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VERIFICATION OF COMPUTER ANALYSIS MODELS FOR SUSPENSION BRIDGES by Masanobu Shinozuka, Distinguished Professor and Chair and Debasis Karmakar, Graduate Student Samit Ray Chaudhuri, Postdoctoral Scholar Ho Lee, Assistant Researcher Department of Civil and Environmental Engineering University of California, Irvine Report No: CA/ UCI VTB 2009 August 2009 Final Report Submitted to the California Department of Transportation under Contract No: RTA 59A0496 VERIFICATION OF COMPUTER ANALYSIS MODELS FOR SUSPENSION BRIDGES Final Report Submitted to the Caltrans under Contract No: RTA 59A0496 by Masanobu Shinozuka, Distinguished Professor and Chair and Debasis Karmakar, Graduate Student Samit Ray Chaudhuri, Postdoctoral Scholar Ho Lee, Assistant Researcher Department of Civil and Environmental Engineering University of California, Irvine Report No: CA/ UCI VTB 2009 August 2009 ii STATE OF CALIFORNIA × DEPARTMENT OF TRASPORTATION TECHNICAL REPORT DOCUMENTAION PAGE TR0003 ( REV. 9/ 99) 1. REPORT NUMBER CA/ UCI VTB 2009 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER 5. REPORT DATE August 2009 4. TITLE AND SUBTITLE VERIFICATION OF COMPUTER ANALYSIS MODELS FOR SUSPENSION BRIDGES 6. PERFORMING ORGANIZATION CODE UC Irvine 7. AUTHOR Masanobu Shinozuka, Debasis Karmakar, Samit Ray Chaudhuri, and Ho Lee 8. PERFORMING ORGANIZATION REPORT NO. 10. WORK UNIT NUMBER 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Civil and Environmental Engineering The Henry Samueli School of Engineering University of California, Irvine Irvine, CA 92697 2175 11. CONTRACT OR GRANT NUMBER RTA 59A0496 13. TYPE OF REPORT AND PERIOD COVERED Final Report 12. SPONSORING AGENCY AND ADDRESS California Department of Transportation ( Caltrans) Division of Research and Innovation 1227 O Street, MS 83 Sacramento, CA 95814 14. SPONSORING AGENT CODE 15. SUPPLEMENTARY NOTES 16. ABSTRACT The Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los Angeles and Long Beach ports, two busiest ports in the west coast of USA. The bridge carries an overwhelming number of traffic with an Annual Average Daily Traffic ( AADT) volume of 45,500, many of which are cargo trucks. Based on the recent finding that the main span of the Vincent Thomas Bridge crosses directly over the Palos Verdes fault, which has the capacity to produce a devastating earthquake, the bridge underwent a major retrofit in spring 2000, mainly using viscoelastic dampers. This study focuses on performance evaluation of the retrofitted bridge under seismic, wind and traffic loads. A member based detailed three dimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by simulating and comparing with the measured dynamic response during two recent earthquakes. FE model is also updated using a sensitivity based parameter updating method. Effect of spatial variability of ground motions on seismic displacement and force demands is investigated. To record actual wind velocity and direction, three anemometers are installed at three different locations of the bridge. Response of the bridge is computed under wind velocity. Finally, analysis of the bridge under traffic load is also carried out. 17. KEYWORDS Suspension Bridge, System Identification, Retrofit, Fragility Curve, Earthquake, Wind, Traffic 18. DISTRIBUTION STATEMENT No restrictions. 19. SECURITY CLASSIFICATION ( of this report) Unclassified 20. NUMBER OF PAGES 178 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 Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los Angeles and Long Beach ports, two busiest ports in the west coast of USA. The bridge carries an overwhelming number of traffic with an Annual Average Daily Traffic ( AADT) volume of 45,500, many of which are cargo trucks. Based on the recent finding that the main span of the Vincent Thomas Bridge crosses directly over the Palos Verdes fault, which has the capacity to produce a devastating earthquake, the bridge underwent a major retrofit in spring 2000, mainly using visco elastic dampers. This study focuses on performance evaluation of the retrofitted bridge under seismic, wind and traffic loads. A member based detailed three dimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by simulating and comparing with the measured dynamic response during two recent earthquakes. Tornado diagram and first order second moment ( FOSM) methods are applied for evaluating the sensitivity of different parameters on the eigenproperties of the FE models. The study indicates that the mass density of deck slab and elastic modulus of bottom chord are very important parameters to control eigenproperties of the models. FE model is also updated using a sensitivity based parameter updating method. Considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed using the FE models developed and seismic fragility curves are derived comparing the ductility demand with the ductility capacity at critical v tower sections. Effect of spatial variability of ground motions on seismic displacement and force demands is also investigated. To generate spatially correlated nonstationary acceleration time histories compatible with design spectrum at each location. A new algorithm is developed involving evolutionary power spectral density function ( PSDF) and with the aid of spectral representation method. It has been found that, in some locations on the bridge deck, the response is higher when the spatially variable ground motion is considered as opposed to the uniform ground motion time histories having the highest ground displacement. To record actual wind velocity and direction, three anemometers are installed at three different locations of the bridge. The fluctuating component of the wind velocity measured at these three locations are found to be non Gaussian. They are used for simulation of fluctuating component of wind velocity throughout the span and along the tower on the basis of three different simulation methods ( i) newly developed non Gaussian conditional method, ( ii) Gaussian conditional method, and ( iii) Gaussian unconditional method. Response of the bridge is computed under wind velocity using these three different methods. It is observed that the non Gaussian conditional simulation technique yields higher response than both Gaussian conditional and Gaussian nonconditional techniques. Finally, analysis of the bridge under traffic load is also carried out and a critical evaluation of shear force in deck shear connectors is performed. vi ACKNOWLEDGEMENT The research presented in this report was sponsored by the California Department of Transportation ( Caltrans) with Dr. Li Hong Sheng as the project manager. The authors are indebted to Caltrans for its support of this project and to Dr. Li Hong Sheng for his helpful comments and suggestions. viii TABLE OF CONTENTS Page LIST OF FIGURES vi LIST OF TABLES xiii ABSTRACT xv CHAPTER 1 Introduction 1 1.1 Background 1 1.2 Literature Survey 3 1.3 Objective and Scope 10 1.4 Dissertation Outline 12 CHAPTER 2 Finite Element Modeling of Vincent Thomas Bridge 13 2.1 Background 13 2.2 Calculation of Dead Weight 13 2.3 Calculation of the Initial Shape of the Cable 14 2.4 Panel Based Simple Model 14 2.4.1 Moment of Inertia ( Iz) 18 2.4.2 Torsional Constant ( J) 19 2.5 Member Based Detail Model 24 2.5.1 Cable Bent 27 2.5.2 Deck Shear Connector 27 2.5.3 Dampers 28 2.5.4 Suspended Truss 28 2.5.5 Suspenders 28 2.6 Eigen Value Analysis 29 ix 2.7 Closure 31 CHAPTER 3. System Identification and Model Verification 33 3.1 Background 33 3.2 Evaluation of Eigenproperties using Ambient Vibration Data 33 3.3 Comparison of System ID Result with Analytical Eigen Properties 37 3.4 Modal Parameter Identification from Chino Hills Earthquake Response 38 3.5 Effect of Parameter Uncertainty on Modal Frequency 40 3.5.1 Soil Spring Modeling 40 3.5.2 Uncertain Parameters Considered 41 3.5.3 Analysis methods 43 3.5.4 Sensitivity of Modal Frequencies 46 3.6 Finite Element Model Updating 55 3.6.1 Sensitivity Based Model Updating 56 3.6.2 Selection of Modes and Parameters 59 3.6.2.1 Selection of Modes 59 3.6.2.2 Selection of Parameters 60 3.6.3 Updated Results 61 3.7 Closure 64 CHAPTER 4 Seismic Analysis 65 4.1 Background 65 4.2 Scope 67 4.3 Response Analysis under Northridge Earthquake 68 4.4 Response Analysis under Chino Hills Earthquake 70 4.5 Generation of Fragility Curves 73 4.6 Simulation of Ground Motion Considering Spatial Variability 78 4.6.1 Generation of Evolutionary PSDF from Given Ground Motion using STFT 78 4.6.2 Generation of Evolutionary PSDF from Given x Ground Motion using Wavelet Transform 79 4.6.3 Simulation of One Dimensional Multi Variate ( 1D mV), Nonstationary Gaussian Stochastic Process 82 4.6.4 Simulation of Seismic Spectrum Compatible Accelrograms 85 4.6.5 Examples of Generated Seismic Ground Motion 90 4.7 Results 99 4.8 Closure 105 CHAPTER 5 Wind Sensor Installation and Wind Speed Measurement 106 5.1 Background 106 5.2 Anemometer and Data Acquisition System 107 5.2.1 Anemometer for Vantage Pro2 107 5.2.2 Anemometer Transmitter with Solar Power 107 5.2.3 Wireless Repeater with Solar Power 107 5.2.4 Wireless Weather Envoy ( Wireless Receiver) 110 5.2.5 WeatherLink Software for Data Collection 110 5.2.6 Data Acquisition Software Developed 110 5.2.7 Experimental Setup 111 5.2.8 Anemometer Installation and Data Acquisition System 112 5.3 WeatherLink Software for Data Collection 114 5.4 Recorded Wind Velocities 115 5.5 Closure 117 CHAPTER 6 Wind Buffeting Analysis 118 6.1 Background 118 6.2 Scope 121 6.3 Conditional Simulation of Gaussian Random Processes 122 6.3.1 Conditional Simulation in Frequency Domain 123 6.4 Conditional Simulation of Non Gaussian Random Processes 124 xi 6.5 Simulation of Spatially Correlated Gaussian Wind Velocity Fluctuations 128 6.6 Conditional Simulation of Gaussian Wind Velocity Fluctuations 135 6.7 Conditional Simulation of non Gaussian Wind Velocity Fluctuations 138 6.8 Buffeting Force Calculation 152 6.9 Buffeting Response of Vincent Thomas Bridge 153 6.10 Closure 157 CHAPTER 7 Traffic Load Analysis 158 7.1 Background 158 7.2 Moving Load Analysis 158 7.3 Closure 165 CHAPTER 8 Conclusions and Future Work 166 8.1 Summary and Conclusions 166 8.2 Future Work 169 REFERENCES 171 xii LIST OF FIGURES Page Figure 2.1 The shape of the initial cable profile under dead load 18 Figure 2.2 Cross section of deck 19 Figure 2.3 Location of stringers in one side of the deck 20 Figure 2.4 Commonly used lateral bracing systems and stiffening girders 21 Figure 2.5 Horizontal system ( K type) 21 Figure 2.6 Vertical web system ( Worren type) 22 Figure 2.7 Different sections of the tower 26 Figure 2.8 Typical tower cross section 26 Figure 2.9 The detailed model of one panel 28 Figure 2.10 Deck shear connector ( before retrofit) 29 Figure 2.11 Deck shear connector ( after retrofit) 29 Figure 2.12 K truss modifications after retrofit 30 Figure 2.13 Suspender modifications after retrofit 30 Figure 2.14 First three mode shapes of the simple model 31 Figure 3.1 Location and direction of sensors installed in the bridge 35 Figure 3.2 Vertical accelerometer data used in the study 36 Figure 3.3 Lateral accelerometer data used in the study 36 Figure 3.4 Plot of SV vs. Frequency 37 Figure 3.5 Detailed model in SAP 2000 with foundation springs 41 Figure 3.6 Tornado diagram considering 19 parameters 51 xiii Figure 3.7 Relative variance contribution ( neglecting correlation terms) from FOSM analysis 54 Figure 3.8 Three dimensional finite element model of Vincent Thomas Bridge 55 Figure 3.9 Procedure for the sensitivity based model updating 58 Figure 3.10 Comparison of frequency differences using the initial and updated FE models 62 Figure 4.1 Location and direction of sensors 69 Figure 4.2 Comparison of measured and calculated longitudinal displacement at channel # 10 location 69 Figure 4.3 Comparison of analytical lateral response at channel 5 due to ground motions at east anchorage, east tower and west tower with field measured response 71 Figure 4.4 Comparison of analytical lateral response at channel 3 due to ground motions at east tower with field measured response 72 Figure 4.5 Comparison of analytical vertical response at channel 17 due to ground motions at east tower with field measured response 72 Figure 4.6 Comparison of analytical longitudinal response at channel 10 due to ground motions at east tower with field measured response 73 Figure 4.7 Before and after retrofit Fragility curves for different damage levels 77 Figure 4.8 Evolutionary PSDF of LA21 earthquake record using STFT method 79 Figure 4.9 Evolutionary PSDF of LA21 earthquake record using wavelet transform 82 Figure 4.10 Iterative scheme to simulate spectrum compatible acceleration time histories 89 Figure 4.11 Different support locations of the bridge 91 Figure 4.12 Acceleration time history of LA 21 scenario earthquake 92 Figure 4.13 Acceleration time history at location 1 93 Figure 4.14 Acceleration time history at location 2 94 xiv Figure 4.15 Acceleration time history at location 3 94 Figure 4.16 Acceleration time history at location 4 95 Figure 4.17 Acceleration time history at location 5 95 Figure 4.18 Acceleration time history at location 6 96 Figure 4.19 Displacement time history at location 3 96 Figure 4.20 Displacement time history at location 6 97 Figure 4.21 Comparison between simulated and design spectra at location 1 using STFT 97 Figure 4.22 Comparison between simulated and design spectra at location 3 using STFT 98 Figure 4.23 Comparison between simulated and design spectra at location 1 using Wavelet 98 Figure 4.24 Comparison between simulated and design spectra at location 3 using Wavelet 99 Figure 4.25 Absolute axial force demand envelope for the bridge girder 101 Figure 4.26 Absolute shear force demand envelope for the bridge girder 101 Figure 4.27 Absolute moment demand envelope for the bridge girder 102 Figure 4.28 Absolute torsional force demand envelope for the bridge girder 102 Figure 4.29 Absolute axial force demand envelope for the east tower of the bridge 103 Figure 4.30 Absolute shear force demand envelope for the east tower of the bridge 103 Figure 4.31 Absolute moment demand envelope for the east tower of the bridge 104 Figure 4.32 Absolute torsional force demand envelope for the east tower of the bridge 104 Figure 5.1 Anemometer 108 Figure 5.2 Anemometer transmitter with solar power 109 xv Figure 5.3 Wireless repeater with solar power 109 Figure 5.4 Wireless Weather Envoy ( Wireless Receiver) 110 Figure 5.5 Layout of the data acquisition system 111 Figure 5.6 Locations of anemometers, transmitters, repeaters and receivers 113 Figure 5.7 Distance between different components 113 Figure 5.8 Installation of anemometers, transmitters and repeaters ( a) Top of the east tower ( b) Vertical post on deck ( c) East tower platform ( d) Anchorage house wall 115 Figure 5.9 Screen shots from Weather Link and data acquisition system ( a) Anemometer # 1 ( b) Anemometer # 2 ( c) Anemometer # 3 ( d) Data acquisition system 116 Figure 5.10 Wind velocity recorded for 24 hrs on April 8, 2009 ( 1 sample/ min) 116 Figure 5.11 Wind velocity recorded for 30 minutes on April 15, 2009 ( 1 sample/ 3s) 117 Figure 6.1 Flow chart of conditional simulation of non Gaussian random processes 127 Figure 6.2 Installed anemometer locations on VTB 133 Figure 6.3 Locations of “ aerodynamic” nodes along the bridge deck 133 Figure 6.4 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 2) in m/ s from Gaussian unconditional simulation 134 Figure 6.5 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 1) in m/ s from Gaussian unconditional simulation 134 Figure 6.6 Horizontal wind velocity fluctuations at two different locations from Gaussian unconditional simulation 135 xvi Figure 6.7 Measured wind velocity fluctuation at anemometer # 1 location 136 Figure 6.8 Measured wind velocity fluctuation at anemometer # 2 location 136 Figure 6.9 Measured wind velocity fluctuation at anemometer # 3 location 136 Figure 6.10 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 2) in m/ s from Gaussian conditional simulation 137 Figure 6.11 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 1) in m/ s from Gaussian conditional simulation 137 Figure 6.12 Horizontal wind velocity fluctuations at two different locations from Gaussian unconditional simulation 138 Figure 6.13 Actual and analytical PDF of wind velocity fluctuation measured at anemometer # 1 location 140 Figure 6.14 Actual and analytical PDF of wind velocity fluctuation measured at anemometer # 2 location 140 Figure 6.15 Actual and analytical PDF of wind velocity fluctuation measured at anemometer # 3 location 141 Figure 6.16 Actual and analytical CDF of wind velocity fluctuation measured at anemometer # 1 location 141 Figure 6.17 Actual and analytical CDF of wind velocity fluctuation measured at anemometer # 2 location 142 Figure 6.18 Actual and analytical CDF of wind velocity fluctuation measured at anemometer # 3 location 142 Figure 6.19 Horizontal wind velocity fluctuations at two different locations from non Gaussian conditional simulation 144 Figure 6.20 Simulated and target CDF of wind velocity fluctuation at point # 10 144 Figure 6.21 Simulated and target CDF of wind velocity fluctuation at point # 16 145 xvii Figure 6.22 Comparison of PSDF from simulated wind velocity fluctuation and target PSDF at point # 10 115 Figure 6.23 Comparison of PSDF from simulated wind velocity fluctuation and target PSDF at point # 16 146 Figure 6.24 Comparison of PSDF from measured velocity fluctuation at anemometer # 1 and assumed analytical PSDF 147 Figure 6.25 Comparison of PSDF from measured velocity fluctuation at anemometer # 2 and assumed analytical PSDF 147 Figure 6.26 Comparison of PSDF from measured velocity fluctuation at anemometer # 3 and assumed analytical PSDF 148 Figure 6.27 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 2) in m/ s from non Gaussian conditional simulation 148 Figure 6.28 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 1) in m/ s from non Gaussian conditional simulation 149 Figure 6.29 Simulated wind velocity fluctuations at location # 10 with three different simulation techniques 149 Figure 6.30 Simulated wind velocity fluctuations at location # 16 with three different simulation techniques 150 Figure 6.31 Horizontal wind velocity fluctuations at different locations along the tower ( around anemometer # 2) in m/ s from Gaussian unconditional simulation 150 Figure 6.32 Horizontal wind velocity fluctuations at different locations along the tower ( around anemometer # 2) in m/ s from Gaussian conditional simulation 151 Figure 6.33 Horizontal wind velocity fluctuations at different locations along the tower ( around anemometer # 2) in m/ s from non Gaussian conditional simulation 151 Figure 6.34 Schematic diagram for aerodynamic forces on bridge deck 155 Figure 6.35 Simulated lateral deck displacements at the center of the mid span 156 Figure 6.36 Simulated vertical deck displacement at the center of the mid span 156 xviii Figure 7.1 Plan view of deck shear connectors before and after retrofit 159 Figure 7.2 Deck shear connector 159 Figure 7.3 Deck shear connector design drawing 160 Figure 7.4 HS20 44 AASTHO traffic loading 161 Figure 7.5 Different traffic load cases 162 Figure 7.6 Axial force in shear connector due to traffic load ( before and after retrofit) 163 Figure 7.7 Vertical shear force in shear connector due to traffic load ( before and after retrofit) 163 Figure 7.8 Longitudinal shear force in shear connector due to traffic load ( before and after retrofit) 164 Figure 7.9 Shear key in east side span 164 xix LIST OF TABLES Page Table 2.1 Calculated dead load ( kip/ ft) along the length of bridge 16 Table 2.2 Calculated nodal coordinates of the cable only system 17 Table 2.3 Calculated sectional properties of panels 23 Table 2.4 Calculated sectional properties of the tower sections ( before retrofit) 27 Table 2.5 Calculated sectional properties of the tower sections ( after retrofit) 27 Table 2.6 Comparison of modal frequencies in Hz ( before retrofit) 31 Table 2.7 Comparison of modal frequencies in Hz ( after retrofit) 32 Table 3.1 Location and direction of accelerometers 35 Table 3.2 Comparison of modal frequencies in Hz ( before retrofit) 39 Table 3.3 Comparison of modal frequencies in Hz ( after retrofit) 39 Table 3.5 Location and number of piles considered 41 Table 3.6 Parameters considered for sensitivity analysis 43 Table 3.7 Comparison of natural frequencies 59 Table 3.8 Parameters selected for adjustment 60 Table 3.9 MAC matrix of updated FE model 62 Table 3.10 Comparison of natural frequencies between baseline and updated FE model 63 Table 3.11 Updated design parameters 63 Table 4.1 Different support motions considered with channel numbers 70 xx Table 4.2 Details of the motions considered in this study for fragility Development 77 Table 4.3 Site coefficient parameters to calculate design spectra at different supports 92 Table 4.4 Displacement demand comparison 100 Table 5.1 Settings of different repeaters 114 Table 5.2 Settings of different receivers 114 Table 6.1 Properties for assumed generalized extreme value distribution 141 Table 7.1 Shear stress developed in shear key bolts 165 1 CHAPTER 1 INTRODUCTION 1.1 Background Throughout the history of suspension bridges, their tendency to vibrate under different dynamic loadings such as wind, earthquake, and traffic loads has been a matter of concern. The failure of the Tacoma Narrows bridge in 1940 has pointed out that the suspension bridges are vulnerable to wind loading ( Rannie 1941). It is now widely accepted that the wind induced vibration of suspension bridges may be significant and should be taken into consideration. Similar conclusions have also been drawn for other dynamic loadings. As a prerequisite to the investigation of aerodynamic stability, traffic impact, soil structure interaction and earthquake resistant design of suspension bridges, it is necessary to know certain dynamic characteristics such as the natural frequencies and the possible modes of vibration. Several investigations have been taken place in recent years to determine the vibrational properties of suspension bridges. However, the complexity of a suspension bridge structure makes the determination of vibrational characteristics difficult. With the advent of computers, non conventional structures like suspension bridges are analyzed with the finite element ( FE) analysis technique. There are several commercially available finite element software packages that are used by practicing 2 engineers as well as researchers, which can evaluate the response of a suspension bridge from operational traffic, wind and earthquake loads taking into account both material and geometric non linear behavior. In addition to analytical modeling and response analysis of suspension bridges, field tests are also very important from the analysis and design point of view. Field test results not only give experimental data but also help us to understand the behavior of the structure and to calibrate the analytical model. To perform field tests, it is necessary to measure, input loadings such as wind velocity at different pints and earthquake ground acceleration at different support locations, and output responses such as acceleration, velocity and displacement time history at different points of the bridge. For predicting response of long span suspension bridges under random wind, the most widely used method is the frequency domain analysis. In theory, the frequency domain solution is accurate, when the load response relationship is linear. Although the structural elements in a suspension bridge generally behave in a linear elastic fashion under normal loading, the overall load displacement relationship exhibits geometrical nonlinearity, particularly when it is subjected to high wind. Therefore, in this case, a frequency domain analysis may not be appropriate. One way in which the limitation of the frequency domain analysis can be overcome is the use of Monte Carlo simulation technique. One of the most important components of the Monte Carlo simulation method is the generation of sample functions of stochastic processes, fields, or waves those are involved in the problem. For buffeting analysis, wind velocity fluctuation in the horizontal and vertical directions needed to be digitally simulated and fed into the equation of motion. Since the length of a modern suspension bridge generally exceeds 1 3 km, the simulated sample functions must accurately describe the probabilistic characteristics not only in terms of temporal variation but also in spatial distribution. Similarly for seismic response, critical members of the bridge may undergo significant nonlinear deformation and a simple response spectrum method for analyzing such response may not be adequate. In addition, there may be significant variation of ground motion from one support of the bridge to the other. 1.2 Literature Survey Theoretical and practical treatises on the vibrational characteristics and the dynamic analysis of suspension bridges, have been developed by many authors, especially after the disastrous collapse of the Tacoma Narrows Bridge in 1940 ( Rennie 1941). Bleich et al., 1950 studied the free vertical and torsional vibration by solving a forth order linearized differential equation. In addition, an approximate method of the Rayleigh Ritz type solution was suggested. However, the procedure is applicable only for calculating the lowest few modes due to the great level of complexity and redundancy of higher modes of suspension bridges. Steinman, 1959 introduced a number of simplified formulas for estimating the natural frequencies and the associated mode shapes of vibration, both vertical and torsional, of suspension bridges. Japanese researchers ( Konishi et al 1965; Konishi and Yamada 1969; Yamada and Takemiya 1969, 1970; Yamada and Goto 1972; and Yamada et al. 1979) performed extensive studies to investigate the vertical and lateral vibration as well as the tower pier system of a three span suspension bridge by using a lumped mass system interconnected by spring elements. In their analysis for the 4 suspended structure, they assumed simple harmonic excitations and applied it separately to each supporting point. They reported that there was a fairly significant contribution from the higher modes to the bending response and a large number of modes should be included to accurately determine the dynamic response of suspension bridges. The geometrically nonlinear behavior of suspension bridges was considered ( Tezcan and Cherry 1969) due to large deflection and presented an iterative technique for the nonlinear static analysis by using tangent stiffness matrices. These matrices are incorporated in obtaining the free vibrational modes of the structure. In their analysis, the bridge was modeled as a three dimensional lumped mass system. They calculated the response of the bridge considering three orthogonal components of uniform ground motion and pointed out that the longitudinal motion of the deck as well as the vertical motion of the tower were small and therefore could be neglected. Major advances in studying the dynamic characteristics of suspension bridges have been achieved through the use of finite element method and linearized deflection theory ( Abdel Ghaffar 1976, 1977, 1978a, 1978b, 1979, 1980 and 1982). Natural frequencies, mode shapes, and energy capacities of the different structural components for vertical, torsional, and lateral vibrations were investigated. Several examples were presented and the applicability of the proposed methods was illustrated by comparing the results obtained from analyzing the Vincent Thomas bridge ( Los Angeles Harbor) with the results of full scale ambient vibration tests ( Abdel Ghaffar 1976, 1978 and Abdel Ghaffar and Housner 1977). Some researchers ( Abdel Ghaffar and Rubin 1983a and Abdel Ghaffar and Rubin 1983b) studied the effect of large amplitude nonlinear free coupled vertical torsional vibrations of suspension bridges using a continuum approach 5 where approximate solutions of the nonlinear coupled equations were conducted. Nonlinearities due to large deflections of cables, the axial stretching of stiffening structure, and the nonlinear curvature of the stiffening structure were considered. It was mentioned that the importance of geometric nonlinearities arises only for very high amplitude vibration. Also, they studied using two dimensional models the directional vertical, torsional, and lateral earthquake response, in both time and frequency domains, of long span suspension bridges subjected to multiple input excitations ( Abdel Ghaffar and Rubin 1982; Abdel Ghaffar and Rubin 1983c and Abdel Ghaffar et al. 1983). In addition, they considered a simplified model for the tower pier system and investigated the longitudinal vibration response taking into account the flexibility and damping characteristics of the underlying and surrounding soil. They applied their procedure to the tower pier system of the Golden Gate bridge ( San Francisco) and different soil conditions were used. The vertical response of suspension bridges has been studied to seismic excitations using a stochastic approach ( Dumanoglu and Severn 1990). They applied their method to three suspension bridges using one set of earthquake records and a filtered white noise as well. They pointed out that the accuracy of that approach, in comparison to the time history approach, depends upon the magnitudes of the fundamental period of the bridge under consideration. They reported that, for long span suspension bridges like the Bosporus ( in Turkey) and Humber ( in England) bridges, the response results of the stochastic approach should be cautiously assessed, especially when the earthquake records are not zero padded. 