure D. 1.9 Variation of the coefficient of lateral stress ratio with depth .................... 218
Figure D. 1.10 Definition of b, Sh, and Sv ...................................................................... 220
Figure D. 1.11. Mechanisms of pullout resistance .......................................................... 222
Figure D. 1.12 Typical values for F* ............................................................................... 227
Figure D. 1.13. Cross section area for strip ..................................................................... 230
Figure D. 1.14 Cross section area for bars ....................................................................... 231
Figure D. 2.1 Wall section with embedded rebar. ........................................................... 239
Figure D. 2.2 Wall face panels and spacing between reinforcements ............................. 239
Figure D. 2.3 Determining F* using interpolation ........................................................... 251
Figure E. 1.1 Example cross- section for the abutment ................................................... 255
x viii
Figure E. 5.1 Summary of permanent Loads ................................................................... 260
Figure E. 8.1 Load diagram for stem design .................................................................... 277
Figure E. 8.2 Location of neutral Axis ............................................................................ 278
Figure E. 10.1 Load diagram for backwall design ........................................................... 284
x ix
List of Tables
Table 4.1.1 Summary of validation check ........................................................................ 46
Table 4.2.1 Peak acceleration of earthquake recording ................................................... 51
Table 4.3.1 Element properties of finite element model of PSO ...................................... 53
Table 4.3.2 Effective stiffness and damping coefficient of PSO boundary ..................... 55
Table 4.3.3 Natural frequency and period of PSO ............................................................ 57
Table 4.4.1 Eigenvalues and natural frequencies of NP- Model of PSO .......................... 61
Table 4.4.2 EMDR of PSO by CMA method .................................................................. 62
Table 4.4.3 Undamped natural frequency and EMDR from CMA method ..................... 62
Table 4.4.4 Modal damping matrix ([ φ ] T [ c][ φ ] ) ............................................................. 64
Table 4.4.5 EMDR from NODE method .......................................................................... 65
Table 4.4.6 Modal coupling parameter ............................................................................ 66
Table 4.4.7 Potential energy ratio in CDR method .......................................................... 68
Table 4.4.8 EMDR from CDR method ............................................................................. 68
Table 4.4.9 EMDR from OPT method in time domain .................................................... 72
Table 4.4.10 Summary of EMDR identified by each method .......................................... 76
Table 4.4.11 Summary of peak acceleration from each method ....................................... 76
Table 4.4.12 Summary of peak displacement from each method ..................................... 76
Table 4.5.1 Acceleration from response spectrum method ( unit : g) .............................. 78
Table 4.5.2 Displacement from response spectrum method ( unit : cm) .......................... 78
Table 4.6.1 List of ground motion parameters ................................................................. 81
Table 4.6.2 Peak acceleration of earthquake and EMDR ................................................ 82
x x
Table 4.6.3 Prediction of EMDR by ground motion parameters ..................................... 82
Table 5.3.1 Element properties of example bridge .......................................................... 95
Table 5.3.2 Characteristic values of isolator..................................................................... 95
Table 5.3.3 Preliminary modal analysis of example bridge ............................................. 98
Table 5.3.4 Description of ground motion group ........................................................... 100
Table 5.4.1 Seismic displacement from Bi- linear model ............................................... 102
Table 5.5.1 RMSE of linearization method .................................................................... 115
Table 5.6.1 Eigenvalues and natural frequencies of NP- Model .................................... 121
Table 5.6.2 EMDR of example bridge by CMA method ............................................... 122
Table 5.6.3 Undamped natural frequency and EMDR from CMA method ................... 122
Table 5.6.4 Modal damping matrix ([ φ ] T [ c][ φ ] ) ........................................................... 123
Table 5.6.5 EMDR from NODE method ........................................................................ 124
Table 5.6.6 Modal coupling parameter .......................................................................... 124
Table 5.6.7 Potential energy ratio in CDR method ........................................................ 128
Table 5.6.8 EMDR from CDR method ........................................................................... 128
Table 5.6.9 EMDR from OPT method in time domain .................................................. 132
Table 5.6.10 Approximation of EMDR base on ductility ratio ...................................... 134
Table 5.7.1 RMSE of modal combination results with Bi- linear Model results ............ 140
Table 5.8.1 Damping coefficient ( AASHTO Guide, 1999) ............................................ 144
Table B. 1.1 Equivalent linearization of isolation bearing by AASHTO method ........... 181
Table B. 1.2 Equivalent linearization of isolation bearing by Caltrans 94 method ......... 182
Table B. 1.3 Equivalent linearization of isolation bearing by Caltrans 96 method ......... 183
Table B. 2.1 Displacement from NP- Model by AASHTO .............................................. 184
Table B. 2.2 Displacement from NP- Model by Caltrans 94 ............................................ 185
Table B. 2.3 Displacement from NP- Model by Caltrans 96 ............................................ 186
x xi
Table D. 1.1 Minimum embedment requirements for MSE walls ................................... 204
Table D. 1.2 Load factors and load combinations .......................................................... 209
Table D. 1.3 Typical values for α ................................................................................. 224
Table D. 1.4. Resistance factors for tensile resistance..................................................... 229
Table D. 1.5 Installation damage reduction factors ......................................................... 234
Table D. 1.6. Creep reduction factors ( RFCR) ................................................................ 235
Table D. 1.7 Aging reduction factors ( RFD) ................................................................... 236
Table D. 2.1 Equivalent height of soil for vehicular loading ( after AASHTO 2007) ..... 240
Table D. 2.2 Unfactored vertical loads and moment arm for design example ................ 242
Table D. 2.3 Unfactored horizontal loads and moment arm for design Example ........... 242
Table D. 2.4 Factored vertical loads and moments .......................................................... 243
Table D. 2.5 Factored horizontal loads and moments ..................................................... 243
Table D. 2.6 Summary for eccentricity check ................................................................. 243
Table D. 2.7 Program results – direct sliding for given layout ........................................ 245
Table D. 2.8 Summary for checking bearing resistance .................................................. 247
Table D. 2.9 Program results – strength with L= 20 ft ..................................................... 251
Table D. 2.10 Program results – pullout with L= 20 ft ..................................................... 252
Table D. 2.11 Program results – pullout with L= 34 ft ..................................................... 253
Table E. 1.1 Material and design parameters ................................................................... 255
Table E. 4.1 Vertical load components and moments about toe of footing ..................... 258
Table E. 4.2 Horizontal load components and moments about bottom of footing .......... 259
Table E. 5.1 Force resultants ........................................................................................... 259
Table E. 5.2 Pile group properties ................................................................................... 259
x xii
Acknowledgement
Financial support for this study was provided by the California Department of
Transportation under Grant RTA- 59A0495. The valuable advices to this study and
review of the report by Dr. Joseph Penzien are greatly appreciated.
1
Chapter 1
INTRODUCTION
This chapter first describes the motivations of this research for the determination of the
effective damping of highway bridges then summarizes the objectives and overall scope
of the research followed by the organization of this report.
1.1 Background
For the seismic design of ordinary bridges, current design specifications require the use
of the modal- superposition- response- spectrum approach. It involves the following steps:
( 1) A three- dimensional space frame model of the bridge is developed with mass and
stiffness matrices assembled. ( 2) Eigen analysis of this model is performed, usually
using finite element analysis software, to obtain the undamped frequencies and mode
shapes of the structure. A minimum of three times the number of spans or 12 modes are
selected. ( 3) Assuming classical ( i. e. proportional) Rayleigh’s viscous damping, the
equations of motion are reduced into individual decoupled modal equations, each of
which can be envisioned as the motion equation for a corresponding single- degree- of-freedom
( SDOF). ( 4) The seismic response for each of the selected modes to the design
earthquake is evaluated using the specified SDOF acceleration response spectrum curve.
( 5) Combine the peak responses of all selected modes using the square- root- of- sum- of-
2
squares ( SRSS) or complete- quadratic- combination ( CQC) rule resulting in maximum
demands that the structure is designed to sustain.
The response spectrum method is based on the assumption of proportional damping
characteristics in the structure with a 5% modal damping ratio for all the selected modes.
However, if a bridge has some components that are expected to have significant damping,
the conventional 5% damping ratio is not likely to be a reasonable assumption.
Therefore, in the cases of short- span bridges under strong ground motion and fully- or
partially- isolated bridges which have isolators with extremely high damping, an
appropriate damping ratio should be determined for each mode to provide a more
economic and accurate design or seismic retrofit plan.
Resulting from several previous seismic observations and studies by other researchers, it
was found that the concrete structure of short- span bridges behaves within the elastic
range and sustains no damage, even under strong earthquakes, which can be attributed to
the significant restraint and energy dissipation at the boundaries of these bridges.
Through the analysis of valuable earthquake response data recorded at several bridge
sites, the energy dissipation capacity of abutment- embankment and column boundaries of
short- span bridges has been highlighted. In many previous studies, damping ratios much
greater than 5% had to be used so that simulated responses would match well with the
recorded ones. Therefore, when short- span brides are designed to sustain strong ground
motion, a rational damping ratio for each mode should be found considering the damping
effects of the bridge boundaries.
A seismically isolated bridge is another type of bridge with high damping components.
In order to prevent damage resulting from seismic hazards, isolation bearing devices have
3
been commonly adopted in highway bridges. The isolation bearings alleviate seismic
damage by shifting the first mode natural period of the original, un- isolated bridge into
the region of lesser spectral acceleration and through the high dissipation of energy in the
isolation bearings. Even for the seismic design of isolated bridges, many design guides
such as the American Association of State Highway and Transportation Officials
( AASHTO) Guide ( 2000), Japan Public Works Research Institute, and California
Department of Transportation ( Caltrans) adopt an equivalent linear analysis procedure
utilizing an equivalent linear system for the isolation bearings and providing appropriate
linear methods for estimating seismic response.
To develop a rational and systematic approach for evaluating modal damping in a
structural system comprised of components with drastically different damping ratios,
there arise a problems of fundamental theoretical interest. It has been well- established
that only when a system is viscously damped with a damping matrix that conforms to the
form identified by Caughey and O’Kelly ( 1965) can the damping matrix be diagonalized
by the mode shape matrix. This system is said to be classically ( or proportionally)
damped for which the classical uncoupled modal superposition method applies.
Unfortunately, the damping matrix of a system consisting of components with
significantly different damping ratios is non- classical, such as the cases of short- span
bridges and seismically isolated bridges. Usually, the embankments of short- span bridges
and the base- isolation devices have equivalent damping ratios as high as 20- 30% under
strong ground motion, while the equivalent damping ratio of the rest of the concrete
structural system can usually be reasonably approximated as 5%. Though the nonlinear
behavior and damping of bridge boundaries and isolation bearings can be approximated
by an equivalent linear system which is composed of effective stiffnesses and effective
4
damping coefficients, the damping matrix of the entire bridge system as described will be
non- classical having important off- diagonal terms that cannot be diagonalized by the
mode shape matrix. Therefore, the response spectrum method cannot be rigorously
applied to non- classically damped systems.
1.2 Effective Modal Damping
To keep the design procedure within the framework of the modal superposition method,
which is the current dynamic design procedure favored by engineers/ designers,
compromise has to be made to approximate the non- classical damping by a classical
damping matrix. A usual approach for this purpose is as follows. Let C = φ TCφ , where
C is the non- classical damping matrix of the system, φ is the mode shape matrix
associated with the undamped system, and φ T is the transposed mode shape matrix. C
can have substantial off- diagonal terms that produce coupling of the normal modes.
