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Evaluation of the effects of nonlinear soil-structure interaction on the inelastic seismic response of… Ghalibafian, Houman 2006

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E V A L U A T I O N OF THE EFFECTS OF NONLINEAR SOIL-STRUCTURE INTERACTION ON THE INELASTIC SEISMIC RESPONSE OF PILE-SUPPORTED BRIDGE PIERS by H O U M A N GHALIBAFIAN B.Sc, University of Tehran, 1993 M.A.Sc., The University of British Columbia, 2001 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Civil Engineering) T H E UNIVERSITY OF BRITISH COLUMBIA November 2006 © Houman Ghalibafian, 2006 A B S T R A C T This dissertation presents an evaluation of the effects of nonlinear soil-structure interaction (SSI) on the inelastic seismic response of pile-supported bridge piers on soft soil. The research was carried out by studying the dynamic responses of prototype soil-foundation-bridge pier systems subjected to earthquake ground motions. The responses were obtained by performing nonlinear dynamic analyses using a commercial finite difference program. The nonlinearities of the soil, the structure, and the soil-structure interface were all accounted for. The numerical analysis method was carefully validated by verifying the modeling of each component of the system and by verifying the modeling of the system as a whole through analyzing an instrumented bridge pier subjected to an actual earthquake. The dynamic responses of the prototype bridge piers were computed with and without consideration of SSI (i.e. flexible-base versus fixed-base piers), and with and without consideration of the inelastic behaviour of the piers. This work explores the efficient implementation and practical application of the direct methods of SSI analysis with a system approach. It presents the seismic demands of the prototype piers and foundations, and provides a quantified picture of the effects of SSI on the ductility and the total displacement demands of the piers as functions of their natural period. This study investigates the effects of the modeling assumption of the structural elements (i.e. elastic versus inelastic behaviour) on the estimated demands, and demonstrates that SSI analyses with elastic structures cannot always provide plausible predictions of the inelastic responses. The effects of SSI on the seismic demands of the bridge piers are also studied probabilistically in order to consider the uncertainties in the system parameters and to account for the dispersions introduced by the variability of input ground motions and soil conditions. Subsequently, SSI modification factors are proposed to estimate the demands of the flexible-base piers from their corresponding fixed-base demands. The proposed method is probabilistic and quantifies the uncertainties involved in computing the modification factors. Finally, this work demonstrates the shortcomings of the nonlinear static pushover analysis for seismic demand estimation of pile-supported bridge piers when SSI is significant. u TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES vii LIST OF FIGURES viii ACKNOWLEDGMENTS xiv DEDICATION xv 1 INTRODUCTION 1 1.1 Background... 1 1.2 Motivations for this Research ...8 1.3 Objectives of the Dissertation 11 1.4 Scope of Work 11 1.5 Organization of the Dissertation 12 2 DEVELOPMENT OF THE NUMERICAL MODELS 14 2.1 Prototype Soil-Foundation-Structure Systems 14 2.2 Modelling of Soil 16 2.2.1 Soil Profile and Properties 16 2.2.1.1 Stiffness properties of Soil 16 2.2.1.2 Strength Properties of Soil 17 2.2.1.3 Nonlinear Behaviour of Soil Elements in F L A C 18 2.2.2 Boundary Conditions 21 2.3 Modeling of Bridge Piers 21 2.3.1 Beam Element Properties 21 2.3.2 Damping 22 2.3.3 Nonlinear Behaviour of the Bridge Piers 23 2.3.3.1 Material Nonlinearity 24 2.3.3.2 Geometric Nonlinearity 26 2.4 Modelling of Foundation 27 iii 2.5 Modeling of Soil-Foundation-Structure System 27 2.5.1 Modeling of Soil-Foundation Interface 29 2.5.1.1 Pile Cap-Soil Interface 29 2.5.1.2 Pile-Soil Interface 30 2.5.2 Modeling of Structural Elements in Plane Strain Analysis 36 2.6 Sequence of Analysis 36 2.7 System Physical Properties 37 2.7.1 Stiffness of Foundation 37 2.7.2 Stiffness of Pile-Supported Piers 39 2.7.3 Natural Period of the Pile-Supported Piers 43 2.8 Summary .' .44 3 ANALYSIS VERIFICATION STUDY: DYNAMIC ANALYSIS OF AN INSTRUMENTED PILE-SUPPORTED BRIDGE PIER 45 3.1 Structure and Instrumentation of the Bridge 46 3.2 Modeling and Verification Procedure 50 3.3 Site Response Analysis 50 3.3.1 Soil Profile and Properties 50 3.3.2 Nonlinear Behaviour of Soil Elements in F L A C 53 3.3.3 Input Rock Motion : 55 3.3.4 Nonlinear Dynamic Analysis 56 3.3.5 Comparison of the Recorded and the Computed Surface Motions 57 3.3.6 Comparison of F L A C and SHAKE 61 3.3.7 Estimation of the Input Motion from the Surface Motion 63 3.4 Analysis of the Fixed-Base Pier 65 3.5 Analysis of Soil-Foundation-Structure Interaction 65 3.6 Response Analysis 67 3.7 Summary 68 4 NONLINEAR DYNAMIC ANALYSIS OF THE PROTOTYPE BRIDGE PIERS 69 4.1 Selection of input Rock motions 69 i v 4.2 Site Response Analysis 73 4.3 Analysis of Fixed-Base Piers 76 4.4 Analysis of Soil-Foundation-Structure Systems 77 4.5 Seismic Behaviour of the System 82 4.6 Summary 90 5 SEISMIC DEMANDS OF BRIDGE PIERS 91 5.1 Definition of Demand Parameters 91 5.2 Seismic Demands of the Fixed-Base Piers 95 5.2.1 Strength Reduction Factors 96 5.2.2 Ductility Demands without P-A Effects 97 5.2.3 Ductility Demands with P-A Effects 98 5.2.4 Comparison of Ductility and Strength Reduction Factors 100 5.3 Seismic Demands of Pile-Supported Piers 102 5.3.1 Strength Reduction Factors 103 5.3.2 Ductility Demands 104 5.3.3 Comparison of Ductility and Strength Reduction Factors 105 5.3.4 Effects of Piers Modeling Assumption on the Ductility Demands 106 5.3.5 Total Displacement Demands 110 5.3.6 Effects of Piers Modeling Assumption on the Total Displacements 113 5.4 Summary 114 6 SEISMIC DEMANDS OF FOUNDATIONS 116 6.1 Translation of Foundation 116 6.2 Rotation of Foundation 118 6.3 Comparison of the Translation and the Rotation of Foundations 119 6.4 Demand of Piles 122 6.5 Effects of Piers Modeling Assumption on the Demands of Foundations 122 6.6 Summary 125 v 7 EFFECTS OF SOIL-STRUCTURE INTERACTION ON THE SEISMIC DEMANDS OF PIERS 126 7.1 Effects of Soil-Structure Interaction on Ductility Demands 127 7.2 Effects of Soil-Structure Interaction on Total Displacement Demands 129 7.3 Comparison of the Effects of Soil-Structure Interaction on Ductility and Total Displacement Demands 132 7.4 Summary 134 8 PROBABILISTIC ASSESSMENT OF THE EFFECTS OF SOIL-STRUCTURE INTERACTION 136 8.1 Methodology 138 8.2 Response Surfaces 140 8.2.1 Regression Models 141 8.2.2 Artificial Neural Networks 144 8.3 Probabilistic Data Processing 145 8.4 Performance-Based Assessment of the Effects of Soil-Structure Interaction on the Seismic Demands of the Piers 150 8.5 Summary.. 160 9 EVALUATION OF NONLINEAR STATIC PUSHOVER ANALYSIS FOR DEMAND ESTIMATION INVOLVING SOIL-STRUCTURE INTERACTION. 