6 Some researchers ( Lin and Imbsen 1990; Ketchum and Seim 1991 and Ketchum and Heledermon 1991) carried out an investigation on the Golden Gate bridge by developing an elaborate 3 D finite element model. The lower wind bracing system of the bridge was considered to carry a light train. They incorporated different elements types and performed a nonlinear static analysis to determine the stiffness of the bridge in its dead load state and used this matrix in the solution for the natural frequencies and mode shapes. Their model is verified by comparing its results with those obtained from previous studies ( Abdel Ghaffer and Scanlan 1985a and Abdel Ghaffer and Scanlan 1985b). They reported that most of the lowest modes involving vibration of cables and torsional motion of the deck are not relevant to the earthquake performance of the bridge. A 3D finite element model was proposed for the Vincent Thomas bridge ( Niazy et al. 1991). They considered geometrical nonlinearities in suspension bridges, and an iterative nonlinear static analysis technique was adopted. The stiffening truss, tower and cable bent elements, were modeled as 3 D frame elements and cable elements were modeled as 3 D truss elements. In their study, 50 lowest natural frequencies and the corresponding mode shapes of the bridge model were determined in its dead load configuration. However, in their modeling they did not consider the actual mass distribution over the length of the bridge. They considered uniform mass distribution over the center span and the side spans. Initial shape of the cable is one of the important parameters in the analysis of suspension bridges. A non linear shape finding analysis was used for a self anchored suspension bridge named Yongjong Grand Bridge ( Kim et al., 2002). The shape finding analysis determines the coordinates of the main cable and 7 initial tension of main cable and hangers, which satisfies the design parameters at the initial equilibrium state under full dead loads. Several models and expressions have been proposed ( Davenport 1968) in relation to spatial variation of wind velocity fluctuation. For a more complete bibliography, the reader is referred to Simiu and Scanlan ( 1996). The analytical work by Beliveau et al., 1977 combined the effect of buffeting and self excited forces. They used a two degrees of freedom mathematical model. Even though simulation techniques have been reported since 1970 ( Shinozuka and Jan 1972), some earlier studies assumed uniformly distributed wind velocity fluctuations for the nonlinear time history analysis of cable supported bridges ( Arzoumanidis 1980). In past decades, a number of researchers reported on efficient methods for generating spatially correlated wind velocity fluctuations ( Li and Kareem 1993; Shinozuka and Deodatis 1996; Deodatis 1996; Facchini 1996; Yang et al. 1997; Paola 1998; Paola and Gullo 2001). As a result of improvements in simulation techniques as well as computational speed, the time domain approach has been utilized more frequently in recent buffeting analyses of long span cable supported bridges to take aerodynamic and/ or geometric nonlinearity into consideration ( Aas Jakobsen and Strømmen 1998, 2001; Minh et al. 1999; Ding and Lee 2000; Chen et al. 2000; Chen and Kareem 2001; Lin et al. 2001). Kareem’s group, in particular, has reported extensively on the line of time domain analysis framework for use in predicting aerodynamic nonlinear responses by incorporating frequency dependent parameters of unsteady aerodynamic forces by utilizing a rational function approximation technique ( Chen and Kareem 2001). This technique is also readily available for the structure originated nonlinearity in buffeting analysis. However, only a few studies utilized a nonlinear analysis procedure 8 for estimating buffeting response using structural nonlinearity, which is potentially involved in long span cable supported bridges, has been taken into consideration ( Ding and Lee 2000; Lin et al. 2001). The spatial variation of earthquake ground motions may have significant effect on the response of long span suspension bridges. Abdel Ghaffar and Rubin ( 1982) and Abdel Ghaffar and Nazmy ( 1988) studied response of suspension and cable stayed bridges under multiple support excitations. Zerva ( 1990) and Harichandran and Wang ( 1990) examined the effect of spatial variable ground motions on different types of bridge models. Harichandran et al. ( 1996) studied the response of long span bridges to spatially varying ground motion. Deodatis et al. ( 2000) and Kim and Feng ( 2003) investigated the effect of spatial variability of ground motions on fragility curves for bridges. Lou and Zerva ( 2005) analyzed the effects of spatially variable ground motions on the seismic response of a skewed, multi span, RC highway bridge. Most of the aforementioned studies dealt with simple FE models of the bridge, as a result response of critical members could not be evaluated. In the present analysis a panel based detailed 3D FE model of a long span suspension bridge is utilized. In this study, an iterative algorithm is proposed to generate spatially variable, design spectrum compatible acceleration time histories at different support locations of the bridge. The proposed algorithm is used to generate synthetic ground motions at six different points on the ground surface. For generating non stationary accelerograms, previously researchers used time dependent envelope function on top of simulated stationary ground motions ( Deodatis 1996). In this study by using evolutionary power spectral density function from the mother accelerogram, a new algorithm has been 9 proposed to simulate spatial variable ground motions. In the simulated acceleration time histories the temporal variations of the frequency content are same as the mother accelerogram. Mukherjee and Gupta ( 2002) proposed a new wavelet based approach to simulate spectrum compatible time histories. But they only considered one design spectrum and simulated one accelrogram from a single mother acceleration time history. Sarkar and Gupta ( 2006) developed a wavelet based approach to simulate spatially correlated and spectrum compatible accelerogram. So far in a broad sense two approaches have been introduced by researchers regarding conditional simulation. The two approaches are based on “ kriging” ( Krige, 1966) ( linear estimation theory applied to random functions) and conditional probability density function. Vanmarcke and Fenton ( 1991) applied conditional simulation of to simulate Fourier coefficients using kriging technique. Kameda and Morikawa ( 1992 and 1994), used an analytical framework based on spectral representation method, derived joint probability density functions of Fourier coefficients obtained from the expansion of conditioned random processes into Fourier series. They calculated conditional expectations and variances of the conditioned random processes and considered their first passage probabilities. Hoshiya ( 1994) considered a conditional random field as a sum of its kriging estimate and the error. He simulated the kriging estimate and the error separately and combined them to get the Gaussian conditionally simulated field. In all the above studies the investigators considered Gaussian processes and Gaussian random fields. Sometimes the assumption of Gaussian wind loading is not correct. In those cases, conditional simulation of non Gaussian wind velocity field should be used. Elishakoff et 10 al. ( 1994) combined the conditional simulation technique of Gaussian random fields by Hoshiya ( 1994) and the iterative procedure for unconditional simulation of non Gaussian random fields by Yamazaki and Shinozuka ( 1988), to conditionally simulate timeindependent non Gaussian random fields. Gurley and Kareem ( 1998) developed a procedure for conditional simulation of multivariate non Gaussian velocity/ pressure fields. For mapping the Gaussian process to non Gaussian process and vice versa, they used modified Hermite transformation using Hermite polynomial function. For buffeting analysis of long span cable supported bridges Chen ( 2001), Kim ( 2004) used time domain analysis to consider the effect of non linearity in the structure. Also they only considered the wind forces on the deck only. They neglected the coupling effect of wind forces on tower and cable. Sun ( 1999) considered the coupling effect of the aeroelastic forces on the bridge deck, towers and cables. But they did not consider a 3D detailed finite element ( FE) model of the bridge. Recently, He ( 2008) considered a detailed 3D model for buffeting analysis. 1.3 Objectives and Scope The main purpose of this research is to evaluate the performance of a long span suspension bridge under seismic, wind, and traffic loads. A member based detailed threedimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by 11 simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino Hills earthquake and comparing with the measured response. Tornado diagram and first order second moment ( FOSM) methods are applied for evaluating the sensitivity of different parameters on the eigenproperties of the FE models. This kind of study will be very helpful in selecting parameters and their variability ranges for FE model updating of suspension bridges. Considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed and the ductility demands of critical sections of the tower are presented in terms of fragility curves. Effect of spatial variability of ground motions on seismic displacement demand and seismic force demand is investigated. To generate spatially correlated design spectrum compatible nonstationary acceleration time histories, a newly developed algorithm using evolutionary power spectral density function ( PSDF) and spectral representation method is used. To simulate the wind velocity field accurately for the bridge site, measurement of the wind velocity is needed at the bridge location. For colleting actual wind data i. e. wind velocity and direction, three anemometers have been installed at three different locations of the bridge, so that the wind velocity field can be simulated in both horizontal and vertical directions. The measured wind velocity fluctuation data have been used for conditional simulation of wind velocity fluctuation field. Finally, response of Vincent Thomas Bridge under conditionally simulated wind velocity field is also presented in this study. A new simulation technique for conditional simulation of non Gaussian wind velocity fluctuation field is proposed and used for 12 buffeting analysis of the bridge under simulated wind load. Analysis of the Vincent Thomas bridge under traffic load is also carried out in this study. 1.4 Dissertation Outline The dissertation contains the following chapters Chapter 2 summarizes the finite element ( FE) modeling of the before and after retrofitting of the bridge. Chapter 3 presents the system Identification results obtained from response of the bridge and compared with modal parameters obtained from analytical model. A sensitivity analysis is also carried out. Chapter 4 proposes a new methodology to simulate spectrum compatible spatial variable ground motions. Response variability due to spatial variation in ground motion is also assessed. Chapter 5 describes the wind sensors installation in the bridge and data collection. Chapter 6 proposes a new methodology to conditionally simulate non Gaussian wind velocity fluctuation profiles using the data collected by anemometers at the bridge site. Wind buffeting analysis also carried out using the simulated wind velocity fluctuation profile. Chapter 7 describes the traffic load analysis. 13 CHAPTER 2 FINITE ELEMENT MODELING OF VINCENT THOMAS BRIDGE 2.1 Background With the advent of high speed computer, major advances in studying the dynamic characteristics of suspension bridges have been achieved through the use of finite element method. In addition, effort has also been given for developing simplified models that can predict response consistent with detailed model. In recent years, several commercially available finite element software packages have been used by practicing engineers as well as researchers to evaluate the response of a suspension bridge from operational traffic, wind and earthquake loads taking into account both material and geometric non linear behavior. This chapter focuses on numerical modeling of the Vincent Thomas Bridge. A member based detailed three dimensional Finite Element ( FE) as well as a panel based simplified model of the Vincent Thomas bridge have been developed for before and after retrofit of the bridge. 2.2 Calculation of Dead Weight The dead load along the length of the bridge has been calculated. Table 1 shows the calculated dead load of the different components of the bridge. It has been found that the weight per unit length of the bridge in the center span is very close to the design value of 14 7.2 kip/ ft indicated in the design drawing. The dead load calculation is also compared with the values reported by Abdel Ghaffer, 1976 shown in Table 2.1. 2.3 Calculation of the Initial Shape of the Cable Initial shape of the cable is one of the important parameters in the modeling of suspension bridges. Initial shapes of the cables of Vincent Thomas Bridge have been calculated using non linear shape finding analysis and subsequently used in the FE model. The shape finding analysis determines the coordinates of the main cable and initial tension of main cable and hangers, which satisfies the design parameters at the initial equilibrium state under full dead loads. Details of the analysis methodology and software are described in Kim et al., 2002. The shape of the initial cable profile in the form of preliminary and final configurations are tabulated in Table 2.2 and the initial cable profile is plotted in Figure 2.1. 2.4 Panel Based Simple Model For simplified panel based modeling, the girders and diaphragms are considered as equivalent 3D frame elements. The cable and suspender are modeled as 3D truss element. Also as in the case of detailed model, truss and cable bent were modeled with frame elements. Dampers are also included in the simplified model only at the tower and girder connections. . FE modeling is done with SUCOT ( Kim, 1993) and SAP 2000 V10 ( Computer and Structures, 2002). Area of the stiffening girder is set equal to the sum of the area of top chord, bottom chord and web. 15 Table 2.1 Calculated dead load ( kip/ ft) along the length of bridge Present study Different components Center span Side span Abdel Ghaffer, 1976 Curb 0.066 0.066 Bracket 0.019 0.019 Crash barrier 0.413 0.413 Sub total 0.498 0.498 0.203 Grating 0.036 0.036 Railing 0.0414 0.0414 Fence 0.131 0.131 Sub total 0.208 0.208 0.199 Lightweight concrete 2.521 2.521 2.592 Reinforcement steel 0.173 0.173 0.173 Stringers 0.544 0.544 0.682 Bracings 0.154 0.154 Sub total 3.392 3.392 3.447 Floor Truss 0.41 0.41 Inspection walkway 0.098 0.098 Inspection rail 0.052 0.052 Wind shoe 0.008 0.008 Bridge floor Sub total 0.568 0.568 0.613 Top chord 0.313 0.313 0.315 Bottom chord 0.307 0.291 0.302 Gusset plate, splice 0.234 0.234 0.124 Web ( diagonal) 0.162 0.166 0.142 Post ( vertical) 0.055 0.055 0.053 Strut, rivet, bolt etc 0.007 0.007 0.007 Stiffening truss Sub total 1.078 1.066 0.943 K truss 0.161 0.154 0.159 Lateral system Sub total 0.161 0.154 0.159 Cable 0.971 0.971 1.025 Suspenders 0.066 0.065 0.054 Cable Sub total 1.037 1.036 1.079 Cable and suspender weight 1.037 1.036 1.079 Suspended structure weight 5.905 5.886 5.564 Total weight 6.942 6.922 7.170 For SI: 1 kip/ ft = 14.593 kN/ m 16 Table 2.2 Calculated nodal coordinates of the cable only system Y ( ft) Z ( ft) Preliminary Final X ( ft) Preliminary Final configuration configuration configuration configuration Remark 1256.500 29.5833 29.5833 163.1400 163.1400 Cable bent 1221.840 29.7626 29.7628 172.5984 172.6094 1190.780 29.9295 29.9297 181.5906 181.6075 1159.720 30.1039 30.1042 191.0963 191.1143 1128.660 30.2867 30.2869 201.1168 201.1324 1097.600 30.4782 30.4784 211.6536 211.6664 1066.540 30.6787 30.6787 222.7081 222.7190 1035.480 30.8882 30.8882 234.2820 234.2919 1004.420 31.1069 31.1069 246.3771 246.3866 973.360 31.3349 31.3349 258.9949 259.0050 942.300 31.5722 31.5723 272.1375 272.1489 911.240 31.8188 31.8190 285.8068 285.8202 880.180 32.0749 32.0751 300.0048 300.0211 849.120 32.3404 32.3407 314.7336 314.7525 818.060 32.6154 32.6157 329.9955 330.0132 787.000 32.8998 32.9000 345.7927 345.8045 750.000 33.2500 33.2500 365.2600 365.2600 Tower 714.380 32.9474 32.9482 351.2700 351.3089 683.320 32.6931 32.6946 339.6119 339.6821 652.260 32.4485 32.4505 328.4872 328.5843 621.200 32.2136 32.2161 317.8940 318.0124 590.140 31.9885 31.9913 307.8302 307.9643 559.080 31.7732 31.7762 298.2938 298.4394 528.020 31.5678 31.5710 289.2832 289.4360 496.960 31.3723 31.3756 280.7965 280.9516 465.900 31.1870 31.1901 272.8323 272.9846 434.840 31.0117 31.0147 265.3889 265.5335 403.780 30.8468 30.8495 258.4649 258.5980 372.720 30.6922 30.6946 252.0591 252.1778 341.660 30.5482 30.5502 246.1702 246.2718 310.600 30.4149 30.4165 240.7971 240.8811 279.540 30.2925 30.2937 235.9389 236.0069 248.480 30.1812 30.1821 231.5945 231.6482 217.420 30.0812 30.0819 227.7631 227.8042 17 Table 2.2 Calculated nodal coordinates of the cable only system ( contd.) Y ( ft) Z ( ft) Preliminary Final X ( ft) Preliminary Final configuration configuration configuration configuration Remark 186.360 29.9930 29.9934 224.4441 224.4743 155.300 29.9168 29.9170 221.6367 221.6577 124.240 29.8531 29.8531 219.3405 219.3539 93.180 29.8024 29.8023 217.5550 217.5626 62.120 29.7655 29.7653 216.2799 216.2833 31.060 29.7429 29.7427 215.5150 215.5158 0.000 29.7353 29.7350 215.2600 215.2600 Center For SI: 1 ft = 0.3048 m 0 50 100 150 200 250 300 350 400  1500  1000  500 0 500 1000 1500 Length ( ft) Height ( ft) Figure 2.1 The shape of the initial cable profile under dead load Calculations of other cross sectional properties of girder ( moment of inertia and torsional constant) are given as follows: z x y 18 2.4.1 Moment of Inertia ( Iz) Moment of inertia of various members is computed from the equations in the table below and their values are given following the table. Chord Slab Stringer ( / 2) 2 2 I = A× e × y 12 3 bh Iy = = = 4 1 2 i i i Iy A d Chord: side span = 2 2 2 2 55.56in × 29.585 × 2ea× 2( both) = 194,520in ft center span = 2 2 2 2 53.78in × 29.585 × 2ea× 2( both) = 188,288in ft Slab: Figure 2.2 shows the cross section of the deck. Figure 2.2 Cross section of deck 4 3 3 6744.9 12 54.5 0.5 12 ft bh Iy = × = = For equilibrant steel section: 4 449.7 15 6744.9 Iy = = ft 27.25 27.25 CL 0.5 19 Stringer: Figure 2.3 shows the location of stingers in one side of the deck. Figure 2.3 Location of stringers in one side of the deck For one side: 2 2 2 2 4 4 1 2 Iz A d 0.1389( 3.5 10.5 17.5 24.5 ) 142.93 ft i i i = = + + + = = So, for one stiffening girder 4 142.93 367.78 2 449.7 Iz = + = ft From ( Abdel Ghaffer, 1976), slab + stringers : ( 105,000+ 290) sq. in. sq. ft./ 144/ 2= 4 365.59 ft 2.4.2 Torsional Constant ( J) Figure 2.4 shows commonly used lateral bracing systems and stiffening girders for suspension bridges. i i i i J = 2 b b d ; i vi i hi i i vi hi i b d b d μ μ μ μ b × + × = 2 2 2 2 2 web : A 0.117 ft ; k truss : A 0.115 ft , Av 0.132 ft d d = − = = b = 59.17 ´ , d= 15 ´ 0.25 2.5 2( 1 ) = \ = + = G E E G μ μ CL 0.5 3.5 7 7 7 20 Figure 2.4 Commonly used lateral bracing systems and stiffening girders ( Abdel Ghaffer 1976) To calculate the torsional constant of the suspension bridge girder two coefficients are used. Here, h μ is the coefficient for horizontal K type system and V μ is for vertical Worren type web system. Figure 2.5 shows the horizontal K type system and Figure 2.6 shows the for vertical Worren type web system. The procedure to calculate those two coefficients are shown below. Figure 2.5 Horizontal system ( K type) 2 59.17 31.08 Ad Av 21 = = ° − ) 43.6 31.08 59.17 / 2 tan ( 1 2 a 0.134 ) 0.082 0.114 sin 43.6 2 0.114 0.082 sin 43.6 cos 43.6 2.5 ( ) sin 2 sin cos ( 3 2 2 3 2 2 2 = + × ° × × × ° × ° = × + × × × × × = a a a μ v d d v A A A A G E h Figure 2.6 Vertical web system ( Worren type) = = ° − ) 44 31.08/ 2 15 tan ( 1 1 a sin cos 2.5 0.137 sin 44 cos44 0.119 2 1 1 2 μ = × × a × a = × × ° × ° = v Ad G E 0.032 59.17 0.119 15 0.134 59.17 15 0.119 0.134 2 2 2 2 = × + × × × × = × + × = i vi i hi i i vi hi i b d b d μ μ μ μ b 4 Ji 2 ibidi 2 0.032 59.17 15 56.416 ft = b = × × × = The sectional properties computed in this section ( Section 1) are summarized in Table 2.3 below for each panel. 15' 31.08' 1 Ad 22 Table 2.3 Calculated sectional properties of panels Area Torsional Constant Iz Iy Panel No. ft 2 ft 4 ft 4 ft 4 1 0.958 24.342 369.010 39.367 2 0.931 23.021 369.010 39.367 3 0.911 22.944 369.010 38.234 4 0.941 21.304 369.010 41.676 5 0.972 21.304 369.010 43.398 6 0.902 16.515 369.010 43.398 7 0.972 21.304 369.010 43.398 8 0.938 19.141 369.010 43.398 9 0.938 19.141 369.010 43.398 10 0.938 19.141 369.010 43.398 11 0.972 21.304 369.010 43.398 12 1.010 23.308 369.010 43.398 13 0.979 23.308 369.010 41.676 14 0.948 24.704 369.010 38.234 15 0.968 24.797 369.010 39.367 16 0.968 24.797 369.010 39.367 17 0.866 19.188 369.010 39.367 18 0.866 19.188 369.010 39.367 19 0.846 19.141 369.010 38.234 20 0.811 16.523 369.010 38.234 21 0.811 16.523 369.010 38.234 22 0.841 16.523 369.010 39.957 23 0.824 15.034 369.010 39.957 24 0.824 15.034 369.010 39.957 25 0.824 15.034 369.010 39.957 26 0.824 15.034 369.010 39.957 27 0.855 15.034 369.010 41.680 28 0.907 19.141 369.010 41.680 29 0.907 19.141 369.010 41.680 30 0.968 19.141 369.010 45.117 31 0.968 19.141 369.010 45.117 32 0.968 19.141 369.010 45.117 33 0.968 19.141 369.010 45.117 34 0.916 15.034 369.010 45.117 35 0.916 15.034 369.010 45.117 23 Table 2.3 Calculated sectional properties of panels ( contd.) Area Torsional Constant Iz Iy Panel No. ft 2 ft 4 ft 4 ft 4 36 0.916 15.034 369.010 45.117 37 0.916 15.034 369.010 45.117 38 0.916 15.034 369.010 45.117 39 0.916 15.034 369.010 45.117 40 0.916 15.034 369.010 45.117 37 0.916 15.034 369.010 45.117 41 0.916 15.034 369.010 45.117 42 0.916 15.034 369.010 45.117 43 0.916 15.034 369.010 45.117 44 0.916 15.034 369.010 45.117 45 0.916 15.034 369.010 45.117 46 0.916 15.034 369.010 45.117 47 0.916 15.034 369.010 45.117 48 0.968 19.141 369.010 45.117 49 0.968 19.141 369.010 45.117 50 0.968 19.141 369.010 45.117 51 0.907 19.141 369.010 41.680 52 0.907 19.141 369.010 41.680 53 0.907 19.141 369.010 41.680 54 0.824 15.034 369.010 39.957 55 0.824 15.034 369.010 39.957 56 0.824 15.034 369.010 39.957 57 0.824 15.034 369.010 39.957 58 0.841 16.523 369.010 39.957 59 0.811 16.523 369.010 38.234 60 0.811 16.523 369.010 38.234 61 0.846 19.141 369.010 38.234 62 0.866 19.188 369.010 39.367 63 0.866 19.188 369.010 39.367 64 0.968 24.797 369.010 39.367 65 0.968 24.797 369.010 39.367 66 0.948 24.704 369.010 38.234 67 0.979 23.308 369.010 41.676 68 1.010 23.308 369.010 43.398 69 0.972 21.304 369.010 43.398 24 Table 2.3 Calculated sectional properties of panels ( contd.) Area Torsional Constant Iz Iy Panel No. ft 2 ft 4 ft 4 ft 4 70 0.938 19.141 369.010 43.398 71 0.938 19.141 369.010 43.398 72 0.938 19.141 369.010 43.398 73 0.972 21.304 369.010 43.398 74 0.902 16.515 369.010 43.398 75 0.972 21.304 369.010 43.398 76 0.941 21.304 369.010 41.676 77 0.911 22.944 369.010 38.234 78 0.931 23.021 369.010 39.367 79 0.958 24.342 369.010 39.367 80 0.958 24.342 369.010 39.367 For SI: 1 ft = 0.3048 m Calculation of tower cross sectional properties: For thin walled closed sections the torsional constant is given by the following formula ( Bredt’s formula): = t ds A J 2 4 Different sections of the tower is shown in Figure 2.7 and a typical plan view of the tower section is shown in Figure 2.8. Table 2.4 and 2.5 show the calculated sectional properties of the tower section at different heights for before and after retrofit models respectively. 2.5 Member Based Detail Model Finite Element modeling of the detailed structure is done with the help of SAP 2000 V10 ( Computer and Structures, 2002). The cables and suspenders are modeled as 3D elastic truss elements. The chords, vertical members and the diagonal members in the stiffening 25 Figure 2.7 Different sections of the tower Figure 2.8 Typical tower cross section X Y 1 2 3 4 42.89 5 52.58 52.58 85.50 85.66 26 Table 2.4 Calculated sectional properties of the tower sections ( before retrofit) Area Ix Iy Torsional Constant Section No. ft 2 ft 4 ft 4 ft 4 1 3.18 20.42 21.07 17.81 2 4.35 42.75 48.25 25.69 3 4.92 57.64 65.06 26.89 4 4.93 60.32 65.86 29.14 5 5.47 76.11 90.37 34.20 For SI: 1 ft = 0.3048 m Table 2.5 Calculated sectional properties of the tower sections ( after retrofit) Area Ix Iy Torsional Constant Section No. ft 2 ft 4 ft 4 ft 4 1 3.66 23.48 24.23 23.55 2 5.00 49.16 55.49 33.98 3 5.66 66.29 74.82 35.57 4 5.67 69.37 75.74 38.54 5 6.29 87.53 103.93 45.23 For SI: 1 ft = 0.3048 m girder are modeled as 3D truss elements. Also members in the diaphragm are modeled as truss elements. The tower, the cable bent leg, and strut members are modeled as frame elements. The reinforced concrete deck is modeled as shell element and the supporting stringers are modeled as beam elements. Hydraulic, viscous dampers between tower and the suspended structure are also modeled according to their properties mentioned in the design drawing. Mass is taken distributed over each and every member. To consider the mass of non structural components, equivalent point mass and mass moment of inertia are distributed at joints in the diaphragm. The most important structural components that are considered for post retrofit modeling are suspended truss system, deck shear connectors, cable bent cross sections, 27 suspenders and dampers installed. Figure 2.9 shows detailed model of one panel and construction drawing. Figure 2.9 The detailed model of one panel 2.5.1 Cable Bent Four feet of stiffening truss in the cable bent was removed to allow free oscillations of the side spans of the bridge. Also, the cable bent cross section was changed. This change in the cross section is considered in the post retrofit modeling of the bridge. Cross sectional properties of the modified sections are calculated and used in the post retrofit analysis. 2.5.2 Deck Shear Connector Deck shear connectors were replaced with new types. Deck shear connectors of the original structure were removed and then a new set was introduced. Figures 2.10 and 2.11 ( taken from Design Drawing) shows the comparison between the shapes of the deck shear connectors before and after retrofit. The FE modeling is done according to this design drawing. ECLAEBMLEENT TSRTIUFSFSENING CABEHLLEAE NNMGOEEDNRETS 28 Figure 2.10 Deck shear connector ( before retrofit) Figure 2.11 Deck shear connector ( after retrofit) 2.5.3 Dampers Total of 48 dampers were installed in the bridge as a retrofit measure with 16 dampers installed in each tower, at the junction between tower and girder connection. In each cable bent, 4 dampers were installed. In the middle of each side span a new diaphragm was inserted. At the location of the inserted diaphragm, 4 more dampers were installed in each side span. These 8 dampers in the side spans were non linear dampers having the form of F = cvn where n = 0.5. For all other dampers, n = 1.0 is used. 2.5.4 Suspended Truss The suspended truss structure was modified by inserting new members and also replacing some members in the K truss in the middle span as well as in the side spans. Figure 2.12 ( taken from Design Drawing) shows the modifications made in the K truss. 2.5.5 Suspenders Some suspenders in the middle span were replaced with new suspenders. Figure 2.13 ( taken from Design Drawing) describes the modified suspenders in the middle span. 29 Figure 2.12 K truss modifications after retrofit Figure 2.13 Suspender modifications after retrofit 2.6 Eigen Value Analysis First 100 eigen vectors were calculated with a convergence tolerance of 1.0 10 5 − × . Table 2.6 shows the comparison of modal frequencies obtained from before retrofit panel based simple model and member based detailed model with analytical eigen properties of the bridge obtained by previous researchers. Table 2.7 shows the aforementioned comparison of results obtained from after retrofit model of the bridge, In the tables the computed modal frequencies were obtained from the FE models by using SUCOT ( Kim, 1993) and 30 SAP 2000 V 10 ( Computers and Structures, 2002). It can be seen from the results that the computed modal frequencies obtained from the SUCOT and the SAP 2000 panel based model are having a good match with the calculated frequencies from finite element models developed by previous researchers. First three modes obtained from the SAP 2000 model are shown in Figure 2.14. Figure 2.14 First three mode shapes of the simple model Table 2.6 Comparison of modal frequencies in Hz ( before retrofit) Present study Panel based simple Memberbased detailed Dominant Motion Abdel Ghaffar, 1976 Niazy et al., 1991 SUCOT SAP 2000 SAP 2000 L * S * 0.173 0.169 0.159 0.152 0.161 V * AS * 0.197 0.201 0.210 0.223 0.221 V S 0.221 0.224 0.232 0.239 0.226 V S 0.348 0.336 0.460 0.384 0.363 V AS 0.346 0.344 0.456 0.495 0.369 L AS 0.565 0.432 0.472 0.448 0.503 T * S 0.449 0.438 0.483 0.482 0.477 V S 0.459 0.442 0.500 0.538 0.479 * L: Lateral, S: Symmetric, V: Vertical, AS: Asymmetric and T: Torsional 31 Table 2.7 Comparison of modal frequencies in Hz ( after retrofit) Ingham Present Study et al., 1997 ( ADINA) Fraser, 2003 ( ADINA) Panel based Simple Memberbased Detailed Dominant motion Simple Detailed Detailed SUCOT SAP 2000 SAP 2000 L S 0.162 0.135 0.130 0.161 0.152 0.160 V AS 0.197 0.171 0.182 0.210 0.218 0.220 V S 0.232 0.229 0.226 0.232 0.235 0.226 V S    0.360 0.369 0.362 V AS    0.453 0.469 0.372 L AS 0.535 0.420 0.409 0.473 0.447 0.494 T S 0.588 0.510 0.511 0.490 0.484 0.482 V S    0.498 0.513 0.486 * L: Lateral, S: Symmetric, V: Vertical, AS: Asymmetric and T: Torsional 2.7 Closure In this chapter numerical modeling has been achieved for Vincent Thomas Bridge. A member based detailed three dimensional Finite Element ( FE) as well as a panel based simplified model of the Vincent Thomas Bridge have been developed for the bridge before and after retrofit. First eight modal frequencies obtained from FE models developed using different commercially available softwares have been compared. The results obtained from this study are also compared with previous results obtained for the bridge. It has been observed that the first lateral modal frequency for the member based detailed model is 20% higher than those presented in previous studies. It is also found 32 that results of panel based simple models are in good agreement with those obtained from the detailed model and those reported in previous similar studies. 