Ignoring the off- diagonal terms results in a classical damping matrix, C′ , whose
elements ij c′ relate to the elements of C , by ii ii c′ = c and ′ = 0 ij c when i ≠ j . This
approximation, which is defined as the neglecting off- diagonal elements ( NODE) method,
has been widely used in many studies.
Veletsos and Ventura ( 1986) proposed a critical and exact approach to generalize the
modal superposition method for evaluating the dynamic response of non- classically
damped linear systems. This approach begins by first rewriting the second order
equation of motion into a first order equation in state space, and then by carrying out a
complex- valued eigen analysis giving complex- valued characteristic values and
characteristic vectors for the system. Examining carefully the physical meaning of each
5
pair of conjugated characteristic values and associated characteristic vectors, the authors
were able to interpret each of these pairs as a mode similar to a SDOF system, except that
the mode shape has different configurations at different times, varying periodically. A
damping ratio was obtained for each of these ‘ modes’, and the dynamic response of the
system was represented in terms of modal superposition. This method is defined as the
complex modal analysis ( CMA) method in this study. A variety of system configurations
were investigated through this method and the results were compared to those from the
NODE method described above. It was concluded that while the agreement between
these two methods is generally reasonable, there can be significant differences in the
damping ratios and dynamic responses, particularly when much higher damping ratios
are present in some components of the complete system.
A semi- empirical and semi- theoretical approach, referred to as the composite damping
rule ( CDR) was suggested by Raggett ( 1975). In this approach, energy dissipation in
different components is estimated empirically under the assumption that the mode shapes
and frequencies of a damped system remain the same as those of the undamped system.
Energy dissipation in different components of a certain mode can be summed up to reach
an estimate of the total energy dissipation of the system in this mode, such that an
effective modal damping ratio ( EMDR) for this mode may be obtained. This method has
been adopted by many other studies ( Lee et al, 2004; Chang et al, 1993; Johnson and
Kienholz, 1982).
6
1.3 Objectives and Scope
Various methods have been studied in the literature for evaluating damping in a complex
structural system, but they have never been compared and evaluated in a systematic way
based on available seismic records. Therefore, the overall objective of this research is to
study the fundamental issue of damping in complex bridge structural systems involving
significantly different damping components ( such as short bridges and fully isolated
bridges) and to develop a more rational damping estimation method for improving
dynamic analysis results and the seismic design of such bridges. Another objective is to
relate the effective system damping with ground motion intensity.
In order to achieve these objectives, selected methods are investigated for their ability to
compute the effective system damping of short- span and seismically isolated bridges.
The detailed explanation of each method is given in Chapter 3 following the literature
review on the damping of such bridges in Chapter 2.
The application of the damping estimating methods to a short- span bridge is investigated
in Chapter 4. The Painter Street Overcrossing ( PSO) was chosen as an example bridge
due to the fact that this bridge has invaluable earthquake response data recorded during
strong earthquakes. Utilizing the measured data, the equivalent linear systems of the
bridge boundaries were identified and then each damping estimating method was applied
to compute the effective system damping of the bridge. The validation of the damping
estimating methods was carried out by comparing the modal combination results with the
recorded bridge response data.
In Chapter 5, the application of the methods to a seismically isolated bridge is
demonstrated. Because of the scarcity of measured data from isolated bridges, an
7
example bridge is assumed in this study. Under many earthquake ground motions, the bi-linear
hysteretic behavior of each isolation bearing is approximated with an equivalent
linear viscoelastic element. Afterwards, the damping estimating methods are applied to
compute the effective system damping of the bridge. These methods are verified by
comparing the results found through the standard response spectrum method with the
results obtained from a non- linear seismic analysis. Also, the effective system damping
is related with the characteristics of ground motions.
Finally, conclusions of this research are presented in Chapter 6 along with recommended
future research.
8
Chapter 2
LITERATURE REVIEW
In this chapter, previous studies to understand the impact of the significant energy
dissipation in short- span bridges and isolated bridges on the dynamic response of the
bridges are reviewed. Also, many attempts to find the effective system damping of such
bridges are also described. From the literature review, several important conclusions are
derived to guide this research.
2.1 Energy Dissipation and EMDR of Short- span
Bridges
A short- span bridge has a superstructure constructed to be connected directly to wing-walls
and an abutment at one or each end of the bridge. It has a relatively long
embankment compared with bridge length. In the 1970s, investigating the influence of
the embankment on the dynamic response of such bridges started ( Tseng and Penzien,
1973; Chen and Penzien, 1975, 1977). It was found that the monolithic type of abutment
and embankment typical of short- span bridges has drastic effects on the bridge behavior
under strong ground motions. Because of a long embankment and relatively small size of
the bridge, most of the input energy is dissipated through the embankment soil during
earthquakes and the bridge behaves essentially as a rigid body in the elastic range of the
9
structure. In modern earthquake engineering, appropriate modeling of bridge boundaries
has become one of the important factors in seismic analysis and many efforts have been
focused on identification of a damping ratio for the soil boundary during strong
earthquakes. However, it is essential that any reasonable estimate of this damping should
be based on recorded earthquake data from similar structures.
One of the most valuable data sets available is from the vibration measurements at the
Meloland Road Overpass ( MRO) during the 1979 Imperial Valley earthquake.
Analyzing the data Werner et al. ( 1987) found that this 2- span RC box- girder, single-columned
short bridge with monolithic abutments exhibited two primary modes: the
vertical mode mainly involved the vertical vibration of the superstructure, having a
damping ratio of 6.5%; the transverse mode mainly involved the horizontal translation of
the abutments and the superstructure, inducing bending in the single- column pier,
coincidently having a damping ratio of 6.5%. These modal damping ratios are slightly
higher than the 5% used in design. However, these modes, especially the transverse
mode, involve substantial movement of the abutments. This further implies that the soil
disturbance and friction between the abutments and the soil most likely may have
contributed a large portion of the energy dissipation, leading to a higher damping ratio.
Another set of important earthquake data was recorded at the Painter Street Overpass
( PSO) from which McCallen and Romstad ( 1994) tried to determine the effective system
damping of the PSO. The authors built, as well as a stick model, a full three dimensional
model of the bridge including abutment, pile foundation, and boundary soil using solid
elements. Based on the CALTRANS method, the effective stiffness for the embankment
soil and pile foundation was computed for their stick model and they tried to simulate the
measured bridge response by updating the EMDR of the entire bridge model. Through
10
extensive trial and error, it was found that the EMDR was 20% and 30% for the
transverse and longitudinal modes, respectively.
Utilizing the same measured data at the PSO, the spring force and damping force of the
abutment of the PSO were identified by Goel and Chopra ( 1995). In their study the
spring force and damping force of the abutment were combined as one force. By drawing
the slope line on the force- displacement diagram acquired through the force identification
procedure, the authors could compute the time variant abutment stiffness. Also, they
found that under the less intense earthquakes the force- displacement diagram showed an
elliptical shape which implies linear viscoelastic behavior of the abutment system,
however, it showed significant nonlinearity of the system under stronger ground motion.
Though the damping effect of the abutment system could be obtained from the force-displacement
diagram, the effective system damping of the entire bridge system was not
studied.
The quantification of the EMDR based on the deformation of the abutment system during
an earthquake was attempted by Goel ( 1997). After observing the relation between the
EMDR and the abutment flexibility, he suggested a simple formula by which the EMDR
could be computed. Using this proposed formula and six earthquake ground motions, he
identified the EMDR of the PSO as ranging from 5 to 12%. However, the upper bound
of the EMDR was limited to 15% in his equation.
Though there have been many studies on the identification of the effective system
damping of short- span bridges under strong ground motions, few studies have been done
on the formulation to compute the effective stiffness and damping of the bridge boundary.
However, Wilson and Tan ( 1990) developed simple explicit formulae to represent the
11
embankment of short- or medium- span bridges with linear springs based on the plane
strain analysis of embankment soil. The spring stiffness per unit length of embankment
was expressed as a function of embankment geometry ( i. e. width, height, and slope) and
the shear modulus of the embankment soil. The total spring stiffness was obtained by
multiplying the embedded length of the wing wall by the unit spring stiffness. The
authors applied the method to the MRO. Utilizing the recorded data, the damping ratio
of the embankment soil was found as 20- 40%, however, the damping ratio of the entire
bridge system was determined to range from 3 to 12%. It should be noted that while an
equivalent spring stiffness was developed to model the embankment only, they used it
for the combined abutment- embankment system.
A comprehensive study on the approximation of an equivalent linear system for an
abutment- embankment system of short- span bridges was done by Zhang and Makris
( 2002). Based on previous research, they suggested a systematic approach to compute
the frequency- independent spring and viscous damping coefficient of embankment and
pile groups at the abutments and bridge bents. In their derivation, the embankment was
represented by a one- dimensional shear beam and the solution of the shear beam model
under harmonic loading was used to compute the spring stiffness and damping coefficient
of the embankment. Applying their method to the PSO and MRO, they found the
equivalent linear system of the bridge boundaries. From the complex modal analysis, the
EMDR was found as 9% ( transverse), and 46% ( longitudinal) for the PSO and 19%
( transverse), and 57% ( longitudinal) for the MRO, respectively.
Kotsoglou and Pantazopoulou ( 2007) established an analytical procedure to evaluate the
dynamic characteristics and dynamic response of an embankment under earthquake
excitation. Instead of using the one dimensional shear beam model used by Zhang and
12
Makris ( 2002), the author developed a two dimensional equation of motion for the
embankment and solved it to investigate the dynamic characteristics of the embankment.
From the application of their method to the PSO embankment, the modal damping ratio
of the embankment was found to be 25% in the transverse direction.
Based on bridge damping data base, Tsai et al. ( 1993) investigated appropriate damping
ratio for design of short- span bridges in Caltrans. Though the data base was composed of
53 bridges including steel and concrete bridges, as indicated by the authors, the identified
damping ratios cannot be adopted for seismic design because most of the data base were
from free or forced vibration excitation with well below 0.1g, except two earthquake
excitation data. The authors recommended to use damping ratio of 7.5% for seismic
design when a SSI parameter satisfies a criterion and to investigate the composite
damping rule method for computing effective system damping ratio of short- span bridges.
2.2 Energy Dissipation and EMDR of Isolated
Bridge
The prevention of seismic hazards in highway bridges by installing isolation bearings is
increasingly adopted now days in construction of new bridges and in seismic retrofit of
old bridges ( Mutobe and Cooper, 1999; Robson et al., 2001; Imbsen, 2001; Dicleli, 2002;
Dicleli et al., 2005). The isolation bearing has relatively smaller stiffness than the bridge
column and decouples the superstructure from the substructure such that the substructure
can be protected from the transfer of inertial force from the massive superstructure. From
the viewpoint of response spectrum analysis, the isolation bearings elongate the natural
period of an isolated bridge so that the spectral acceleration which the isolated bridge
13
should sustain becomes less than that of the un- isolated bridge. Among many types of
isolation bearing devices such as rubber bearing, lead rubber bearing, high damping
rubber bearing, friction pendulum bearing, rolling type bearing, and so on, the most
commonly used isolation bearing is the lead rubber bearing ( LRB). In North America,
154 bridges out of the 208 isolated bridges are installed with LRBs ( Buckle et al., 2006).