161 9.1 Base Shear Demands 162 9.2 Global Ductility Demands 167 9.3 Summary 170 10 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 172 REFERENCES 180 APPENDIX A 186 vi LIST OF TABLES Table 2.1: Summary of the yield properties of the piers Table 2.2: The Fixed-base and the flexible-base stiffnesses of the prototype bridge piers Table 2.3: Natural periods of the pile-supported piers and the period elongations... Table 3.1: Site's nearby rock outcrop motions Table 3.2: Components of the recorded ground motions Table 4.1: Input rock motions Table 6.1: Mean ratio of foundation demands estimated from analyses with elastic piers to those estimated with inelastic piers 124 Table 8.1: Summary of the statistics of results 141 vn LIST OF FIGURES Figure 2.1: Schematic description of the prototype soil-foundation-structure system 15 Figure 2.2: Distributions of Gma Xand Su to the depth of 100 m of the prototype soil layer 18 Figure 2.3: Shear modulus reduction curve of clay 19 Figure 2.4: Equivalent viscous damping exhibited by a single soil element of F L A C .... 20 Figure 2.5: An example of the stress-strain relationship of a soil element at the depth of. 40 m when the soil layer was subjected to an earthquake ground motion 20 Figure 2.6: Comparison of the elastic response of the pier obtained from F L A C with non-viscous damping and SAP2000 with 5% viscous damping 23 Figure 2.7: Moment-curvature curve obtained from the cross section analysis of the piers 25 Figure 2.8: A sample load-displacement curve of a pier when P-A effects included 26 Figure 2.9: Moment-curvature curve obtained from the cross section analysis of the piles 27 Figure 2.10: Geometry of the F L A C model (top 30 m of soil) 28 Figure 2.11: Interface elements in F L A C (Itasca 2005) 29 Figure 2.12: Pile-soil coupling springs in F L A C 30 Figure 2.13: Shear coupling springs yield criterion 31 Figure 2.14: Normal coupling springs yield criterion 32 Figure 2.15: The displacement of the pile with respect to the soil and the vectors of the soil incremental displacements in the horizontal plane 34 Figure 2.16: An example load-displacement curve of the pile at a depth of 9.2 m 35 Figure 2.17: Validation of the normal coupling springs at a depth of 9.2 m 35 Figure 2.18: Translational stiffness of foundation 38 Figure 2.19: Rotational stiffness of foundation 38 Figure 2.20: Stiffness of the pile-supported pier with fixed-base period of T=0.3 s 40 Figure 2.21: Stiffness of the pile-supported pier with fixed-base period of T=0.6 s 40 Figure 2.22: Stiffness of the pile-supported pier with fixed-base period of T=0.8 s 41 Figure 2.23: Stiffness of the pile-supported pier with fixed-base period of T=1.0 s 41 vin Figure 2.24: Stiffness of the pile-supported pier with fixed-base period of T—1.5 s 42 Figure 2.25: Stiffness of the pile-supported pier with fixed-base period of T=2.0 s 42 Figure 3.1: Photo of the Hayward BART Elevated Section 46 Figure 3.2: Geometry and instrumentation of the bridge (After Shakal et al. 1989) 48 Figure 3.3:. Foundation plan and pile pattern (per as-built drawings) 49 Figure 3.4: Cross section of the foundation (per as-built drawings) 49 Figure 3.5: Distribution of G m a x 52 Figure 3.6: Soil shear modulus reduction curves 53 Figure 3.7: Shear modulus reduction exhibited by a single element in F L A C 54 Figure 3.8: Equivalent viscous damping exhibited by a single element in F L A C 54 Figure 3.9: Relative location and coordinates of the epicentre, the outcrops recording stations, and the bridge site free field recording station 56 Figure 3.10: Comparison of the recorded and the computed response spectra 59 Figure 3.11: Comparison of transfer functions between input and recorded surface motions and between input and computed surface motions 59 Figure 3.12: Comparison of the computed and the recorded surface response time histories 61 Figure 3.13: Comparison of spectral surface responses obtained from S H A K E and F L A C 63 Figure 3.14: Surface motion computed by F L A C in response to the input motion generated by S H A K E 65 Figure 3.15: F L A C model (top 30 m of soil) of the instrumented bridge pier system 66 Figure 3.16: Recorded and computed acceleration response spectra of the pier ; 68 Figure 4.1: Acceleration response spectra of the input rock motions 71 Figure 4.2: Site acceleration amplification obtained from analysis 73 Figure 4.3: Site acceleration amplification from Idriss (1990) 74 Figure 4.4: Acceleration response spectra of the computed surface motions 75 Figure 4.5: Comparison of the acceleration response spectra of the input and the surface motions 75 Figure 4.6: Sample displacement response at the centre of mass of a fixed-based pier with and without P-A effects 77 ix Figure 4.7: An example of the response of a bridge pier and its foundation in time domain (T=0.8 s, TSjte=1.0 s, input motion: Nahanni Site 2) 78 Figure 4.8: Comparison of the surface spectral response of the 40 m soil layer obtained from the analysis of a 40 m soil column and a 30 m soil column 80 Figure 4.9: Comparison of the surface spectral response of the 100 m soil layer obtained from the analysis of a 100 m soil column and a 30 m soil column 81 Figure 4.10: Example of the response spectra of the soil-foundation-pier system (T=0.3 s (f=3.33 Hz), Ts l te=1.0 s, input motion: Nahanni Site 2) 84 Figure 4.11: Example of the response spectra of the soil-foundation-pier system (T=0.6 s (f=1.67 Hz), Ts i te=1.0 s, input motion: Nahanni Site 2) 85 Figure 4.12: Example of the response spectra of the soil-foundation-pier system (T=0.8 s (f=1.25 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 86 Figure 4.13: Example of the response spectra of the soil-foundation-pier system (T=1.0 s (f=1.0 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 87 Figure 4.14: Example of the response spectra of the soil-foundation-pier system (T=1.5 s (f=0.67 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 88 Figure 4.15: Example of the response spectra of the soil-foundation-pier system (T=2.0 s (f=0.50 Hz), Ts i te=1.0 s, input motion: Nahanni Site 2) 89 Figure 4.16: Translation response of the foundation supporting a massless pier (kinematic interaction) 90 Figure 5.1: Force-deformation relationship of an elastoplastic single-degree-of-freedom system 94 Figure 5.2: Demand parameters of the idealized fixed-base bridge piers 94 Figure 5.3: Demand parameters of the idealized flexible-base bridge piers 95 Figure 5.4: Distribution of the strength reduction factor R 97 Figure 5.5: Distribution of the ductility demands without P-A effects 98 Figure 5.6: Distribution of the ductility demands with P-A effect 99 Figure 5.7: Comparison of the ductility demands with P-A effect to those without 100 Figure 5.8: Distribution of the inelastic deformation ratios without P-A effect 101 Figure 5.9: Distribution of the inelastic deformation ratios with P-A effect 101 Figure 5.10: R-p relationship of the fixed-base piers without P-A effects 102 x Figure 5.11: Distribution of the strength reduction factors 103 Figure 5.12: Distribution of the ductility demands of the pile-supported bridge piers... 