33 CHAPTER 3 SYSTEM IDENTIFICATION AND MODEL VERIFICATION 3.1 Background To ensure the validity of the analytical finite element model of a massive structure like a suspension bridge, the eigenproperties obtained from the analytical model should be compared with the identified modal frequencies. Modal identification can be done by using measured ambient vibration response or strong motion earthquake response of the bridge. The response of the bridge, under external excitations, is measured with the help of acceleration measuring sensors installed at different locations of the bridge. 3.2 Evaluation of Eigenproperties using Ambient Vibration Data Experimental modal analysis has drawn significant attention from structural engineers for updating the analysis model and estimating the present state of structural integrity. Forced vibration tests such as impact tests can be carried out to this end. However, it is usually restricted to small size structures or to their components. For large structures such as dams, and long span bridges, ambient vibration tests under wind, wave, or traffic loadings are the effective alternatives. In this study, modal parameters were obtained using the frequency domain decomposition technique ( Otte et al, 1990 and Brincker et al., 2000) which is one of the frequency domain methods without using input information. It is very difficult, if not impossible, to identify closely spaced modes using the 34 peak picking ( PP) method. In this case, the frequency domain decomposition ( FDD) method that utilizes the singular value decomposition of the PSD matrix may be used to separate close modes ( Brincker et al., 2000). The method was originally used to extract the operational deflection shapes in mechanical vibrating systems ( Otte et al, 1990). The natural frequencies are estimated from the peaks of the PSD functions in the PP method. On the other hand, they are evaluated from singular value ( SV) functions of the PSD matrix in the FDD method. ( ) ( ) ( ) ( ) T S yy w = U w s w V w ( 3.1) where ( ) m m N N yy S R × w Î is the PSD matrix for output responses ( ) m N y t Î R , ( ) m m N N s R × w Î is a diagonal matrix containing the singular values of its PSD matrix, and, U ( w ) , ( ) m m N N V R × w Î are corresponding unitary matrices. m N is the number of measuring points. The general multi DOF system can be transformed to the single DOF system nearby its natural frequencies by singular value decomposition. The mode shape can be estimated as the first column vector of the unitary matrix of U since the first singular value may include the structural mode nearby its natural frequencies. However in the closely spaced modes, the peak of largest singular values at one natural frequency indicates the structural mode and adjacent second singular value may indicate the close mode. Figure 3.1 shows the layout of the acceleration sensors installed in the bridge site. Table 3.1 describes the location and direction of all the accelerometers present in the bridge site. Figure 3.2 shows the vertical accelerometers and Figure 3.3 shows the lateral 35 accelerometers used in the modal identification of the bridge structure. Figure 3.4 shows the plot of SV vs. frequency for the acceleration data obtained from vertical channels. Figure 3.1 Location and direction of sensors installed in the bridge Table 3.1 Location and direction of accelerometers Sensor Number Sensor Location Sensor Direction 22, 15, 16, 17, 18, 21 Truss top/ Deck Vertical 2, 4, 5, 6, 7 Truss top/ Deck Lateral 12 Truss top/ Deck Longitudinal 3 Truss bottom Lateral 8 Tower Lateral 10, 11 Tower Longitudinal 14, 19, 20 Tower base Vertical 1, 9 Tower base Lateral 13, 23 Tower base Longitudinal 26 Anchorage Vertical 24 Anchorage Lateral 25 Anchorage Longitudinal 36 Figure 3.2 Vertical accelerometer data used in the study Figure 3.3 Lateral accelerometer data used in the study 37 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 x 10 4 Frequency ( Hz) SV SV of vertical acc.( CH 15 22) Figure 3.4 Plot of SV vs. Frequency 3.3 Comparison of System ID Result with Analytical Eigen Properties In this study, modal parameters have been obtained using the frequency domain decomposition ( FDD) technique ( Brincker et al. 2000) which is one of the frequency domain methods without using in put information. The method utilizes the singular value decomposition of the PSD matrix and may be used to separate close modes. Total 15 ambient vibration recording has been used for this purpose from the installed sensors. The data were recorded from April, 2003 to October, 2004, over 1 year 6 months record has been considered for system identification analysis. Average identified modal frequencies obtained from 15 data set are considered as final identified modal frequencies from the ambient vibration data. Figure 1 shows previously installed sensor locations on the bridge. For system ID from ambient vibration data, vertical sensors 15, 16, 17, 18, 21, 22 and lateral sensors 4, 5, 6, 7 are used. Sensor # 3 in the lateral direction is excluded because it provided some noisy data. Table 3.2 and 3.3 below shows the comparison of modal frequencies before and after retrofitting of the bridge. Modal identification results from ambient vibration data are also tabulated in Table 3.3. It can be seen from Table 3.3 that in the first mode of 38 vibration, the structure is a little bit stiffer in the simple model rather than detailed model. In case of first mode of vibration the system ID result matches with the frequency obtained from the detailed model. Also, from the second mode and above both the analytical and system ID results shows pretty good match. On an average sense, it can be seen from Table 3.3 that system ID results show pretty good match with detailed model. 3.4 Modal Parameter Identification from Chino Hills Earthquake Response Chino Hills earthquake data recorded at the bridge site are also used in the modal identification. Chino Hills earthquake occurred on July 29, 2008, in Southern California. The epicenter of the magnitude 5.4 earthquake was in Chino Hills, approximately 45 km east southeast of downtown Los Angeles. Table 3.4 compares the modal frequencies of the bridge obtained from ambient vibration and Chino Hills earthquake data. These two identified frequencies matches very well. Note also that the two other previous studies ( Ingham et al. 1997 and Fraser 2003) involving detailed models under predict modal frequencies significantly for the first two modes. Results from these two studies are also tabulated in Table 3.4. 39 Table 3.2 Comparison of modal frequencies in Hz ( before retrofit) Identified ( System ID) Computed Present Study Panel based Simple Memberbased Detailed Dominant Motion Abdel Ghaffar and Housner, 1977 ( Ambient) Niazy et al., 1991 ( Whittier) Ingham et al., 1997 ( Northridge) Abdel Ghaffar, 1976 Niazy et al., 1991 SUCOT SAP 2000 SAP 2000 L * S * 0.168 0.149 0.145 0.173 0.169 0.159 0.152 0.161 V * AS * 0.216 0.209  0.197 0.201 0.210 0.223 0.221 V S 0.234 0.224 0.222 0.221 0.224 0.232 0.239 0.226 V S 0.366 0.363 0.370 0.348 0.336 0.460 0.384 0.363 V AS  0.373  0.346 0.344 0.456 0.495 0.369 L AS 0.623 0.459 0.417 0.565 0.432 0.472 0.448 0.503 T * S 0.494 0.513 0.556 0.449 0.438 0.483 0.482 0.477 V S 0.487 0.448  0.459 0.442 0.500 0.538 0.479 * L: Lateral, S: Symmetric, V: Vertical, AS: Anti Symmetric, T: Torsional Table 3.3 Comparison of modal frequencies in Hz ( after retrofit) Identified ( System ID) Computed Ingham Present Study et al., 1997 ( ADINA) Fraser, 2003 ( ADINA) Panel based Simple Memberbased Detailed Dominant Motion Fraser, 2003 He et al., 2008 Present Study ( Ambient) Simple Detailed Detailed SUCOT SAP 2000 SAP 2000 L S 0.150  0.162 0.162 0.135 0.130 0.161 0.152 0.160 V AS  0.168 0.219 0.197 0.171 0.182 0.210 0.218 0.220 V S 0.233 0.224 0.229 0.232 0.229 0.226 0.232 0.235 0.226 V S 0.367 0.356 0.369    0.360 0.369 0.362 V AS       0.453 0.469 0.372 L AS   0.534 0.535 0.420 0.409 0.473 0.447 0.494 T S  0.483 0.471 0.588 0.510 0.511 0.490 0.484 0.482 V S       0.498 0.513 0.486 * L: Lateral, S: Symmetric, V: Vertical, AS: Anti Symmetric, T: Torsional 40 Table 3.4 Comparison of modal frequencies ( in Hz) of the Vincent Thomas Bridge Identified ( System ID) Computed Mode Number Dominant Motion Ambient Vibration Chino Hills Earthquake SAP 2000 ( Present Study) Ingham et al., 1997 Fraser, 2003 1 L * S * 1 0.162 0.168 0.160 0.135 0.130 2 V * AS * 1 0.219  0.220 0.171 0.182 3 V S1 0.229 0.228 0.222 0.229 0.226 4 V S2 0.369 0.362 0.362   5 V AS2  0.467 0.372   6 T * S1 0.471 0.491 0.478 0.510 0.511 7 V S3   0.483   8 L AS1 0.534  0.491 0.420 0.409 * L: Lateral, S: Symmetric, V: Vertical, AS: Anti Symmetric, T: Torsional 3.5 Effect of Parameter Uncertainty on Modal Frequency 3.5.1 Soil Spring Modeling To consider the effect of soil structure interaction kinematic three translational and three rotational soil springs with their coupling effects are considered at the foundations of east tower, west tower, east cable bent, west cable bent, east anchorage and west anchorage. The stiffness of the soil springs are calculated from the equivalent pile group stiffness at the foundations discussed earlier. Table 3.5 gives the number of piles at different foundations considered for the FE model of the bridge. Figure 3.5 shows the finite element model of the bridge with foundation springs. 41 Table 3.5 Location and number of piles considered Location Number of piles East tower 167 West tower 167 East cable bent 48 Wast cable bent 48 East anchorage 188 West anchorage 188 Figure 3.5 Detailed model in SAP 2000 with foundation springs 3.5.2 Uncertain Parameters Considered For model updating purpose, in ideal case, all parameters related to elastic, inertial properties and boundary conditions should be considered. However, if too many parameters are considered for model updating then chances of obtaining unreliable model increases ( Zhang et al., 2001). For this reason, parameter selection is a very important East tower West tower East cable bent East anchorage 42 task in model updating process. Practically if the parameters considered do not have much effect on the modal frequencies and mode shapes, then they should be excluded from the list. Therefore a comprehensive eigenvalue sensitivity study is performed to figure out the most sensitive parameters to be considered for suspension bridge finite element model calibration. Total 19 parameters are considered for the sensitivity analysis. The selection of these parameters is based on the outcome of previous research ( Zhang et al., 2001) and engineering judgments. Elastic modulus and mass density of different set of structural members, boundary conditions ( deck and tower connection and deck and cable bent connection) and stiffness of the soil springs are considered as variable parameters. However the cable and the concrete deck have homogeneous properties, but due to corrosion the structural strength may get decreased over the service life of the bridge. To capture that effect, elastic modulus and mass density of cable and concrete deck is considered as variable parameters in the analysis. Also, for the generality of the analysis kinematic spring stiffnesses ( soil spring stiffness) are also considered as variable parameters in the analysis. Since there was no tower dominant mode in the considered first 8 mode shapes, therefore, the stiffness and inertial properties of the tower is not considered as a variable parameter in the present study. For evaluating the effect of uncertainty in the modal parameters of Vincent Thomas Bridge, uncertainty associated with elastic and inertial property of different members is represented by assigning a mean and standard deviation in terms of coefficient of variation for each parameter. The mean values considered here are calculated based on the design drawing of the bridge. Table 3.6 lists these parameters with their mean values. 43 To asses the sensitivity, coefficients of variation ( COV) of all the parameters are considered as 10%. In the analysis, all the 36 values of the spring stiffness matrices are varied by 10% for the case of east tower, west tower, east cable bent and west cable bent. For the first order second moment ( FOSM) analysis only lateral translational stiffness of each foundation spring is considered. Table 3.6 Parameters considered for sensitivity analysis Serial Number Parameters Mean Value 1 Side link elastic modulus 2.00 × 10 8 kPa 2 Cable bent and girder connection elastic modulus 2.00 × 10 8 kPa 3 Top Chord Elastic Modulus 2.00 × 10 8 kPa 4 Top Chord Mass Density 7.85 kg/ m 3 5 Bottom Chord Elastic Modulus 2.00 × 10 8 kPa 6 Bottom Chord Mass Density 7.85 kg/ m 3 7 Stringer Elastic Modulus 2.00 × 10 8 kPa 8 Stringer Mass Density 7.85 kg/ m 3 9 Deck Slab Elastic Modulus 2.48 × 10 7 kPa 10 Deck Slab Mass Density 1.48 kg/ m 3 11 Main Cable Elastic Modulus 1.66 × 10 8 kPa 12 Main Cable Mass Density 8.37 kg/ m 3 13 Suspender Elastic Modulus 1.38 × 10 8 kPa 14 Suspender Mass Density 7.85 kg/ m 3 15 Wind Shoe Elastic Modulus 2.00 × 10 8 kPa 16 East Tower Spring 1.30 × 10 6 kPa 17 East Cable Bent Spring 7.35 × 10 6 kPa 18 West Tower Spring 1.19× 10 6 kPa 19 West Cable Bent Spring 4.65× 10 6 kPa 3.5.3 Analysis methods Reduction of the number of uncertain parameters cuts down the computational effort and cost. One way of doing this is to identify those parameters with associated ranges of uncertainty that lead to relatively insignificant variability in response and then treating these as deterministic parameters by fixing their values at their best estimate, such as the mean. For ranking uncertain parameters according to their sensitivity to desired response 44 parameters, there are various methods such as tornado diagram analysis, first order second moment ( FOSM) analysis, and Monte Carlo simulation ( Porter et al. 2002, Lee and Mosalam 2006). Monte Carlo simulation, which is computationally demanding due to the requirement of a large number of simulations, especially for a model consisting of a large number of degrees of freedom as in the case here, is not used in this study because of these practical considerations. Instead, the tornado diagram analysis and the FOSM analysis have been used here due to their simplicity and efficiency to identify sensitivity of uncertain parameters. For the tornado diagram analysis, all uncertain parameters are assumed as random variables, and for each of these random variables, two extreme values the 84 th percentile and 16 th percentile corresponding to assumed upper and lower bounds, respectively, of its probability distribution have been selected. One can observe that these extreme values come from the normal distribution assumption, mean + standard deviation and mean – standard deviation, respectively representing their upper and lower bounds. Using these two extreme values for a certain selected random variable, the modal frequencies of the model has been evaluated for both cases, while all other random variables have been assumed to be deterministic parameter with values equal to their mean value. The absolute difference of these two modal frequency values corresponding to the two extreme values of that random variable, which is termed as swing of the modal frequency corresponding to the selected random variable, is calculated. This calculation procedure has then been repeated for all random variables in question. Finally, these swings have been plotted in a figure from the top to the bottom in a descending order according to their size to demonstrate the relative contribution of each 45 variable to the specific mode under question. It is noteworthy that longer swing implies that the corresponding variable has larger effect on the modal frequency than those with shorter swing. For the FOSM analysis, the modal frequency has been considered as a random variable Y, which has been expressed as the function of random variables, Xi ( for i = 1 to N) denoting uncertain parameters and Y is given by ( , ,..., ) 1 2 N Y = g X X X ( 3.2) Let Xi has been characterized by mean μX and variance s X 2 . Now, the derivatives of g( X) with respect to Xi , one can express Y by expanding Eq. ( 3.2) in Taylor series as LLL K + − − + = + − = = = i j N j i X j X N i i N i X X X i X X X g X X X g Y g X i j N i d d d μ μ d d μ μ μ μ 2 1 1 1 ( )( ) 2! 1 ( ) 1! 1 ( , , , ) 1 2 ( 3.3) Considering only the first order terms of Eq. ( 3.3) and ignoring higher order terms Y can be approximated as i N i X X X i X X g Y g X N i d d μ μ μ μ = » + − 1 ( ) 1! 1 ( , , , ) 1 2 K ( 3.4) Taking expectation of both sides, the mean of Y, μY can be expressed as ( , , , ) Y X1 X2 X N μ g μ μ μ K » ( 3.5) Utilizing the second moment of Y as expressed in Eq. ( 3.4) and simplifying, the variance of Y, s Y 2 can be derived as = = » N i N j j N i N Y i j X g X X X X g X X X X X 1 1 2 1 2 1 2 ( , ,..., ) ( , ,..., ) cov( , ) d d d d s 46 j N i N N i N j i X X i N X N i X g X X X X g X X X X g X X X i i j d d d d r d d s ( , ,..., ) ( , ,..., ) ( , ,..., ) 1 2 1 2 1 2 2 1 2 1 = = ¹ + » ( 3.6) where i j X X r denotes correlation coefficient for random values Xi and Xj ( i. e., coefficient defining the degree to which one variable is related to another). The partial derivative of ( , ,..., ) 1 2 N g X X X with respect to Xi has been calculated numerically using the finite difference equation given below i i i N i i N i N x g x x x x g x x x x X g X X X D + D − − D = 2 ( , ,..., ) ( , ,..., , ) ( , ,..., , ) 1 2 1 2 1 2 μ μ d d ( 3.7) In this case, a large number of simulations were performed varying each input parameter individually to approximate the partial derivatives as given in Eq. ( 3.7). For these calculations, the mean and the standard deviation values given in Table 3.6 are used. For these sensitivity analyses, at first, the reference model with mean parameters of each 19 random variable considered in this study is analyzed. Then the analyses have been carried out using their lower and then upper bounds. Altogether 39 cases of modal analysis are performed for each set of parameters, modal frequencies expressed as ( , ,..., ) 1 2 N Y = g X X X is observed. 3.5.4 Sensitivity of Modal Frequencies For tornado diagram analysis, all the 19 parameters shown in Table 3.6 are used for total 8 mode shapes. Figures 3.6 ( a h) show tornado diagrams for 8 modes developed according to the procedure in section 3.5.3. The vertical line in the middle of tornado 47 diagrams indicates modal frequency value calculated for a certain mode considering only the mean values of all random variables and the length of each swing ( horizontal bar) represents the variation in the modal frequency due to the variation in the respective random variable. It is clear from Figures 3.6 ( a e) that, up to mode # 5 deck slab mass density and bottom chord elastic modulus have almost the largest contribution in response variability. In mode numbers 2, 3, 6, and 7, mostly vertical and torsional modes, main cable elastic modulus is significant contributor of the response variability. One can also notice from Figures 3.6 ( a h) that couple of swings are asymmetric about the vertical line. This skew of the modal frequency distributions implies that the problem is highly nonlinear. In other words, the same amount of a positive and a negative change in these parameters does not produce the same amount of variation in modal frequency. This skewness is very clear for 2 nd mode in case of main cable elastic modulus variation. Since the 2 nd mode is vertical antisymmetric, increase is main cable elastic modulus does not have much effect on increase in frequency but decrease in the stiffness of main cable decreases the frequency by 8% from the base model frequency. Interestingly, deck slab stiffness has most contribution in the 1 st mode, but it does not have any contribution in rest of the modes except the 8 th mode. Most of the boundary condition ( P1, P2, and P15) and soil spring ( P16, P17, P18, and P19) related parameters have very insignificant effect on response variability. For FOSM method, analyses have been carried out to determine the sensitivity of modal frequencies to the uncertainty in each random variable. Focus has been placed on the variance of modal frequency when considering uncertainties of 19 input parameters. 48 Figures 3.7 ( a h) show relative variance contributions of each parameter to the modal frequency when the correlation, as given in the second term of Eq. ( 3.6), is neglected. From this figure, it can be observed that the uncertainties in the deck slab mass density and bottom chord elastic modulus contribute mostly to the variance of modal frequencies. This is the same trend as observed from the tornado diagram analysis for all the 8 modes considered. 49 0.156 0.157 0.158 0.159 0.160 0.161 0.162 0.163 0.164 Modal Frequency ( Hz) : Mode 1 ( LS 1) P 10 P 9 P 5 P 3 P 12 P 7 P 8 P 4 P 6 P 11 P 14 P 18 P 13 P 1 P 2 P 15 P 16 P 17 P 19 0.216 0.217 0.218 0.219 0.220 0.221 0.222 0.223 0.224 0.225 Modal Frequency ( Hz) : Mode 2 ( VAS 1) P 1 0 P 5 P 1 2 P 1 1 P 8 P 4 P 6 P 3 P 9 P 1 4 P 7 P 1 3 P 1 6 P 1 P 2 P 1 5 P 1 7 P 1 8 P 1 9 ( a) ( b) P10 P5 P12 P11 P8 P4 P6 P3 P9 P14 P7 P13 P16 P1 P2 P15 P17 P18 P19 50 0.218 0.220 0.222 0.224 0.226 0.228 Modal Frequency ( Hz) : Mode 3 ( VS 1) P 1 0 P 1 1 P 5 P 1 2 P 8 P 4 P 6 P 3 P 9 P 1 4 P 7 P 1 3 P 1 P 2 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.354 0.356 0.358 0.360 0.362 0.364 0.366 0.368 0.370 Modal Frequency ( Hz) : Mode 4 ( VS 2) P 1 0 P 5 P 1 2 P 3 P 8 P 4 P 6 P 9 P 1 4 P 7 P 1 3 P 1 6 P 1 P 1 1 P 1 8 P 2 P 1 5 P 1 7 P 1 9 0.364 0.366 0.368 0.370 0.372 0.374 0.376 0.378 0.380 Modal Frequency ( Hz) : Mode 5 ( VAS 2) P 1 0 P 5 P 1 2 P 3 P 8 P 4 P 6 P 9 P 1 4 P 7 P 1 1 P 1 3 P 1 8 P 1 6 P 1 P 2 P 1 5 P 1 7 P 1 9 P10 P11 P5 P12 P8 P4 P6 P3 P9 P14 P7 P13 P1 P2 P15 P16 P17 P18 P19 P10 P5 P12 P3 P8 P4 P6 P9 P14 P7 P13 P16 P1 P11 P18 P2 P15 P17 P19 P10 P5 P12 P3 P8 P4 P6 P9 P14 P7 P11 P13 P18 P16 P1 P2 P15 P17 P19 ( c) ( e) ( d) 51 0.468 0.470 0.472 0.474 0.476 0.478 0.480 0.482 0.484 0.486 0.488 Modal Frequency ( Hz) : Mode 6 ( TS 1) P 1 1 P 1 2 P 1 0 P 6 P 4 P 9 P 3 P 8 P 5 P 7 P 1 4 P 1 3 P 1 6 P 1 P 1 8 P 1 5 P 1 7 P 1 9 0.474 0.476 0.478 0.480 0.482 0.484 0.486 0.488 0.490 0.492 Modal Frequency ( Hz) : Mode 7 ( VS 3) P 1 0 P 1 1 P 1 2 P 5 P 8 P 3 P 4 P 6 P 9 P 1 4 P 7 P 1 3 P 1 5 P 1 6 P 1 8 P 1 P 2 P 1 7 P 1 9 0.479 0.482 0.485 0.488 0.491 0.494 0.497 0.500 0.503 Modal Frequency ( Hz) : Mode 8 ( LAS 1) P 1 2 P 1 0 P 9 P 5 P 7 P 3 P 8 P 1 4 P 4 P 1 1 P 6 P 1 8 P 1 6 P 1 P 2 P 1 3 P 1 5 P 1 7 P 1 9 P11 P12 P10 P6 P4 P9 P3 P8 P5 P7 P14 P13 P16 P1 P2 P18 P15 P17 P19 P10 P11 P12 P5 P8 P3 P4 P6 P9 P14 P7 P13 P15 P16 P18 P1 P2 P17 P19 P12 P10 P9 P5 P7 P3 P8 P14 P4 P11 P6 P18 P16 P1 P2 P13 P15 P17 P19 Figure 3.6 Tornado diagram considering 19 parameters ( f) ( g) ( h) 52 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.050 0.002 0.475 0.236 0.021 0.029 0.006 0.096 0.009 0.075 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 1 ( LS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.091 0.055 0.601 0.002 0.027 0.000 0.009 0.197 0.009 0.009 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative variance Modal Frequency ( Hz) : Mode 2 ( VAS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.259 0.528 0.001 0.023 0.000 0.009 0.099 0.009 0.005 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 3 ( VS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 ( a) ( b) ( c) P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 53 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.064 0.000 0.467 0.003 0.020 0.000 0.006 0.406 0.007 0.025 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 4 ( VS 2) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.088 0.000 0.523 0.004 0.020 0.000 0.007 0.322 0.008 0.027 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 5 ( VAS 2) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.237 0.480 0.155 0.027 0.007 0.003 0.040 0.003 0.029 0.018 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 6 ( TS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 ( d) ( e) ( f) P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 54 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.098 0.372 0.423 0.004 0.017 0.000 0.006 0.059 0.007 0.014 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 7 ( VS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.003 0.000 0.774 0.001 0.099 0.076 0.004 0.013 0.001 0.020 0.002 0.006 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative variance Modal Frequency ( Hz) : Mode 8 ( LAS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 ( h) ( g) Figure 3.7 Relative variance contribution ( neglecting correlation terms) from FOSM analysis 55 3.6 Finite Element Model Updating A detailed three dimensional finite element ( FE) model of Vincent Thomas Bridge was developed using the finite element analysis code ADINA 8.3. This finite element model is composed of 3D elastic truss elements to represent the main cables and suspenders, 2D shell elements to model the bridge deck and beam elements to model the stiffening trusses and tower shafts. The ADINA bridge model is shown in Figure 3.8. Figure 3.8 Three dimensional finite element model of Vincent Thomas Bridge For updating the original ADINA model an improved sensitivity based parameter updating method is employed ( Zhang et al., 2001). The method is based on the eigen value sensitivity to some selected structural parameters that are assumed to be bounded within some prescribed regions according to the degrees of uncertainty and variation existing in the parameters, together with engineering judgment. The changes of these parameters are found by solving a quadratic programming problem. 56 3.6.1 Sensitivity Based Model Updating The structural parameters affecting the natural frequencies are selected to construct the design parameter vector a P . The eigenvalue vector based on the designed parameters is denoted as a l , while the measured eigenvalue vector as m l . The error vector is defined as m a d l = l − l . The updating process minimizes the error vector by changing the design parameter vector a P . The variation of design parameter vector d p can be determined by d l = S d p ( 3.8) where S is the sensitivity matrix that represents the variation of natural frequencies of the model due to the variation of design parameter vector. The solution of Eq. ( 3.8) can be solved by the following iterative updating procedures. p p p k k = + d + 1 ( 3.9) l = l + d l + k a k a 1 ( 3.10) where k a k p , l are the parameter vector and eigenvalue vector of FE model, respectively, at the k th updating step. The iterative updating is repeated until the updated eigenvalue vector k a l converges to the measured eigenvalue vector m l . 57 The criteria of convergence are used as tolerance f f f m i m i k a i i £ − , , , max ( 3.11) tolerance k a i k a i k a i i £ − − − 1 , 1 , , max l l l ( 3.12) where k a i f , and k a, i l are the i th natural frequency and corresponding eigenvalue at k th update, and m i f , the measured i th natural frequency. The following optimization problem is applied to determine d p in Eq. ( 3.8) ( Friswell and Mottershead, 1994). J J ( S p ) W ( S p ) p W p p T e T d l d d l d d d 2 1 2 1 min 1 2 + = − − + F ( 3.13) subject to l u b £ d p £ b The first term in right hand side of Eq. ( 3.13) represents the objective function to minimize the error vector, while the second term to minimize the variation of design parameter vector. e W and p W are weighting functions. The constrained optimization solutions as outlined in Eq. ( 3.13) are incorporated into an iterative procedure as shown in Figure 3.9 for the model updating Vincent Thomas Bridge. 58 Input: u l b , b a k p p k = = 0, Yes Convergence Criterion Satisfy? Compute: u Constrained Optimization Determine d p 1 1 = + = + + k k p p p k k k d STOP No a p e p , W , W k k a k S p , FE model ® l ® k l k l k u k b b p b b p = − = − k Figure 3.9 Procedure for the sensitivity based model updating 59 3.6.2 Selection of Modes and Parameters 3.6.2.1 Selection of Modes Average values of the identified modal frequencies obtained from 14 different ambient vibration data recorded at the bridge site are considered as target frequencies for further ADINA model updating. Those 14 ambient vibration data were recorded from April, 2003 to October 2004. In the study, it is decided to select 8 modes to be matched between the updated FE analysis and the measured results. These include five vertical dominant; two lateral dominant; one torsional dominant modes of the deck. Table 3.7 shows the modal frequencies and percentage error in modal frequencies of Initial ( original) and Baseline ADINA model results with respect to identified frequencies obtained from the ambient vibration measurement data. Table 3.7 Comparison of natural frequencies Mode Measured frequency Updated FE model Type ( Hz) Initial Baseline Initial Baseline 1 L S 0.161 0.131 0.148  18.63  7.83 2 V AS1 0.221 0.206 0.210  6.79  5.02 3 V S1 0.233 0.226 0.227  3.00  2.66 4 V S2 0.374 0.363 0.371  2.94  0.86 5 V S3 0.474 0.460 0.470  2.95  0.78 6 L AS 0.476 0.411 0.462  13.66  2.90 7 T S 0.538 0.500 0.506  7.06  6.02 8 V AS2 0.568 0.568 0.583 0.00 2.66 Mode no. Finite element analyzed frequencies err.