To approximate the mechanical behavior of an isolation bearing, the Bouc- Wen model
( Wen, 1976; Baber and Wen, 1981; Wong et al., 1994a, 1994b, Marano and Sgobba,
2007) and the bi- linear model ( Stehmeyer and Rizos, 2007; Lin et al., 1992; Roussie et
al., 2003; Jangid, 2007; Katsaras et al., 2008; Warn and Whittaker, 2006) have been most
commonly used. In contrast to the bi- linear model, the Bouc- Wen model can simulate
the smooth transition from elastic to plastic behavior and many kinds of hysteretic loops
can be generated using different combinations of model parameters. While the bi- linear
model can be thought of as one special case of the Bouc- Wen model, it can easily model
any type of isolation bearing ( Naeim and Kelly, 1999).
Turkington et al. ( 1989) suggested a design procedure for isolated bridges. In their
procedure, the EMDR of an isolated bridge is computed by simply adding together the
damping ratios of the isolation bearing and the concrete structure. The damping ratio of
the isolation bearing is found using the bi- linear model and 5% is assigned for the
concrete structure.
Hwang and Sheng ( 1993) suggested an empirical formulation to compute the effective
period and effective damping ratio of individual isolation bearings represented by the bi-linear
model. Their method is based on the work of Iwan and Gates ( 1979) which
indicates that the maximum inelastic displacement response spectrum can be
14
approximated by using the elastic response spectrum and adopting an effective period
shift and effective damping ratio of the inelastic SDOF system. The work of the authors
was extended to compute the effective linear stiffness and effective system damping ratio
of an isolator- bridge column system ( Hwang et al., 1994). To compute the EMDR of the
isolator- bridge column system, they applied the composite damping rule method.
However, the original work of Iwan and Gates was developed for ductility ratios of 2, 4,
and 8 which is too small for isolated bridges under strong earthquakes.
Considering the large ductility ratio of isolation bearings, Hwang et al. ( 1996) proposed a
semi- empirical formula to approximate the equivalent linear system of isolation bearings.
The suggested equations, which were modified from the AASHTO method, were found
by optimizing the effective stiffness and damping ratio under 20 ground motions using
the same algorithm by Iwan and Gates.
A comprehensive study for the equivalent linear approximation of hysteretic materials
was done by Kwan and Billington ( 2003). The authors considered six types of hysteretic
loops and proposed a formula to compute the effective linear system based on Iwan’s
approach ( Iwan, 1980). In their study, the effective period shift was assumed to be
related to the ductility ratio, and the effective damping ratio to both the effective period
shift and ductility ratio. However, since only a small range of ductility ratios ( i. e. from 2
to 8) was considered, which is too low for isolated bridges, it should be verified that this
method is applicable to isolation bearings. The important finding from this study was
that the effective damping ratio of a hysteretic material increases with increase of the
ductility ratio, even in the case of no hysteretic loop. This observation shows that the
direct summation of the damping ratios of the isolation bearing and concrete structure
might be incorrect.
15
Dall’Asta and Ragni ( 2008) approximated a non- linear, high damping rubber with an
effective linear system during both stationary and transient excitation. The effective
stiffness of the linear system was estimated from the secant stiffness at the maximum
displacement of the force- displacement plot and the effective damping ratio was found by
equating the dissipated energy from the non- linear system and the effective linear system.
Regarding soil- structure interaction in isolated bridges, there is relatively little literature;
however, several published papers have investigated this effect. Tongaonkar and Jangid
( 2003) studied the influence of the SSI on the seismic response of three- span isolated
bridges considering four different soil types ( soft, medium, hard, and rigid). In their
simulation, the soil- pile foundation was modeled with a frequency independent spring-viscous
damping- mass system. The authors concluded that the SSI increases the
displacement of the isolation bearing located at the abutments only, while it decreases
other responses such as deck acceleration, pier base shear, and isolation bearing
displacement at the piers.
Ucak and Tsopelas ( 2008) investigated the effect of the SSI on two types of isolated
bridges, one being a typical stiff freeway overcrossing and the other a typical flexible
multispan highway bridge, under near fault and far field ground motions. From their
results, the consideration of the SSI does not have much affect on either isolator or pier
response of the stiff freeway overcrossing except for isolator drift under far field ground
motions. In the multispan highway bridge case, the consideration of the SSI was
conservative for the design of the isolator system, but not for the pier design.
16
2.3 Summary
From the literature review on short- span bridges, summaries and conclusions are drawn
as follows:
( 1) From research based on recorded earthquake data, bridge boundary soil was found to
have non- linearity during earthquakes. Though the soil non- linearity can be
represented by a non- linear spring or a frequency- dependent spring and a damping
model, these elements cannot be used directly in the current response- spectrum-based
design method. To be applicable in this response spectrum method, these
elements must be approximated in equivalent linear forms. Therefore, in this
research the bridge boundary is modeled with an equivalent linear system composed
of an elastic spring and viscous damping.
( 2) The results of the EMDR of short- span bridges are quite dependent on how the
bridge and boundaries are modeled and which system identification method is
applied. All previous research was conducted utilizing not only its own bridge
modeling technique but also its own system identification method. That is why the
EMDR from previous studies is not consistent, even for the same bridge under the
same earthquake. Thus, if the identified EMDR of short- span bridges is going to be
used for new design or retrofit planning of such bridges, the modeling of bridges
used in the identification of the EMDR should be consistent with the one used in the
current design practice. In this study, the finite element modeling of a short- span
bridge is established based on the current design practice.
( 3) The inherent damping ratio of the concrete structure of bridges is assumed to be
constant regardless of ground motion intensity but the boundary soil damping
17
changes depending ground motion characteristics. The shear modulus and damping
characteristics of the boundary soil varies with the soil strain. Under relatively
strong earthquakes, soil strain becomes large resulting in small shear modulus and
large damping, and vice versa under weak earthquakes. Considering that the bridge
boundary soil damping varies with the characteristics of the exciting ground motion,
the EMDR is represented as being related with the ground motion intensity in this
research.
From the literature review on isolated bridges, summaries and conclusions are drawn as
follows:
( 1) In many studies, the bi- linear hysteretic model has generally been used to
represent the mechanical behavior of the isolation bearing. Although the Bouc-
Wen model has greater capability than this, it is chosen in this study because it
can be applied to any type of isolation bearing and, more importantly, because
most design specifications ( Guide, 2000; Manual, 1992; Hwang et al., 1994,
1996) make use of it.
( 2) Two different levels of equivalent linearization are involved in isolated bridges:
i) equivalent linearization of the isolation bearing unit, and ii) equivalent
linearization of the entire isolated bridge. So far, most of the previous research
has focused on the development of the equivalent linear system of the isolation
bearing. When there has been a need to compute the EMDR of an entire bridge
system, only the composite damping rule method was adopted. In this study, not
only the composite damping rule method but also other methods are applied and
verified in the framework of the response spectrum method.
18
( 3) The enclosed area of a bi- linear hysteretic loop of the isolation bearing is the
dissipated energy which depends on the maximum displacement of the bearing.
Therefore, the effective damping of an isolation bearing varies depending on the
characteristics of the exciting ground motion. Thus, as in the case of short- span
bridges, the EMDR of an isolated bridge is related to the ground motion
parameters.
19
Chapter 3
EFFECTIVE SYSTEM DAMPING
ESTIMATING METHODS
This chapter describes the basic principles of four effective system damping estimating
methods ( complex modal analysis method, neglecting off- diagonal element in damping
matrix method, optimization method, composite damping rule method) for non-proportionally
damped systems. At the end of this chapter, the pros and cons of each
method are discussed.
3.1 Complex Modal Analysis ( CMA) Method
Depending on the damping characteristics of a system, the mode shapes and natural
frequencies of the system are determined as having either real or complex values. If the
damping is classical ( i. e. proportional), the modal properties are real- valued, otherwise
they are complex- valued. In this method the EMDR is directly computed from the
complex- valued eigenvalue of each mode.
3.1.1 Normal Modal Analysis
The equation of motion of a viscously damped multi- degree- of- freedom ( MDOF) system
excited by ground motion is represented by the equation
20
[ m]{ x( t)} [ c]{ x( t)} [ k]{ x( t)} [ m]{ i} x ( t) && + & + = − && g ( 3- 1)
in which [ m], [ c] and [ k] are the mass, damping, and stiffness matrices of the MDOF
system; { x( t)} is the column vector of the displacement of nodes relative to ground
motion; the dots denote differentiation with respect to time, t ; { i} is the influence
vector ; and x ( t) && g is the acceleration ground motion.
The damping of a MDOF system is defined as proportional damping if and only if it
satisfies the following Caughey criterion ( Caughey and O’Kelley, 1965).
[ c][ m]− 1[ k] = [ k][ m]− 1[ c] ( 3- 2)
For a proportionally damped system, the coupled Eq. ( 3- 1) can be decoupled into single-degree-
of- freedom ( SDOF) systems using normal modal analysis. The solution of each
decoupled SDOF system is computed in modal coordinates and the total solution is
obtained by combining all the individual responses, which is known as the modal
superposition method.
The solution of Eq. ( 3- 1) has the form of { x( t)} = [ Φ]{ q( t)} where [ Φ] is mass
normalized mode shape matrix. Substituting this form into Eq. ( 3- 1) and pre- multiplying
both sides by [ Φ] T goes
[ ] [ m][ ]{ q} [ ] [ c][ ]{ q} [ ] [ k][ ]{ q} [ ] [ m]{ i} x ( t) g
Φ T Φ && + Φ T Φ & + Φ T Φ = − Φ T && ( 3- 3)
Using modal orthoonality relation, Eq. ( 3- 3) can be rewritten for the nth SDOF equation
as
21
q ( t) 2 q ( t) 2q ( t) f ( t) n n n n n n n && + ξ ω & + ω = n = 1, 2, .... ( 3- 4)
in which n ω
is the natural frequency of the nth mode; n ξ is the modal damping ratio of
the nth mode; and f ( t) n is modal force ( ( ) { } [ ]{ } ( ) { } [ ]{ }) n
T
g n
T
n n f t = φ m i & x& t φ m φ .
Thus, the nth mode frequency and damping ratio are
{ } [ ]{ }
{ } [ ]{ }
n
T
n
n
T
n
n m
k
φ φ
φ φ
ω = ( 3- 5)
2 { } [ ]{ }
{ } [ ]{ }
n
T
n n
n
T
n
n m
c
ω φ φ
φ φ
ξ = ( 3- 6)
3.1.2 Complex Modal Analysis and EMDR Estimation
For the proportionally damped system the modal analysis and identification of the
damping ratio is straightforward as illustrated above. However, a non- proportionally
damped system which does not satisfy Eq. ( 3- 2) has complex- valued eigenvectors and
eigenvalues. Because the eigenvectors have different phase at each node of the system,
the maximum amplitude at each node does not occur simultaneously.
Modal analysis is still applicable to the non- proportionally damped system; however, it is
in the modal domain with complex numbers. Veletsos and Ventura ( 1986) generalized
the modal analysis which is applicable to both proportionally and non- proportionally
damped system. In the case of the non- proportionally damped system, Eq. ( 3- 1) can be
decoupled using the complex modal analysis by introducing the state space variables
⎭ ⎬ ⎫
⎩ ⎨ ⎧ =
{ }
{ }
{ }
x
x
z
&
. Equation ( 3- 1) can be transformed to
22
[ A]{ z&}+ [ B]{ z} = { Y ( t)} ( 3- 7)
in which [ A] and [ B] are 2n by 2n real matrices as shown below and { Y( t)} is a 2n
component vector.