104 Figure 5.13: Distribution of the inelastic deformation ratios of the pile-supported structure 105 Figure 5.14: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 0.6 s 108 Figure 5.15: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 0.8 s 108 Figure 5.16: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 1.0 s 109 Figure 5.17: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 1.5 s 109 Figure 5.18: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 2.0 s 110 Figure 5.19: Distribution of the total displacement of the piers normalized with respect to their height I l l Figure 5.20: Distribution of the local drift of the piers 112 Figure 5.21: Distribution of the piers portion of the total displacements 112 Figure 5.22: Distribution of the ratio of total displacements (SSI with elastic piers to SSI with inelastic piers) 113 Figure 6.1: Distribution of peak translation of foundation 117 Figure 6.2: Contribution of the foundation translation to the total displacements of the piers 117 Figure 6.3: Distribution of peak rotation of foundation 118 Figure 6.4: Contribution of the foundation rotation to the total displacements of the piers 119 Figure 6.5: Comparison of the contribution of the foundation translation and rotation to the total displacements of the piers 121 Figure 6.6: Ratio of foundation translation demands estimated from analyses with elastic piers to those estimated with inelastic piers 123 xi Figure 6.7: Ratio of foundation rotation demands estimated from analyses with elastic piers to those estimated with inelastic piers 124 Figure 7.1: Distributions of pier ductility demand ratios (DDR) 128 Figure 7.2: Mean values of pier ductility demand ratios (DDR) 129 Figure 7.3: Distributions of pier total displacement demand ratios (TDR) 131 Figure 7.4: Mean values of pier total displacement demand ratios (TDR) 131 Figure 7.5: Mean response ratios (DDR and TDR) 132 Figure 7.6: Mean demand ratios (DDR and TDR) as functions of the period ratio T s y s / T 134 Figure 8.1: Nonlinear regression fit to mean response ratios (DDR and TDR) as functions of the piers fixed-base period T 142 Figure 8.2: Nonlinear regression fit to standard deviations of response ratios (DDR and TDR) as functions of the piers fixed-base period T 142 Figure 8.3: Linear regression fit to mean response ratios (DDR and TDR) as functions of the period ratio T s y s / T 143 Figure 8.4: Linear regression fit to standard deviations of response ratios (DDR and TDR) as functions of the period ratio T s y s / T . . . . 143 Figure 8.5: Probability of DDR>1 as a function of the piers fixed-base period T 147 Figure 8.6: Probability of TDR>1 as a function of the piers fixed-base period T 148 Figure 8.7: Probability of DDR>1 as a function of the period ratio T s y s / T 149 Figure 8.8: Probability of TDR>1 as a function of the period ratio T s y s / T 149 Figure 8.9: Probability of DDR>r as a function T s y s / T for r values from 0.8 to 1.5 151 Figure 8.10: Probability of TDR>r as a function of T s y s / T for r values from 1.0 to 1.5. 151 Figure 8.11: Probability of DDR>r as a function of r for T s y s / T of 1.02 to 1.3 152 Figure 8.12: Probability of TDR>r as a function of r for T s y s / T of 1.02 to 1.3 153 Figure 8.13: A bridge example with simply supported spans and pile-supported piers on soft soil 154 Figure 8.14: Correlation between DDR and site surface spectral acceleration at the natural period of the fixed-base pier 157 Figure 8.15: Correlation between DDR and site surface spectral velocity at the natural period of the fixed-base pier 158 xii Figure 8.16: Correlation between DDR and site surface spectral displacement at the natural period of the fixed-base pier 159 Figure 9.1: Numerical model for pushover analysis 163 Figure 9.2: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.3 s 164 Figure 9.3: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.6 s 164 Figure 9.4: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.8 s 165 Figure 9.5: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 1.0 s 165 Figure 9.6: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 1.5 s 166 Figure 9.7: Comparison of force-displacement relationships with SSI obtained from pushover and nonlinear dynamic analyses of the pier with T = 2.0 s 166 Figure 9.8: Mean ratios of the global ductility demands to the local ductility demands 168 Figure 9.9: Comparison of the pier deformations with total displacements obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.3 s 169 Figure 9.10: Comparison of the pier deformations with total displacements obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.6 s 170 xiii A C K N O W L E D G M E N T S I would like to gratefully thank Professor Carlos Ventura for his guidance throughout the course of my research and for trusting me with my vision and research interests. He provided me with the environment and the research tools I needed for the success of this work. My great appreciation extends to my advisory committee members. The expertise of Professor Peter Byrne was instrumental in my understanding of the geotechnical aspects of the work. Professor Ricardo Foschi shared his inspiring ideas and expertise on the application of the probabilistic methods which greatly contributed to the success of this work, and Professor Robert Sexsmith provided valuable inputs. I am grateful for the involvement of Professor Liam Finn upon his return to UBC and I sincerely appreciate his invaluable advice and support. I would also like to thank Dr. Steve Zhu of Buckland & Taylor Ltd. and Dr. John Cassidy of the Geological Survey of Canada for their inputs. Mr. Tom Horton of San Francisco Bay Area Rapid Transit District (BART) is gratefully acknowledged for providing the soil and the structure data of the Hayward BART elevated section used as a case study in this research. My financial support came from different sources including the Postgraduate Scholarship of the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant Supplement Award of the Faculty of Applied Science, and the University Graduate Fellowship (UGF) of the University of British Columbia. I am indebt to Professors Sexsmith and Foschi for trusting me with the success of this work and for providing me with crucial financial support from their NSERC grants during the course of this research. The additional fund from Professor Ventura supported me in the final stages of this research which is greatly appreciated. My deep gratitude goes to my parents and my sisters for the support and the peace of mind that they provided me. The pressure over the course of this work would be unbearable without the joyful times that I spent with my dear friends and I would like to extend my appreciation to them for all their support. xiv To My Beloved Parents... X V 1 INTRODUCTION 1.1 Background With the evolution of performance based earthquake engineering methodologies, there is an ever increasing need for more accurate and quantifiable assessment of the complex behaviour of structures in response to earthquake loadings. While the objective of seismic design codes has been to provide life safety, the recent methodologies place emphasize on various performance objectives and require the estimation of seismic demands for various levels of expected earthquake motions. Seismic demand estimations must account for the response of systems in the inelastic range, so that measures of damage to the systems could be obtained. Another aspect of performance-based design methodologies is the explicit consideration of the uncertainties in both capacity and demand estimations in order to optimize the design to meet various performance criteria with various levels of reliability (e.g. Bertero and Bertero 2002, Krawinkler and Miranda 2004). Seismic demand estimations involve many challenges, one of which is the evaluation of the effects of seismic soil-structure interaction (SSI) on the inelastic response of structures. Seismic SSI is potentially a highly nonlinear phenomenon which causes the structural response to differ from that of the ideal structure with rigid base, often assumed by engineers. SSI modifies the system response in two different ways, namely kinematic interaction and inertial interaction. Kinematic interaction occurs when the deformation of the foundation does not fully conform to the deformation of its surrounding soil when seismic waves propagate through the soil. This modifies the incident seismic wave as it reaches to the