(%) 60 3.6.2.2 Selection of Parameters All possible parameters relating to the geometric, structural properties as well as the boundary conditions should be considered for adjustment in the updating procedure. However, if the parameters are found to have little or no effect on the targeted vibration modes, then they can be excluded from parameters list. After removing those parameters with very small sensitivities, total 17 different parameters are considered for this analysis. For this purpose, a sensitivity study is done and is explained in Section 3.5. They are summarized in Table 3.8 together with their initial estimates. Table 3.8 Parameters selected for adjustment Structure parameters Variations in % Stiffening truss Top chord Elastic modulus 29000 kip/ in 2 15 Mass density 8.71E 07 kip/ in 3 15 Bottom chord Elastic modulus 29000 kip/ in 2 20 Mass density 8.71E 07 kip/ in 3 15 Diagonal Mass density 8.71E 07 kip/ in 3 20 lateral brace ( k truss) Elastic modulus 29000 kip/ in 2 10 Mass density 1.35E 06 kip/ in 3 20 Stringers Elastic modulus 29000 kip/ in 2 20 Mass density 9.02E 07 kip/ in 3 10 Deck Slab Elastic modulus 2825 kip/ in 2 30 Mass density 2.01E 07 kip/ in 3 5 Cable Main cable Initial strain 1 20 Elastic modulus 29000 kip/ in 2 20 Mass density 7.71E 07 kip/ in 3 15 Suspender Mass density 7.65E 07 kip/ in 3 15 Tower Elastic modulus 29000 kip/ in 2 15 Mass density 7.62E 07 kip/ in 3 15 Initial estimation ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 61 3.6.3 Updated Results The allowable errors permitted for the check of natural frequency convergence was applied 6% for the general modes, while 3% for the first and second modes. If the ratio of variation for the eigenvalue is lower than 0.1%, then the iteration is also ended. For the cable supported bridge of which modes are closely spaced, the disorder between adjacent modes should be critically checked. The following MAC ( Modal Assurance Criteria) is applied to the each set of two updated natural modes ( Friswell and Mottershead, 1994). ( ) ( ) 0 1 1 , , 1 , , 2 , , = £ £ = = = MAC MAC p l a l j a l j p l e l i e l i p t l a l j e l i f f f f f f ( 3.14) If the two shape vectors a e f , f to be compared are identical, then MAC becomes 1, while if the two shape vectors are orthogonal, MAC becomes 0. Therefore, MAC can be utilized to prevent disorder between the calculated and measured frequency. MAC also provides the criteria for the reliability of the developed model after model updating. The MACs are listed in Table 3.9. The differences between the measured and the calculated frequencies for the initial and the final updated FE modes are showed in Figure 3.10. Table 3.10 shows the natural frequencies of the baseline model and updated model. For most of the modes, the discrepancies between measured frequencies and updated frequencies decreased less than 3%, while a few modes such as the first lateral frequency shows about 4% discrepancy. However, the discrepancy between measured and baseline model was about 19% and the current updating decreases the error in amount of 4%. 62 Table 3.9 MAC matrix of updated FE model # 1 2 3 4 5 6 7 8 1 0.542 0.015 0.007 0.000 0.000 0.000 0.000 0.000 2 0.001 0.531 0.000 0.000 0.000 0.000 0.000 0.000 3 0.003 0.009 0.490 0.000 0.000 0.000 0.000 0.000 4 0.000 0.000 0.000 0.500 0.000 0.000 0.000 0.000 5 0.000 0.000 0.000 0.000 0.500 0.000 0.000 0.000 6 0.000 0.000 0.000 0.000 0.000 0.538 0.000 0.000 7 0.090 0.005 0.001 0.003 0.006 0.094 0.541 0.000 8 0.001 0.003 0.008 0.113 0.002 0.000 0.000 0.486  20  18  15  13  10  8  5  3 0 3 5 L S1 V AS1 V S1 V S2 V S3 L AS1 T S1 V AS2 Frequency differences(%) Modes Initial FE model Baseline FE model Updated FE model Figure 3.10 Comparison of frequency differences using the initial and updated FE models 63 Table 3.10 Comparison of natural frequencies between baseline and updated FE model Frequency ( Hz) Frequency ( Hz) Err.(%) Frequency ( Hz) Err.(%) Frequency ( Hz) Err.(%) 1 L S 0.161 0.131  18.63 0.148  7.83 0.155  4.04 2 V AS1 0.221 0.206  6.79 0.210  5.02 0.215  2.90 3 V S1 0.233 0.226  3.00 0.227  2.66 0.233  0.09 4 V S2 0.374 0.363  2.94 0.371  0.86 0.373  0.19 5 V S3 0.474 0.460  2.95 0.470  0.78 0.478 0.80 6 L AS 0.476 0.411  13.66 0.462  2.90 0.487 2.25 7 T S 0.538 0.500  7.06 0.506  6.02 0.538  0.04 8 V AS2 0.568 0.568 0.00 0.583 2.66 0.587 3.31 Mode no. Finite element analyzed frequencies Mode Type Initial Identified Baseline Updated The variations of design parameters are also important to estimate reliability and effectiveness of updating results. The variations of design parameters are well limited in permitted arrange that can be regarded as reasonable as shown in Table 3.11. Table 3.11 Updated design parameters Structure parameters Initial estimation Updated value Percent changes Stiffening truss Top chord Elastic modulus( kip/ in 2 ) 29000 30815 6.3 Mass density( kip/ in 3 ) 8.71E 07 7.85E 07  9.8 Bottom chord Elastic modulus( kip/ in 2 ) 29000 33350 15.0 Mass density( kip/ in 3 ) 8.71E 07 7.58E 07  13.0 Diagonal Mass density( kip/ in 3 ) 8.71E 07 7.49E 07  14.0 lateral brace ( k truss) Elastic modulus( kip/ in 2 ) 29000 29442 1.5 Mass density( kip/ in 3 ) 1.35E 06 1.14E 06  15.0 Stringers Elastic modulus( kip/ in 2 ) 29000 24650  15.0 Mass density( kip/ in 3 ) 9.02E 07 8.16E 07  9.5 Deck Slab Elastic modulus( kip/ in 2 ) 2825 3390 20.0 Mass density( kip/ in 3 ) 2.01E 07 1.82E 07  9.2 Cable Main cable Initial strain 1.00 1.15 15.0 Elastic modulus( kip/ in 2 ) 29000 24650  15.0 Mass density( kip/ in 3 ) 7.71E 07 7.45E 07  3.3 Suspender Mass density( kip/ in3) 7.65E 07 8.41E 07 10.0 Tower Elastic modulus( kip/ in 2 ) 29000 27931  3.7 Mass density( kip/ in 3 ) 7.62454E 07 7.87E 07 3.3 64 3.7 Closure To demonstrate the appropriateness of the bridge models developed in the previous chapter, eigen properties of the models are evaluated in this chapter and compared with those of the system identification results obtained using frequency domain decomposition technique on ambient vibration and recorded earthquake response data. After that, a comprehensive sensitivity analysis is performed considering 19 different structural and soil spring parameters. First eight modal frequencies are considered for the sensitivity study. Tornado diagram and FOSM methods are applied for the sensitivity study. It is observed that the mass density of deck slab and elastic modulus of bottom chord contributes most to the modal frequencies of the bridge. This kind of study will be very helpful in selecting parameters and their variability ranges for FE model updating of suspension bridges. In this study, a sensitivity based automatic model updating procedure is presented, which solves an optimization problem for model error minimization. Four vertical vibration modes, two lateral modes, one torsional mode and 17 design parameters are selected for the problem. Updated results show that the model error could be reduced from 0~ 18% to 0~ 4% in terms of modal frequency ratio. During the optimization procedure, the target error bounds were 3% for the lower vertical modes and 6% for the horizontal modes. In order to prevent mode interchange due to the closely spaced frequencies of the three dimensional FE model, MACs are introduced to verify the updated results through the optimization procedure. 65 CHAPTER 4 SEISMIC ANALYSIS 4.1 Background The Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los Angeles and Long Beach ports, two of the busiest ports in the west coast of USA. Thus, the bridge carries an overwhelming number of traffic with an Annual Average Daily Traffic ( AADT) volume of 100,000, many of which are cargo trucks. Based on the recent finding that the main span of the Vincent Thomas Bridge crosses directly over the Palos Verdes fault, which has the capacity to produce a devastating earthquake, in spring 2000, the bridge underwent a major retrofit using visco elastic dampers. This study focuses on seismic vulnerability of the retrofitted bridge. A member based detailed threedimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino Hills earthquake and comparing with the measured response. Finally, considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed and the ductility demands of critical sections are presented in terms of fragility curves. The study shows that a ground motion with PGA of 0.9g or greater will result in plastic hinge formation at one or more locations with a probability of exceedance of 50%. 66 Also, it is found that the effect of damper is minimal for low to moderate earthquakes and high for strong earthquakes. The spatial variation of earthquake ground motions may have significant effect on the response of long span suspension bridges. Abdel Ghaffar and Rubin ( 1982) and Abdel Ghaffar and Nazmy ( 1988) studied response of suspension and cable stayed bridges under multiple support excitations. Zerva ( 1990) and Harichandran and Wang ( 1990) examined the effect of spatial variable ground motions on different types of bridge models. Harichandran et al. ( 1996) studied the response of long span bridges to spatially varying ground motion. Deodatis et al. ( 2000) and Kim and Feng ( 2003) investigated the effect of spatial variability of ground motions on fragility curves for bridges. Lou and Zerva ( 2005) analyzed the effects of spatially variable ground motions on the seismic response of a skewed, multi span, RC highway bridge. Most of the aforementioned studies dealt with simple FE models of the bridge, as a result response of critical members could not be evaluated. In the present analysis a panel based detailed 3D FE model of a long span suspension bridge is utilized. For design purpose of important structures in a site, U. S. Geological Survey ( USGS) provides a set of scenario earthquakes specified for a site. To consider spatial variability of ground motions one needs to know the ground excitations at different supports of a long span suspension bridge. For generating spatial variable ground motions from a scenario earthquake compatible to different design spectra for different supports ( as the local soil conditions will be different for different supports) a new algorithm is proposed using evolutionary power spectral density function ( PSDF) of the scenario earthquake specified for the site. Evolutionary PSDF of LA21 scenario earthquake is 67 estimated by using short time Fourier transform ( STFT) and wavelet transform ( WT) methods. Two evolutionary PSDFs thus developed maintain the same total energy possessed by the time history data. Using the evolutionary 20 sets of simulated ground motions for six different spatially correlated supports are generated. Ensemble average of 5% damped spectral acceleration response spectra obtained from simulated earthquake time histories are compared with the design response spectra for all the support locations. Good match has been found with the target design acceleration response spectra with the simulated one. Simulated spatially variable ground motions are used in calculating the response of the bridge. In addition to spatial variable seismic ground motions, two uniform ground motions are also considered for comparison purpose. The seismic responses of the bridge deck and the east tower are calculated using those three different cases and compared in both seismic displacement demand and seismic force demand. 4.2 Scope FE model validation of the bridge is also performed by simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino Hills earthquake and comparing with the measured response from installed acceleration sensors. Considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed and ductility demands of critical tower section are presented in terms of seismic fragility curves. Effect of spatial variability of ground motions on seismic displacement demand and seismic force demand is investigated. To generate spatially 68 correlated spectrum compatible nonstationary acceleration time histories, a newly developed algorithm using evolutionary PSDF and spectral representation method is used. 4.3 Response Analysis under Northridge Earthquake To validate the developed numerical models ( discussed in Chapter 2), time history analysis is performed using the 1994 Northridge earthquake ( Mw = 6.7) ground motions recorded at the bridge sites. Newmark Beta method is used with g = 0.5 and b = 0.25 for this purpose. The ground motions and the bridge response during the Northridge earthquake are collected from the sensors installed at the bridge site ( California Strong Motion Instrumentation Program ( CSMIP) [ http:// www. strongmotioncenter. org/]). Since the earthquake occurred before the retrofit, detailed model before the retrofit is used here. To consider the effect of spatial variation, different ground motions are considered at different support locations, wherever possible. In some cases, due to the unavailability of recorded support motions, ground motions recorded at the nearest support is considered. Figure 4.1 shows the location of sensors and Table 4.1 illustrates the list of supports on which ground motions are applied for this analysis. Figure 4.2 shows comparison of measured and calculated longitudinal displacement at the top of the east tower location ( channel # 10) of the bridge. The plot shows good match between the calculated and field measured responses. 69 Figure 4.1 Location and direction of sensors  8  4 0 4 8 0 20 40 60 80 100 120 Time ( Sec) Displacement ( cm) Measured Calculated Figure 4.2 Comparison of measured and calculated longitudinal displacement at channel # 10 location Vertical Lateral Longitudinal East Anchorage West Tower 70 Table 4.1 Different support motions considered with channel numbers Location Longitudinal Lateral Vertical East Anchorage Ch. 25 Ch. 24 Ch. 26 East Cable Bent* Ch. 13 Ch. 9 Ch. 19 East Tower Ch. 13 Ch. 9 Ch. 19 West Anchorage* Ch. 23 Ch. 1 Ch. 14 West Cable Bent* Ch. 23 Ch. 1 Ch. 14 West Tower Ch. 23 Ch. 1 Ch. 14 * No recording at these locations 4.4 Response Analysis under Chino Hills Earthquake To study the developed numerical model, time history analysis is performed using the 2008 Chino Hills earthquake ( Mw = 5.4) ground motions recorded at the bridge sites. Newmark Beta method is used with g = 0.5 and b = 0.25 for this purpose with time step equal to 0.01 sec. The ground motions and the bridge response during the Chino Hills earthquake are collected from the sensors installed at the bridge site ( California Strong Motion Instrumentation Program ( CSMIP) [ http:// www. strongmotioncenter. org/]). Figure 1 shows the location of sensors already installed in the bridge. Since the earthquake occurred after the retrofit, detailed model after the retrofit is used here. Three directional components of ground motions recorded at east anchorage, east tower and west tower are applied uniformly over all the supports to study which set of ground motions will give much more accurate results. Figure 4.3 shows the comparison of analytical lateral response at channel 5 due to ground motions at east anchorage, east tower, west tower and considering spatial variation in ground motion with field measured response. It can be seen from figure 4.3 that the analytical response due the ground motion recorded at east tower matches well with the measured response. Figures 4.4, 4.5 and 4.6 show 71 comparison of analytical lateral, vertical and longitudinal responses at different channels due to ground motions at east tower with field measured response. These plots shows good match between the analytical and field measured responses.  0.8  0.6  0.4  0.2 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed_ East Anchorage Computed_ East Tower Computed_ West Tower Computed_ Spatial Figure 4.3 Comparison of analytical lateral response at channel 5 due to ground motions at east anchorage, east tower and west tower with field measured response 72  1.2  1  0.8  0.6  0.4  0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed based on east tower data Figure 4.4 Comparison of analytical lateral response at channel 3 due to ground motions at east tower with field measured response  1.5  1  0.5 0 0.5 1 1.5 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed based on east tower data Figure 4.5 Comparison of analytical vertical response at channel 17 due to ground motions at east tower with field measured response 73  0.8  0.6  0.4  0.2 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed based on east tower data Figure 4.6 Comparison of analytical longitudinal response at channel 10 due to ground motions at east tower with field measured response 4.5 Generation of Fragility Curves It is clear from the previous literature, especially those studies in the aftermath of 1995 Kobe ( Hyogo ken Nanbu) earthquake at Japan that the bridge deck and cables of suspension bridges are less vulnerable under strong earthquake ground motion ( remain elastic) while the tower is the most vulnerable part. In order to simplify analysis, in this study, only the towers are modeled as nonlinear elements. Remaining elements of the bridge are considered as linear. Each tower leg is constructed with members of 5 different cross sections. A total of 40 plastic hinges are introduced at all four tower legs. An elastoplastic behavior with 3% strain hardening is considered for the material models of these plastic hinges. 2% Raleigh damping is used for the first and tenth modes. Forty ground motions representing 2% in 50 years and 10% in 50 years of hazard level as specified by 74 FEMA/ SAC are used for evaluating seismic vulnerability of the retrofitted bridge. The motions cover wide range peak characteristics with Peak Ground Acceleration ( PGA) ranging from 0.42 to 1.30g. Note also that these motions include expected motions from Palos Verdes fault, the fault crossing the main span of this bridge. For nonlinear time history analysis, direct time integration is used in the framework of SAP 2000. Motions are applied in the lateral direction of the bridge and no spatial variation is considered. After performing the nonlinear time history analysis, the ductility demands of all the critical sections are evaluated and the maximum ductility demand is noted for each motion. Considering all these motions, the maximum ductility demand is found to be 6.23, which is from LA 36 motion ( with a PGA of 1.1g) and for the plastic hinge at the base of the tower. In this study, fragility curves corresponding to different damage states are developed following Shinozuka et al., 2000. For a given damage state, the fragility curves are expressed in terms of lognormal distribution. PGA is considered as Ground motion intensity. Two fragility parameters, median ( c) and log standard deviation ( z ) are estimated through a maximum likelihood method such that fragility curves at different damage levels do not intersect each other. Therefore, a common z is needed to satisfy this criterion. Although this method can be used for any number of damage states, for the ease of demonstration of analytical procedure it is assumed here that there are three states of bridge damage. Therefore, a family of three fragility curves exists in this case for damage states of ‘ Level I’, ‘ Level II’, and ‘ Level III’ identified by k = 1, 2, and 3. Under this lognormal assumption, the analytical form of the fragility function F(•) for the state of damage k is, 75 ( ) ( ) = F z z i k i k a c F a c ln / , , ( 4.1) where ck is median of the fragility function associated with damage state k, z is the common log standard deviation, ai is the PGA value to which the bridge is subjected and F [•] is the standardized normal distribution function. The fragility parameters are computed by maximizing the likelihood function, L which is given by Eq. ( 4.2), where xik is 1 or 0, depending on whether or not the bridge sustains damage state k under ai, and n is the total number of ground motions under which the analysis is performed. Pik is the probability that the example bridge will suffer from a damage state k when subjected to ai and is expressed as 1 ( , , z ) 0 1 P F a c i i = − ( , , z ) ( , , z ) 1 1 2 P F a c F a c i i i = − ( , , z ) ( , , z ) 2 2 3 P F a c F a c i i i = − ( , , z ) 3 3 P F a c i i = ( ) [ ] Õ Õ = = = 3 1 1 1 2 3 , , , k n i x ik ik L c c c z P ( 4.2) ( 4.3) ( 4.4) ( 4.5) ( 4.6) 76 Fragility parameters are obtained by solving the Eq. ( 4.7), by implementing a straightforward optimization algorithm. ( ) ( ) 0 ln , , , ln , , , 1 2 3 1 2 3 = ¶ ¶ = ¶ ¶ z z L c c c z c L c c c k for k = 1,2,3 For the fragility curves, this study proposes performance levels in terms of ductility demands of critical tower sections, since the damage states related to expected performance level of suspension bridge is not clearly defined in the literature. Three different damage states are considered in this study in terms of the maximum ductility demands of all the critical tower sections. They are ( 1) Level I ( plastic hinge formation, ductility > 1) ( 2) Level II ( ductility ³ 2) and ( 3) Level III ( ductility ³ 4). Figure 4.5 shows the fragility curves considering these damage states and for before and after retrofitting of the bridge. One can observe from this figure that for a PGA of 0.9g, the probability of exceedance corresponding to damage Level I ( i. e., plastic hinge formation at one or more locations) is 50%. Similarly, for the same probability of exceedance, a ground motion with PGA of 1.05g or greater will cause a damage of Level II. PGA of 1.82g was recorded at the Tarzana Station during the main shock of the 1994 Northridge earthquake. For that PGA the probability of exceedance to damage Level II is 90%. The bridge was retrofitted with total 48 dampers and from the fragility curves it is clear that the effect of dampers are minimal for low to moderate earthquake and high for strong earthquake. ( 4.7) 77 Table 4.2 Details of the motions considered in this study for fragility development SAC Earthquake Distance Scale dt Duration PGA PGV PGD Name Magnitude ( km) Factor ( sec) ( sec) ( g) ( cm/ sec) ( cm) LA21 1995 Kobe 6.9 3.4 1.15 0.02 59.98 1.28 142.70 37.81 LA22 1995 Kobe 6.9 3.4 1.15 0.02 59.98 0.92 123.16 34.22 LA23 1989 Loma Prieta 7 3.5 0.82 0.01 24.99 0.42 73.75 23.07 LA24 1989 Loma Prieta 7 3.5 0.82 0.01 24.99 0.47 136.88 58.85 LA25 1994 Northridge 6.7 7.5 1.29 0.005 14.945 0.87 160.42 29.31 LA26 1994 Northridge 6.7 7.5 1.29 0.005 14.945 0.94 163.72 42.93 LA27 1994 Northridge 6.7 6.4 1.61 0.02 59.98 0.93 130.46 28.27 LA28 1994 Northridge 6.7 6.4 1.61 0.02 59.98 1.33 193.52 43.72 LA29 1974 Tabas 7.4 1.2 1.08 0.02 49.98 0.81 71.20 34.58 LA30 1974 Tabas 7.4 1.2 1.08 0.02 49.98 0.99 138.68 93.43 LA31 Elysian Park ( simulated) 7.1 17.5 1.43 0.01 29.99 1.30 119.97 36.17 LA32 Elysian Park ( simulated) 7.1 17.5 1.43 0.01 29.99 1.19 141.12 45.80 LA33 Elysian Park ( simulated) 7.1 10.7 0.97 0.01 29.99 0.78 111.03 50.61 LA34 Elysian Park ( simulated) 7.1 10.7 0.97 0.01 29.99 0.68 108.44 50.12 LA35 Elysian Park ( simulated) 7.1 11.2 1.1 0.01 29.99 0.99 222.78 89.88 LA36 Elysian Park ( simulated) 7.1 11.2 1.1 0.01 29.99 1.10 245.41 82.94 LA37 Palos Verdes ( simulated) 7.1 1.5 0.9 0.02 59.98 0.71 177.47 77.38 LA38 Palos Verdes ( simulated) 7.1 1.5 0.9 0.02 59.98 0.78 194.07 92.56 LA39 Palos Verdes ( simulated) 7.1 1.5 0.88 0.02 59.98 0.50 85.50 22.64 LA40 Palos Verdes ( simulated) 7.1 1.5 0.88 0.02 59.98 0.63 169.30 67.84 Record Figure 4.7 Before and after retrofit Fragility curves for different damage levels Probability of Exceeding a Damage State PGA ( g) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Level I_ Before Level II_ Before Level III_ Before Level I_ After Level II_ After Level III_ After 78 4.6 Simulation of Ground Motion Considering Spatial Variability 4.6.1 Generation of Evolutionary PSDF from Given Ground Motion using STFT This section briefly reviews the work done by Liang et al. ( 2007). The STFT F ( t, w ) of a function f ( t ) is expressed by the convolution integral in the following form: ( ) ( ) ( ) ¥ − ¥ − w = t − t t w t F t f h t e d i , ( 4.8) where h ( t ) is an appropriate time window. The evolutionary PSDF S ( t w ) f f , 0 0 can be written as ( ) ( ) ( ) ( ) ( ) ¥ − ¥ − − ¥ − ¥ = − − 1 2 1 2 1 2 2 , w t t t t 1 2 t t w t w t F t f f h t h t e e d d i i ( 4.9) The total energy of f ( t ) can be estimated as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ − − ¥ − ¥ ¥ − ¥ ¥ − ¥ = − = − − f h t d dt f f h t h t e d d dtd F t dtd i t t t t t t t t t w w w w t t 2 2 1 2 1 2 1 2 2 1 2 , ( 4.10) For the derivation of Eq. ( 4.10), the following equation is used: ( ) ( ) ¥ − ¥ − − = − 1 2 1 2 w d t t w t t e d i ( 4.11) If h ( t ) = d ( t ) 2 , the total energy in Eq. ( xx) is ( ) ( ) ¥ − ¥ ¥ − ¥ ¥ − ¥ F t dtd = f t dt 2 2 , w w ( 4.12) 79 This implies that the time window should be chosen such that it satisfies the following condition ( ) ¥ − ¥ = 1 2 h t dt ( 4.13) The total energy can be kept identical ( Perseval’s identity) in estimating evolutionary PSDF. Here a Gaussian time window squared with standard deviation s = 0.25 s, is used. It satisfies the condition in Eq. ( 4.13). The time window function has the following form, ( ) ( 0.25 ) 2 1 2 2 2 2 = = − s s p t s h t e ( 4.14) Figure 4.8 shows the evolutionary PSDF of LA21 scenario earthquake record estimated using STFT ( Gaussian window). Figure 4.8 Evolutionary PSDF of LA21 earthquake record using STFT method 80 4.6.2 Generation of Evolutionary PSDF from Given Ground Motion using Wavelet Transform This section briefly reviews the work done by Liang et al. ( 2007). The wavelet transform ( WT) of a function f ( R ) 2 Î L ( finite energy function f ( t ) dt < + ¥ 2 ) at time u and scale s , and the corresponding inverse relationship are given by Daubechies ( 1992) ( ) ( ) dt u s R s t u f t s W f u s Î − = ¥ − ¥ * , , 1 , y y ( 4.15) and ( ) ( ) ( ) ¥ − ¥ ¥ − ¥ Î − = duds u s R s t u W f u s f t C s f t , , , 1 2 1 2 y p y y ( 4.16) where ( ) = < ¥ ¥ − ¥ w w y w y C d 2 ˆ ( 4.17) In Equations ( 4.15) – ( 4.17), the wavelet function ( R ) 2 y Î L known as ‘ mother” wavelet with average value equal to zero, ( ) ¥ − ¥ y t dt = 0 ( 4.18) and is centered in the neighborhood of t = 0, and as normalized y = 1. y ˆ ( w ) denotes the Fourier transform of y ( t ) and is given by ( ) ( ) ¥ − ¥ − = t e dt i w t y p y w 2 1 ˆ ( 4.19) It may be noted that the WT decomposes signal f ( t ) over dilated and translated wavelets. As W f ( u, s ) y is convolution of f ( t ) with ( 1 s ) ( t s ) , W f ( u, s ) * y y − 81 represents the contribution of the function f ( t ) in the neighborhood of t = u and in the frequency band corresponding to scale s . It can be shown that ( Daubechies, 1992) ( ) W f ( u s ) duds C s f t dt 2 2 2 , 1 2 1 y y p ¥ − ¥ ¥ − ¥ ¥ − ¥ = ( 4.20) Now, if any wavelet function satisfies the condition ( ) ¥ − ¥ ˆ = 1 2 , y w d w u s ( 4.21) Then Equation ( 4.20) can be written as ( ) ( ) ( ) ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ × = y w w p y y W f u s duds d C s f t dt u s 2 , 2 2 2 , ˆ 1 2 1 ( 4.22) In Equations ( 4.21) and ( 4.22), y ( w ) u, s ˆ represents the Fourier transform of − s t u y and can be expressed as ( ) ( ) i u u s s s e w y ˆ w y ˆ w , = . Then, using Perseval’s identity, one can write ( ) W f ( u s ) ( ) duds C s F u s 2 , 2 2 2 , ˆ 1 2 1 y w p w y y ¥ − ¥ ¥ − ¥ = ( 4.23) where F ( w ) = Fourier transform of f ( t ) . As the wavelet coefficient W f ( u, s ) y provides the localized information of signal f ( t ) at t = u , from Equation ( 4.23) the Evolutionary PSDF ( , w ) 0 0 S t f f can be expressed as ( ) ( ) ( ) ¥ − ¥ = W f t s ds C s F t t s 2 , 2 2 2 , ˆ 1 2 1 , y w p w y y ( 4.24) It may be noted that the expression of evolutionary PSDF given in Equation ( 4.24) obeys total energy equilibrium. Therefore, any wavelet basis can be used which satisfies Equation ( 4.21), for generation of evolutionary PSDF [ e. g., modified Littlewood Paley basis proposed by Basu and Gupta ( 1998)] that maintains total energy. Figure 4.9 shows 82 the evolutionary PSDF of LA21 scenario earthquake record estimated using STFT ( Gaussian window). Figure 4.9 Evolutionary PSDF of LA21 earthquake record using wavelet transform 4.6.3 Simulation of One Dimensional Multi Variate ( 1D mV), Nonstationary Gaussian Stochastic Process To generate sample functions of stochastic processes, the spectral representation method developed by Shinozuka and Jan ( 1972) appears to be most versatile and widely used today. Spectral representation based algorithm to simulate one dimensional multi variate nonstationary Gaussian stochastic process developed by Deodatis ( 1996b) is used in this study and described as follows. 83 Consider a one dimensional, n variate ( 1D nV) non stationary stochastic vector process with components ( ) , ( ) ,........., ( ) , 0 0 2 0 1 f t f t f t n
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Title  Verification of computer analysis models for suspension bridges 
Subject  Suspension bridgesPerformanceCaliforniaMeasurement.; Suspension bridgesCaliforniaTestingComputer simulation.; Suspension bridgesEarthquake effectsCaliforniaComputer simulation.; BridgesLive loadsCaliforniaTesting.; Vincent Thomas Bridge (Los Angeles, Calif.) 
Description  Title from PDF title page (viewed on February 15, 2011).; "August 2009."; Includes bibliographical references (p. 171178).; Final report.; Text document (PDF).; Performed for California Dept. of Transportation under contract no. 
Publisher  University of California, Irvine, Dept. of Civil and Environmental Engineering 
Contributors  Shinozuka, Masanobu.; California. Dept. of Transportation.; University of California, Irvine. Dept. of Civil and Environmental Engineering. 
Type  Text 
Identifier  http://www.dot.ca.gov/hq/esc/earthquake_engineering/Research_Reports/vendor/uc_irvine/2009002/UCI_0902Verification_of_Computer_Analysis_Models_for_Suspension_Bridges.pdf 
Language  eng 
Relation  http://worldcat.org/oclc/701909049/viewonline 
DateIssued  [2009] 
FormatExtent  xx, 178 p. : digital, PDF file (4.5 MB) with ill., charts. 