⎥⎦
⎤
⎢⎣
⎡
=
[ ] [ ]
[ 0] [ ]
[ ]
m c
m
A , ⎥⎦
⎤
⎢⎣
⎡−
=
[ 0] [ ]
[ ] [ 0]
[ ]
k
m
B , { }
⎭ ⎬ ⎫
⎩ ⎨ ⎧
−
=
[ ]{ } ( )
{ 0}
( )
m i x t
Y t
&& g
The homogeneous solution of Eq. ( 3- 7) is { x} = { ϕ} es t and its characteristic equation
becomes
s([ A] + [ B]){ z} = { 0} ( 3- 8)
The eigenvalues and eigenvectors of Eq. ( 3- 8) are complex conjugate pairs as given by
Eq. ( 3- 9) and ( 3- 10), respectively.
1 2 D n n n n
n
n i i
s
s
= − σ ± ω = − ω ξ ± ω − ξ
⎭ ⎬ ⎫
( 3- 9)
{ } { }
{ }
{ }
n n
n
n ϕ i χ
ψ
ψ
= ±
⎭ ⎬ ⎫
( 3- 10)
Finally, the natural frequency and EMDR of a non- proportionally damped system is
obtained from Eq. ( 3- 9) as
( Re( )) 2 ( Im( )) 2 n n n ω = s + s ( 3- 11)
n
n
n
s
ω
ξ = Re( ) ( 3- 12)
23
where, Re( ) n s and Im( ) n s are the real and imaginary parts of n s .
3.1.3 Procedure of CMA Method
The steps of applying the CMA method are as follows:
Step 1. Establish mass, stiffness, and damping matrix of a bridge system.
Step 2. Compute [ A] and [ B] matrix from Eq. ( 3- 7).
Step 3. Obtain eigenvalues of the characteristic equation shown in Eq.( 3- 8).
Step 4. Compute natural frequency of each mode from corresponding eigenvalue using
Eq. ( 3- 11).
Step 5. Compute effective damping ratio of each mode from real part of eigenvalue and
natural frequency of corresponding mode using Eq. ( 3- 12).
24
3.2 Neglecting Off- Diagonal Elements ( NODE)
Method
In the modal superposition method, the equation of motion of a MDOF system is
transformed into modal coordinates so that the coupled equation may be decoupled
allowing the solution of the MDOF system to be reduced to the solution of many SDOF
systems. However, if the damping matrix is non- proportional, the equation of motion
cannot be decoupled by pre- and post- multiplication by undamped normal mode shapes.
If the off- diagonal elements in this damping matrix are neglected, the MDOF equation of
motion becomes uncoupled allowing the EMDR to be computed from the diagonal
elements.
3.2.1 Basic Principles
The coupled matrix equation of motion, Eq. ( 3- 1), is decoupled by transforming the
original equation into modal coordinates. If the damping of a system is proportional, pre-and
post- multiplication by the mode shape matrix decomposes the damping matrix as
shown in Eq. ( 3- 13)
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
=
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
Φ Φ =
O
O
O
O
0
2
0
0
{ } [ ]{ }
0
[ ] [ ][ ] i i i
T
i
T c φ c φ ω ξ ( 3- 13)
in which [ Φ] is the normal mode shape matrix. From Eq. ( 3- 13) the EMDR for each
mode can be calculated as shown in Eq. ( 3- 6).
25
However, if the damping matrix consists of proportional damping from structure and
local damping from the system boundaries or other damping components, as shown in
Fig. 3.2.1, the overall damping matrix becomes non- proportional and the MDOF equation
cannot be decoupled.
Figure 3.2.1 Non- proportional damping of short- span bridge
As shown in Eq. ( 3- 14), the proportional damping matrix of structure is diagonalized, but
the damping matrix composed of boundary damping can not be diagonalized.
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
+
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
Φ Φ = Φ Φ + Φ Φ =
n n n
i i
n
local i i
T
str
T T
c c
c
c c
c c c c
,1 ,
,
1,1 1,
*
,
0
0
[ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ]
L
M M
L
O
O
( 3- 14)
If the damping effect from the off- diagonal elements in Eq. ( 3- 14) on overall dynamic
response is small, the off- diagonal elements can be neglected and Eq. ( 3- 14) is reduced to
Eq. ( 3- 15). From Eq. ( 3- 15), the EMDR of each mode can be computed as Eq. ( 3- 16).
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
Φ Φ = +
O
O
0
0
[ ] [ ][ ] , ,
*
i i i i
T c c c ( 3- 15)
26
i i
T
i
i i i i
i m
c c
φ φ ω
ξ
2{ } [ ]{ }
, ,
* +
= ( 3- 16)
where i ξ
is the i th EMDR.
3.2.2 Error Criteria of NODE Method
The accuracy of the NODE method depends on the significance of the neglected elements
on overall dynamic response. Equation ( 3- 17) shows the generalized damping matrix
having off- diagonal terms.
⎥ ⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
Φ Φ =
{ } [ ]{ } { } [ ]{ }
{ } [ ]{ } { } [ ]{ }
[ ] [ ][ ]
1
1 1 1
n
T
n
T
n
n
T T
T
c c
c c
c
φ φ φ φ
φ φ φ φ
L
M O M
L
( 3- 17)
Warburton and Soni ( 1977) proposed a parameter i j e , to quantify the modal coupling for
the NODE method as shown below ;
, 2 2
{ } [ ]{ }
i j
j i
T
i
i j
c
e
ω ω
φ φ ω
−
= ( 3- 18)
A small i j e , , less than 1, indicates little modal coupling of the ith and jth modes. If i j e , is
small enough relative to unity for all pairs of modes, the NODE method is thought to
yield accurate results.
3.2.3 Procedure of NODE Method
The steps of applying the NODE method are as follows:
27
Step 1. Establish mass, stiffness, and damping matrix of a bridge system.
Step 2. Compute undamped mode shape and natural frequency of each mode from mass
and stiffness matrix.
Step 3. Obtain modal damping matrix by pre- and post- multiplying mode shape matrix to
damping matrix as shown in Eq.( 3- 15).
Step 4. Compute effective damping ratio of each mode from Eq. ( 3- 16) ignoring off-diagonal
elements of modal damping matrix.
Step 5. Check error criteria using Eq. ( 3- 18). If a parameter from Eq. ( 3- 18) of any two
modes is greater than unity, change to other methods.
28
3.3 Optimization ( OPT) Method
The optimization method in both time domain and frequency domain is used to compute
the EMDR. In this method, the damping of a non- proportionally damped model ( NP-Model)
is approximated by the EMDR for an equivalent proportionally damped model
( P- Model) to produce the same damping effect through model iterations.
3.3.1 Basic Principle
In the OPT method, the damping ratio of equivalent P- Model is searched through
iteration so that the dynamic responses from P- Model and NP- Model close to each other.
The damping matrix of the NP- Model shown in Eq.( 3- 19) is composed of the damping
matrix of concrete structure [ ] str c , which is assumed as the Rayleigh damping, and the
damping matrix from other damping components [ ] local c . The damping matrix of the
equivalent P- Model shown in Eq. ( 3- 20) is assumed as the Rayleigh damping with
coefficients α and β to have the same damping effect of the NP- Model.
[ ] [ ] [ ] str . local c = c + c NP- Model ( 3- 19)
[ c] = α [ m] + β [ k] Equivalent P- Model ( 3- 20)
The coefficients α and β can be computed from specified damping ratios i ξ
and j ξ
for
the i th and j th modes, respectively, as shown in Eq. ( 3- 21)
29
( )
( i i j j)
i j
i j j i
i j
i j
ω ξ ω ξ
ω ω
β
ω ξ ω ξ
ω ω
ω ω
α
−
−
=
−
−
=
2 2
2 2
2
2
( 3- 21)
where i ω
and j ω
are natural frequency of the i th and j th modes, respectively.
The damping ratio of n th mode can be determined by Eq. ( 3- 22).
β
ω
α
ω
ξ
2 2
1 n
n
n = + ( 3- 22)
The optimization method is conducted in both time domain and frequency domain. In the
time domain, a time history response from the equivalent P- Model is compared with that
of the original NP- Model, while in the frequency domain the frequency response
functions of both systems are used in the optimization algorithm.
3.3.2 Time Domain
Figure 3.3.1 ( a) shows the flow chart of the optimization method in time domain. The
procedures are explained as follows: ( 1) the initial EMDR of the equivalent
proportionally damped system is assumed, ( 2) time history analysis of both models under
a ground motion is performed, ( 3) an objective function is made by mean- square- error of
results from the NP- Model and P- Model as shown in Eq. ( 3- 23), ( 4) check criterion, ( 5) if
the criterion is not satisfied, the EMDR is updated to minimize the objective function, ( 6)
repeat procedure ( 2) to ( 5) until the criteria is satisfied.
( )⎥⎦
⎤
⎢⎣
⎡
− = Σ=
N
i
p
i
np
i x x
N
F
1
min 1 2 ( 3- 23)
30
where the superscript np and p represent the NP- Model and P- Model, respectively, N is
total number of analysis time step, and np
i x and p
i x are the response at the ith time step of
the NP- Model and P- Model, respectively.
3.3.3 Frequency Domain
The optimization method in time domain requires the application of a direct numerical
integration method such as the Newmark method to compute the dynamic response from
both the non- proportionally and proportionally damped models. However, the time
history analysis can be avoided in frequency domain by establishing the objective
function as being composed of the frequency response function of both models.
The optimization method in the frequency domain, shown in Fig. 3.3.1 ( b), is almost the
same as that in time domain. However, instead of computing the response time history,
the frequency response functions of both models are utilized in this method. The
frequency response function is defined by Eq. ( 3- 24) where X ( jω ) is the Fourier
Transform of the response; F( jω ) is the Fourier Transform of the input force.
( )
( ) ( )
ω
ω
ω
F j
H j = X j ( 3- 24)
The equation of motion of a MDOF system subject to ground motion is shown in Eq. ( 3-
25). The Fourier Transform of the second- order equation of motion reduces the original
problem into a linear algebraic problem as shown in Eq. ( 3- 26) where j is − 1 ;
X ( jω ) and X ( jω ) G are the Fourier Transforms of the response and ground motion
accelerations, respectively; and ω is circular frequency in rad/ sec.
31
[ m]{ x( t)} [ c]{ x( t)} [ k]{ x( t)} [ m]{ i} x ( t) && + & + = − && g ( 3- 25)
[[ k] ω 2 [ m] jω[ c]]{ X ( jω )} [ m]{ i} X ( jω ) G − + = − ( 3- 26)
The frequency response function is expressed by Eq. ( 3- 27)
[ ]
[ ] [ ] [ ]
1
( )
( ) ( ) F j k 2 m j c
H j X j
ω ω ω
ω
ω
− +
= = ( 3- 27)
where F( jω ) is [ m]{ i} X ( jω ) G − .
The mass and stiffness matrices of the frequency response function of both models are
the same but the damping matrix of both models is different. The damping matrix of the
non- proportionally damped and the proportionally damped model in Eq. ( 3- 27) are
expressed as Eq. ( 3- 19) and ( 3- 20).
The objective function in the frequency domain is composed of frequency response
functions of both P- Model and NP- Model shown in Eq. ( 3- 28).