foundation and causes the foundation motion to differ from the free field motion, which is normally assumed to be the input to the structure when SSI is not 1 accounted for. The inertial interaction is due to the reaction forces caused by the dynamic response of the structure. These forces are transmitted to the foundation and to the soil subsequently, further displacing the foundation and its surrounding soil and thus further modifying the motion of the foundation with respect to the free field which in return further modifies the response of the structure (Gazetas and Mylonakis 1998). The flexibility at the base of the structure increases the number of the degrees of freedom of the system and lowers its stiffness causing the natural period of the system to elongate. SSI dissipates the energy of vibration through hysteretic behaviour of the foundation's surrounding soil (material damping), and by radiating the seismic waves into the soil continuum away from the structure (radiation damping). SSI can particularly play a significant role in the response of bridge structures due to their relatively simple structural form and the low degree of redundancy of these structures that make them sensitive to the effects of SSI and SSI induced displacements. This is particularly true for bridges with single-column piers and simply supported spans. SSI may significantly affect the ductility demands of the bridge piers or cause large differential displacements between the piers. This can be detrimental to displacement-sensitive components of bridges and may compromise the bridges structural integrity. Displacement-sensitive components could be both structural and non-structural. Examples of structural components include bearing seat width, restrainers or expansion joints, and examples of non-structural elements include pipe, electrical and telephone lines. Evaluation of SSI has remained as one of the most challenging aspects of seismic demand evaluation, especially for pile foundations, and there is still a gap between the practice of geotechnical and structural engineering. While geotechnical engineers commonly focus more on the complex soil behaviour with simplifications in representing the structure, structural engineers concentrate more on the behaviour of the structure and favour simplifications in modeling of the soil and the soil-foundation interface as much as possible. This has been due to the complex nature of the problem and the practical limitations of the available tools that do not allow practicing engineers to efficiently 2 account for all different aspects of SSI, such as the nonlinearity of soil, the nonlinearity of structure, and the nonlinearity of soil-foundation interaction, at the same time. Particularly, the SSI of pile foundations is a very complicated phenomenon to model due to the complex behaviour of individual piles and the pile-soil interaction, and the complex behaviour of piles in a group of piles and the pile-soil-pile interaction. As a result of the complexities, the significance of the role of SSI on the seismic performance of structures is not yet clearly understood. This is despite wealth of research that has gone into several aspects of seismic SSI. The past research on SSI has largely been component-oriented and focused on the evaluation of the complex soil-foundation interaction in response to earthquake loading with less attention on the effects of this interaction on the overall system response. On the other hand, researches that have dealt with the effect of SSI on the overall system response, required many simplifications in regard to the behaviour of soil, or structure, or soil-foundation interface, in order to reduce the size and complexity of the problem so that it could be managed with the available computing power at the time of each research. Performance-based design of pile-supported bridges requires rigorous estimation of the effects of SSI on the response of the piers, so that more accurate estimations can be made and better understanding of the role of SSI can be achieved. The focus of performance-based design, on the other hand, is on estimating the nonlinear displacements, as a better indicator of damage in the inelastic range, rather than forces obtained from linear elastic analyses that have traditionally been used in structural and foundation design (displacement-based design versus force-based design). Therefore, estimation of SSI for performance-based design requires accurate estimation of the effects of SSI on the nonlinear displacements of systems, which in return requires simultaneous consideration of nonlinear soil behaviour, nonlinear soil-foundation interaction, and nonlinear structural behaviour, so that an accurate estimation of the interaction of all various components of soil-foundation-structure systems can be achieved. The need for better understanding of the effects of SSI on seismic response of bridges and buildings and the resulting need for 3 improved methods of SSI evaluation have been emphasized on and demonstrated by researchers in recent years. Gazetas and Mylonakis (1998) provide an overview of the methods of SSI analysis and re-explore the role of SSI on the seismic response of pile-supported bridge piers by studying the failure of the Hanshin Expressway Route 3 in Higashi-nada, with single circular columns supported on pile foundations, during the 1995 Great Hanshin Earthquake in Japan (Kobe Earthquake). They found that SSI could have contributed into the collapse of this bridge by elongating the natural period of the system which resulted in higher spectral accelerations at the shifted period. Mylonakis and Gazetas (2000) further explore the effects of SSI by discussing the misconceptions in regard to ignoring SSI as a conservative assumption and explain how the spectral characteristics of ground motions, which depend on both seismic input and soil conditions, can result in higher seismic demands at the elongated natural periods of structures when SSI is accounted for, despite possible increase in damping due to SSI. They demonstrate that SSI may result in increased ductility demands of bridge piers on soft soil and conclude that this may not be revealed if conventional code design spectra, as opposed to actual response spectra, are used in seismic demand estimations including SSI. The possibility of increased response due to SSI was also stated previously by Jennings and Bielak (1973), and Veletsos (1993) and has been recently demonstrated by other researchers such as Sextos et al. (2002) and Jeremic et al. (2004). Martin and Lam (2000) explain that the traditional force-based design of structures assumes foundation elements are rigid or elastic and ensures structural load demands are less than the capacity of foundation elements. They mention that a major change in conventional design philosophy in geotechnical engineering is the concept of allowing mobilization of ultimate capacity of foundations during earthquakes and state that design procedures need to account for foundation performance and its effect on the overall bridge response. Martin and Lam (2000) report that in collaboration with Fenves (1998), they conducted sensitivity studies of the dynamic response of a global bridge model with improved representation of the foundation characteristics by using nonlinear springs with 4 gapping elements and they conclude that the focus of future research should be better understanding and modeling of nonlinear foundation behaviour and to integrate such models with nonlinear structural