RelationRequires  Mode of access: World Wide Web. 
Transcript  VERIFICATION OF COMPUTER ANALYSIS MODELS FOR SUSPENSION BRIDGES by Masanobu Shinozuka, Distinguished Professor and Chair and Debasis Karmakar, Graduate Student Samit Ray Chaudhuri, Postdoctoral Scholar Ho Lee, Assistant Researcher Department of Civil and Environmental Engineering University of California, Irvine Report No: CA/ UCI VTB 2009 August 2009 Final Report Submitted to the California Department of Transportation under Contract No: RTA 59A0496 VERIFICATION OF COMPUTER ANALYSIS MODELS FOR SUSPENSION BRIDGES Final Report Submitted to the Caltrans under Contract No: RTA 59A0496 by Masanobu Shinozuka, Distinguished Professor and Chair and Debasis Karmakar, Graduate Student Samit Ray Chaudhuri, Postdoctoral Scholar Ho Lee, Assistant Researcher Department of Civil and Environmental Engineering University of California, Irvine Report No: CA/ UCI VTB 2009 August 2009 ii STATE OF CALIFORNIA × DEPARTMENT OF TRASPORTATION TECHNICAL REPORT DOCUMENTAION PAGE TR0003 ( REV. 9/ 99) 1. REPORT NUMBER CA/ UCI VTB 2009 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER 5. REPORT DATE August 2009 4. TITLE AND SUBTITLE VERIFICATION OF COMPUTER ANALYSIS MODELS FOR SUSPENSION BRIDGES 6. PERFORMING ORGANIZATION CODE UC Irvine 7. AUTHOR Masanobu Shinozuka, Debasis Karmakar, Samit Ray Chaudhuri, and Ho Lee 8. PERFORMING ORGANIZATION REPORT NO. 10. WORK UNIT NUMBER 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Civil and Environmental Engineering The Henry Samueli School of Engineering University of California, Irvine Irvine, CA 92697 2175 11. CONTRACT OR GRANT NUMBER RTA 59A0496 13. TYPE OF REPORT AND PERIOD COVERED Final Report 12. SPONSORING AGENCY AND ADDRESS California Department of Transportation ( Caltrans) Division of Research and Innovation 1227 O Street, MS 83 Sacramento, CA 95814 14. SPONSORING AGENT CODE 15. SUPPLEMENTARY NOTES 16. ABSTRACT The Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los Angeles and Long Beach ports, two busiest ports in the west coast of USA. The bridge carries an overwhelming number of traffic with an Annual Average Daily Traffic ( AADT) volume of 45,500, many of which are cargo trucks. Based on the recent finding that the main span of the Vincent Thomas Bridge crosses directly over the Palos Verdes fault, which has the capacity to produce a devastating earthquake, the bridge underwent a major retrofit in spring 2000, mainly using viscoelastic dampers. This study focuses on performance evaluation of the retrofitted bridge under seismic, wind and traffic loads. A member based detailed three dimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by simulating and comparing with the measured dynamic response during two recent earthquakes. FE model is also updated using a sensitivity based parameter updating method. Effect of spatial variability of ground motions on seismic displacement and force demands is investigated. To record actual wind velocity and direction, three anemometers are installed at three different locations of the bridge. Response of the bridge is computed under wind velocity. Finally, analysis of the bridge under traffic load is also carried out. 17. KEYWORDS Suspension Bridge, System Identification, Retrofit, Fragility Curve, Earthquake, Wind, Traffic 18. DISTRIBUTION STATEMENT No restrictions. 19. SECURITY CLASSIFICATION ( of this report) Unclassified 20. NUMBER OF PAGES 178 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 Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los Angeles and Long Beach ports, two busiest ports in the west coast of USA. The bridge carries an overwhelming number of traffic with an Annual Average Daily Traffic ( AADT) volume of 45,500, many of which are cargo trucks. Based on the recent finding that the main span of the Vincent Thomas Bridge crosses directly over the Palos Verdes fault, which has the capacity to produce a devastating earthquake, the bridge underwent a major retrofit in spring 2000, mainly using visco elastic dampers. This study focuses on performance evaluation of the retrofitted bridge under seismic, wind and traffic loads. A member based detailed three dimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by simulating and comparing with the measured dynamic response during two recent earthquakes. Tornado diagram and first order second moment ( FOSM) methods are applied for evaluating the sensitivity of different parameters on the eigenproperties of the FE models. The study indicates that the mass density of deck slab and elastic modulus of bottom chord are very important parameters to control eigenproperties of the models. FE model is also updated using a sensitivity based parameter updating method. Considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed using the FE models developed and seismic fragility curves are derived comparing the ductility demand with the ductility capacity at critical v tower sections. Effect of spatial variability of ground motions on seismic displacement and force demands is also investigated. To generate spatially correlated nonstationary acceleration time histories compatible with design spectrum at each location. A new algorithm is developed involving evolutionary power spectral density function ( PSDF) and with the aid of spectral representation method. It has been found that, in some locations on the bridge deck, the response is higher when the spatially variable ground motion is considered as opposed to the uniform ground motion time histories having the highest ground displacement. To record actual wind velocity and direction, three anemometers are installed at three different locations of the bridge. The fluctuating component of the wind velocity measured at these three locations are found to be non Gaussian. They are used for simulation of fluctuating component of wind velocity throughout the span and along the tower on the basis of three different simulation methods ( i) newly developed non Gaussian conditional method, ( ii) Gaussian conditional method, and ( iii) Gaussian unconditional method. Response of the bridge is computed under wind velocity using these three different methods. It is observed that the non Gaussian conditional simulation technique yields higher response than both Gaussian conditional and Gaussian nonconditional techniques. Finally, analysis of the bridge under traffic load is also carried out and a critical evaluation of shear force in deck shear connectors is performed. vi ACKNOWLEDGEMENT The research presented in this report was sponsored by the California Department of Transportation ( Caltrans) with Dr. Li Hong Sheng as the project manager. The authors are indebted to Caltrans for its support of this project and to Dr. Li Hong Sheng for his helpful comments and suggestions. viii TABLE OF CONTENTS Page LIST OF FIGURES vi LIST OF TABLES xiii ABSTRACT xv CHAPTER 1 Introduction 1 1.1 Background 1 1.2 Literature Survey 3 1.3 Objective and Scope 10 1.4 Dissertation Outline 12 CHAPTER 2 Finite Element Modeling of Vincent Thomas Bridge 13 2.1 Background 13 2.2 Calculation of Dead Weight 13 2.3 Calculation of the Initial Shape of the Cable 14 2.4 Panel Based Simple Model 14 2.4.1 Moment of Inertia ( Iz) 18 2.4.2 Torsional Constant ( J) 19 2.5 Member Based Detail Model 24 2.5.1 Cable Bent 27 2.5.2 Deck Shear Connector 27 2.5.3 Dampers 28 2.5.4 Suspended Truss 28 2.5.5 Suspenders 28 2.6 Eigen Value Analysis 29 ix 2.7 Closure 31 CHAPTER 3. System Identification and Model Verification 33 3.1 Background 33 3.2 Evaluation of Eigenproperties using Ambient Vibration Data 33 3.3 Comparison of System ID Result with Analytical Eigen Properties 37 3.4 Modal Parameter Identification from Chino Hills Earthquake Response 38 3.5 Effect of Parameter Uncertainty on Modal Frequency 40 3.5.1 Soil Spring Modeling 40 3.5.2 Uncertain Parameters Considered 41 3.5.3 Analysis methods 43 3.5.4 Sensitivity of Modal Frequencies 46 3.6 Finite Element Model Updating 55 3.6.1 Sensitivity Based Model Updating 56 3.6.2 Selection of Modes and Parameters 59 3.6.2.1 Selection of Modes 59 3.6.2.2 Selection of Parameters 60 3.6.3 Updated Results 61 3.7 Closure 64 CHAPTER 4 Seismic Analysis 65 4.1 Background 65 4.2 Scope 67 4.3 Response Analysis under Northridge Earthquake 68 4.4 Response Analysis under Chino Hills Earthquake 70 4.5 Generation of Fragility Curves 73 4.6 Simulation of Ground Motion Considering Spatial Variability 78 4.6.1 Generation of Evolutionary PSDF from Given Ground Motion using STFT 78 4.6.2 Generation of Evolutionary PSDF from Given x Ground Motion using Wavelet Transform 79 4.6.3 Simulation of One Dimensional Multi Variate ( 1D mV), Nonstationary Gaussian Stochastic Process 82 4.6.4 Simulation of Seismic Spectrum Compatible Accelrograms 85 4.6.5 Examples of Generated Seismic Ground Motion 90 4.7 Results 99 4.8 Closure 105 CHAPTER 5 Wind Sensor Installation and Wind Speed Measurement 106 5.1 Background 106 5.2 Anemometer and Data Acquisition System 107 5.2.1 Anemometer for Vantage Pro2 107 5.2.2 Anemometer Transmitter with Solar Power 107 5.2.3 Wireless Repeater with Solar Power 107 5.2.4 Wireless Weather Envoy ( Wireless Receiver) 110 5.2.5 WeatherLink Software for Data Collection 110 5.2.6 Data Acquisition Software Developed 110 5.2.7 Experimental Setup 111 5.2.8 Anemometer Installation and Data Acquisition System 112 5.3 WeatherLink Software for Data Collection 114 5.4 Recorded Wind Velocities 115 5.5 Closure 117 CHAPTER 6 Wind Buffeting Analysis 118 6.1 Background 118 6.2 Scope 121 6.3 Conditional Simulation of Gaussian Random Processes 122 6.3.1 Conditional Simulation in Frequency Domain 123 6.4 Conditional Simulation of Non Gaussian Random Processes 124 xi 6.5 Simulation of Spatially Correlated Gaussian Wind Velocity Fluctuations 128 6.6 Conditional Simulation of Gaussian Wind Velocity Fluctuations 135 6.7 Conditional Simulation of non Gaussian Wind Velocity Fluctuations 138 6.8 Buffeting Force Calculation 152 6.9 Buffeting Response of Vincent Thomas Bridge 153 6.10 Closure 157 CHAPTER 7 Traffic Load Analysis 158 7.1 Background 158 7.2 Moving Load Analysis 158 7.3 Closure 165 CHAPTER 8 Conclusions and Future Work 166 8.1 Summary and Conclusions 166 8.2 Future Work 169 REFERENCES 171 xii LIST OF FIGURES Page Figure 2.1 The shape of the initial cable profile under dead load 18 Figure 2.2 Cross section of deck 19 Figure 2.3 Location of stringers in one side of the deck 20 Figure 2.4 Commonly used lateral bracing systems and stiffening girders 21 Figure 2.5 Horizontal system ( K type) 21 Figure 2.6 Vertical web system ( Worren type) 22 Figure 2.7 Different sections of the tower 26 Figure 2.8 Typical tower cross section 26 Figure 2.9 The detailed model of one panel 28 Figure 2.10 Deck shear connector ( before retrofit) 29 Figure 2.11 Deck shear connector ( after retrofit) 29 Figure 2.12 K truss modifications after retrofit 30 Figure 2.13 Suspender modifications after retrofit 30 Figure 2.14 First three mode shapes of the simple model 31 Figure 3.1 Location and direction of sensors installed in the bridge 35 Figure 3.2 Vertical accelerometer data used in the study 36 Figure 3.3 Lateral accelerometer data used in the study 36 Figure 3.4 Plot of SV vs. Frequency 37 Figure 3.5 Detailed model in SAP 2000 with foundation springs 41 Figure 3.6 Tornado diagram considering 19 parameters 51 xiii Figure 3.7 Relative variance contribution ( neglecting correlation terms) from FOSM analysis 54 Figure 3.8 Three dimensional finite element model of Vincent Thomas Bridge 55 Figure 3.9 Procedure for the sensitivity based model updating 58 Figure 3.10 Comparison of frequency differences using the initial and updated FE models 62 Figure 4.1 Location and direction of sensors 69 Figure 4.2 Comparison of measured and calculated longitudinal displacement at channel # 10 location 69 Figure 4.3 Comparison of analytical lateral response at channel 5 due to ground motions at east anchorage, east tower and west tower with field measured response 71 Figure 4.4 Comparison of analytical lateral response at channel 3 due to ground motions at east tower with field measured response 72 Figure 4.5 Comparison of analytical vertical response at channel 17 due to ground motions at east tower with field measured response 72 Figure 4.6 Comparison of analytical longitudinal response at channel 10 due to ground motions at east tower with field measured response 73 Figure 4.7 Before and after retrofit Fragility curves for different damage levels 77 Figure 4.8 Evolutionary PSDF of LA21 earthquake record using STFT method 79 Figure 4.9 Evolutionary PSDF of LA21 earthquake record using wavelet transform 82 Figure 4.10 Iterative scheme to simulate spectrum compatible acceleration time histories 89 Figure 4.11 Different support locations of the bridge 91 Figure 4.12 Acceleration time history of LA 21 scenario earthquake 92 Figure 4.13 Acceleration time history at location 1 93 Figure 4.14 Acceleration time history at location 2 94 xiv Figure 4.15 Acceleration time history at location 3 94 Figure 4.16 Acceleration time history at location 4 95 Figure 4.17 Acceleration time history at location 5 95 Figure 4.18 Acceleration time history at location 6 96 Figure 4.19 Displacement time history at location 3 96 Figure 4.20 Displacement time history at location 6 97 Figure 4.21 Comparison between simulated and design spectra at location 1 using STFT 97 Figure 4.22 Comparison between simulated and design spectra at location 3 using STFT 98 Figure 4.23 Comparison between simulated and design spectra at location 1 using Wavelet 98 Figure 4.24 Comparison between simulated and design spectra at location 3 using Wavelet 99 Figure 4.25 Absolute axial force demand envelope for the bridge girder 101 Figure 4.26 Absolute shear force demand envelope for the bridge girder 101 Figure 4.27 Absolute moment demand envelope for the bridge girder 102 Figure 4.28 Absolute torsional force demand envelope for the bridge girder 102 Figure 4.29 Absolute axial force demand envelope for the east tower of the bridge 103 Figure 4.30 Absolute shear force demand envelope for the east tower of the bridge 103 Figure 4.31 Absolute moment demand envelope for the east tower of the bridge 104 Figure 4.32 Absolute torsional force demand envelope for the east tower of the bridge 104 Figure 5.1 Anemometer 108 Figure 5.2 Anemometer transmitter with solar power 109 xv Figure 5.3 Wireless repeater with solar power 109 Figure 5.4 Wireless Weather Envoy ( Wireless Receiver) 110 Figure 5.5 Layout of the data acquisition system 111 Figure 5.6 Locations of anemometers, transmitters, repeaters and receivers 113 Figure 5.7 Distance between different components 113 Figure 5.8 Installation of anemometers, transmitters and repeaters ( a) Top of the east tower ( b) Vertical post on deck ( c) East tower platform ( d) Anchorage house wall 115 Figure 5.9 Screen shots from Weather Link and data acquisition system ( a) Anemometer # 1 ( b) Anemometer # 2 ( c) Anemometer # 3 ( d) Data acquisition system 116 Figure 5.10 Wind velocity recorded for 24 hrs on April 8, 2009 ( 1 sample/ min) 116 Figure 5.11 Wind velocity recorded for 30 minutes on April 15, 2009 ( 1 sample/ 3s) 117 Figure 6.1 Flow chart of conditional simulation of non Gaussian random processes 127 Figure 6.2 Installed anemometer locations on VTB 133 Figure 6.3 Locations of “ aerodynamic” nodes along the bridge deck 133 Figure 6.4 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 2) in m/ s from Gaussian unconditional simulation 134 Figure 6.5 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 1) in m/ s from Gaussian unconditional simulation 134 Figure 6.6 Horizontal wind velocity fluctuations at two different locations from Gaussian unconditional simulation 135 xvi Figure 6.7 Measured wind velocity fluctuation at anemometer # 1 location 136 Figure 6.8 Measured wind velocity fluctuation at anemometer # 2 location 136 Figure 6.9 Measured wind velocity fluctuation at anemometer # 3 location 136 Figure 6.10 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 2) in m/ s from Gaussian conditional simulation 137 Figure 6.11 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 1) in m/ s from Gaussian conditional simulation 137 Figure 6.12 Horizontal wind velocity fluctuations at two different locations from Gaussian unconditional simulation 138 Figure 6.13 Actual and analytical PDF of wind velocity fluctuation measured at anemometer # 1 location 140 Figure 6.14 Actual and analytical PDF of wind velocity fluctuation measured at anemometer # 2 location 140 Figure 6.15 Actual and analytical PDF of wind velocity fluctuation measured at anemometer # 3 location 141 Figure 6.16 Actual and analytical CDF of wind velocity fluctuation measured at anemometer # 1 location 141 Figure 6.17 Actual and analytical CDF of wind velocity fluctuation measured at anemometer # 2 location 142 Figure 6.18 Actual and analytical CDF of wind velocity fluctuation measured at anemometer # 3 location 142 Figure 6.19 Horizontal wind velocity fluctuations at two different locations from non Gaussian conditional simulation 144 Figure 6.20 Simulated and target CDF of wind velocity fluctuation at point # 10 144 Figure 6.21 Simulated and target CDF of wind velocity fluctuation at point # 16 145 xvii Figure 6.22 Comparison of PSDF from simulated wind velocity fluctuation and target PSDF at point # 10 115 Figure 6.23 Comparison of PSDF from simulated wind velocity fluctuation and target PSDF at point # 16 146 Figure 6.24 Comparison of PSDF from measured velocity fluctuation at anemometer # 1 and assumed analytical PSDF 147 Figure 6.25 Comparison of PSDF from measured velocity fluctuation at anemometer # 2 and assumed analytical PSDF 147 Figure 6.26 Comparison of PSDF from measured velocity fluctuation at anemometer # 3 and assumed analytical PSDF 148 Figure 6.27 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 2) in m/ s from non Gaussian conditional simulation 148 Figure 6.28 Horizontal wind velocity fluctuations at different locations along the deck ( around anemometer # 1) in m/ s from non Gaussian conditional simulation 149 Figure 6.29 Simulated wind velocity fluctuations at location # 10 with three different simulation techniques 149 Figure 6.30 Simulated wind velocity fluctuations at location # 16 with three different simulation techniques 150 Figure 6.31 Horizontal wind velocity fluctuations at different locations along the tower ( around anemometer # 2) in m/ s from Gaussian unconditional simulation 150 Figure 6.32 Horizontal wind velocity fluctuations at different locations along the tower ( around anemometer # 2) in m/ s from Gaussian conditional simulation 151 Figure 6.33 Horizontal wind velocity fluctuations at different locations along the tower ( around anemometer # 2) in m/ s from non Gaussian conditional simulation 151 Figure 6.34 Schematic diagram for aerodynamic forces on bridge deck 155 Figure 6.35 Simulated lateral deck displacements at the center of the mid span 156 Figure 6.36 Simulated vertical deck displacement at the center of the mid span 156 xviii Figure 7.1 Plan view of deck shear connectors before and after retrofit 159 Figure 7.2 Deck shear connector 159 Figure 7.3 Deck shear connector design drawing 160 Figure 7.4 HS20 44 AASTHO traffic loading 161 Figure 7.5 Different traffic load cases 162 Figure 7.6 Axial force in shear connector due to traffic load ( before and after retrofit) 163 Figure 7.7 Vertical shear force in shear connector due to traffic load ( before and after retrofit) 163 Figure 7.8 Longitudinal shear force in shear connector due to traffic load ( before and after retrofit) 164 Figure 7.9 Shear key in east side span 164 xix LIST OF TABLES Page Table 2.1 Calculated dead load ( kip/ ft) along the length of bridge 16 Table 2.2 Calculated nodal coordinates of the cable only system 17 Table 2.3 Calculated sectional properties of panels 23 Table 2.4 Calculated sectional properties of the tower sections ( before retrofit) 27 Table 2.5 Calculated sectional properties of the tower sections ( after retrofit) 27 Table 2.6 Comparison of modal frequencies in Hz ( before retrofit) 31 Table 2.7 Comparison of modal frequencies in Hz ( after retrofit) 32 Table 3.1 Location and direction of accelerometers 35 Table 3.2 Comparison of modal frequencies in Hz ( before retrofit) 39 Table 3.3 Comparison of modal frequencies in Hz ( after retrofit) 39 Table 3.5 Location and number of piles considered 41 Table 3.6 Parameters considered for sensitivity analysis 43 Table 3.7 Comparison of natural frequencies 59 Table 3.8 Parameters selected for adjustment 60 Table 3.9 MAC matrix of updated FE model 62 Table 3.10 Comparison of natural frequencies between baseline and updated FE model 63 Table 3.11 Updated design parameters 63 Table 4.1 Different support motions considered with channel numbers 70 xx Table 4.2 Details of the motions considered in this study for fragility Development 77 Table 4.3 Site coefficient parameters to calculate design spectra at different supports 92 Table 4.4 Displacement demand comparison 100 Table 5.1 Settings of different repeaters 114 Table 5.2 Settings of different receivers 114 Table 6.1 Properties for assumed generalized extreme value distribution 141 Table 7.1 Shear stress developed in shear key bolts 165 1 CHAPTER 1 INTRODUCTION 1.1 Background Throughout the history of suspension bridges, their tendency to vibrate under different dynamic loadings such as wind, earthquake, and traffic loads has been a matter of concern. The failure of the Tacoma Narrows bridge in 1940 has pointed out that the suspension bridges are vulnerable to wind loading ( Rannie 1941). It is now widely accepted that the wind induced vibration of suspension bridges may be significant and should be taken into consideration. Similar conclusions have also been drawn for other dynamic loadings. As a prerequisite to the investigation of aerodynamic stability, traffic impact, soil structure interaction and earthquake resistant design of suspension bridges, it is necessary to know certain dynamic characteristics such as the natural frequencies and the possible modes of vibration. Several investigations have been taken place in recent years to determine the vibrational properties of suspension bridges. However, the complexity of a suspension bridge structure makes the determination of vibrational characteristics difficult. With the advent of computers, non conventional structures like suspension bridges are analyzed with the finite element ( FE) analysis technique. There are several commercially available finite element software packages that are used by practicing 2 engineers as well as researchers, which can evaluate the response of a suspension bridge from operational traffic, wind and earthquake loads taking into account both material and geometric non linear behavior. In addition to analytical modeling and response analysis of suspension bridges, field tests are also very important from the analysis and design point of view. Field test results not only give experimental data but also help us to understand the behavior of the structure and to calibrate the analytical model. To perform field tests, it is necessary to measure, input loadings such as wind velocity at different pints and earthquake ground acceleration at different support locations, and output responses such as acceleration, velocity and displacement time history at different points of the bridge. For predicting response of long span suspension bridges under random wind, the most widely used method is the frequency domain analysis. In theory, the frequency domain solution is accurate, when the load response relationship is linear. Although the structural elements in a suspension bridge generally behave in a linear elastic fashion under normal loading, the overall load displacement relationship exhibits geometrical nonlinearity, particularly when it is subjected to high wind. Therefore, in this case, a frequency domain analysis may not be appropriate. One way in which the limitation of the frequency domain analysis can be overcome is the use of Monte Carlo simulation technique. One of the most important components of the Monte Carlo simulation method is the generation of sample functions of stochastic processes, fields, or waves those are involved in the problem. For buffeting analysis, wind velocity fluctuation in the horizontal and vertical directions needed to be digitally simulated and fed into the equation of motion. Since the length of a modern suspension bridge generally exceeds 1 3 km, the simulated sample functions must accurately describe the probabilistic characteristics not only in terms of temporal variation but also in spatial distribution. Similarly for seismic response, critical members of the bridge may undergo significant nonlinear deformation and a simple response spectrum method for analyzing such response may not be adequate. In addition, there may be significant variation of ground motion from one support of the bridge to the other. 1.2 Literature Survey Theoretical and practical treatises on the vibrational characteristics and the dynamic analysis of suspension bridges, have been developed by many authors, especially after the disastrous collapse of the Tacoma Narrows Bridge in 1940 ( Rennie 1941). Bleich et al., 1950 studied the free vertical and torsional vibration by solving a forth order linearized differential equation. In addition, an approximate method of the Rayleigh Ritz type solution was suggested. However, the procedure is applicable only for calculating the lowest few modes due to the great level of complexity and redundancy of higher modes of suspension bridges. Steinman, 1959 introduced a number of simplified formulas for estimating the natural frequencies and the associated mode shapes of vibration, both vertical and torsional, of suspension bridges. Japanese researchers ( Konishi et al 1965; Konishi and Yamada 1969; Yamada and Takemiya 1969, 1970; Yamada and Goto 1972; and Yamada et al. 1979) performed extensive studies to investigate the vertical and lateral vibration as well as the tower pier system of a three span suspension bridge by using a lumped mass system interconnected by spring elements. In their analysis for the 4 suspended structure, they assumed simple harmonic excitations and applied it separately to each supporting point. They reported that there was a fairly significant contribution from the higher modes to the bending response and a large number of modes should be included to accurately determine the dynamic response of suspension bridges. The geometrically nonlinear behavior of suspension bridges was considered ( Tezcan and Cherry 1969) due to large deflection and presented an iterative technique for the nonlinear static analysis by using tangent stiffness matrices. These matrices are incorporated in obtaining the free vibrational modes of the structure. In their analysis, the bridge was modeled as a three dimensional lumped mass system. They calculated the response of the bridge considering three orthogonal components of uniform ground motion and pointed out that the longitudinal motion of the deck as well as the vertical motion of the tower were small and therefore could be neglected. Major advances in studying the dynamic characteristics of suspension bridges have been achieved through the use of finite element method and linearized deflection theory ( Abdel Ghaffar 1976, 1977, 1978a, 1978b, 1979, 1980 and 1982). Natural frequencies, mode shapes, and energy capacities of the different structural components for vertical, torsional, and lateral vibrations were investigated. Several examples were presented and the applicability of the proposed methods was illustrated by comparing the results obtained from analyzing the Vincent Thomas bridge ( Los Angeles Harbor) with the results of full scale ambient vibration tests ( Abdel Ghaffar 1976, 1978 and Abdel Ghaffar and Housner 1977). Some researchers ( Abdel Ghaffar and Rubin 1983a and Abdel Ghaffar and Rubin 1983b) studied the effect of large amplitude nonlinear free coupled vertical torsional vibrations of suspension bridges using a continuum approach 5 where approximate solutions of the nonlinear coupled equations were conducted. Nonlinearities due to large deflections of cables, the axial stretching of stiffening structure, and the nonlinear curvature of the stiffening structure were considered. It was mentioned that the importance of geometric nonlinearities arises only for very high amplitude vibration. Also, they studied using two dimensional models the directional vertical, torsional, and lateral earthquake response, in both time and frequency domains, of long span suspension bridges subjected to multiple input excitations ( Abdel Ghaffar and Rubin 1982; Abdel Ghaffar and Rubin 1983c and Abdel Ghaffar et al. 1983). In addition, they considered a simplified model for the tower pier system and investigated the longitudinal vibration response taking into account the flexibility and damping characteristics of the underlying and surrounding soil. They applied their procedure to the tower pier system of the Golden Gate bridge ( San Francisco) and different soil conditions were used. The vertical response of suspension bridges has been studied to seismic excitations using a stochastic approach ( Dumanoglu and Severn 1990). They applied their method to three suspension bridges using one set of earthquake records and a filtered white noise as well. They pointed out that the accuracy of that approach, in comparison to the time history approach, depends upon the magnitudes of the fundamental period of the bridge under consideration. They reported that, for long span suspension bridges like the Bosporus ( in Turkey) and Humber ( in England) bridges, the response results of the stochastic approach should be cautiously assessed, especially when the earthquake records are not zero padded. 