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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 shortspan 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 offdiagonal 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 SHORTSPAN 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 offreedom ( 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 bilinear 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 wingwalls 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, singlecolumned 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 forcedisplacement 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 bilinear 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 springviscous 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 spectrumbased 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 nonproportionally 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 singledegree 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, preand 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 offdiagonal 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 ( NPModel) 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 nonproportionally 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 NPModel 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 NPModel 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 degreesof 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 NPModel 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 offdiagonal 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 NPModel 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 PModel 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
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Title  Determining the effective system damping of highway bridges 
Subject  Concrete bridgesEarthquake effects.; Damping (Mechanics)Testing.; Seismic wavesDamping.; Earthquake engineering. 
Description  Title from PDF title page (viewed on February 15, 2011).; "June 2009."; Includes bibliographical references (p. 169173).; Final report.; Text document (PDF).; Performed for California Dept. of Transportation under contract no. 
Creator  Feng, Maria Q. 
Publisher  University of California, Irvine, Dept. of Civil & Environmental Engineering 
Contributors  Lee, Sung Chil, 1957; California. Dept. of Transportation.; University of California, Irvine. Dept. of Civil and Environmental Engineering. 
Type  Text 
Language  eng 
Relation  http://www.dot.ca.gov/hq/esc/earthquake_engineering/Research_Reports/vendor/uc_irvine/2009001/UCI_0901Determining_the_Effective_System_Damping_of_Highway_Bridges.pdf; http://worldcat.org/oclc/701908900/viewonline 
DateIssued  [2009] 
FormatExtent  xxii, 288 p. : digital, PDF file (2 MB) with col. ill., col. charts. 
RelationRequires  Mode of access: World Wide Web. 
Transcript  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 shortspan 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 offdiagonal 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 SHORTSPAN 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 offreedom ( 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 bilinear 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 wingwalls 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, singlecolumned 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 forcedisplacement 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 bilinear 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 springviscous 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 spectrumbased 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 nonproportionally 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 singledegree 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, preand 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 offdiagonal 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 ( NPModel) 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 nonproportionally 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 NPModel 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 NPModel 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 degreesof 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 NPModel 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 offdiagonal 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 NPModel 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 PModel 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 



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