analyses to allow overall performance assessment. Finn (2004a, 2004b) reviews the state of practice for characterization of the actions of foundations on the response of structures and evaluates the effectiveness of various approximate approaches. He states that a weakness in modeling of pile-supported structures is inadequate representation of foundation by ignoring the coupling between the translational and rotational stiffnesses of the foundation represented by single valued springs. He adds that most of the approximate methods in use for evaluating foundation stiffnesses are based on single pile analysis and further assumptions are made to account for the group response. He states that many factors such as soil nonlinearity, kinematic interaction between piles and soil, inertial interaction of the structure with soil and piles, dynamic interaction between piles themselves, and seismically induced pore water pressures, must be taken simultaneously into account so that a complete picture of the effects of foundation on the seismic response of structures can be obtained. He presents an investigation into the reliability of approximate methods for representing the rotational and translational stiffnesses of pile foundations in numerical models of pile-supported bridge piers by using a pseudo-3-dimensional nonlinear continuum soil model (Wu and Finn 1997a and 1997b, Thavaraj and Finn 2001), and highlights the importance of the relative stiffness of structure and foundation for interaction, and of including both kinematic and inertial interaction in analyses. Crouse and McGuire (2001) discuss the effects of SSI from the energy dissipation standpoint. They review the general state of SSI practice by structural engineers in regard to energy dissipation and state that energy dissipation in SSI is usually ignored or misapplied in structural engineering practice. They discuss the knowledge gap in the practice of SSI, and present practical systems-identification methods for estimating composite modal damping ratios for the significant modes of vibration of structures to account for both material and radiation damping of SSI. 5 Kim and Roesset (2004) studied the importance of accounting for the nonlinear soil behaviour in the evaluation of the effects of SSI on the inelastic response of structures. They demonstrated significant difference between the response with elastic soil and the response with inelastic soil and showed the importance of accounting for the nonlinearity of soil response especially for pile foundations. The review of literature also reveals that there is a lack of sufficient statistical description of the effects of SSI on structural response to demonstrate the effects of SSI quantitatively, rather than merely qualitatively. For bridge structures, there are a few published studies on the effects of SSI on the response of the bridge piers with a statistical approach, but they are different in scope and certainly not enough to provide a complete picture. For instance, Ciampoli and Pinto (1995) studied the effects of SSI on the inelastic response of bridge piers supported on spread footings without consideration of the nonlinearity of soil. Hutchinson et al. (2004a, 2004b) performed nonlinear dynamic analyses to estimate the inelastic response of the extended pile-shaft-supported bridge piers. These foundations are different than pile groups and the results of these studies cannot be extended to bridges supported on pile groups. In summary, the above background remarks the need for the accurate evaluation of the effects of SSI on seismic response of pile-supported bridge piers on soft soil and identifies the following: • There is a need for a quantified picture of the effects of SSI on the response of the pile-supported bridge piers, so that the circumstances under which these effects are significant can be identified, and it can be quantified what the effects of SSI are and how they can affect the overall bridge performance. • Accurate estimation of SSI requires a system approach to SSI analysis with fully coupled representation of the main components of the system and with proper consideration of the nonlinearity of soil, structure and soil-structure interface. Attention should also be paid to appropriate consideration of the radiation damping of the system. 6 • Assessment of the effects of SSI must account for various performance objectives and various levels of earthquake ground motions. • Evaluation of SSI must be in line with displacement based design methodologies, i.e. with the goal of estimating the inelastic displacements of piers and foundations as the primary demand parameters. • Estimation of SSI must account for uncertainties involved in input ground motions, soil conditions, and system properties, so that the design of the system can be optimized to meet different performance criteria with different levels of reliability tailored to a specific project. Moreover, it appears that a great number of structural engineers have not been able to take the most advantage of the available results of research on SSI due to the scatter of the current published literature and due to many instances of the lack of a link between the geotechnical aspects of the available knowledge and the practice of structural engineering. Hence, the effects of SSI need to be presented in a format familiar to structural engineers and in accord with the fundamentals of structural analysis and design. In practice, the structural analysis is normally performed by analyzing the elastic system to obtain the elastic displacements, and the corresponding inelastic displacements are obtained by employing natural-period-dependent relationships between the elastic and the inelastic displacements obtained by parametric studies of single degree of freedom (SDOF) systems. The relationship of the elastic and the corresponding inelastic displacements are expressed by the inelastic deformation ratios (Cu). The inelastic deformation ratio is a function of the elastic natural period (T), and can be expressed in terms of the ductility factor (u) of the system, defined as the ratio of the inelastic displacement of the system to the yield displacement of the system, and in terms of the strength reduction factor (R), defined as the strength required for the structure to remain elastic divided by the yield strength of the structure. It can be shown that Cu=u./R (see Chapter 5 ) , therefore, the study of the relationship between C u and T becomes equivalent to the study of the relations between \i, R, and T commonly referred to as R-U.-T relations. 7 The often used "equal displacement rule" which assumes that the elastic and inelastic displacements are equivalent (i.e. C u=l or u=R), is a result of the R-u-T relations proposed by Veletsos and Newmark (1960). The equal displacement rule is valid for most practical ranges of natural period, but not for all periods. Veletsos and Newmark (1960) also proposed the equal energy rule for the period range other than that of the equal displacement rule (see Chopra and Chintanapakdee 2004 and Chopra 2005 for an overview). More refined estimation of the relationships between C u and T and their application in structural demand estimation have also been the subject of recent research due to their relevance to the displacement based design of structures with the need for more accurate estimation of the relationship between the elastic and inelastic displacements of structures (e.g. Miranda 2001, Farrow and Kurama 2003, Chopra and Chintanapakdee 2004, Ruiz-Garcia and Miranda 2004). The C u - T or R-u-T relations are central to structural demand estimation and therefore consideration of these relations must be included in a study of the effects of SSI. A rational approach to the presentation the effects of SSI on the response of structures would be to obtain these effects as functions of the structures natural periods so that a link between structural analysis and SSI analysis can be established. 