[ ]⎥⎦
⎤
⎢⎣
⎡
− = Σ=
M
i
i
p
i
H np j H j
M
F
1
min 1 ( ω ) ( ω ) 2 ( 3- 28)
where ( ) i
H np jω and ( ) i
H p jω are the frequency response functions of the non-proportionally
and equivalent proportionally damped systems at i ω , respectively, and
M is total number of frequencies considered.
If the damping of the non- proportionally damped system is hysteretic, the frequency
response function in Eq. ( 3- 27) changes to
32
[ ] [ ] [ ˆ ]
( ) 1 k 2 m j k
H j
− +
=
ω
ω ( 3- 29)
where, [ k ˆ ] is a stiffness matrix for the entire system obtained by assembling individual
finite- element stiffness matrices [ k ˆ ( m) ] of the form ( superscript m denotes element m )
[ k ˆ ( m) ] = 2ξ ( m)[ k ( m) ] ( 3- 30)
in which [ k ( m) ] denotes the individual elastic stiffness matrix for an element m as used in
the assembly process to obtain the stiffness matrix [ k] for the entire system; and ξ ( m) is a
damping ratio selected to be appropriate for the material used in element m .
The frequency response function of a proportionally damped system is shown in Fig.
3.3.2 for several different values of the EMDR. The peaks of the frequency response
function correspond to the natural frequencies of the system. As seen in these figures,
the overlapping of frequency response function with adjacent modes increases as the
EMDR increases. However, the natural frequencies ( i. e. peaks) do not move by
increasing the EMDR but they change in the non- proportionally damped system with
increases in the damping of local damping components. Thus, if the undamped and
damped natural frequencies of a non- proportionally damped system are not close to each
other, the accuracy of the frequency domain optimization is not guaranteed.
3.3.4 Procedure of OPT Method
The steps of applying the OPT method in time domain are as follows:
Step 1. Establish mass, stiffness, and damping matrix of a bridge system.
33
Step 2. Compute undamped natural frequencies.
Step 3. Specify damping ratios of two modes of P- Model as Rayleigh damping and
compute α and β using Eq. ( 3- 21).
Step 4. Compute damping matrix of P- Model as shown in Eq. ( 3- 20).
Step 5. Compute seismic responses of both NP- Model and P- Model through time history
analysis.
Step 6. Evaluate objective function of Eq. ( 3- 23). If a value from objective function is
smaller than criterion, go to step 8.
Step 7. Repeat from step 3 to step 6.
Step 8. Compute damping ratios of other modes using Eq. ( 3- 22).
The procedure of the OPT method in frequency domain is as follows:
Step 1 to step 4 are the same as those in time domain method above.
Step 5. Compute frequency response function of both NP- Model and P- Model using Eq.
( 3- 27)
Step 6. Evaluate objective function of Eq. ( 3- 28). If a value from objective function is
smaller than criterion, go to step 8.
Step 7. Repeat from step 3 to step 6.
34
Step 8. Compute damping ratios of other modes using Eq. ( 3- 22).
35
( a) Time domain
( b) Frequency domain
Figure 3.3.1 Flow chart of optimization method
36
( a) EMDR= 0% ( a) EMDR= 2%
( c) EMDR= 5% ( d) EMDR= 10%
Figure 3.3.2 Frequency response function for different EMDR
37
3.4 Composite Damping Rule ( CDR) Method
The composite damping rule was suggested by Raggett ( 1975) for calculation of the
EMDR of building structures with different damping components. This method is based
on the assumption of viscous damping of the components. Hwang and Tseng ( 2005)
applied this method to compute the EMDR for the design of viscous dampers to reduce
the seismic hazard of highway bridges. The basic principle of this method is described
here.
3.4.1 Basic Principle
The total dissipated energy of a linear system with different damping components is the
sum of the dissipated energy of each component as shown in Eq. ( 3- 31).
= Σ
i
t i E E ( 3- 31)
If equivalent viscous damping is assumed for each component, the dissipated energy
from each component is
i i i E = 4π ξ U ( 3- 32)
where, i ξ
is the component energy dissipation ratio; i U is the peak component energy
per cycle of motion.
From Eqs. ( 3- 31) and ( 3- 32), the total dissipated energy is
t t
i
t i i E = 4π Σξ U = 4π ξ U ( 3- 33)
38
where, t ξ
is the total modal viscous damping ratio ( EMDR); t U is the total peak
potential energy per cycle of motion. From Eq. ( 3- 33), the EMDR is
= Σ
i t
i
t i U
ξ ξ U ( 3- 34)
Equation ( 3- 34) shows that the EMDR is equal to the sum of the damping ratios of each
component weighted by the ratio of the components potential energy to the total potential
energy. The potential energy of the total system and of each component are computed by
Eq. ( 3- 35) and after substituting { x} = { φ} q( t) into the potential energy ratio ( i t U / U ) is
given by ( 3- 36).
{ } [ ]{ }
2
U 1 x k x t
T
t =
( 3- 35)
{ } [ ]{ }
2
U 1 x k x i
T
i =
{ } [ ]{ }
{ } [ ]{ }
φ φ
φ φ
t
T
i
T
t
i
k
k
U
U = ( 3- 36)
where { φ} is the mode shape; [ ] t k is the system stiffness matrix of the entire system;
[ ] i k is the system stiffness matrix having all zero elements except for the stiffness of the
ith component.
Another method which is conceptually very similar to the composite damping rule
method is the modal strain energy method. This method was developed by Johnson and
39
Kienhholz ( 1982). Though it was first developed for aerospace structures with
viscoelastic material, this method has been applied to concrete and steel frames with
viscoelastic dampers ( Shen and Soong, 1995; Chang et al, 1995).
The dissipated energy per cycle through viscous damping is proportional to response
frequency. However, many tests indicate that the energy loss is essentially independent
of frequency ( Clough and Penzien, 1993). Therefore, hysteretic damping, in which the
damping force is proportional to the displacement amplitude and in phase with velocity,
is used in the modal strain energy method.
The damping force of hysteretic damping is expressed as
f ( t) i k x( t) d = η ( 3- 37)
where i is − 1 which puts the damping force in phase with the velocity; η is the
hysteretic damping coefficient; k is the elastic stiffness of a component; and x( t) is the
displacement of the component.
Considering the complex damping force combined with the elastic force, the equation of
motion of free vibration is expressed as
[ m]{& x&}+ [ k + iη k ]{ x}= { 0} ( 3- 38)
If a system has different damping components such as an embankment at short- span
bridges or isolation bearings at isolated bridges, the complex- stiffness in Eq. ( 3- 38) is
comprised of two parts as shown in Eq. ( 3- 39).
40
[ ] [ ] [ ] ( [ ] [ ]) 1 2 1 1 2 2 k + iη k = k + k + i η k + η k ( 3- 39)
where, [ ] 1 k is the elastic stiffness matrix of the structures concrete components; [ ] 2 k is
the elastic stiffness matrix of the soil boundary components; 1 η
and 2 η are the hysteretic
damping coefficients corresponding to [ ] 1 k and [ ] 2 k , respectively.
The total hysteretic damping energy of the system is the sum of the hysteretic damping
energies of all components.
{ } ([ ] [ ]){ } { } [ ]{ } { } [ ]{ } i
T
i i
T
i i
T
i eq i η φ k k φ η φ k φ η φ k φ , 1 2 1 1 2 2 + = + ( 3- 40)
where, i, eq η is the EMDR of the ith mode; { } i φ
is the ith mode shape.
Therefore, the EMDR of the ith mode is
{ } [ ]{ } { } [ ]{ }
{ } ([ ] [ ]){ } i
T
i
i
T
i i
T
i
i eq k k
k k
φ φ
η φ φ η φ φ
η
1 2
1 1 2 2
, +
+
= ( 3- 41)
Replacing [ ] 2 k in Eq. ( 3- 41) with [ ] [ ] [ ] [ ] 2 1 2 1 k = k + k − k , Eq. ( 3- 41) becomes
{ } [ ]{ }
{ } ([ ] [ ]){ }
{ } [ ]{ }
{ } [ ] [ ] ( ){ }⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
+
+ −
+
=
i
T
i
i
T
i
i
T
i
i
T
i
i eq k k
k
k k
k
φ φ
φ φ
η
φ φ
η φ φ
η
1 2
1
2
1 2
1 1
, 1 ( 3- 42)
Finally, the EMDR of the ith mode of the modal strain energy method is
( ) { } [ ]{ }
{ } ([ ] [ ]){ } i
T
i
i
T
i
i eq k k
k
φ φ
φ φ
η η η η
1 2
1
, 2 2 1 +
= − − ( 3- 43)
41
Or, applying the relation { } ( [ ] [ ] ){ } 2
1 2 i i
T
i φ k + k φ = ω , Eq. ( 3- 43) can be simplified to Eq.
( 3- 44).
( ){ } [ ]{ }
2
1
, 2 2 1
i
i
T
i
i eq
k
ω
φ φ
η = η − η − η ( 3- 44)
In the modal strain energy method, the true damped mode shapes are approximated by
undamped normal mode shapes. From Eq. ( 3- 44), the EMDR of a bridge system ( ) , eq i η
is
always lower than the highest damping ratio of components ( ) 2 η .
3.4.2 Procedure of CDR Method
The steps of the CDR method are as follows:
Step 1. Establish mass and stiffness matrix of a bridge system.
Step 2. Obtain undamped mode shapes.
Step 3. Compute potential energy ratio of each component for each mode using Eq. ( 3-
36).
Step 4. Compute EMDR using Eq. ( 3- 34).
42
3.5 Summary
The four damping estimating methods ( CMA, NODE, OPT, and CDR) used to
approximate the EMDR of non- proportionally damped systems are discussed above.
Each method has pros and cons. Some important features of each when they are applied
to compute the EMDR are described here.
The CMA method is thought to be an exact solution of the EMDR. This method involves
establishing the mass, stiffness, and damping matrices of a system and carrying out a
complex modal analysis to determine the eigenvalues of the system. However, the
EMDR of each mode is determined easily once the eigenvalue of each mode is attained.
In the NODE method, the damping coefficient values are needed as in the CMA method
for the computation of the EMDR. The mode shapes from an undamped system should
be computed from normal modal analysis to get the generalized damping matrix. This
method is very easy to implement, however, the accuracy depends on the significance of
modal coupling between the modes ( Warburton and Soni, 1977) and also the location of
the different damping components as shown in Veletsos and Ventura ( 1986).
The OPT method in the time domain is the only method which requires response time
history analysis to compute the EMDR among the proposed methods. As applied in the
frequency domain, a frequency response function is utilized to establish the objective
function. The unique advantage of the frequency domain method is that the complex
frequency- dependent stiffness can be accommodated easily, which is very difficult in the
other methods. Damping coefficients are required to compute the EMDR in both the
time and frequency domain methods .
43
Instead of damping coefficient values, the CDR method needs the damping ratio of the
individual structural components. For the computation of the EMDR, the potential
energy of an entire system and each component of the system should be computed. This
process also requires mode shapes of the system.
44
Chapter 4
APPLICATION TO SHORT- SPAN
BRIDGE
In this chapter, the four methods described for estimating the damping in the previous
chapter are applied to a short- span bridge. The Painter Street Overcrossing ( PSO),
which has strong earthquake recordings, is chosen as an example bridge. The whole
analysis procedure is explained first and then the description of the bridge and the finite
element modeling of the bridge are presented, followed by the application and results of
each method.