6 Some researchers ( Lin and Imbsen 1990; Ketchum and Seim 1991 and Ketchum and Heledermon 1991) carried out an investigation on the Golden Gate bridge by developing an elaborate 3 D finite element model. The lower wind bracing system of the bridge was considered to carry a light train. They incorporated different elements types and performed a nonlinear static analysis to determine the stiffness of the bridge in its dead load state and used this matrix in the solution for the natural frequencies and mode shapes. Their model is verified by comparing its results with those obtained from previous studies ( Abdel Ghaffer and Scanlan 1985a and Abdel Ghaffer and Scanlan 1985b). They reported that most of the lowest modes involving vibration of cables and torsional motion of the deck are not relevant to the earthquake performance of the bridge. A 3D finite element model was proposed for the Vincent Thomas bridge ( Niazy et al. 1991). They considered geometrical nonlinearities in suspension bridges, and an iterative nonlinear static analysis technique was adopted. The stiffening truss, tower and cable bent elements, were modeled as 3 D frame elements and cable elements were modeled as 3 D truss elements. In their study, 50 lowest natural frequencies and the corresponding mode shapes of the bridge model were determined in its dead load configuration. However, in their modeling they did not consider the actual mass distribution over the length of the bridge. They considered uniform mass distribution over the center span and the side spans. Initial shape of the cable is one of the important parameters in the analysis of suspension bridges. A non linear shape finding analysis was used for a self anchored suspension bridge named Yongjong Grand Bridge ( Kim et al., 2002). The shape finding analysis determines the coordinates of the main cable and 7 initial tension of main cable and hangers, which satisfies the design parameters at the initial equilibrium state under full dead loads. Several models and expressions have been proposed ( Davenport 1968) in relation to spatial variation of wind velocity fluctuation. For a more complete bibliography, the reader is referred to Simiu and Scanlan ( 1996). The analytical work by Beliveau et al., 1977 combined the effect of buffeting and self excited forces. They used a two degrees of freedom mathematical model. Even though simulation techniques have been reported since 1970 ( Shinozuka and Jan 1972), some earlier studies assumed uniformly distributed wind velocity fluctuations for the nonlinear time history analysis of cable supported bridges ( Arzoumanidis 1980). In past decades, a number of researchers reported on efficient methods for generating spatially correlated wind velocity fluctuations ( Li and Kareem 1993; Shinozuka and Deodatis 1996; Deodatis 1996; Facchini 1996; Yang et al. 1997; Paola 1998; Paola and Gullo 2001). As a result of improvements in simulation techniques as well as computational speed, the time domain approach has been utilized more frequently in recent buffeting analyses of long span cable supported bridges to take aerodynamic and/ or geometric nonlinearity into consideration ( Aas Jakobsen and Strømmen 1998, 2001; Minh et al. 1999; Ding and Lee 2000; Chen et al. 2000; Chen and Kareem 2001; Lin et al. 2001). Kareem’s group, in particular, has reported extensively on the line of time domain analysis framework for use in predicting aerodynamic nonlinear responses by incorporating frequency dependent parameters of unsteady aerodynamic forces by utilizing a rational function approximation technique ( Chen and Kareem 2001). This technique is also readily available for the structure originated nonlinearity in buffeting analysis. However, only a few studies utilized a nonlinear analysis procedure 8 for estimating buffeting response using structural nonlinearity, which is potentially involved in long span cable supported bridges, has been taken into consideration ( Ding and Lee 2000; Lin et al. 2001). The spatial variation of earthquake ground motions may have significant effect on the response of long span suspension bridges. Abdel Ghaffar and Rubin ( 1982) and Abdel Ghaffar and Nazmy ( 1988) studied response of suspension and cable stayed bridges under multiple support excitations. Zerva ( 1990) and Harichandran and Wang ( 1990) examined the effect of spatial variable ground motions on different types of bridge models. Harichandran et al. ( 1996) studied the response of long span bridges to spatially varying ground motion. Deodatis et al. ( 2000) and Kim and Feng ( 2003) investigated the effect of spatial variability of ground motions on fragility curves for bridges. Lou and Zerva ( 2005) analyzed the effects of spatially variable ground motions on the seismic response of a skewed, multi span, RC highway bridge. Most of the aforementioned studies dealt with simple FE models of the bridge, as a result response of critical members could not be evaluated. In the present analysis a panel based detailed 3D FE model of a long span suspension bridge is utilized. In this study, an iterative algorithm is proposed to generate spatially variable, design spectrum compatible acceleration time histories at different support locations of the bridge. The proposed algorithm is used to generate synthetic ground motions at six different points on the ground surface. For generating non stationary accelerograms, previously researchers used time dependent envelope function on top of simulated stationary ground motions ( Deodatis 1996). In this study by using evolutionary power spectral density function from the mother accelerogram, a new algorithm has been 9 proposed to simulate spatial variable ground motions. In the simulated acceleration time histories the temporal variations of the frequency content are same as the mother accelerogram. Mukherjee and Gupta ( 2002) proposed a new wavelet based approach to simulate spectrum compatible time histories. But they only considered one design spectrum and simulated one accelrogram from a single mother acceleration time history. Sarkar and Gupta ( 2006) developed a wavelet based approach to simulate spatially correlated and spectrum compatible accelerogram. So far in a broad sense two approaches have been introduced by researchers regarding conditional simulation. The two approaches are based on “ kriging” ( Krige, 1966) ( linear estimation theory applied to random functions) and conditional probability density function. Vanmarcke and Fenton ( 1991) applied conditional simulation of to simulate Fourier coefficients using kriging technique. Kameda and Morikawa ( 1992 and 1994), used an analytical framework based on spectral representation method, derived joint probability density functions of Fourier coefficients obtained from the expansion of conditioned random processes into Fourier series. They calculated conditional expectations and variances of the conditioned random processes and considered their first passage probabilities. Hoshiya ( 1994) considered a conditional random field as a sum of its kriging estimate and the error. He simulated the kriging estimate and the error separately and combined them to get the Gaussian conditionally simulated field. In all the above studies the investigators considered Gaussian processes and Gaussian random fields. Sometimes the assumption of Gaussian wind loading is not correct. In those cases, conditional simulation of non Gaussian wind velocity field should be used. Elishakoff et 10 al. ( 1994) combined the conditional simulation technique of Gaussian random fields by Hoshiya ( 1994) and the iterative procedure for unconditional simulation of non Gaussian random fields by Yamazaki and Shinozuka ( 1988), to conditionally simulate timeindependent non Gaussian random fields. Gurley and Kareem ( 1998) developed a procedure for conditional simulation of multivariate non Gaussian velocity/ pressure fields. For mapping the Gaussian process to non Gaussian process and vice versa, they used modified Hermite transformation using Hermite polynomial function. For buffeting analysis of long span cable supported bridges Chen ( 2001), Kim ( 2004) used time domain analysis to consider the effect of non linearity in the structure. Also they only considered the wind forces on the deck only. They neglected the coupling effect of wind forces on tower and cable. Sun ( 1999) considered the coupling effect of the aeroelastic forces on the bridge deck, towers and cables. But they did not consider a 3D detailed finite element ( FE) model of the bridge. Recently, He ( 2008) considered a detailed 3D model for buffeting analysis. 1.3 Objectives and Scope The main purpose of this research is to evaluate the performance of a long span suspension bridge under seismic, wind, and traffic loads. A member based detailed threedimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by 11 simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino Hills earthquake and comparing with the measured response. Tornado diagram and first order second moment ( FOSM) methods are applied for evaluating the sensitivity of different parameters on the eigenproperties of the FE models. This kind of study will be very helpful in selecting parameters and their variability ranges for FE model updating of suspension bridges. Considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed and the ductility demands of critical sections of the tower are presented in terms of fragility curves. Effect of spatial variability of ground motions on seismic displacement demand and seismic force demand is investigated. To generate spatially correlated design spectrum compatible nonstationary acceleration time histories, a newly developed algorithm using evolutionary power spectral density function ( PSDF) and spectral representation method is used. To simulate the wind velocity field accurately for the bridge site, measurement of the wind velocity is needed at the bridge location. For colleting actual wind data i. e. wind velocity and direction, three anemometers have been installed at three different locations of the bridge, so that the wind velocity field can be simulated in both horizontal and vertical directions. The measured wind velocity fluctuation data have been used for conditional simulation of wind velocity fluctuation field. Finally, response of Vincent Thomas Bridge under conditionally simulated wind velocity field is also presented in this study. A new simulation technique for conditional simulation of non Gaussian wind velocity fluctuation field is proposed and used for 12 buffeting analysis of the bridge under simulated wind load. Analysis of the Vincent Thomas bridge under traffic load is also carried out in this study. 1.4 Dissertation Outline The dissertation contains the following chapters Chapter 2 summarizes the finite element ( FE) modeling of the before and after retrofitting of the bridge. Chapter 3 presents the system Identification results obtained from response of the bridge and compared with modal parameters obtained from analytical model. A sensitivity analysis is also carried out. Chapter 4 proposes a new methodology to simulate spectrum compatible spatial variable ground motions. Response variability due to spatial variation in ground motion is also assessed. Chapter 5 describes the wind sensors installation in the bridge and data collection. Chapter 6 proposes a new methodology to conditionally simulate non Gaussian wind velocity fluctuation profiles using the data collected by anemometers at the bridge site. Wind buffeting analysis also carried out using the simulated wind velocity fluctuation profile. Chapter 7 describes the traffic load analysis. 13 CHAPTER 2 FINITE ELEMENT MODELING OF VINCENT THOMAS BRIDGE 2.1 Background With the advent of high speed computer, major advances in studying the dynamic characteristics of suspension bridges have been achieved through the use of finite element method. In addition, effort has also been given for developing simplified models that can predict response consistent with detailed model. In recent years, several commercially available finite element software packages have been used by practicing engineers as well as researchers to evaluate the response of a suspension bridge from operational traffic, wind and earthquake loads taking into account both material and geometric non linear behavior. This chapter focuses on numerical modeling of the Vincent Thomas Bridge. A member based detailed three dimensional Finite Element ( FE) as well as a panel based simplified model of the Vincent Thomas bridge have been developed for before and after retrofit of the bridge. 2.2 Calculation of Dead Weight The dead load along the length of the bridge has been calculated. Table 1 shows the calculated dead load of the different components of the bridge. It has been found that the weight per unit length of the bridge in the center span is very close to the design value of 14 7.2 kip/ ft indicated in the design drawing. The dead load calculation is also compared with the values reported by Abdel Ghaffer, 1976 shown in Table 2.1. 2.3 Calculation of the Initial Shape of the Cable Initial shape of the cable is one of the important parameters in the modeling of suspension bridges. Initial shapes of the cables of Vincent Thomas Bridge have been calculated using non linear shape finding analysis and subsequently used in the FE model. The shape finding analysis determines the coordinates of the main cable and initial tension of main cable and hangers, which satisfies the design parameters at the initial equilibrium state under full dead loads. Details of the analysis methodology and software are described in Kim et al., 2002. The shape of the initial cable profile in the form of preliminary and final configurations are tabulated in Table 2.2 and the initial cable profile is plotted in Figure 2.1. 2.4 Panel Based Simple Model For simplified panel based modeling, the girders and diaphragms are considered as equivalent 3D frame elements. The cable and suspender are modeled as 3D truss element. Also as in the case of detailed model, truss and cable bent were modeled with frame elements. Dampers are also included in the simplified model only at the tower and girder connections. . FE modeling is done with SUCOT ( Kim, 1993) and SAP 2000 V10 ( Computer and Structures, 2002). Area of the stiffening girder is set equal to the sum of the area of top chord, bottom chord and web. 15 Table 2.1 Calculated dead load ( kip/ ft) along the length of bridge Present study Different components Center span Side span Abdel Ghaffer, 1976 Curb 0.066 0.066 Bracket 0.019 0.019 Crash barrier 0.413 0.413 Sub total 0.498 0.498 0.203 Grating 0.036 0.036 Railing 0.0414 0.0414 Fence 0.131 0.131 Sub total 0.208 0.208 0.199 Lightweight concrete 2.521 2.521 2.592 Reinforcement steel 0.173 0.173 0.173 Stringers 0.544 0.544 0.682 Bracings 0.154 0.154 Sub total 3.392 3.392 3.447 Floor Truss 0.41 0.41 Inspection walkway 0.098 0.098 Inspection rail 0.052 0.052 Wind shoe 0.008 0.008 Bridge floor Sub total 0.568 0.568 0.613 Top chord 0.313 0.313 0.315 Bottom chord 0.307 0.291 0.302 Gusset plate, splice 0.234 0.234 0.124 Web ( diagonal) 0.162 0.166 0.142 Post ( vertical) 0.055 0.055 0.053 Strut, rivet, bolt etc 0.007 0.007 0.007 Stiffening truss Sub total 1.078 1.066 0.943 K truss 0.161 0.154 0.159 Lateral system Sub total 0.161 0.154 0.159 Cable 0.971 0.971 1.025 Suspenders 0.066 0.065 0.054 Cable Sub total 1.037 1.036 1.079 Cable and suspender weight 1.037 1.036 1.079 Suspended structure weight 5.905 5.886 5.564 Total weight 6.942 6.922 7.170 For SI: 1 kip/ ft = 14.593 kN/ m 16 Table 2.2 Calculated nodal coordinates of the cable only system Y ( ft) Z ( ft) Preliminary Final X ( ft) Preliminary Final configuration configuration configuration configuration Remark 1256.500 29.5833 29.5833 163.1400 163.1400 Cable bent 1221.840 29.7626 29.7628 172.5984 172.6094 1190.780 29.9295 29.9297 181.5906 181.6075 1159.720 30.1039 30.1042 191.0963 191.1143 1128.660 30.2867 30.2869 201.1168 201.1324 1097.600 30.4782 30.4784 211.6536 211.6664 1066.540 30.6787 30.6787 222.7081 222.7190 1035.480 30.8882 30.8882 234.2820 234.2919 1004.420 31.1069 31.1069 246.3771 246.3866 973.360 31.3349 31.3349 258.9949 259.0050 942.300 31.5722 31.5723 272.1375 272.1489 911.240 31.8188 31.8190 285.8068 285.8202 880.180 32.0749 32.0751 300.0048 300.0211 849.120 32.3404 32.3407 314.7336 314.7525 818.060 32.6154 32.6157 329.9955 330.0132 787.000 32.8998 32.9000 345.7927 345.8045 750.000 33.2500 33.2500 365.2600 365.2600 Tower 714.380 32.9474 32.9482 351.2700 351.3089 683.320 32.6931 32.6946 339.6119 339.6821 652.260 32.4485 32.4505 328.4872 328.5843 621.200 32.2136 32.2161 317.8940 318.0124 590.140 31.9885 31.9913 307.8302 307.9643 559.080 31.7732 31.7762 298.2938 298.4394 528.020 31.5678 31.5710 289.2832 289.4360 496.960 31.3723 31.3756 280.7965 280.9516 465.900 31.1870 31.1901 272.8323 272.9846 434.840 31.0117 31.0147 265.3889 265.5335 403.780 30.8468 30.8495 258.4649 258.5980 372.720 30.6922 30.6946 252.0591 252.1778 341.660 30.5482 30.5502 246.1702 246.2718 310.600 30.4149 30.4165 240.7971 240.8811 279.540 30.2925 30.2937 235.9389 236.0069 248.480 30.1812 30.1821 231.5945 231.6482 217.420 30.0812 30.0819 227.7631 227.8042 17 Table 2.2 Calculated nodal coordinates of the cable only system ( contd.) Y ( ft) Z ( ft) Preliminary Final X ( ft) Preliminary Final configuration configuration configuration configuration Remark 186.360 29.9930 29.9934 224.4441 224.4743 155.300 29.9168 29.9170 221.6367 221.6577 124.240 29.8531 29.8531 219.3405 219.3539 93.180 29.8024 29.8023 217.5550 217.5626 62.120 29.7655 29.7653 216.2799 216.2833 31.060 29.7429 29.7427 215.5150 215.5158 0.000 29.7353 29.7350 215.2600 215.2600 Center For SI: 1 ft = 0.3048 m 0 50 100 150 200 250 300 350 400  1500  1000  500 0 500 1000 1500 Length ( ft) Height ( ft) Figure 2.1 The shape of the initial cable profile under dead load Calculations of other cross sectional properties of girder ( moment of inertia and torsional constant) are given as follows: z x y 18 2.4.1 Moment of Inertia ( Iz) Moment of inertia of various members is computed from the equations in the table below and their values are given following the table. Chord Slab Stringer ( / 2) 2 2 I = A× e × y 12 3 bh Iy = = = 4 1 2 i i i Iy A d Chord: side span = 2 2 2 2 55.56in × 29.585 × 2ea× 2( both) = 194,520in ft center span = 2 2 2 2 53.78in × 29.585 × 2ea× 2( both) = 188,288in ft Slab: Figure 2.2 shows the cross section of the deck. Figure 2.2 Cross section of deck 4 3 3 6744.9 12 54.5 0.5 12 ft bh Iy = × = = For equilibrant steel section: 4 449.7 15 6744.9 Iy = = ft 27.25 27.25 CL 0.5 19 Stringer: Figure 2.3 shows the location of stingers in one side of the deck. Figure 2.3 Location of stringers in one side of the deck For one side: 2 2 2 2 4 4 1 2 Iz A d 0.1389( 3.5 10.5 17.5 24.5 ) 142.93 ft i i i = = + + + = = So, for one stiffening girder 4 142.93 367.78 2 449.7 Iz = + = ft From ( Abdel Ghaffer, 1976), slab + stringers : ( 105,000+ 290) sq. in. sq. ft./ 144/ 2= 4 365.59 ft 2.4.2 Torsional Constant ( J) Figure 2.4 shows commonly used lateral bracing systems and stiffening girders for suspension bridges. i i i i J = 2 b b d ; i vi i hi i i vi hi i b d b d μ μ μ μ b × + × = 2 2 2 2 2 web : A 0.117 ft ; k truss : A 0.115 ft , Av 0.132 ft d d = − = = b = 59.17 ´ , d= 15 ´ 0.25 2.5 2( 1 ) = \ = + = G E E G μ μ CL 0.5 3.5 7 7 7 20 Figure 2.4 Commonly used lateral bracing systems and stiffening girders ( Abdel Ghaffer 1976) To calculate the torsional constant of the suspension bridge girder two coefficients are used. Here, h μ is the coefficient for horizontal K type system and V μ is for vertical Worren type web system. Figure 2.5 shows the horizontal K type system and Figure 2.6 shows the for vertical Worren type web system. The procedure to calculate those two coefficients are shown below. Figure 2.5 Horizontal system ( K type) 2 59.17 31.08 Ad Av 21 = = ° − ) 43.6 31.08 59.17 / 2 tan ( 1 2 a 0.134 ) 0.082 0.114 sin 43.6 2 0.114 0.082 sin 43.6 cos 43.6 2.5 ( ) sin 2 sin cos ( 3 2 2 3 2 2 2 = + × ° × × × ° × ° = × + × × × × × = a a a μ v d d v A A A A G E h Figure 2.6 Vertical web system ( Worren type) = = ° − ) 44 31.08/ 2 15 tan ( 1 1 a sin cos 2.5 0.137 sin 44 cos44 0.119 2 1 1 2 μ = × × a × a = × × ° × ° = v Ad G E 0.032 59.17 0.119 15 0.134 59.17 15 0.119 0.134 2 2 2 2 = × + × × × × = × + × = i vi i hi i i vi hi i b d b d μ μ μ μ b 4 Ji 2 ibidi 2 0.032 59.17 15 56.416 ft = b = × × × = The sectional properties computed in this section ( Section 1) are summarized in Table 2.3 below for each panel. 15' 31.08' 1 Ad 22 Table 2.3 Calculated sectional properties of panels Area Torsional Constant Iz Iy Panel No. ft 2 ft 4 ft 4 ft 4 1 0.958 24.342 369.010 39.367 2 0.931 23.021 369.010 39.367 3 0.911 22.944 369.010 38.234 4 0.941 21.304 369.010 41.676 5 0.972 21.304 369.010 43.398 6 0.902 16.515 369.010 43.398 7 0.972 21.304 369.010 43.398 8 0.938 19.141 369.010 43.398 9 0.938 19.141 369.010 43.398 10 0.938 19.141 369.010 43.398 11 0.972 21.304 369.010 43.398 12 1.010 23.308 369.010 43.398 13 0.979 23.308 369.010 41.676 14 0.948 24.704 369.010 38.234 15 0.968 24.797 369.010 39.367 16 0.968 24.797 369.010 39.367 17 0.866 19.188 369.010 39.367 18 0.866 19.188 369.010 39.367 19 0.846 19.141 369.010 38.234 20 0.811 16.523 369.010 38.234 21 0.811 16.523 369.010 38.234 22 0.841 16.523 369.010 39.957 23 0.824 15.034 369.010 39.957 24 0.824 15.034 369.010 39.957 25 0.824 15.034 369.010 39.957 26 0.824 15.034 369.010 39.957 27 0.855 15.034 369.010 41.680 28 0.907 19.141 369.010 41.680 29 0.907 19.141 369.010 41.680 30 0.968 19.141 369.010 45.117 31 0.968 19.141 369.010 45.117 32 0.968 19.141 369.010 45.117 33 0.968 19.141 369.010 45.117 34 0.916 15.034 369.010 45.117 35 0.916 15.034 369.010 45.117 23 Table 2.3 Calculated sectional properties of panels ( contd.) Area Torsional Constant Iz Iy Panel No. ft 2 ft 4 ft 4 ft 4 36 0.916 15.034 369.010 45.117 37 0.916 15.034 369.010 45.117 38 0.916 15.034 369.010 45.117 39 0.916 15.034 369.010 45.117 40 0.916 15.034 369.010 45.117 37 0.916 15.034 369.010 45.117 41 0.916 15.034 369.010 45.117 42 0.916 15.034 369.010 45.117 43 0.916 15.034 369.010 45.117 44 0.916 15.034 369.010 45.117 45 0.916 15.034 369.010 45.117 46 0.916 15.034 369.010 45.117 47 0.916 15.034 369.010 45.117 48 0.968 19.141 369.010 45.117 49 0.968 19.141 369.010 45.117 50 0.968 19.141 369.010 45.117 51 0.907 19.141 369.010 41.680 52 0.907 19.141 369.010 41.680 53 0.907 19.141 369.010 41.680 54 0.824 15.034 369.010 39.957 55 0.824 15.034 369.010 39.957 56 0.824 15.034 369.010 39.957 57 0.824 15.034 369.010 39.957 58 0.841 16.523 369.010 39.957 59 0.811 16.523 369.010 38.234 60 0.811 16.523 369.010 38.234 61 0.846 19.141 369.010 38.234 62 0.866 19.188 369.010 39.367 63 0.866 19.188 369.010 39.367 64 0.968 24.797 369.010 39.367 65 0.968 24.797 369.010 39.367 66 0.948 24.704 369.010 38.234 67 0.979 23.308 369.010 41.676 68 1.010 23.308 369.010 43.398 69 0.972 21.304 369.010 43.398 24 Table 2.3 Calculated sectional properties of panels ( contd.) Area Torsional Constant Iz Iy Panel No. ft 2 ft 4 ft 4 ft 4 70 0.938 19.141 369.010 43.398 71 0.938 19.141 369.010 43.398 72 0.938 19.141 369.010 43.398 73 0.972 21.304 369.010 43.398 74 0.902 16.515 369.010 43.398 75 0.972 21.304 369.010 43.398 76 0.941 21.304 369.010 41.676 77 0.911 22.944 369.010 38.234 78 0.931 23.021 369.010 39.367 79 0.958 24.342 369.010 39.367 80 0.958 24.342 369.010 39.367 For SI: 1 ft = 0.3048 m Calculation of tower cross sectional properties: For thin walled closed sections the torsional constant is given by the following formula ( Bredt’s formula): = t ds A J 2 4 Different sections of the tower is shown in Figure 2.7 and a typical plan view of the tower section is shown in Figure 2.8. Table 2.4 and 2.5 show the calculated sectional properties of the tower section at different heights for before and after retrofit models respectively. 2.5 Member Based Detail Model Finite Element modeling of the detailed structure is done with the help of SAP 2000 V10 ( Computer and Structures, 2002). The cables and suspenders are modeled as 3D elastic truss elements. The chords, vertical members and the diagonal members in the stiffening 25 Figure 2.7 Different sections of the tower Figure 2.8 Typical tower cross section X Y 1 2 3 4 42.89 5 52.58 52.58 85.50 85.66 26 Table 2.4 Calculated sectional properties of the tower sections ( before retrofit) Area Ix Iy Torsional Constant Section No. ft 2 ft 4 ft 4 ft 4 1 3.18 20.42 21.07 17.81 2 4.35 42.75 48.25 25.69 3 4.92 57.64 65.06 26.89 4 4.93 60.32 65.86 29.14 5 5.47 76.11 90.37 34.20 For SI: 1 ft = 0.3048 m Table 2.5 Calculated sectional properties of the tower sections ( after retrofit) Area Ix Iy Torsional Constant Section No. ft 2 ft 4 ft 4 ft 4 1 3.66 23.48 24.23 23.55 2 5.00 49.16 55.49 33.98 3 5.66 66.29 74.82 35.57 4 5.67 69.37 75.74 38.54 5 6.29 87.53 103.93 45.23 For SI: 1 ft = 0.3048 m girder are modeled as 3D truss elements. Also members in the diaphragm are modeled as truss elements. The tower, the cable bent leg, and strut members are modeled as frame elements. The reinforced concrete deck is modeled as shell element and the supporting stringers are modeled as beam elements. Hydraulic, viscous dampers between tower and the suspended structure are also modeled according to their properties mentioned in the design drawing. Mass is taken distributed over each and every member. To consider the mass of non structural components, equivalent point mass and mass moment of inertia are distributed at joints in the diaphragm. The most important structural components that are considered for post retrofit modeling are suspended truss system, deck shear connectors, cable bent cross sections, 27 suspenders and dampers installed. Figure 2.9 shows detailed model of one panel and construction drawing. Figure 2.9 The detailed model of one panel 2.5.1 Cable Bent Four feet of stiffening truss in the cable bent was removed to allow free oscillations of the side spans of the bridge. Also, the cable bent cross section was changed. This change in the cross section is considered in the post retrofit modeling of the bridge. Cross sectional properties of the modified sections are calculated and used in the post retrofit analysis. 2.5.2 Deck Shear Connector Deck shear connectors were replaced with new types. Deck shear connectors of the original structure were removed and then a new set was introduced. Figures 2.10 and 2.11 ( taken from Design Drawing) shows the comparison between the shapes of the deck shear connectors before and after retrofit. The FE modeling is done according to this design drawing. ECLAEBMLEENT TSRTIUFSFSENING CABEHLLEAE NNMGOEEDNRETS 28 Figure 2.10 Deck shear connector ( before retrofit) Figure 2.11 Deck shear connector ( after retrofit) 2.5.3 Dampers Total of 48 dampers were installed in the bridge as a retrofit measure with 16 dampers installed in each tower, at the junction between tower and girder connection. In each cable bent, 4 dampers were installed. In the middle of each side span a new diaphragm was inserted. At the location of the inserted diaphragm, 4 more dampers were installed in each side span. These 8 dampers in the side spans were non linear dampers having the form of F = cvn where n = 0.5. For all other dampers, n = 1.0 is used. 2.5.4 Suspended Truss The suspended truss structure was modified by inserting new members and also replacing some members in the K truss in the middle span as well as in the side spans. Figure 2.12 ( taken from Design Drawing) shows the modifications made in the K truss. 2.5.5 Suspenders Some suspenders in the middle span were replaced with new suspenders. Figure 2.13 ( taken from Design Drawing) describes the modified suspenders in the middle span. 29 Figure 2.12 K truss modifications after retrofit Figure 2.13 Suspender modifications after retrofit 2.6 Eigen Value Analysis First 100 eigen vectors were calculated with a convergence tolerance of 1.0 10 5 − × . Table 2.6 shows the comparison of modal frequencies obtained from before retrofit panel based simple model and member based detailed model with analytical eigen properties of the bridge obtained by previous researchers. Table 2.7 shows the aforementioned comparison of results obtained from after retrofit model of the bridge, In the tables the computed modal frequencies were obtained from the FE models by using SUCOT ( Kim, 1993) and 30 SAP 2000 V 10 ( Computers and Structures, 2002). It can be seen from the results that the computed modal frequencies obtained from the SUCOT and the SAP 2000 panel based model are having a good match with the calculated frequencies from finite element models developed by previous researchers. First three modes obtained from the SAP 2000 model are shown in Figure 2.14. Figure 2.14 First three mode shapes of the simple model Table 2.6 Comparison of modal frequencies in Hz ( before retrofit) Present study Panel based simple Memberbased detailed Dominant Motion Abdel Ghaffar, 1976 Niazy et al., 1991 SUCOT SAP 2000 SAP 2000 L * S * 0.173 0.169 0.159 0.152 0.161 V * AS * 0.197 0.201 0.210 0.223 0.221 V S 0.221 0.224 0.232 0.239 0.226 V S 0.348 0.336 0.460 0.384 0.363 V AS 0.346 0.344 0.456 0.495 0.369 L AS 0.565 0.432 0.472 0.448 0.503 T * S 0.449 0.438 0.483 0.482 0.477 V S 0.459 0.442 0.500 0.538 0.479 * L: Lateral, S: Symmetric, V: Vertical, AS: Asymmetric and T: Torsional 31 Table 2.7 Comparison of modal frequencies in Hz ( after retrofit) Ingham Present Study et al., 1997 ( ADINA) Fraser, 2003 ( ADINA) Panel based Simple Memberbased Detailed Dominant motion Simple Detailed Detailed SUCOT SAP 2000 SAP 2000 L S 0.162 0.135 0.130 0.161 0.152 0.160 V AS 0.197 0.171 0.182 0.210 0.218 0.220 V S 0.232 0.229 0.226 0.232 0.235 0.226 V S    0.360 0.369 0.362 V AS    0.453 0.469 0.372 L AS 0.535 0.420 0.409 0.473 0.447 0.494 T S 0.588 0.510 0.511 0.490 0.484 0.482 V S    0.498 0.513 0.486 * L: Lateral, S: Symmetric, V: Vertical, AS: Asymmetric and T: Torsional 2.7 Closure In this chapter numerical modeling has been achieved for Vincent Thomas Bridge. A member based detailed three dimensional Finite Element ( FE) as well as a panel based simplified model of the Vincent Thomas Bridge have been developed for the bridge before and after retrofit. First eight modal frequencies obtained from FE models developed using different commercially available softwares have been compared. The results obtained from this study are also compared with previous results obtained for the bridge. It has been observed that the first lateral modal frequency for the member based detailed model is 20% higher than those presented in previous studies. It is also found 32 that results of panel based simple models are in good agreement with those obtained from the detailed model and those reported in previous similar studies. 