1.2 Motivations for this Research There are a number of motivations behind this research. One is to explore the system approach to SSI analysis of pile-supported bridge piers with a) fully coupled representation of soils, foundations, and piers, b) proper consideration of the nonlinearity of soil, structure and soil-structure interface, and c) appropriate consideration of the radiation damping of the system. The goal is to demonstrate the feasibility and practical application of such SSI simulation for seismic demand estimation of structures by performing nonlinear dynamic analyses. The SSI analysis methods with system approach are the dynamic beam on a nonlinear Winkler foundation (dynamic p-y springs) methods, and the direct methods by 8 performing finite element or finite difference analyses. In the beam on a nonlinear Winkler foundation method, the nonlinear stiffness and damping characteristics of the soil-foundation system are accounted for through a series of nonlinear springs and dashpots distributed along the piles and the free field motion is applied to the ends of the springs (Gazetas and Mylonakis 1998). In the direct methods, finite element or finite difference models of the soil continuum, the foundation and the structure are made all in one model. The material damping is accounted for by inelastic behaviour of the system and the radiation damping is considered by using non-reflecting boundaries for the soil continuum. The input motion in this case is applied at the base of the soil layer. Compared to the p-y springs methods, direct methods have the advantages of being the most complete and the most transparent representation of the system. Direct methods may be perceived as the more complex method. However, they deal with basic physical properties of the system as apposed to the Winker foundation approach that requires complex springs representing the combination of a number of physical properties of the system with properties that may be very difficult to estimate. Thus direct methods are not necessarily the most complicated ones. In addition, direct methods can model the pile cap and automatically account for the pile group effects with no further approximations. They can also include large deformations due to liquefaction of soil which is a great advantage for the analysis of liquefiable soils. The main disadvantage of the direct methods is the required long analysis runtime which can be prohibitive and can make the SSI analysis seem impractical. Despite this disadvantage, with the ever increasing speed of personal computers and improving available programs, direct methods are becoming more practical and cost-effective tools for engineering applications. Successful applications, however, require careful understanding of the modeling issues and the practical limitations of the employed programs. This understanding can be gained only through the repeated use of the direct methods to solve practical problems and by performing verification studies. Thus, a direct method of SSI analysis was chosen for this research by employing a commercial program 9 to demonstrate the practicality of the method and its advantages for practical earthquake engineering, and to highlight its limitations and the needs for future improvements. Another motivation for this work is to present a quantitative picture of the effects of nonlinear SSI on the seismic response of pile-supported bridge piers in a format useful to structural engineers, i.e. as functions of structural period. The questions to address are: 1) when SSI should be accounted for (i.e. when is SSI significant); 2) how does the foundation contribute in the overall system response; and 3) how the response of the system is modified by accounting for SSI. The approach to follow is to use the fully coupled, fully nonlinear numerical models developed in the first stage of this study to simulate the seismic response of prototype pile-supported bridge piers (including SSI) and by comparing the results with the results of the dynamic analyses of the same piers without accounting for SSI (i.e. fixed-base piers). The goal is to study the role of SSI by providing statistical description of the response of various prototype systems with various natural periods subjected to various input ground motions with different characteristics. The statistics obtained can further be processed probabilistically to include the uncertainties of the system parameters and to account for the dispersions introduced by input ground motions and soil conditions. Representation of results suitable for performance-based evaluation of the effects of SSI can also be explored so that it can be used for future estimation of SSI for similar systems. By using the statistics obtained from the dynamic analyses, it is of interest to examine some aspects of the methods of structural analysis currently used in practice. This includes: 1) examining the accuracy of the inelastic displacements estimated from elastic displacements of the SSI system by using inelastic displacement ratios that are obtained from the analysis of SDOF systems; 2) examining the change in the foundation demands when SSI analysis is performed with an elastic structure (i.e. not accounting for the inelastic behaviour of the piers); and 3) to examine the accuracy of nonlinear static pushover analysis in demand estimation and in capturing the salient features of dynamic response of SSI systems when SSI is significant. 10 1.3 Obj ectives of the Dissertation The objectives of this dissertation are as follows: • To explore efficient implementation and practical application of the direct methods of nonlinear SSI analysis with a system approach. • To evaluate the effects of nonlinear SSI on the inelastic seismic demands of pile-supported bridge piers, and to provide a quantified picture of these effects as functions of the piers natural periods. • To investigate the effects of the modeling assumption of the piers (i.e. elastic or inelastic behaviour) on the estimated demands of the piers and on the estimated demands of the foundations, and to examine whether analysis with an elastic structure can provide plausible prediction of the inelastic demands when SSI is significant. • To provide a probabilistic representation of the effects of SSI on the seismic demands of bridge piers by including the uncertainties of the system parameters and by accounting for the dispersions introduced by the variability of input ground motions and soil conditions. • By employing statistics of seismic responses, explore simplified methods for evaluating the effects of SSI on the response of piers from their estimated response without SSI. • To examine the accuracy of the nonlinear static pushover analysis for seismic demand estimation of pile-supported bridge piers when SSI is significant. 