4.1 Analysis Procedure
The application and verification of the four damping estimating methods is summarized
in Fig. 4.1.1. The finite element model of the PSO was established first. The damping of
the finite element model is composed of two components: i) damping from the concrete
structure part which is assumed as Rayleigh damping, and ii) damping from the bridge
boundary which is assumed as viscous damping. The boundary condition of the bridge
under strong earthquake was modeled with a viscoelastic element. The linear elastic
stiffness and viscous damping coefficient of the element were determined by utilizing the
recorded data through optimization. Because of the viscous damping at the boundary, the
45
finite element model of the bridge is a non- proportionally damped model which is
denoted as NP- Model in Fig. 4.1.1.
After establishing the NP- Model, the damping of the NP- Model is approximated with
the EMDR for each mode, applying each damping estimating method. Therefore, the
EMDR is thought to have the same damping effect as the NP- Model. The NP- Model is
changed to an equivalent proportionally damped model ( P- Model) with the previously
determined EMDR. The mass and stiffness matrices of the NP- Model and P- Model are
the same but only the damping of the NP- Model is approximated with the EMDR. Now,
based on the P- Model, the mode shapes and undamped natural frequencies which will be
used in the modal combination can be computed. The computed responses from the NP-Model
and P- Model under a strong ground motion are termed ‘ Computed response 1’ and
‘ Computed response 2’, respectively, in Fig. 4.1.1. The time history response of the NP-Model
was computed by the Newmark integration method; however, that of the P- Model
was calculated by the modal superposition method using the EMDR of each mode.
With the mode shapes, natural frequencies, and the EMDR of each mode, the response
spectrum method can be applied to compute the seismic demand on the bridge. The
modal combination results are termed ‘ Computed response 3’ in Fig. 4.1.1. Instead of
using a constant modal damping ratio for all modes considered, different modal damping
ratios from the EMDR estimation methods are used for the modal combination results.
The finite element modeling of the bridge with the boundary elements is validated by
comparing the ‘ Measured response’ and ‘ Computed response 1’ and the accuracy of the
EMDR by each method is verified by comparing the ‘ Computed response 1’ and
‘ Computed response 2’. Finally, the application of the response spectrum method with
46
the approximated mode shapes, natural frequencies, and EMDR to compute the seismic
demand of the NP- Model is verified by comparing the peak value of the ‘ Measured
response’ and ‘ Computed response 3’. The validation lists and comparable responses are
summarized in Table 4.1.1.
Table 4.1.1 Summary of validation check
Validation Comparable responses
FE modeling and
boundary condition
Measured & Computed response 1
Estimation of EMDR Computed response 1 & 2
Response spectrum method
for NP- Model
Computed response 1 & 3
Overall performance Measured & Computed response 3
47
Equivalent
linear system
of boundary
Real Bridge
( PSO)
Measured
response
Non- proportionally
damped model
( NP- Model)
Computed
response 1
Proportionally
damped model
( P- Model)
Modal
combination
( Keq, Ceq) boundary
Computation
of EMDR
1. CMA
2. NODE
3. OPT
4. CDR
Computed
response 2
Computed
response 3
ξ EMDR
F. E. Model
( PSO)
Response
spectrum
Figure 4.1.1 Analysis procedure
48
4.2 Example Bridge and Earthquake Recordings
The description of the PSO and the sensor locations of the monitoring system are
presented in this section. In addition, the recorded free field ground motions are shown.
4.2.1 Description of Painter St. Overpass
The PSO, shown in Fig. 4.2.1, is located in Rio Dell, California. The bridge consists of a
continuous reinforced concrete, multi- cell, box- girder deck and is supported on integral
abutments at both ends and a two- column center bent. It has two unequal spans of 119
and 146 ft. Both abutments are skewed at an angle of 38.9°. The east abutment is
monolithically connected to the deck, but the west abutment contains a thermal
expansion joint between the abutment diaphragm and the pile cap of the abutment.
4.2.2 Recorded Earthquakes and Dynamic Responses
To date, the monitoring system installed at the PSO has recorded 9 sets of earthquake
data. Among them, 6 earthquakes were selected for use in this study based on the
availability of all channel data. The peak ground acceleration ( PGA) and bridge
response are summarized in Table 4.2.1. The free field ground motion acceleration,
velocity, and displacement time histories of the six earthquakes are shown in Appendix A.
The recorded PGA varied from 0.06g to 0.54g in the transverse direction. In this chapter,
the results of the analysis under only the Cape Mendocino/ Petrolina Earthquake in 1992,
which is the strongest earthquake, are presented.
49
( a) Elevation view
( b) Plan view
( c) Section at bent 2
Figure 4.2.1 Description of PSO and sensor locations
50
Figure 4.2.2 shows the free field ground motion of the Cape/ Mendocino Earthquake in
the transverse direction. The PGA was 0.54g and the dominant frequency of the ground
motion was found as 2- 2.5Hz as shown in Fig. 4.2.2. The acceleration response at the top
of both embankments are displayed in Fig. 4.2.3 along with the free field ground motion.
The PGA of 0.54g was amplified to 1.34g and 0.78g at the West and East embankments,
respectively. The different amplification effect is attributed to the different conditions of
the abutment- deck connection. The deck of the East side is monolithic with the pile
foundation cap, however, the deck is resting on a neoprene pad on the West side. The
effect of the different boundary conditions on the acceleration response at both ends of
the deck is shown in Fig. 4.2.4.
3 4 5 6 7 8 9 10 11 12
- 0.5
0.0
0.5
Time ( sec)
Acceleration ( g)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.00
0.01
0.02
0.03
0.04
0.05
Frequency ( Hz)
PSD
Figure 4.2.2 Cape Mendocino/ Petrolina Earthquake in 1992
51
3 4 5 6 7 8 9 10 11 12
- 1.0
- 0.5
0.0
0.5
1.0
1.5
Time ( sec)
Acceleration ( g)
Emb.- West
Free field
Emb.- East
Figure 4.2.3 Acceleration response at embankment of PSO
3 4 5 6 7 8 9 10 11 12
- 1.0
- 0.5
0.0
0.5
1.0
1.5
Time ( sec)
Acceleration ( g)
Deck- West
Bent- top
Deck- East
Figure 4.2.4 Acceleration response at deck of PSO
Table 4.2.1 Peak acceleration of earthquake recording
Earthquake
Maximum acceleration ( g) in transverse
direction
Free
field
Deck Embankment
East West Cente
r East West
Cape Mendocino ( 1986) 0.15 0.22 0.16 0.25 0.22 0.30
Aftershock 0.12 0.30 0.21 0.35 0.22 0.22
Cape Mendocino ( 1987) 0.09 0.24 0.17 0.33 0.15 0.23
Cape Mendocino/ Petrolina ( 1992) 0.54 0.69 1.09 0.86 0.78 1.34
Aftershock 1 0.52 0.60 0.76 0.62 0.72 0.83
Aftershock 2 0.20 0.26 0.31 0.30 0.31 0.32
52
4.3 Seismic Response from NP- Model
In order to include the energy dissipation from boundary soil, the bridge boundaries were
modeled with equivalent viscoelastic elements. The effective elastic stiffness and
damping of the elements were estimated to minimize the error between the simulated and
measured response.
4.3.1 Modeling of Concrete Structure
Figure 4.3.1 shows the finite element model of the PSO. The deck and bent are
composed of 10 and 4 elements, respectively. Each node was assumed to have 2 degrees-of-
freedom, i. e. displacement in the Y direction and rotation about the Z- axis for deck
elements and displacement in the Y direction and rotation about the X- axis for column
elements. In totality, the finite element model has 30 degrees- of- freedom. The original
two columns of the center bent of the bridge were combined as one equivalent member in
the finite element model for simplification. The effective viscoelastic elements at the
bridge boundaries were assumed to act only in the transverse direction. The rotational
degree- of- freedom at the bottom of the bent was assumed to be fixed.
Table 4.3.1 shows the element properties used in the finite element model. The Young’s
modulus of concrete was assumed to be 80% of its initial value after considering the
ageing effect ( Zhang and Makris, 2002). The damping ratio of the concrete structure part
of the system was assigned 5% Rayleigh damping. The mass of the deck and bent was
lumped at each node and rotational mass was not considered.
53
Figure 4.3.1 Finite element model of PSO
Table 4.3.1 Element properties of finite element model of PSO
Properties Deck Column
Mass density ( ρ ) 2,400kg/ m3 2,400kg/ m3
Young’s modulus ( Ec) 22GPa 22GPa
Sectional area ( A) 8.29m2 1.92m2
Moment of inertia ( I) 153.90m4 0.29m4
4.3.2 Estimation of Boundary Condition and Response from NP- Model
The shear modulus and damping characteristics of the boundary soils change depending
on soil properties ( Seed and Idriss, 1970). In the previous study ( Goel, 1997), the natural
frequency of the PSO was observed to vary according to the ground motion intensity
during the Cape Mendocino/ Petrolina earthquake, which indicates the nonlinearity of the
boundary soils. However, instead of a non- linear model, an equivalent viscoelastic
54
model composed of linear elastic stiffness and viscous damping was adopted to represent
the bridge boundaries for application of the damping estimating methods.
The effective stiffness and the damping coefficients were estimated by minimizing the
square error between the measured and computed response. The optimization procedure
is shown in Fig. 4.3.2. The free field ground motion in the transverse direction was used
as an input ground motion and the response at the top of the bent was chosen for
comparison. The objective function was constituted by the sum of squares of the
difference between the measured and computed response and the power spectral density
as shown in Eq. ( 4- 1) ( Li and Mau, 1991).
( )
( )
{ }
Σ{ }
Σ
Σ
Σ −
+
−
=
j
meas
j
j
comp
j
meas
j
i
meas
i
i
comp
i
meas
i
p
p p
x
x x
F 2
2
2
2
( )
( ) ( )
ω
ω ω
( 4- 1)
where, F is the objective function; i x is the response at the ith time step; ( ) j p ω is the
power spectral density of response at frequency j ω; superscript meas and comp means
‘ Measured response’ and ‘ Computed response’, respectively.
Table 4.3.2 shows the final identified results for the equivalent viscoelastic model of the
bridge boundaries. Figures 4.3.3 and 4.3.4 show the response time histories and power
spectral densities of the PSO at the top of the bent obatained from both measurement and
simulation under the Cape Mendocino/ Petrolina Earthquake in 1992. From the figures,
both in time domain and frequency domain, the computed response shows good
agreement with the measured response.
55
Figure 4.3.2 Optimization algorithm for estimating boundary condition
Table 4.3.2 Effective stiffness and damping coefficient of PSO boundary
Boundary Identified
Spring stiffness
( MN/ m)
East Abutment 78
West Abutment 78
Bent 642
Damping coefficient
( MN · sec/ m)
East Abutment 5
West Abutment 5
Bent 5
56
3 4 5 6 7 8 9 10 11 12
- 1.0
- 0.5
0.0
0.5
1.0
1.5
Time ( sec)
Acceleration( g)
Meas.
NP- Model
Figure 4.3.3 Comparison of response time history
1 1.5 2 2.5 3 3.5 4 4.5 5
0.00
0.05
0.10
0.15
0.20
0.25
Frequency ( Hz)
PSD
Meas.