33 CHAPTER 3 SYSTEM IDENTIFICATION AND MODEL VERIFICATION 3.1 Background To ensure the validity of the analytical finite element model of a massive structure like a suspension bridge, the eigenproperties obtained from the analytical model should be compared with the identified modal frequencies. Modal identification can be done by using measured ambient vibration response or strong motion earthquake response of the bridge. The response of the bridge, under external excitations, is measured with the help of acceleration measuring sensors installed at different locations of the bridge. 3.2 Evaluation of Eigenproperties using Ambient Vibration Data Experimental modal analysis has drawn significant attention from structural engineers for updating the analysis model and estimating the present state of structural integrity. Forced vibration tests such as impact tests can be carried out to this end. However, it is usually restricted to small size structures or to their components. For large structures such as dams, and long span bridges, ambient vibration tests under wind, wave, or traffic loadings are the effective alternatives. In this study, modal parameters were obtained using the frequency domain decomposition technique ( Otte et al, 1990 and Brincker et al., 2000) which is one of the frequency domain methods without using input information. It is very difficult, if not impossible, to identify closely spaced modes using the 34 peak picking ( PP) method. In this case, the frequency domain decomposition ( FDD) method that utilizes the singular value decomposition of the PSD matrix may be used to separate close modes ( Brincker et al., 2000). The method was originally used to extract the operational deflection shapes in mechanical vibrating systems ( Otte et al, 1990). The natural frequencies are estimated from the peaks of the PSD functions in the PP method. On the other hand, they are evaluated from singular value ( SV) functions of the PSD matrix in the FDD method. ( ) ( ) ( ) ( ) T S yy w = U w s w V w ( 3.1) where ( ) m m N N yy S R × w Î is the PSD matrix for output responses ( ) m N y t Î R , ( ) m m N N s R × w Î is a diagonal matrix containing the singular values of its PSD matrix, and, U ( w ) , ( ) m m N N V R × w Î are corresponding unitary matrices. m N is the number of measuring points. The general multi DOF system can be transformed to the single DOF system nearby its natural frequencies by singular value decomposition. The mode shape can be estimated as the first column vector of the unitary matrix of U since the first singular value may include the structural mode nearby its natural frequencies. However in the closely spaced modes, the peak of largest singular values at one natural frequency indicates the structural mode and adjacent second singular value may indicate the close mode. Figure 3.1 shows the layout of the acceleration sensors installed in the bridge site. Table 3.1 describes the location and direction of all the accelerometers present in the bridge site. Figure 3.2 shows the vertical accelerometers and Figure 3.3 shows the lateral 35 accelerometers used in the modal identification of the bridge structure. Figure 3.4 shows the plot of SV vs. frequency for the acceleration data obtained from vertical channels. Figure 3.1 Location and direction of sensors installed in the bridge Table 3.1 Location and direction of accelerometers Sensor Number Sensor Location Sensor Direction 22, 15, 16, 17, 18, 21 Truss top/ Deck Vertical 2, 4, 5, 6, 7 Truss top/ Deck Lateral 12 Truss top/ Deck Longitudinal 3 Truss bottom Lateral 8 Tower Lateral 10, 11 Tower Longitudinal 14, 19, 20 Tower base Vertical 1, 9 Tower base Lateral 13, 23 Tower base Longitudinal 26 Anchorage Vertical 24 Anchorage Lateral 25 Anchorage Longitudinal 36 Figure 3.2 Vertical accelerometer data used in the study Figure 3.3 Lateral accelerometer data used in the study 37 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 x 10 4 Frequency ( Hz) SV SV of vertical acc.( CH 15 22) Figure 3.4 Plot of SV vs. Frequency 3.3 Comparison of System ID Result with Analytical Eigen Properties In this study, modal parameters have been obtained using the frequency domain decomposition ( FDD) technique ( Brincker et al. 2000) which is one of the frequency domain methods without using in put information. The method utilizes the singular value decomposition of the PSD matrix and may be used to separate close modes. Total 15 ambient vibration recording has been used for this purpose from the installed sensors. The data were recorded from April, 2003 to October, 2004, over 1 year 6 months record has been considered for system identification analysis. Average identified modal frequencies obtained from 15 data set are considered as final identified modal frequencies from the ambient vibration data. Figure 1 shows previously installed sensor locations on the bridge. For system ID from ambient vibration data, vertical sensors 15, 16, 17, 18, 21, 22 and lateral sensors 4, 5, 6, 7 are used. Sensor # 3 in the lateral direction is excluded because it provided some noisy data. Table 3.2 and 3.3 below shows the comparison of modal frequencies before and after retrofitting of the bridge. Modal identification results from ambient vibration data are also tabulated in Table 3.3. It can be seen from Table 3.3 that in the first mode of 38 vibration, the structure is a little bit stiffer in the simple model rather than detailed model. In case of first mode of vibration the system ID result matches with the frequency obtained from the detailed model. Also, from the second mode and above both the analytical and system ID results shows pretty good match. On an average sense, it can be seen from Table 3.3 that system ID results show pretty good match with detailed model. 3.4 Modal Parameter Identification from Chino Hills Earthquake Response Chino Hills earthquake data recorded at the bridge site are also used in the modal identification. Chino Hills earthquake occurred on July 29, 2008, in Southern California. The epicenter of the magnitude 5.4 earthquake was in Chino Hills, approximately 45 km east southeast of downtown Los Angeles. Table 3.4 compares the modal frequencies of the bridge obtained from ambient vibration and Chino Hills earthquake data. These two identified frequencies matches very well. Note also that the two other previous studies ( Ingham et al. 1997 and Fraser 2003) involving detailed models under predict modal frequencies significantly for the first two modes. Results from these two studies are also tabulated in Table 3.4. 39 Table 3.2 Comparison of modal frequencies in Hz ( before retrofit) Identified ( System ID) Computed Present Study Panel based Simple Memberbased Detailed Dominant Motion Abdel Ghaffar and Housner, 1977 ( Ambient) Niazy et al., 1991 ( Whittier) Ingham et al., 1997 ( Northridge) Abdel Ghaffar, 1976 Niazy et al., 1991 SUCOT SAP 2000 SAP 2000 L * S * 0.168 0.149 0.145 0.173 0.169 0.159 0.152 0.161 V * AS * 0.216 0.209  0.197 0.201 0.210 0.223 0.221 V S 0.234 0.224 0.222 0.221 0.224 0.232 0.239 0.226 V S 0.366 0.363 0.370 0.348 0.336 0.460 0.384 0.363 V AS  0.373  0.346 0.344 0.456 0.495 0.369 L AS 0.623 0.459 0.417 0.565 0.432 0.472 0.448 0.503 T * S 0.494 0.513 0.556 0.449 0.438 0.483 0.482 0.477 V S 0.487 0.448  0.459 0.442 0.500 0.538 0.479 * L: Lateral, S: Symmetric, V: Vertical, AS: Anti Symmetric, T: Torsional Table 3.3 Comparison of modal frequencies in Hz ( after retrofit) Identified ( System ID) Computed Ingham Present Study et al., 1997 ( ADINA) Fraser, 2003 ( ADINA) Panel based Simple Memberbased Detailed Dominant Motion Fraser, 2003 He et al., 2008 Present Study ( Ambient) Simple Detailed Detailed SUCOT SAP 2000 SAP 2000 L S 0.150  0.162 0.162 0.135 0.130 0.161 0.152 0.160 V AS  0.168 0.219 0.197 0.171 0.182 0.210 0.218 0.220 V S 0.233 0.224 0.229 0.232 0.229 0.226 0.232 0.235 0.226 V S 0.367 0.356 0.369    0.360 0.369 0.362 V AS       0.453 0.469 0.372 L AS   0.534 0.535 0.420 0.409 0.473 0.447 0.494 T S  0.483 0.471 0.588 0.510 0.511 0.490 0.484 0.482 V S       0.498 0.513 0.486 * L: Lateral, S: Symmetric, V: Vertical, AS: Anti Symmetric, T: Torsional 40 Table 3.4 Comparison of modal frequencies ( in Hz) of the Vincent Thomas Bridge Identified ( System ID) Computed Mode Number Dominant Motion Ambient Vibration Chino Hills Earthquake SAP 2000 ( Present Study) Ingham et al., 1997 Fraser, 2003 1 L * S * 1 0.162 0.168 0.160 0.135 0.130 2 V * AS * 1 0.219  0.220 0.171 0.182 3 V S1 0.229 0.228 0.222 0.229 0.226 4 V S2 0.369 0.362 0.362   5 V AS2  0.467 0.372   6 T * S1 0.471 0.491 0.478 0.510 0.511 7 V S3   0.483   8 L AS1 0.534  0.491 0.420 0.409 * L: Lateral, S: Symmetric, V: Vertical, AS: Anti Symmetric, T: Torsional 3.5 Effect of Parameter Uncertainty on Modal Frequency 3.5.1 Soil Spring Modeling To consider the effect of soil structure interaction kinematic three translational and three rotational soil springs with their coupling effects are considered at the foundations of east tower, west tower, east cable bent, west cable bent, east anchorage and west anchorage. The stiffness of the soil springs are calculated from the equivalent pile group stiffness at the foundations discussed earlier. Table 3.5 gives the number of piles at different foundations considered for the FE model of the bridge. Figure 3.5 shows the finite element model of the bridge with foundation springs. 41 Table 3.5 Location and number of piles considered Location Number of piles East tower 167 West tower 167 East cable bent 48 Wast cable bent 48 East anchorage 188 West anchorage 188 Figure 3.5 Detailed model in SAP 2000 with foundation springs 3.5.2 Uncertain Parameters Considered For model updating purpose, in ideal case, all parameters related to elastic, inertial properties and boundary conditions should be considered. However, if too many parameters are considered for model updating then chances of obtaining unreliable model increases ( Zhang et al., 2001). For this reason, parameter selection is a very important East tower West tower East cable bent East anchorage 42 task in model updating process. Practically if the parameters considered do not have much effect on the modal frequencies and mode shapes, then they should be excluded from the list. Therefore a comprehensive eigenvalue sensitivity study is performed to figure out the most sensitive parameters to be considered for suspension bridge finite element model calibration. Total 19 parameters are considered for the sensitivity analysis. The selection of these parameters is based on the outcome of previous research ( Zhang et al., 2001) and engineering judgments. Elastic modulus and mass density of different set of structural members, boundary conditions ( deck and tower connection and deck and cable bent connection) and stiffness of the soil springs are considered as variable parameters. However the cable and the concrete deck have homogeneous properties, but due to corrosion the structural strength may get decreased over the service life of the bridge. To capture that effect, elastic modulus and mass density of cable and concrete deck is considered as variable parameters in the analysis. Also, for the generality of the analysis kinematic spring stiffnesses ( soil spring stiffness) are also considered as variable parameters in the analysis. Since there was no tower dominant mode in the considered first 8 mode shapes, therefore, the stiffness and inertial properties of the tower is not considered as a variable parameter in the present study. For evaluating the effect of uncertainty in the modal parameters of Vincent Thomas Bridge, uncertainty associated with elastic and inertial property of different members is represented by assigning a mean and standard deviation in terms of coefficient of variation for each parameter. The mean values considered here are calculated based on the design drawing of the bridge. Table 3.6 lists these parameters with their mean values. 43 To asses the sensitivity, coefficients of variation ( COV) of all the parameters are considered as 10%. In the analysis, all the 36 values of the spring stiffness matrices are varied by 10% for the case of east tower, west tower, east cable bent and west cable bent. For the first order second moment ( FOSM) analysis only lateral translational stiffness of each foundation spring is considered. Table 3.6 Parameters considered for sensitivity analysis Serial Number Parameters Mean Value 1 Side link elastic modulus 2.00 × 10 8 kPa 2 Cable bent and girder connection elastic modulus 2.00 × 10 8 kPa 3 Top Chord Elastic Modulus 2.00 × 10 8 kPa 4 Top Chord Mass Density 7.85 kg/ m 3 5 Bottom Chord Elastic Modulus 2.00 × 10 8 kPa 6 Bottom Chord Mass Density 7.85 kg/ m 3 7 Stringer Elastic Modulus 2.00 × 10 8 kPa 8 Stringer Mass Density 7.85 kg/ m 3 9 Deck Slab Elastic Modulus 2.48 × 10 7 kPa 10 Deck Slab Mass Density 1.48 kg/ m 3 11 Main Cable Elastic Modulus 1.66 × 10 8 kPa 12 Main Cable Mass Density 8.37 kg/ m 3 13 Suspender Elastic Modulus 1.38 × 10 8 kPa 14 Suspender Mass Density 7.85 kg/ m 3 15 Wind Shoe Elastic Modulus 2.00 × 10 8 kPa 16 East Tower Spring 1.30 × 10 6 kPa 17 East Cable Bent Spring 7.35 × 10 6 kPa 18 West Tower Spring 1.19× 10 6 kPa 19 West Cable Bent Spring 4.65× 10 6 kPa 3.5.3 Analysis methods Reduction of the number of uncertain parameters cuts down the computational effort and cost. One way of doing this is to identify those parameters with associated ranges of uncertainty that lead to relatively insignificant variability in response and then treating these as deterministic parameters by fixing their values at their best estimate, such as the mean. For ranking uncertain parameters according to their sensitivity to desired response 44 parameters, there are various methods such as tornado diagram analysis, first order second moment ( FOSM) analysis, and Monte Carlo simulation ( Porter et al. 2002, Lee and Mosalam 2006). Monte Carlo simulation, which is computationally demanding due to the requirement of a large number of simulations, especially for a model consisting of a large number of degrees of freedom as in the case here, is not used in this study because of these practical considerations. Instead, the tornado diagram analysis and the FOSM analysis have been used here due to their simplicity and efficiency to identify sensitivity of uncertain parameters. For the tornado diagram analysis, all uncertain parameters are assumed as random variables, and for each of these random variables, two extreme values the 84 th percentile and 16 th percentile corresponding to assumed upper and lower bounds, respectively, of its probability distribution have been selected. One can observe that these extreme values come from the normal distribution assumption, mean + standard deviation and mean – standard deviation, respectively representing their upper and lower bounds. Using these two extreme values for a certain selected random variable, the modal frequencies of the model has been evaluated for both cases, while all other random variables have been assumed to be deterministic parameter with values equal to their mean value. The absolute difference of these two modal frequency values corresponding to the two extreme values of that random variable, which is termed as swing of the modal frequency corresponding to the selected random variable, is calculated. This calculation procedure has then been repeated for all random variables in question. Finally, these swings have been plotted in a figure from the top to the bottom in a descending order according to their size to demonstrate the relative contribution of each 45 variable to the specific mode under question. It is noteworthy that longer swing implies that the corresponding variable has larger effect on the modal frequency than those with shorter swing. For the FOSM analysis, the modal frequency has been considered as a random variable Y, which has been expressed as the function of random variables, Xi ( for i = 1 to N) denoting uncertain parameters and Y is given by ( , ,..., ) 1 2 N Y = g X X X ( 3.2) Let Xi has been characterized by mean μX and variance s X 2 . Now, the derivatives of g( X) with respect to Xi , one can express Y by expanding Eq. ( 3.2) in Taylor series as LLL K + − − + = + − = = = i j N j i X j X N i i N i X X X i X X X g X X X g Y g X i j N i d d d μ μ d d μ μ μ μ 2 1 1 1 ( )( ) 2! 1 ( ) 1! 1 ( , , , ) 1 2 ( 3.3) Considering only the first order terms of Eq. ( 3.3) and ignoring higher order terms Y can be approximated as i N i X X X i X X g Y g X N i d d μ μ μ μ = » + − 1 ( ) 1! 1 ( , , , ) 1 2 K ( 3.4) Taking expectation of both sides, the mean of Y, μY can be expressed as ( , , , ) Y X1 X2 X N μ g μ μ μ K » ( 3.5) Utilizing the second moment of Y as expressed in Eq. ( 3.4) and simplifying, the variance of Y, s Y 2 can be derived as = = » N i N j j N i N Y i j X g X X X X g X X X X X 1 1 2 1 2 1 2 ( , ,..., ) ( , ,..., ) cov( , ) d d d d s 46 j N i N N i N j i X X i N X N i X g X X X X g X X X X g X X X i i j d d d d r d d s ( , ,..., ) ( , ,..., ) ( , ,..., ) 1 2 1 2 1 2 2 1 2 1 = = ¹ + » ( 3.6) where i j X X r denotes correlation coefficient for random values Xi and Xj ( i. e., coefficient defining the degree to which one variable is related to another). The partial derivative of ( , ,..., ) 1 2 N g X X X with respect to Xi has been calculated numerically using the finite difference equation given below i i i N i i N i N x g x x x x g x x x x X g X X X D + D − − D = 2 ( , ,..., ) ( , ,..., , ) ( , ,..., , ) 1 2 1 2 1 2 μ μ d d ( 3.7) In this case, a large number of simulations were performed varying each input parameter individually to approximate the partial derivatives as given in Eq. ( 3.7). For these calculations, the mean and the standard deviation values given in Table 3.6 are used. For these sensitivity analyses, at first, the reference model with mean parameters of each 19 random variable considered in this study is analyzed. Then the analyses have been carried out using their lower and then upper bounds. Altogether 39 cases of modal analysis are performed for each set of parameters, modal frequencies expressed as ( , ,..., ) 1 2 N Y = g X X X is observed. 3.5.4 Sensitivity of Modal Frequencies For tornado diagram analysis, all the 19 parameters shown in Table 3.6 are used for total 8 mode shapes. Figures 3.6 ( a h) show tornado diagrams for 8 modes developed according to the procedure in section 3.5.3. The vertical line in the middle of tornado 47 diagrams indicates modal frequency value calculated for a certain mode considering only the mean values of all random variables and the length of each swing ( horizontal bar) represents the variation in the modal frequency due to the variation in the respective random variable. It is clear from Figures 3.6 ( a e) that, up to mode # 5 deck slab mass density and bottom chord elastic modulus have almost the largest contribution in response variability. In mode numbers 2, 3, 6, and 7, mostly vertical and torsional modes, main cable elastic modulus is significant contributor of the response variability. One can also notice from Figures 3.6 ( a h) that couple of swings are asymmetric about the vertical line. This skew of the modal frequency distributions implies that the problem is highly nonlinear. In other words, the same amount of a positive and a negative change in these parameters does not produce the same amount of variation in modal frequency. This skewness is very clear for 2 nd mode in case of main cable elastic modulus variation. Since the 2 nd mode is vertical antisymmetric, increase is main cable elastic modulus does not have much effect on increase in frequency but decrease in the stiffness of main cable decreases the frequency by 8% from the base model frequency. Interestingly, deck slab stiffness has most contribution in the 1 st mode, but it does not have any contribution in rest of the modes except the 8 th mode. Most of the boundary condition ( P1, P2, and P15) and soil spring ( P16, P17, P18, and P19) related parameters have very insignificant effect on response variability. For FOSM method, analyses have been carried out to determine the sensitivity of modal frequencies to the uncertainty in each random variable. Focus has been placed on the variance of modal frequency when considering uncertainties of 19 input parameters. 48 Figures 3.7 ( a h) show relative variance contributions of each parameter to the modal frequency when the correlation, as given in the second term of Eq. ( 3.6), is neglected. From this figure, it can be observed that the uncertainties in the deck slab mass density and bottom chord elastic modulus contribute mostly to the variance of modal frequencies. This is the same trend as observed from the tornado diagram analysis for all the 8 modes considered. 49 0.156 0.157 0.158 0.159 0.160 0.161 0.162 0.163 0.164 Modal Frequency ( Hz) : Mode 1 ( LS 1) P 10 P 9 P 5 P 3 P 12 P 7 P 8 P 4 P 6 P 11 P 14 P 18 P 13 P 1 P 2 P 15 P 16 P 17 P 19 0.216 0.217 0.218 0.219 0.220 0.221 0.222 0.223 0.224 0.225 Modal Frequency ( Hz) : Mode 2 ( VAS 1) P 1 0 P 5 P 1 2 P 1 1 P 8 P 4 P 6 P 3 P 9 P 1 4 P 7 P 1 3 P 1 6 P 1 P 2 P 1 5 P 1 7 P 1 8 P 1 9 ( a) ( b) P10 P5 P12 P11 P8 P4 P6 P3 P9 P14 P7 P13 P16 P1 P2 P15 P17 P18 P19 50 0.218 0.220 0.222 0.224 0.226 0.228 Modal Frequency ( Hz) : Mode 3 ( VS 1) P 1 0 P 1 1 P 5 P 1 2 P 8 P 4 P 6 P 3 P 9 P 1 4 P 7 P 1 3 P 1 P 2 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.354 0.356 0.358 0.360 0.362 0.364 0.366 0.368 0.370 Modal Frequency ( Hz) : Mode 4 ( VS 2) P 1 0 P 5 P 1 2 P 3 P 8 P 4 P 6 P 9 P 1 4 P 7 P 1 3 P 1 6 P 1 P 1 1 P 1 8 P 2 P 1 5 P 1 7 P 1 9 0.364 0.366 0.368 0.370 0.372 0.374 0.376 0.378 0.380 Modal Frequency ( Hz) : Mode 5 ( VAS 2) P 1 0 P 5 P 1 2 P 3 P 8 P 4 P 6 P 9 P 1 4 P 7 P 1 1 P 1 3 P 1 8 P 1 6 P 1 P 2 P 1 5 P 1 7 P 1 9 P10 P11 P5 P12 P8 P4 P6 P3 P9 P14 P7 P13 P1 P2 P15 P16 P17 P18 P19 P10 P5 P12 P3 P8 P4 P6 P9 P14 P7 P13 P16 P1 P11 P18 P2 P15 P17 P19 P10 P5 P12 P3 P8 P4 P6 P9 P14 P7 P11 P13 P18 P16 P1 P2 P15 P17 P19 ( c) ( e) ( d) 51 0.468 0.470 0.472 0.474 0.476 0.478 0.480 0.482 0.484 0.486 0.488 Modal Frequency ( Hz) : Mode 6 ( TS 1) P 1 1 P 1 2 P 1 0 P 6 P 4 P 9 P 3 P 8 P 5 P 7 P 1 4 P 1 3 P 1 6 P 1 P 1 8 P 1 5 P 1 7 P 1 9 0.474 0.476 0.478 0.480 0.482 0.484 0.486 0.488 0.490 0.492 Modal Frequency ( Hz) : Mode 7 ( VS 3) P 1 0 P 1 1 P 1 2 P 5 P 8 P 3 P 4 P 6 P 9 P 1 4 P 7 P 1 3 P 1 5 P 1 6 P 1 8 P 1 P 2 P 1 7 P 1 9 0.479 0.482 0.485 0.488 0.491 0.494 0.497 0.500 0.503 Modal Frequency ( Hz) : Mode 8 ( LAS 1) P 1 2 P 1 0 P 9 P 5 P 7 P 3 P 8 P 1 4 P 4 P 1 1 P 6 P 1 8 P 1 6 P 1 P 2 P 1 3 P 1 5 P 1 7 P 1 9 P11 P12 P10 P6 P4 P9 P3 P8 P5 P7 P14 P13 P16 P1 P2 P18 P15 P17 P19 P10 P11 P12 P5 P8 P3 P4 P6 P9 P14 P7 P13 P15 P16 P18 P1 P2 P17 P19 P12 P10 P9 P5 P7 P3 P8 P14 P4 P11 P6 P18 P16 P1 P2 P13 P15 P17 P19 Figure 3.6 Tornado diagram considering 19 parameters ( f) ( g) ( h) 52 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.050 0.002 0.475 0.236 0.021 0.029 0.006 0.096 0.009 0.075 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 1 ( LS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.091 0.055 0.601 0.002 0.027 0.000 0.009 0.197 0.009 0.009 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative variance Modal Frequency ( Hz) : Mode 2 ( VAS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.259 0.528 0.001 0.023 0.000 0.009 0.099 0.009 0.005 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 3 ( VS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 ( a) ( b) ( c) P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 53 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.064 0.000 0.467 0.003 0.020 0.000 0.006 0.406 0.007 0.025 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 4 ( VS 2) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.088 0.000 0.523 0.004 0.020 0.000 0.007 0.322 0.008 0.027 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 5 ( VAS 2) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.237 0.480 0.155 0.027 0.007 0.003 0.040 0.003 0.029 0.018 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 6 ( TS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 ( d) ( e) ( f) P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 54 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.098 0.372 0.423 0.004 0.017 0.000 0.006 0.059 0.007 0.014 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative variance Modal Frequency ( Hz) : Mode 7 ( VS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 0.000 0.000 0.000 0.000 0.000 0.003 0.000 0.774 0.001 0.099 0.076 0.004 0.013 0.001 0.020 0.002 0.006 0.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative variance Modal Frequency ( Hz) : Mode 8 ( LAS 1) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 0 P 1 1 P 1 2 P 1 3 P 1 4 P 1 5 P 1 6 P 1 7 P 1 8 P 1 9 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 ( h) ( g) Figure 3.7 Relative variance contribution ( neglecting correlation terms) from FOSM analysis 55 3.6 Finite Element Model Updating A detailed three dimensional finite element ( FE) model of Vincent Thomas Bridge was developed using the finite element analysis code ADINA 8.3. This finite element model is composed of 3D elastic truss elements to represent the main cables and suspenders, 2D shell elements to model the bridge deck and beam elements to model the stiffening trusses and tower shafts. The ADINA bridge model is shown in Figure 3.8. Figure 3.8 Three dimensional finite element model of Vincent Thomas Bridge For updating the original ADINA model an improved sensitivity based parameter updating method is employed ( Zhang et al., 2001). The method is based on the eigen value sensitivity to some selected structural parameters that are assumed to be bounded within some prescribed regions according to the degrees of uncertainty and variation existing in the parameters, together with engineering judgment. The changes of these parameters are found by solving a quadratic programming problem. 56 3.6.1 Sensitivity Based Model Updating The structural parameters affecting the natural frequencies are selected to construct the design parameter vector a P . The eigenvalue vector based on the designed parameters is denoted as a l , while the measured eigenvalue vector as m l . The error vector is defined as m a d l = l − l . The updating process minimizes the error vector by changing the design parameter vector a P . The variation of design parameter vector d p can be determined by d l = S d p ( 3.8) where S is the sensitivity matrix that represents the variation of natural frequencies of the model due to the variation of design parameter vector. The solution of Eq. ( 3.8) can be solved by the following iterative updating procedures. p p p k k = + d + 1 ( 3.9) l = l + d l + k a k a 1 ( 3.10) where k a k p , l are the parameter vector and eigenvalue vector of FE model, respectively, at the k th updating step. The iterative updating is repeated until the updated eigenvalue vector k a l converges to the measured eigenvalue vector m l . 57 The criteria of convergence are used as tolerance f f f m i m i k a i i £ − , , , max ( 3.11) tolerance k a i k a i k a i i £ − − − 1 , 1 , , max l l l ( 3.12) where k a i f , and k a, i l are the i th natural frequency and corresponding eigenvalue at k th update, and m i f , the measured i th natural frequency. The following optimization problem is applied to determine d p in Eq. ( 3.8) ( Friswell and Mottershead, 1994). J J ( S p ) W ( S p ) p W p p T e T d l d d l d d d 2 1 2 1 min 1 2 + = − − + F ( 3.13) subject to l u b £ d p £ b The first term in right hand side of Eq. ( 3.13) represents the objective function to minimize the error vector, while the second term to minimize the variation of design parameter vector. e W and p W are weighting functions. The constrained optimization solutions as outlined in Eq. ( 3.13) are incorporated into an iterative procedure as shown in Figure 3.9 for the model updating Vincent Thomas Bridge. 58 Input: u l b , b a k p p k = = 0, Yes Convergence Criterion Satisfy? Compute: u Constrained Optimization Determine d p 1 1 = + = + + k k p p p k k k d STOP No a p e p , W , W k k a k S p , FE model ® l ® k l k l k u k b b p b b p = − = − k Figure 3.9 Procedure for the sensitivity based model updating 59 3.6.2 Selection of Modes and Parameters 3.6.2.1 Selection of Modes Average values of the identified modal frequencies obtained from 14 different ambient vibration data recorded at the bridge site are considered as target frequencies for further ADINA model updating. Those 14 ambient vibration data were recorded from April, 2003 to October 2004. In the study, it is decided to select 8 modes to be matched between the updated FE analysis and the measured results. These include five vertical dominant; two lateral dominant; one torsional dominant modes of the deck. Table 3.7 shows the modal frequencies and percentage error in modal frequencies of Initial ( original) and Baseline ADINA model results with respect to identified frequencies obtained from the ambient vibration measurement data. Table 3.7 Comparison of natural frequencies Mode Measured frequency Updated FE model Type ( Hz) Initial Baseline Initial Baseline 1 L S 0.161 0.131 0.148  18.63  7.83 2 V AS1 0.221 0.206 0.210  6.79  5.02 3 V S1 0.233 0.226 0.227  3.00  2.66 4 V S2 0.374 0.363 0.371  2.94  0.86 5 V S3 0.474 0.460 0.470  2.95  0.78 6 L AS 0.476 0.411 0.462  13.66  2.90 7 T S 0.538 0.500 0.506  7.06  6.02 8 V AS2 0.568 0.568 0.583 0.00 2.66 Mode no. Finite element analyzed frequencies err.