1.4 Scope of Work This research was carried out by performing nonlinear dynamic analyses of prototype pile supported bridge piers on soft soils subjected to a set of input ground motions selected from historic ground motions. The analysis methodology was carefully verified: the component responses were verified individually and the system response was verified by a case history analysis of an instrumented pile-supported bridge pier. The analyses of the prototype piers were performed with and without accounting for SSI. In each case, the analyses were carried out with assuming both inelastic and elastic behaviour of the piers. The statistics of responses were summarized and further validated against available 11 knowledge where possible. Comparisons were made between the demands of the elastic piers and the demands of the inelastic piers to make conclusions in regard to the modeling assumption of the piers. Comparisons were made between the demands with SSI and the demands without SSI to draw conclusions in regard to the effects of SSI. Statistics of demands were further processed probabilistically by performing reliability analyses, and the performance-based assessment of the effects of SSI using response statistics was explored. Finally, pushover analyses were carried out and the resulting demands were compared to the predicted demands by the nonlinear dynamic analyses. 1.5 Organization of the Dissertation This dissertation represents four phases of work. The first phase deals with the numerical analysis and includes Chapters 2 and 3. The second phase involves application of the numerical analysis techniques discussed in the first phase to obtain the statistics of the response of SSI systems and to study the effects of SSI on the response of bridge piers. This phase is presented in Chapters 4 to 7. The third phase, which is presented in Chapter 8, involves the probabilistic analyses of the results of the numerical analyses and their application for performance-based assessment of the effects of SSI on the response of bridge piers. The fourth phase, presented in Chapter 9, is in regard to pushover analysis. In Chapter 2, after a description of the prototype systems, the construction of the numerical models is presented by explaining and verifying the modeling of various components of the system, and by discussing issues related to modeling of the system as a whole. This chapter ends by using the constructed numerical models to obtain the system properties. Chapter 3 discusses a verification study of the modeling technique presented in Chapter 2 by performing a case history analysis of an instrumented bridge pier with recorded motions from an actual earthquake. This chapter particularly verifies the capability of the numerical analysis in representing the overall system response. 12 The selection of input ground motions is presented in Chapter 4. This chapter also describes various stages of dynamic analysis including site response analysis, analysis of the piers without SSI, and analysis of the system as a whole including SSI. The primary goal of this chapter is to examine the behaviour of the system rather qualitatively by observing the salient characteristics of the dynamic response of the system, without emphasizing on the statistics of response. Chapter 5 introduces the demand parameters and presents the statistics of the seismic demands of the piers. The behaviour of the piers is studied in this chapter by means of the statistical distributions of demands. Chapter 6 presents the seismic demands of the foundations and their relation to the demands of the piers in order to better understand the role of the foundation response in the overall system response. In Chapter 7, the effects of SSI on the response of the piers are discussed by directly comparing the demands of the piers with SSI to the demands of the piers without SSI. This chapter discusses the circumstances under which SSI must be accounted for, and demonstrates how SSI modifies the response of the piers. Chapter 8 presents the probabilistic assessment of the effects of SSI on the demands of the piers to account for the dispersions of demands that are related to the input ground motions and soil profiles, and to account for the uncertainties of the system properties. This chapter explains how results can be used to estimate the effects of SSI on the performance of the piers with given target reliabilities and demonstrates how to use them to modify the response of the piers obtained from the analyses without SSI to account for the effects of SSI. In Chapter 9, the accuracy of nonlinear pushover analyses in estimating seismic demands of pile-supported piers is examined. Finally, Chapter 10 presents a summary of the conclusions and the recommendations for future research. 13 2 D E V E L O P M E N T OF T H E NUMERICAL MODELS This chapter presents the details of the numerical modeling of the prototype pile-supported bridge systems that were considered in this research. The commercial program F L A C (Itasca 2005) was employed for the numerical analysis. F L A C which stands for Fast Lagrangian Analysis of Continua, is a two dimensional explicit finite difference program for engineering mechanics computation. It simulates the behaviour of soil, rock, steel, concrete and other materials that may undergo plastic flow when their yield limits are reached. Materials are represented by elements or zones which form a grid in the shape of the object to be modeled. The material can yield and flow according to several built-in constitutive models of the program or constitutive models defined by the user with the help of FISH, the built-in programming language of F L A C . A useful feature of F L A C for the study of soil-structure interaction is the availability of interface elements which simulate distinct planes along which slip and/or separation can occur. 2.1 Pro to type So i l -Founda t ion -S t ruc tu re Systems The prototype soil-foundation-structure systems used in this research are illustrated in Figure 2.1. Each prototype system consists of a bridge pier, a pile foundation, and a soil layer. The soil layer consists of saturated soft clay with average shear wave velocity in the upper 30 m of Vs3Q = 145 m/s and average strength to the depth of 30 m of Su30 = 40 kPa. The resulting prototype site is classified as Site Class E or F (NEHRP 2004, ATC-49 2003, NBCC 2005). The height of the site soil layer is 21.0 m, 40.0 m, and 100.0 m so that site periods (at low amplitude of motion) of 0.6, 1.0 and 2.0 seconds were obtained. The foundation consist of a 6><6 pile group with 0.3 m square piles spaced at 1.25 m, driven to the depth of 15 m below the 7.5x7.5x1.5 m pile cap. The pier has a typical 14 circular cross section with a diameter of 1.5 m and with 1% longitudinal reinforcement. The height of the pier varies from 3.5 m to 12.1 m to provide first mode fixed-base elastic natural periods of 0.3, 0.6, 0.8, 1.0, 1.5, and 2.0 seconds (see Table 2.1). A 3500 kN load was applied on the top of the piers to represent the weight of the bridges superstructures. Details of the system properties and modelling of the individual components of the system and the system as a whole are explained in the following sections. L = 3.5 to 12.1 m T s t r = 0.3 to 2.0 s 3500 kN CM 7 Circular Concrete Section with 1% Longitudinal Reinforcement H—H 1.5 m 7.5x7.5x1.5 m Pile Cap H = 21.0 to 100.0 m T s i t e = 0.65 to 2.0 s Typical 0.3x0.3x15.0 m Precast Reinforced Concrete Square Piles Soil Type: Clay J/ 3 0 = 145 m/s S„30 = 40kPa Input Motion < • Figure 2.1: Schematic description of the prototype soil-foundation-structure system 15 2.2 Modelling of Soil 2.2.1 Soil Profile and Properties The stiffness, strength and mass properties of the prototype soft soil layers were chosen so that the prototype sites are classified as Site Class E or Site Class F as defined by current seismic design codes such as NEHRP (2004), ATC-49 (2003), and NBCC (2005). The material of soil was assumed to be soft clay; therefore liquefaction did not play a role in the analyses. The water table was assumed to be at the surface of the soil layer. The Mohr-Coulomb constitutive model (Itasca 2005) was used for the soil. The parameters required by this constitutive model are stiffness parameters defined by the shear and bulk modulus of soil and by strength parameters determined by the friction angle and the cohesion of soil as described in the following subsections. 