NP- Model
Figure 4.3.4 Comparison of power spectral density
4.3.3 Natural Frequencies and Mode Shapes
The natural frequencies and mode shapes obtained from the eigen analysis of the
undamped model of the PSO are given in Table 4.3.3 and Fig. 4.3.5. The mode shapes in
Fig. 4.3.5 are the mass normalized mode shapes. The first and second mode frequencies
were computed as 1.696 and 2.643 Hz, respectively, which are in the dominant frequency
range of the earthquake as shown in Fig. 4.2.2.
The boundary springs at both ends of the deck deform in the same direction as the bent in
the first mode, but they deform in opposite directions in the third mode. In the second
mode, the bent does not deform much, but the boundary springs at both ends of the deck
exhibit large deformations in opposite directions to each other.
57
Table 4.3.3 Natural frequency and period of PSO
No. of mode
Painter St. Overpassing
Frequency ( Hz) Period ( sec)
1 1.648 0.606
2 2.643 0.378
3 7.329 0.136
4 18.832 0.053
5 23.762 0.042
58
1 2 3 4 5 6 7 8 9 10 11
- 0.1
- 0.09
- 0.08
- 0.07
Mode 1
1 2 3 4 5 6 7 8 9 10 11
- 0.2
0
0.2
Transverse Displacement
Mode 2
1 2 3 4 5 6 7 8 9 10 11
- 0.2
0
0.2
Deck Node No.
Mode 3
( a) Deck
- 0.2 - 0.15 - 0.1 - 0.05 0 0.05 0.1 0.15 0.2
12
12.5
13
13.5
14
14.5
15
Transverse Displacement
Bent Node No.
Mode1
Mode2
Mode3
( b) Bent
Figure 4.3.5 Mode shape of PSO
59
4.4 EMDR Estimation
Based on the NP- Model of the PSO, each damping estimating method is applied to
determine the EMDR of each mode. The responses from the NP- Model, P- Model, and
the measured response at the top of the bent are compared to verify each method. The
results of the EMDR from each method is summarized in Table 4.4.10, and the
comparison of the peak response values from each method is given in Table 4.4.11 and
4.4.12.
4.4.1 Complex Modal Analysis ( CMA) Method
The NP- Model of the PSO is analyzed using the CMA method to obtain the EMDR. The
procedure of the CMA method is explained as follows:
Step 1. Establish mass, stiffness, and damping matrix of a bridge system.
􀁺 The element used for deck and bent of the PSO is shown in Fig. 4.4.1. The lumped
mass matrix [ me ] of the element which has 2 degrees- of- freedom is represented as
Eq. ( 4- 2). The matrix has half of the element mass at each translational nodal
degree- of- freedom. In Eq. ( 4- 2), ρ is the mass density of concrete, A is the area
of element section, and l is element length. The global mass matrix of whole
bridge system is obtained by assembling each element mass matrix.
⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢
⎣
⎡
=
0 0 0 0
0 0 1 0
0 0 0 0
1 0 0 0
2
[ me ] ρ Al ( 4- 2)
60
􀁺 The stiffness matrix of the element shown in Fig. 4.4.1 is shown in Eq. ( 4- 3)
⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢
⎣
⎡
−
− − −
−
−
=
2 2
2 2
3
6 2 6 4
12 6 12 6
6 4 6 2
12 6 12 6
[ ]
l l l l
l l
l l l l
l l
l
k e EI
( 4- 3)
where E is Young’s modulus of element and I is the moment of inertia. Global
stiffness matrix is obtained by assembling each element stiffness matrix. The
boundary spring stiffnesses found in section 4.3.2 are directly added to
corresponding degree- of- freedom elements in the global stiffness matrix.
Figure 4.4.1 Element degree- of- freedom
􀁺 The damping of concrete structure of the PSO is assumed as 5% Rayleigh damping.
The damping coefficient α and β can be found by specifying 5% damping ratio to
any two modes. In this research, it was assigned to the first and third mode. So, the
damping matrix of the concrete structure is [ c ] [ m] [ k] str = α + β . The damping
matrix of the bridge boundary [ ] local c has zero elements except the corresponding
degree- of- freedom elements of boundary damping found in section 4.3.2. The
global damping matrix is obtained by adding [ ] str c and [ ] local c
Step 2. Obtain [ A] and [ B] matrix using Eq. ( 3- 7).
61
Step 3. Compute eigenvalues of the characteristic equation shown in Eq.( 3- 8).
􀁺 The dimension of the matrix [ A] and [ B] is 2n × 2n ( n is the total number of
degree- of- freedom), and 2n conjugate eigenvalues are obtained from eigen
analysis. The second column of Table 4.4.1 shows the eigenvalues of the NP-Model
of the PSO from the complex modal analysis.
Table 4.4.1 Eigenvalues and natural frequencies of NP- Model of PSO
Mode Eigenvalues Natural frequency ( rad/ sec)
1 2.674 – 10.291i 2.6742 + 10.2912 = 10.633
2 9.288 – 14.054i 9.2882 + 14.0542 = 16.845
3 13.930 – 42.939i 13.9302 + 42.9392 = 45.142
4 22.866 – 114.754i 22.8662 + 114.7542 = 117.010
5 50.547 – 161.112i 50.5472 + 161.1122 = 168.855
Step 4. Compute natural frequency of each mode from corresponding eigenvalue using
Eq. ( 3- 11).
􀁺 The third column of Table 4.4.1 shows the natural frequency computed using the
eigenvalues of the second column of Table 4.4.1.
Step 5. Compute effective damping ratio of each mode from real part of eigenvalue and
natural frequency of corresponding mode using Eq. ( 3- 12).
62
􀁺 Table 4.4.2 shows the final results of EMDR from the CMA method. The first and
second modal damping ratios are found as 25% and 55%, respectively. Table 4.4.3
compares the undamped natural frequency of NP- Model and P- Model of the PSO.
Table 4.4.2 EMDR of PSO by CMA method
Mode EMDR
1 2.674 10.633 = 0.251
2 9.288 16.845 = 0.551
3 13.930 45.142 = 0.324
4 22.866 117.010 = 0.195
5 50.547 168.855 = 0.299
The acceleration and displacement time history from the NP- Model and P- Model at the
top of the bent are drawn along with the measured time history in Fig. 4.4.2. The
response of the P- Model shows good agreement with the NP- Model response as well as
the measured response. The summary of peak response values from the measurement,
NP- Model, and P- Model are presented in Table 4.4.11 and 4.4.12. The relative error of
the P- Model with the NP- Model and measurement is within 10% and 2%, respectively.
63
Table 4.4.3 Undamped natural frequency and EMDR from CMA method
Mode
Undamped Natural Frequency ( Hz)
EMDR
NP- Model P- Model
1 1.742 1.648 0.251
2 2.681 2.643 0.551
3 7.194 7.329 0.324
4 18.624 18.832 0.195
5 27.166 23.762 0.299
3 4 5 6 7 8 9 10 11 12
- 1.0
- 0.5
0.0
0.5
1.0
1.5
Time ( sec)
Acceleration( g)
Meas
NP- Model
P- Model
( a) Acceleration
3 4 5 6 7 8 9 10 11 12
- 6
- 4
- 2
0
2
4
6
Time ( sec)
Displacement( cm)
Meas
NP- Model
P- Model
( b) Displacement
Figure 4.4.2 Response time history from CMA method
64
4.4.2 Neglecting Off- Diagonal Elements ( NODE) Method
The step 1 of the NODE method is the same as in the CMA method.
Step 2. Compute undamped mode shape and natural frequency of each mode from mass
and stiffness matrix.
􀁺 The undamped mode shapes and natural frequencies are obtained in Table 4.3.3 and
Fig. 4.3.5.
Step 3. Obtain modal damping matrix by pre- and post- multiplying mode shape matrix to
damping matrix as shown in Eq.( 3- 15).
􀁺 Table 4.4.4 shows the results of pre- and post- multiplication of the normal mode
shapes to the damping matrix of the NP- Model up to the fifth mode.
Table 4.4.4 Modal damping matrix ([ φ ] T [ c][ φ ] )
Mode 1 2 3 4 5
1 5.141 0.196 - 9.722 0.484 2.294
2 0.196 18.068 0.200 18.390 1.692
3 - 9.722 0.200 27.708 - 0.255 1.938
4 0.484 - 18.394 - 0.255 46.057 1.566
5 2.294 1.692 1.938 1.566 107.145
Step 4. Compute effective damping ratio of each mode from Eq. ( 3- 16) ignoring off-diagonal
elements of modal damping matrix.
65
􀁺 If the mode shapes are mass normalized ones, the term { } [ ]{ } i
T
i φ m φ in the
denominator of Eq. ( 3- 16) is unity and the EMDR of i th mode becomes
2( 2 )
,
i
i i
i f
c
π
ξ = ( 4- 4)
where i f is undamped natural frequency ( Hz) of i th mode. Table 4.4.5 shows the
EMDR of each mode computed by Eq. ( 4- 4).
Step 5. Check error criteria using Eq. ( 3- 18). If a parameter from Eq. ( 3- 18) of any two
modes is greater than unity, change to other methods.
􀁺 The accuracy of the NODE method can be assessed by modal coupling parameters
which are shown in Table 4.4.6. Though the shaded off- diagonal elements in Table
4.4.4 are significant compared with the elements in the diagonal line, the modal
coupling parameters of the off- diagonal elements in Table 4.4.6 are much less than
unity.
Table 4.4.5 EMDR from NODE method
Mode EMDR
1 5.141 /( 2 × 2π ×1.648) = 0.248
2 18.068 /( 2 × 2π × 2.643) = 0.544
3 27.708/( 2 × 2π × 7.329) = 0.301
4 46.057 /( 2 × 2π ×18.832) = 0.195
5 107.145 /( 2 × 2π × 23.762) = 0.359
Table 4.4.6 Modal coupling parameter
66
Mode 1 2 3 4 5
1 - 0.012 - 0.050 0.000 0.001
2 0.019 - 0.002 0.022 0.001
3 - 0.222 0.005 - - 0.001 0.004
4 0.004 0.158 - 0.002 - 0.022
5 - 0.015 0.011 0.014 - 0.028 -
The computed responses from the NP- Model and P- Model are shown in Fig. 4.4.3 along
with the measured response. In Table 4.4.10 the EMDR from this method is very close to
the result from the complex modal analysis method in all modes. The relative error of the
P- Model with NP- Model and measurement is less than 10% and 3%, respectively, in
Table 4.4.11 and 4.4.12.
3 4 5 6 7 8 9 10 11 12
- 1
- 0.5
0
0.5
1
1.5
Time ( sec)
Acceleration( g)
Meas
NP- Model
P- Model
( a) Acceleration
3 4 5 6 7 8 9 10 11 12
- 10
- 5
0
5
10
Time ( sec)
Displacement( cm)
Meas
NP- Model
P- Model
( b) Displacement
Figure 4.4.3 Response time history from NODE method
67
4.4.3 Composite Damping Rule ( CDR) Method
To apply the CDR method, the PSO was divided into two components: i) concrete
structure component which includes deck and bent, and ii) bridge boundary component.
The damping ratios were assumed as 25% and 5% for the boundary component
( Kotsoglou and Pantazopoulou, 2007) and the concrete structure component, respectively.
The procedure of the CDR method is as follows:
Step 1. Establish mass and stiffness matrix of a bridge system. This step is the same as
in the CMA method.