(%) 60 3.6.2.2 Selection of Parameters All possible parameters relating to the geometric, structural properties as well as the boundary conditions should be considered for adjustment in the updating procedure. However, if the parameters are found to have little or no effect on the targeted vibration modes, then they can be excluded from parameters list. After removing those parameters with very small sensitivities, total 17 different parameters are considered for this analysis. For this purpose, a sensitivity study is done and is explained in Section 3.5. They are summarized in Table 3.8 together with their initial estimates. Table 3.8 Parameters selected for adjustment Structure parameters Variations in % Stiffening truss Top chord Elastic modulus 29000 kip/ in 2 15 Mass density 8.71E 07 kip/ in 3 15 Bottom chord Elastic modulus 29000 kip/ in 2 20 Mass density 8.71E 07 kip/ in 3 15 Diagonal Mass density 8.71E 07 kip/ in 3 20 lateral brace ( k truss) Elastic modulus 29000 kip/ in 2 10 Mass density 1.35E 06 kip/ in 3 20 Stringers Elastic modulus 29000 kip/ in 2 20 Mass density 9.02E 07 kip/ in 3 10 Deck Slab Elastic modulus 2825 kip/ in 2 30 Mass density 2.01E 07 kip/ in 3 5 Cable Main cable Initial strain 1 20 Elastic modulus 29000 kip/ in 2 20 Mass density 7.71E 07 kip/ in 3 15 Suspender Mass density 7.65E 07 kip/ in 3 15 Tower Elastic modulus 29000 kip/ in 2 15 Mass density 7.62E 07 kip/ in 3 15 Initial estimation ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 61 3.6.3 Updated Results The allowable errors permitted for the check of natural frequency convergence was applied 6% for the general modes, while 3% for the first and second modes. If the ratio of variation for the eigenvalue is lower than 0.1%, then the iteration is also ended. For the cable supported bridge of which modes are closely spaced, the disorder between adjacent modes should be critically checked. The following MAC ( Modal Assurance Criteria) is applied to the each set of two updated natural modes ( Friswell and Mottershead, 1994). ( ) ( ) 0 1 1 , , 1 , , 2 , , = £ £ = = = MAC MAC p l a l j a l j p l e l i e l i p t l a l j e l i f f f f f f ( 3.14) If the two shape vectors a e f , f to be compared are identical, then MAC becomes 1, while if the two shape vectors are orthogonal, MAC becomes 0. Therefore, MAC can be utilized to prevent disorder between the calculated and measured frequency. MAC also provides the criteria for the reliability of the developed model after model updating. The MACs are listed in Table 3.9. The differences between the measured and the calculated frequencies for the initial and the final updated FE modes are showed in Figure 3.10. Table 3.10 shows the natural frequencies of the baseline model and updated model. For most of the modes, the discrepancies between measured frequencies and updated frequencies decreased less than 3%, while a few modes such as the first lateral frequency shows about 4% discrepancy. However, the discrepancy between measured and baseline model was about 19% and the current updating decreases the error in amount of 4%. 62 Table 3.9 MAC matrix of updated FE model # 1 2 3 4 5 6 7 8 1 0.542 0.015 0.007 0.000 0.000 0.000 0.000 0.000 2 0.001 0.531 0.000 0.000 0.000 0.000 0.000 0.000 3 0.003 0.009 0.490 0.000 0.000 0.000 0.000 0.000 4 0.000 0.000 0.000 0.500 0.000 0.000 0.000 0.000 5 0.000 0.000 0.000 0.000 0.500 0.000 0.000 0.000 6 0.000 0.000 0.000 0.000 0.000 0.538 0.000 0.000 7 0.090 0.005 0.001 0.003 0.006 0.094 0.541 0.000 8 0.001 0.003 0.008 0.113 0.002 0.000 0.000 0.486  20  18  15  13  10  8  5  3 0 3 5 L S1 V AS1 V S1 V S2 V S3 L AS1 T S1 V AS2 Frequency differences(%) Modes Initial FE model Baseline FE model Updated FE model Figure 3.10 Comparison of frequency differences using the initial and updated FE models 63 Table 3.10 Comparison of natural frequencies between baseline and updated FE model Frequency ( Hz) Frequency ( Hz) Err.(%) Frequency ( Hz) Err.(%) Frequency ( Hz) Err.(%) 1 L S 0.161 0.131  18.63 0.148  7.83 0.155  4.04 2 V AS1 0.221 0.206  6.79 0.210  5.02 0.215  2.90 3 V S1 0.233 0.226  3.00 0.227  2.66 0.233  0.09 4 V S2 0.374 0.363  2.94 0.371  0.86 0.373  0.19 5 V S3 0.474 0.460  2.95 0.470  0.78 0.478 0.80 6 L AS 0.476 0.411  13.66 0.462  2.90 0.487 2.25 7 T S 0.538 0.500  7.06 0.506  6.02 0.538  0.04 8 V AS2 0.568 0.568 0.00 0.583 2.66 0.587 3.31 Mode no. Finite element analyzed frequencies Mode Type Initial Identified Baseline Updated The variations of design parameters are also important to estimate reliability and effectiveness of updating results. The variations of design parameters are well limited in permitted arrange that can be regarded as reasonable as shown in Table 3.11. Table 3.11 Updated design parameters Structure parameters Initial estimation Updated value Percent changes Stiffening truss Top chord Elastic modulus( kip/ in 2 ) 29000 30815 6.3 Mass density( kip/ in 3 ) 8.71E 07 7.85E 07  9.8 Bottom chord Elastic modulus( kip/ in 2 ) 29000 33350 15.0 Mass density( kip/ in 3 ) 8.71E 07 7.58E 07  13.0 Diagonal Mass density( kip/ in 3 ) 8.71E 07 7.49E 07  14.0 lateral brace ( k truss) Elastic modulus( kip/ in 2 ) 29000 29442 1.5 Mass density( kip/ in 3 ) 1.35E 06 1.14E 06  15.0 Stringers Elastic modulus( kip/ in 2 ) 29000 24650  15.0 Mass density( kip/ in 3 ) 9.02E 07 8.16E 07  9.5 Deck Slab Elastic modulus( kip/ in 2 ) 2825 3390 20.0 Mass density( kip/ in 3 ) 2.01E 07 1.82E 07  9.2 Cable Main cable Initial strain 1.00 1.15 15.0 Elastic modulus( kip/ in 2 ) 29000 24650  15.0 Mass density( kip/ in 3 ) 7.71E 07 7.45E 07  3.3 Suspender Mass density( kip/ in3) 7.65E 07 8.41E 07 10.0 Tower Elastic modulus( kip/ in 2 ) 29000 27931  3.7 Mass density( kip/ in 3 ) 7.62454E 07 7.87E 07 3.3 64 3.7 Closure To demonstrate the appropriateness of the bridge models developed in the previous chapter, eigen properties of the models are evaluated in this chapter and compared with those of the system identification results obtained using frequency domain decomposition technique on ambient vibration and recorded earthquake response data. After that, a comprehensive sensitivity analysis is performed considering 19 different structural and soil spring parameters. First eight modal frequencies are considered for the sensitivity study. Tornado diagram and FOSM methods are applied for the sensitivity study. It is observed that the mass density of deck slab and elastic modulus of bottom chord contributes most to the modal frequencies of the bridge. This kind of study will be very helpful in selecting parameters and their variability ranges for FE model updating of suspension bridges. In this study, a sensitivity based automatic model updating procedure is presented, which solves an optimization problem for model error minimization. Four vertical vibration modes, two lateral modes, one torsional mode and 17 design parameters are selected for the problem. Updated results show that the model error could be reduced from 0~ 18% to 0~ 4% in terms of modal frequency ratio. During the optimization procedure, the target error bounds were 3% for the lower vertical modes and 6% for the horizontal modes. In order to prevent mode interchange due to the closely spaced frequencies of the three dimensional FE model, MACs are introduced to verify the updated results through the optimization procedure. 65 CHAPTER 4 SEISMIC ANALYSIS 4.1 Background The Vincent Thomas Bridge, connecting Terminal Island with San Pedro, serves both Los Angeles and Long Beach ports, two of the busiest ports in the west coast of USA. Thus, the bridge carries an overwhelming number of traffic with an Annual Average Daily Traffic ( AADT) volume of 100,000, many of which are cargo trucks. Based on the recent finding that the main span of the Vincent Thomas Bridge crosses directly over the Palos Verdes fault, which has the capacity to produce a devastating earthquake, in spring 2000, the bridge underwent a major retrofit using visco elastic dampers. This study focuses on seismic vulnerability of the retrofitted bridge. A member based detailed threedimensional Finite Element ( FE) as well as panel based simplified models of the bridge are developed. In order to show the appropriateness of these models, eigenproperties of the bridge models are evaluated and compared with the system identification results obtained using ambient vibration. In addition, model validation is also performed by simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino Hills earthquake and comparing with the measured response. Finally, considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed and the ductility demands of critical sections are presented in terms of fragility curves. The study shows that a ground motion with PGA of 0.9g or greater will result in plastic hinge formation at one or more locations with a probability of exceedance of 50%. 66 Also, it is found that the effect of damper is minimal for low to moderate earthquakes and high for strong earthquakes. The spatial variation of earthquake ground motions may have significant effect on the response of long span suspension bridges. Abdel Ghaffar and Rubin ( 1982) and Abdel Ghaffar and Nazmy ( 1988) studied response of suspension and cable stayed bridges under multiple support excitations. Zerva ( 1990) and Harichandran and Wang ( 1990) examined the effect of spatial variable ground motions on different types of bridge models. Harichandran et al. ( 1996) studied the response of long span bridges to spatially varying ground motion. Deodatis et al. ( 2000) and Kim and Feng ( 2003) investigated the effect of spatial variability of ground motions on fragility curves for bridges. Lou and Zerva ( 2005) analyzed the effects of spatially variable ground motions on the seismic response of a skewed, multi span, RC highway bridge. Most of the aforementioned studies dealt with simple FE models of the bridge, as a result response of critical members could not be evaluated. In the present analysis a panel based detailed 3D FE model of a long span suspension bridge is utilized. For design purpose of important structures in a site, U. S. Geological Survey ( USGS) provides a set of scenario earthquakes specified for a site. To consider spatial variability of ground motions one needs to know the ground excitations at different supports of a long span suspension bridge. For generating spatial variable ground motions from a scenario earthquake compatible to different design spectra for different supports ( as the local soil conditions will be different for different supports) a new algorithm is proposed using evolutionary power spectral density function ( PSDF) of the scenario earthquake specified for the site. Evolutionary PSDF of LA21 scenario earthquake is 67 estimated by using short time Fourier transform ( STFT) and wavelet transform ( WT) methods. Two evolutionary PSDFs thus developed maintain the same total energy possessed by the time history data. Using the evolutionary 20 sets of simulated ground motions for six different spatially correlated supports are generated. Ensemble average of 5% damped spectral acceleration response spectra obtained from simulated earthquake time histories are compared with the design response spectra for all the support locations. Good match has been found with the target design acceleration response spectra with the simulated one. Simulated spatially variable ground motions are used in calculating the response of the bridge. In addition to spatial variable seismic ground motions, two uniform ground motions are also considered for comparison purpose. The seismic responses of the bridge deck and the east tower are calculated using those three different cases and compared in both seismic displacement demand and seismic force demand. 4.2 Scope FE model validation of the bridge is also performed by simulating the dynamic response during the 1994 Northridge earthquake and 2008 Chino Hills earthquake and comparing with the measured response from installed acceleration sensors. Considering a set of strong ground motions in the Los Angeles area, nonlinear time history analyses are performed and ductility demands of critical tower section are presented in terms of seismic fragility curves. Effect of spatial variability of ground motions on seismic displacement demand and seismic force demand is investigated. To generate spatially 68 correlated spectrum compatible nonstationary acceleration time histories, a newly developed algorithm using evolutionary PSDF and spectral representation method is used. 4.3 Response Analysis under Northridge Earthquake To validate the developed numerical models ( discussed in Chapter 2), time history analysis is performed using the 1994 Northridge earthquake ( Mw = 6.7) ground motions recorded at the bridge sites. Newmark Beta method is used with g = 0.5 and b = 0.25 for this purpose. The ground motions and the bridge response during the Northridge earthquake are collected from the sensors installed at the bridge site ( California Strong Motion Instrumentation Program ( CSMIP) [ http:// www. strongmotioncenter. org/]). Since the earthquake occurred before the retrofit, detailed model before the retrofit is used here. To consider the effect of spatial variation, different ground motions are considered at different support locations, wherever possible. In some cases, due to the unavailability of recorded support motions, ground motions recorded at the nearest support is considered. Figure 4.1 shows the location of sensors and Table 4.1 illustrates the list of supports on which ground motions are applied for this analysis. Figure 4.2 shows comparison of measured and calculated longitudinal displacement at the top of the east tower location ( channel # 10) of the bridge. The plot shows good match between the calculated and field measured responses. 69 Figure 4.1 Location and direction of sensors  8  4 0 4 8 0 20 40 60 80 100 120 Time ( Sec) Displacement ( cm) Measured Calculated Figure 4.2 Comparison of measured and calculated longitudinal displacement at channel # 10 location Vertical Lateral Longitudinal East Anchorage West Tower 70 Table 4.1 Different support motions considered with channel numbers Location Longitudinal Lateral Vertical East Anchorage Ch. 25 Ch. 24 Ch. 26 East Cable Bent* Ch. 13 Ch. 9 Ch. 19 East Tower Ch. 13 Ch. 9 Ch. 19 West Anchorage* Ch. 23 Ch. 1 Ch. 14 West Cable Bent* Ch. 23 Ch. 1 Ch. 14 West Tower Ch. 23 Ch. 1 Ch. 14 * No recording at these locations 4.4 Response Analysis under Chino Hills Earthquake To study the developed numerical model, time history analysis is performed using the 2008 Chino Hills earthquake ( Mw = 5.4) ground motions recorded at the bridge sites. Newmark Beta method is used with g = 0.5 and b = 0.25 for this purpose with time step equal to 0.01 sec. The ground motions and the bridge response during the Chino Hills earthquake are collected from the sensors installed at the bridge site ( California Strong Motion Instrumentation Program ( CSMIP) [ http:// www. strongmotioncenter. org/]). Figure 1 shows the location of sensors already installed in the bridge. Since the earthquake occurred after the retrofit, detailed model after the retrofit is used here. Three directional components of ground motions recorded at east anchorage, east tower and west tower are applied uniformly over all the supports to study which set of ground motions will give much more accurate results. Figure 4.3 shows the comparison of analytical lateral response at channel 5 due to ground motions at east anchorage, east tower, west tower and considering spatial variation in ground motion with field measured response. It can be seen from figure 4.3 that the analytical response due the ground motion recorded at east tower matches well with the measured response. Figures 4.4, 4.5 and 4.6 show 71 comparison of analytical lateral, vertical and longitudinal responses at different channels due to ground motions at east tower with field measured response. These plots shows good match between the analytical and field measured responses.  0.8  0.6  0.4  0.2 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed_ East Anchorage Computed_ East Tower Computed_ West Tower Computed_ Spatial Figure 4.3 Comparison of analytical lateral response at channel 5 due to ground motions at east anchorage, east tower and west tower with field measured response 72  1.2  1  0.8  0.6  0.4  0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed based on east tower data Figure 4.4 Comparison of analytical lateral response at channel 3 due to ground motions at east tower with field measured response  1.5  1  0.5 0 0.5 1 1.5 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed based on east tower data Figure 4.5 Comparison of analytical vertical response at channel 17 due to ground motions at east tower with field measured response 73  0.8  0.6  0.4  0.2 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 Time ( sec) Displacement ( cm) Measured Computed based on east tower data Figure 4.6 Comparison of analytical longitudinal response at channel 10 due to ground motions at east tower with field measured response 4.5 Generation of Fragility Curves It is clear from the previous literature, especially those studies in the aftermath of 1995 Kobe ( Hyogo ken Nanbu) earthquake at Japan that the bridge deck and cables of suspension bridges are less vulnerable under strong earthquake ground motion ( remain elastic) while the tower is the most vulnerable part. In order to simplify analysis, in this study, only the towers are modeled as nonlinear elements. Remaining elements of the bridge are considered as linear. Each tower leg is constructed with members of 5 different cross sections. A total of 40 plastic hinges are introduced at all four tower legs. An elastoplastic behavior with 3% strain hardening is considered for the material models of these plastic hinges. 2% Raleigh damping is used for the first and tenth modes. Forty ground motions representing 2% in 50 years and 10% in 50 years of hazard level as specified by 74 FEMA/ SAC are used for evaluating seismic vulnerability of the retrofitted bridge. The motions cover wide range peak characteristics with Peak Ground Acceleration ( PGA) ranging from 0.42 to 1.30g. Note also that these motions include expected motions from Palos Verdes fault, the fault crossing the main span of this bridge. For nonlinear time history analysis, direct time integration is used in the framework of SAP 2000. Motions are applied in the lateral direction of the bridge and no spatial variation is considered. After performing the nonlinear time history analysis, the ductility demands of all the critical sections are evaluated and the maximum ductility demand is noted for each motion. Considering all these motions, the maximum ductility demand is found to be 6.23, which is from LA 36 motion ( with a PGA of 1.1g) and for the plastic hinge at the base of the tower. In this study, fragility curves corresponding to different damage states are developed following Shinozuka et al., 2000. For a given damage state, the fragility curves are expressed in terms of lognormal distribution. PGA is considered as Ground motion intensity. Two fragility parameters, median ( c) and log standard deviation ( z ) are estimated through a maximum likelihood method such that fragility curves at different damage levels do not intersect each other. Therefore, a common z is needed to satisfy this criterion. Although this method can be used for any number of damage states, for the ease of demonstration of analytical procedure it is assumed here that there are three states of bridge damage. Therefore, a family of three fragility curves exists in this case for damage states of ‘ Level I’, ‘ Level II’, and ‘ Level III’ identified by k = 1, 2, and 3. Under this lognormal assumption, the analytical form of the fragility function F(•) for the state of damage k is, 75 ( ) ( ) = F z z i k i k a c F a c ln / , , ( 4.1) where ck is median of the fragility function associated with damage state k, z is the common log standard deviation, ai is the PGA value to which the bridge is subjected and F [•] is the standardized normal distribution function. The fragility parameters are computed by maximizing the likelihood function, L which is given by Eq. ( 4.2), where xik is 1 or 0, depending on whether or not the bridge sustains damage state k under ai, and n is the total number of ground motions under which the analysis is performed. Pik is the probability that the example bridge will suffer from a damage state k when subjected to ai and is expressed as 1 ( , , z ) 0 1 P F a c i i = − ( , , z ) ( , , z ) 1 1 2 P F a c F a c i i i = − ( , , z ) ( , , z ) 2 2 3 P F a c F a c i i i = − ( , , z ) 3 3 P F a c i i = ( ) [ ] Õ Õ = = = 3 1 1 1 2 3 , , , k n i x ik ik L c c c z P ( 4.2) ( 4.3) ( 4.4) ( 4.5) ( 4.6) 76 Fragility parameters are obtained by solving the Eq. ( 4.7), by implementing a straightforward optimization algorithm. ( ) ( ) 0 ln , , , ln , , , 1 2 3 1 2 3 = ¶ ¶ = ¶ ¶ z z L c c c z c L c c c k for k = 1,2,3 For the fragility curves, this study proposes performance levels in terms of ductility demands of critical tower sections, since the damage states related to expected performance level of suspension bridge is not clearly defined in the literature. Three different damage states are considered in this study in terms of the maximum ductility demands of all the critical tower sections. They are ( 1) Level I ( plastic hinge formation, ductility > 1) ( 2) Level II ( ductility ³ 2) and ( 3) Level III ( ductility ³ 4). Figure 4.5 shows the fragility curves considering these damage states and for before and after retrofitting of the bridge. One can observe from this figure that for a PGA of 0.9g, the probability of exceedance corresponding to damage Level I ( i. e., plastic hinge formation at one or more locations) is 50%. Similarly, for the same probability of exceedance, a ground motion with PGA of 1.05g or greater will cause a damage of Level II. PGA of 1.82g was recorded at the Tarzana Station during the main shock of the 1994 Northridge earthquake. For that PGA the probability of exceedance to damage Level II is 90%. The bridge was retrofitted with total 48 dampers and from the fragility curves it is clear that the effect of dampers are minimal for low to moderate earthquake and high for strong earthquake. ( 4.7) 77 Table 4.2 Details of the motions considered in this study for fragility development SAC Earthquake Distance Scale dt Duration PGA PGV PGD Name Magnitude ( km) Factor ( sec) ( sec) ( g) ( cm/ sec) ( cm) LA21 1995 Kobe 6.9 3.4 1.15 0.02 59.98 1.28 142.70 37.81 LA22 1995 Kobe 6.9 3.4 1.15 0.02 59.98 0.92 123.16 34.22 LA23 1989 Loma Prieta 7 3.5 0.82 0.01 24.99 0.42 73.75 23.07 LA24 1989 Loma Prieta 7 3.5 0.82 0.01 24.99 0.47 136.88 58.85 LA25 1994 Northridge 6.7 7.5 1.29 0.005 14.945 0.87 160.42 29.31 LA26 1994 Northridge 6.7 7.5 1.29 0.005 14.945 0.94 163.72 42.93 LA27 1994 Northridge 6.7 6.4 1.61 0.02 59.98 0.93 130.46 28.27 LA28 1994 Northridge 6.7 6.4 1.61 0.02 59.98 1.33 193.52 43.72 LA29 1974 Tabas 7.4 1.2 1.08 0.02 49.98 0.81 71.20 34.58 LA30 1974 Tabas 7.4 1.2 1.08 0.02 49.98 0.99 138.68 93.43 LA31 Elysian Park ( simulated) 7.1 17.5 1.43 0.01 29.99 1.30 119.97 36.17 LA32 Elysian Park ( simulated) 7.1 17.5 1.43 0.01 29.99 1.19 141.12 45.80 LA33 Elysian Park ( simulated) 7.1 10.7 0.97 0.01 29.99 0.78 111.03 50.61 LA34 Elysian Park ( simulated) 7.1 10.7 0.97 0.01 29.99 0.68 108.44 50.12 LA35 Elysian Park ( simulated) 7.1 11.2 1.1 0.01 29.99 0.99 222.78 89.88 LA36 Elysian Park ( simulated) 7.1 11.2 1.1 0.01 29.99 1.10 245.41 82.94 LA37 Palos Verdes ( simulated) 7.1 1.5 0.9 0.02 59.98 0.71 177.47 77.38 LA38 Palos Verdes ( simulated) 7.1 1.5 0.9 0.02 59.98 0.78 194.07 92.56 LA39 Palos Verdes ( simulated) 7.1 1.5 0.88 0.02 59.98 0.50 85.50 22.64 LA40 Palos Verdes ( simulated) 7.1 1.5 0.88 0.02 59.98 0.63 169.30 67.84 Record Figure 4.7 Before and after retrofit Fragility curves for different damage levels Probability of Exceeding a Damage State PGA ( g) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Level I_ Before Level II_ Before Level III_ Before Level I_ After Level II_ After Level III_ After 78 4.6 Simulation of Ground Motion Considering Spatial Variability 4.6.1 Generation of Evolutionary PSDF from Given Ground Motion using STFT This section briefly reviews the work done by Liang et al. ( 2007). The STFT F ( t, w ) of a function f ( t ) is expressed by the convolution integral in the following form: ( ) ( ) ( ) ¥ − ¥ − w = t − t t w t F t f h t e d i , ( 4.8) where h ( t ) is an appropriate time window. The evolutionary PSDF S ( t w ) f f , 0 0 can be written as ( ) ( ) ( ) ( ) ( ) ¥ − ¥ − − ¥ − ¥ = − − 1 2 1 2 1 2 2 , w t t t t 1 2 t t w t w t F t f f h t h t e e d d i i ( 4.9) The total energy of f ( t ) can be estimated as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ − − ¥ − ¥ ¥ − ¥ ¥ − ¥ = − = − − f h t d dt f f h t h t e d d dtd F t dtd i t t t t t t t t t w w w w t t 2 2 1 2 1 2 1 2 2 1 2 , ( 4.10) For the derivation of Eq. ( 4.10), the following equation is used: ( ) ( ) ¥ − ¥ − − = − 1 2 1 2 w d t t w t t e d i ( 4.11) If h ( t ) = d ( t ) 2 , the total energy in Eq. ( xx) is ( ) ( ) ¥ − ¥ ¥ − ¥ ¥ − ¥ F t dtd = f t dt 2 2 , w w ( 4.12) 79 This implies that the time window should be chosen such that it satisfies the following condition ( ) ¥ − ¥ = 1 2 h t dt ( 4.13) The total energy can be kept identical ( Perseval’s identity) in estimating evolutionary PSDF. Here a Gaussian time window squared with standard deviation s = 0.25 s, is used. It satisfies the condition in Eq. ( 4.13). The time window function has the following form, ( ) ( 0.25 ) 2 1 2 2 2 2 = = − s s p t s h t e ( 4.14) Figure 4.8 shows the evolutionary PSDF of LA21 scenario earthquake record estimated using STFT ( Gaussian window). Figure 4.8 Evolutionary PSDF of LA21 earthquake record using STFT method 80 4.6.2 Generation of Evolutionary PSDF from Given Ground Motion using Wavelet Transform This section briefly reviews the work done by Liang et al. ( 2007). The wavelet transform ( WT) of a function f ( R ) 2 Î L ( finite energy function f ( t ) dt < + ¥ 2 ) at time u and scale s , and the corresponding inverse relationship are given by Daubechies ( 1992) ( ) ( ) dt u s R s t u f t s W f u s Î − = ¥ − ¥ * , , 1 , y y ( 4.15) and ( ) ( ) ( ) ¥ − ¥ ¥ − ¥ Î − = duds u s R s t u W f u s f t C s f t , , , 1 2 1 2 y p y y ( 4.16) where ( ) = < ¥ ¥ − ¥ w w y w y C d 2 ˆ ( 4.17) In Equations ( 4.15) – ( 4.17), the wavelet function ( R ) 2 y Î L known as ‘ mother” wavelet with average value equal to zero, ( ) ¥ − ¥ y t dt = 0 ( 4.18) and is centered in the neighborhood of t = 0, and as normalized y = 1. y ˆ ( w ) denotes the Fourier transform of y ( t ) and is given by ( ) ( ) ¥ − ¥ − = t e dt i w t y p y w 2 1 ˆ ( 4.19) It may be noted that the WT decomposes signal f ( t ) over dilated and translated wavelets. As W f ( u, s ) y is convolution of f ( t ) with ( 1 s ) ( t s ) , W f ( u, s ) * y y − 81 represents the contribution of the function f ( t ) in the neighborhood of t = u and in the frequency band corresponding to scale s . It can be shown that ( Daubechies, 1992) ( ) W f ( u s ) duds C s f t dt 2 2 2 , 1 2 1 y y p ¥ − ¥ ¥ − ¥ ¥ − ¥ = ( 4.20) Now, if any wavelet function satisfies the condition ( ) ¥ − ¥ ˆ = 1 2 , y w d w u s ( 4.21) Then Equation ( 4.20) can be written as ( ) ( ) ( ) ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ ¥ − ¥ × = y w w p y y W f u s duds d C s f t dt u s 2 , 2 2 2 , ˆ 1 2 1 ( 4.22) In Equations ( 4.21) and ( 4.22), y ( w ) u, s ˆ represents the Fourier transform of − s t u y and can be expressed as ( ) ( ) i u u s s s e w y ˆ w y ˆ w , = . Then, using Perseval’s identity, one can write ( ) W f ( u s ) ( ) duds C s F u s 2 , 2 2 2 , ˆ 1 2 1 y w p w y y ¥ − ¥ ¥ − ¥ = ( 4.23) where F ( w ) = Fourier transform of f ( t ) . As the wavelet coefficient W f ( u, s ) y provides the localized information of signal f ( t ) at t = u , from Equation ( 4.23) the Evolutionary PSDF ( , w ) 0 0 S t f f can be expressed as ( ) ( ) ( ) ¥ − ¥ = W f t s ds C s F t t s 2 , 2 2 2 , ˆ 1 2 1 , y w p w y y ( 4.24) It may be noted that the expression of evolutionary PSDF given in Equation ( 4.24) obeys total energy equilibrium. Therefore, any wavelet basis can be used which satisfies Equation ( 4.21), for generation of evolutionary PSDF [ e. g., modified Littlewood Paley basis proposed by Basu and Gupta ( 1998)] that maintains total energy. Figure 4.9 shows 82 the evolutionary PSDF of LA21 scenario earthquake record estimated using STFT ( Gaussian window). Figure 4.9 Evolutionary PSDF of LA21 earthquake record using wavelet transform 4.6.3 Simulation of One Dimensional Multi Variate ( 1D mV), Nonstationary Gaussian Stochastic Process To generate sample functions of stochastic processes, the spectral representation method developed by Shinozuka and Jan ( 1972) appears to be most versatile and widely used today. Spectral representation based algorithm to simulate one dimensional multi variate nonstationary Gaussian stochastic process developed by Deodatis ( 1996b) is used in this study and described as follows. 83 Consider a one dimensional, n variate ( 1D nV) non stationary stochastic vector process with components ( ) , ( ) ,........., ( ) , 0 0 2 0 1 f t f t f t n 



B 

C 

I 

S 