Step 2. Obtain undamped mode shapes. Based on the mass and stiffness matrix of the
NP- Model of the PSO, undamped mode shapes are computed as shown in Fig.
4.3.5.
Step 3. Compute potential energy ratio of each component for each mode using Eq. ( 3-
36).
􀁺 The computed potential energy of each component is given in Table 4.4.7. In the
table the potential energy ratio of the boundary component is 72% and 97% for the
first and second mode, respectively, and it becomes smaller at the third mode.
Considering that the first two modes are in the dominant frequency range of the
earthquake, the potential energy ratio implies that most of the input energy will be
dissipated from the boundary component rather than the concrete structure
component.
Step 4. Compute EMDR using Eq. ( 3- 34).
68
􀁺 Based on Eq. ( 3- 34), the EMDR of each mode is computed as in Table 4.4.8.
Table 4.4.7 Potential energy ratio in CDR method
Mode
Potential energy Energy ratio
Total
( Utotal)
Structure
( Ustr)
Boundary
( Ubnd)
Ustr / Utotal Ubnd / Utotal
1 0.54E2 0.16E2 0.38E2 0.293 0.707
2 1.38E2 0.04E2 1.34E2 0.026 0.974
3 10.60E2 8.69E2 1.92E2 0.819 0.181
4 70.00E2 68.23E2 1.77E2 0.975 0.025
5 111.46E2 55.52E2 55.93E2 0.498 0.502
Table 4.4.8 EMDR from CDR method
Mode EMDR
1 ( 0.05)( 0.293) + ( 0.25)( 0.707) = 0.191
2 ( 0.05)( 0.026) + ( 0.25)( 0.974) = 0.245
3 ( 0.05)( 0.819) + ( 0.25)( 0.181) = 0.086
4 ( 0.05)( 0.975) + ( 0.25)( 0.025) = 0.055
5 ( 0.05)( 0.498) + ( 0.25)( 0.502) = 0.150
The EMDR from the composite damping rule was computed as 19% and 24% for the first
and second mode, respectively. As the modal potential energy ratio of the concrete
structure component increases after the third mode, the EMDR decreases consequently.
69
Figure 4.4.4 shows the response time history of the P- Model compared with the NP-Model
and with measured response. The relative error from the composite damping rule
method was computed as less than 6% and 14% when compared with the NP- Model and
with measured response, respectively, as shown in Tables 4.4.11 and 4.4.12.
3 4 5 6 7 8 9 10 11 12
- 1.5
- 1
- 0.5
0
0.5
1
1.5
Time ( sec)
Acceleration( g)
Meas
NP- Model
P- Model
( a) Acceleration
3 4 5 6 7 8 9 10 11 12
- 10
- 5
0
5
10
Time ( sec)
Displacement( cm)
Meas
NP- Model
P- Model
( b) Displacement
Figure 4.4.4 Response time from CDR method
70
4.4.4 Optimization ( OPT) Method
Using the optimization algorithm shown in Fig. 3.3.1, the EMDR of the NP- Model was
estimated. The procedure of the OPT method is explained below.
OPT method in time domain
Step 1 is the same as in the CMA method.
Step 2. Compute undamped natural frequencies.
􀁺 The undamped natural frequencies were computed based on the mass and stiffness
matrix of the NP- Model of the PSO.
Step 3. Specify damping ratios of two modes of P- Model as Rayleigh damping and
compute α and β using Eq. ( 3- 21).
􀁺 Initial damping ratio of 5% is assumed for the first and third modes to compute
Rayleigh damping coefficient α and β . From Eq. ( 3- 21), α and β are computed as
0.845
1.648 7.329
( 0.05) 2( 2 )( 1.648)( 7.329) =
+
=
π
α
0.002
( 2 )( 1.648 7.329)
( 0.05) 2 =
+
=
π
β
􀁺 From the next iteration, damping ratio is searched by optimization algorithm. After
damping ratio is determined, new α and β values are computed.
Step 4. Compute damping matrix of P- Model as shown in Eq. ( 3- 20).
71
􀁺 Using α and β values, the damping matrix of the P- Model is constructed as
[ c] = 0.865[ m] + 0.002[ k]
Step 5. Compute seismic responses of both NP- Model and P- Model through time history
analysis.
􀁺 For time history analysis, any ground motion can be used. In this research, Cape
Mendocino/ Petrolina earthquake ( 1992) was used and Newmark direct integration
method was adopted.
Step 6. Evaluate objective function of Eq. ( 3- 23). If a value from objective function is
smaller than criterion, go to step 8.
Step 7. Repeat from step 3 to step 6.
Step 8. Compute damping ratios of other modes using Eq. ( 3- 22).
􀁺 From optimization, the damping ratio was obtained as 0.255. α and β values
corresponding to the damping ratio are
4.311
1.648 7.329
( 0.255) 2( 2 )( 1.648)( 7.329) =
+
=
π
α
0.009
( 2 )( 1.648 7.329)
( 0.255) 2 =
+
=
π
β
􀁺 Damping ratios of other modes are computed using Eq. ( 3- 22) and shown in Table
4.4.9.
72
Table 4.4.9 EMDR from OPT method in time domain
Mode EMDR
1 ( 0.009) 0.255
2
( 4.311) ( 2 )( 1.648)
2( 2 )( 1.648)
1 + =
π
π
2 ( 0.009) 0.205
2
( 4.311) ( 2 )( 2.643)
2( 2 )( 2.643)
1 + =
π
π
3 ( 0.009) 0.255
2
( 4.311) ( 2 )( 7.329)
2( 2 )( 7.329)
1 + =
π
π
4 ( 0.009) 0.553
2
( 4.311) ( 2 )( 18.823)
2( 2 )( 18.823)
1 + =
π
π
5 ( 0.009) 0.689
2
( 4.311) ( 2 )( 23.762)
2( 2 )( 23.762)
1 + =
π
π
OPT method in frequency domain
Step 1 to step 4 are the same as those in time domain method above.
Step 5. Compute frequency response function of both NP- Model and P- Model using Eq.
( 3- 27).
􀁺 The frequency response function at the top of the bent ( i H 6, ) was chosen for
objective function of optimization. The only difference of frequency response
function of the NP- Model and P- Model is damping matrix of both models. The
damping matrix of the NP- Model is [ ] [ ] [ ] NP str local c = c + c , while that of the P-Model
is [ c ] [ m] [ k] P = α + β . α and β values are updated for every iteration.
Step 6. Evaluate objective function of Eq. ( 3- 28). If a value from objective function is
smaller than criterion, go to step 8.
Step 7. Repeat from step 3 to step 6.
73
Step 8. Compute damping ratios of other modes using Eq. ( 3- 22).
Figure 4.4.5 shows the frequency response function of the NP- Model and P- Model after
optimization. The frequency response function of the P- Model shows a good agreement
with that of the NP- Model.
The EMDR from the optimization method in time domain and frequency domain are
summarized in Table 4.4.10. From Table 4.4.10 it can be seen that the EMDR from the
optimization method is very close to the results from the CMA method. It should be
noted in Table 4.4.10 that the large EMDR after the fourth mode from the optimization
method is attributed to the assumption of Rayleigh damping for the P- Model. However,
because of little contribution from the higher modes, the overall time history responses
are very similar to the results from the complex modal analysis method.
Figure 4.4.6 and 4.4.7 show the time history response of the P- Model with the NP- Model
and measurement. The comparison of the peak values of the measured and computed
response is given in Tables 4.4.11 and 4.4.12. The relative error of the P- Model with the
NP- Model and measurement is less than 10% and 4% for acceleration and displacement,
respectively.
74
0 2 4 6 8 10
0
1
x 10 - 4
Frequency ( Hz)
TF
NP- Model
P- Model
Figure 4.4.5 FRF of NP- Model and P- Model after optimization
75
3 4 5 6 7 8 9 10 11 12
- 1
- 0.5
0
0.5
1
1.5
Time ( sec)
Acceleration( g)
Meas
NP- Model
P- Model
( a) Acceleration
3 4 5 6 7 8 9 10 11 12
- 10
- 5
0
5
10
Time ( sec)
Displacement( cm)
Meas
NP- Model
P- Model
( b) Displacement
Figure 4.4.6 Response time history from OPT method in time domain
3 4 5 6 7 8 9 10 11 12
- 1
- 0.5
0
0.5
1
1.5
Time ( sec)
Acceleration( g)
Meas
NP- Model
P- Model
( a) Acceleration
3 4 5 6 7 8 9 10 11 12
- 10
- 5
0
5
10
Time ( sec)
Displacement( cm)
Meas
NP- Model
P- Model
( b) Displacement
Figure 4.4.7 Response time history from OPT method in frequency domain
76
Table 4.4.10 Summary of EMDR identified by each method
Mode CMA NODE
OPT
Time CDR
Domain
Frequency
Domain
1 0.251 0.248 0.255 0.242 0.191
2 0.551 0.544 0.205 0.197 0.245
3 0.308 0.301 0.255 0.242 0.086
4 0.195 0.195 0.553 0.522 0.055
5 0.299 0.359 0.689 0.651 0.150
Table 4.4.11 Summary of peak acceleration from each method
Method Measured
( g)
Computed ( g) Relative Error
Response 1
( NP- Model)
Response 2
( P- Model) ( P − NP) P ( P − Meas) P
CMA
0.942 1.031
0.941 - 9% - 1%
NODE 0.947 - 9% 1%
OPT* 0.937 - 10% - 1%
OPT** 0.954 - 8% 1%
CDR 1.068 4% 11%
OPT* & OPT** : optimization method in time domain and frequency domain, respectively
Meas : Measured response, NP : results from NP- Model, P : results from P- Model
Table 4.4.12 Summary of peak displacement from each method
Method Measured
( cm)
Computed ( cm) Relative Error
Response 1
( NP- Model)
Response 2
( P- Model) ( P − NP) P ( P − Meas) P
CMA
5.553 6.098
5.662 - 8% 2%
NODE 5.706 - 7% 3%
OPT* 5.622 - 8% 1%
OPT** 5.758 - 6% 4%
CDR 6.478 6% 14%
OPT* & OPT** : optimization method in time domain and frequency domain, respectively
Meas : Measured response, NP : results from NP- Model, P : results from P- Model
77
4.5 Comparison with Current Design Method
Seismic response of the PSO was computed based on the response spectrum method.
The normal mode shapes and natural periods from the P- Model, and the EMDR of each
mode in Table 4.4.10 were used for the computation. The three modal combination rules
such as the absolute sum ( ABSSUM), square- root- of- sum- of- squares ( SRSS), and
complete quadratic combination ( CQC) methods were applied in the response spectrum
method.
Tables 4.5.1 and 4.5.2 summarize the response spectrum analysis results for each
damping estimating method. The last row of Tables 4.5.1 and 4.5.2 show the modal
combination results when the conventional 5% damping ratio was used for all the modes.
The computed response with the 5% damping ratio is nearly twice that of the measured
response. From the tables it is concluded that the conventional 5% damping ratio is too
conservative for the seismic design of short- span bridges under strong earthquakes. Also,
in these tables it can be seen that the result from each modal combination rule is very
similar to each other, which is attributed to the well- separated modes of the P- Model.
Figure 4.5.1 shows the relative error of the results from the response spectrum method
with the peak values of measured response at the top of bent. Except for the composite
damping rule method, the relative error of each damping estimating method is less than
5% and 10% for acceleration and displacement, re