2.2.1.1 Stiffness properties of Soil The initial shear modulus of soil ( G m a x ) was determined such that the resulting average shear wave velocity in the upper 30 m of soil {Vm) was 145 m/s. Vs30 is used by many design codes and guidelines such as NEHRP (2004), ATC-49 (2003), and N B C C (2005) for the purpose of site classification for seismic design of structures. The values of G m a x were proportional to the square root of the mean normal effective stress of soil within the soil layer. In other words, the distribution of G m a x was proportional to Jcr'm I PA where <y'm is the mean normal effective stress and PA is the atmospheric pressure (100 kPa). The G m a x at the mean normal effective stress of 100 kPa (the atmospheric pressure) was (Gmax)ioo = 43 MPa. The mean normal effective stresses were obtained by performing static analysis of the soil layer under its self weight and by updating the stresses in the soil before performing the dynamic analysis. The G m a x profile thus obtained is depicted in Figure 2.2. The value of Vs30 can be calculated from the Equation 2.1 (e.g. NEHRP 2004). In Equation 2.1, di is the thickness of any layer i between 0 and 30 m and Vsi is the shear wave velocity of the layer i obtained from the Equation 2.2; where p is the mass density and Vs is the shear wave velocity of the soil layer. 16 14 (2.1) G, P max (2.2) The Vs30 thus calculated for the soil layer with the G m a x profile of Figure 2.2, and for the assumed mass density of p = 1700 kg/m3 was 145 m/s. The bulk modulus was assumed to be ten times the shear modulus (corresponding to a Poisson ratio of 0.45) for the static analysis. For the dynamic analysis, the bulk modulus was assumed to be 50 times the shear modulus for the saturated soil below the water table. The nonlinear hysteretic behaviour of soil prior to yield was based on the well known modulus reduction curve proposed by Seed and Sun (1989) for clays (upper range). This modulus reduction curve is shown in Figure 2.3. 2.2.1.2 St rength Proper t ies of So i l The shear strength of clay is defined by its cohesive strength. The average cohesive strength in upper 30 m of the prototype clay of this study (Su30) was taken equal to 40 kPa (within the limiting values of Su for site classes E and F). The cohesive strength Su was assumed to be distributed linearly with the effective vertical stress (<rv0) within the soil layer. For the prototype clay layer, the ratio Su / <r'v0 was assumed to be 0.381, which corresponds to a plasticity index of 73% according to the equation proposed by Skempton (Craig 1987) described by Equation 2.3, where PI is the plasticity index. The profile of Su along the height of the soil layer is shown in Figure 2.2. S, u = 0.11 + 0.0037P7 (2.3) 17 Gmax (M Pa) Su (kPa) Figure 2.2: Distributions of G m a x and Su to the depth of 100 m of the prototype soil layer 2.2.1.3 Nonlinear Behaviour of Soil Elements in F L A C To verify the nonlinear behaviour of soil prior to yielding when modeled in F L A C , cyclic loading of a soil element was simulated. Cyclic shear loads were applied incrementally on the element and the hysteretic response of the element was tracked. Resulting hysteresis loops were then used to estimate the equivalent stiffness of the soil element corresponding to each level of the applied shear strain. Then the modulus reduction curve (GI G m a x curve) of the element was constructed as a function of the shear strain. The equivalent viscous damping representing the energy dissipated through the hysteresis loops was also estimated (see Chopra 1995 for methodology). Since this single element does not represent all soil elements in the model, the result of this simulation is solely for an overall verification of the numerical model. The modulus reduction curve and equivalent viscous damping exhibited by the soil element undergoing the cyclic loading are presented in Figures 2.3 and 2.4. Figure 2.3 shows that the stiffness degradation of the soil, as exhibited by the soil element, matches 18 the target curve. Figure 2.4 illustrates a comparison of the computed equivalent viscous damping of the soil element with a number of damping curves of clay available in the literature. It can be observed that in comparison with the available damping curves, the soil element of F L A C was under-damped at low strains and was over damped at high strains. However, Figure 2.4 shows that the damping exhibited by the element is within the bounds of the damping curves obtained from the literature. Moreover, no conclusions can be drawn as to which one of these curves is the most appropriate one for the prototype soil of this study and therefore the damping exhibited by the soil element is acceptable. To compensate for the low damping at small strains, Rayleigh stiffness damping was added to the numerical model. It is pointed out that when a soil element yields, the stiffness degradation curve of Figure 2.3 is no longer used by the program and the stress-strain relationship of the soil element is calculated based on the yield criterion defined by the Mohr-Coulomb constitutive model. Figure 2.5 shows an example of the stress-strain relationship of a soil element at the depth of 40 m when the soil layer was subjected to an earthquake ground motion (see Chapter 4). This figure illustrates the hysteretic behaviour of the soil as was captured by F L A C . Figure 2.3: Shear modulus reduction curve of clay 19 Figure 2.4: Equivalent viscous damping exhibited by a single soil element of F L A C o. -100 -I r— —r— —r— —I— ! 1 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Shear Strain (%) Figure 2.5: An example of the stress-strain relationship of a soil element at the depth of 40 m when the soil layer was subjected to an earthquake ground motion 20 2.2.2 Boundary Conditions The lateral boundaries of the soil model must represent the infinite extent of the soil layer in the horizontal direction and therefore they should not reflect the incident waves reaching to them. A non-reflecting boundary option of F L A C known as the "free field boundary" (Itasca 2005) was used in the numerical models. Since these boundaries may not fully absorb the incident waves, they are placed at enough distance from the foundation so that the material damping prevents the reflected wave from returning to the vicinity of the foundation. 2.3 Modeling of Bridge Piers The bridge piers were modeled using beam elements. The inelastic behaviour of the pier was accounted for by yielding of the plastic hinge at the bottom of the pier. It is pointed out that a typical reinforced concrete column can have various modes of failure, including shear failure mode, flexural failure mode or a combination of shear and flexural failure modes (Priestly et al. 1996). In this work, only the flexural failure mode is considered by accounting for the hysteretic behaviour of the plastic hinges. Modeling of the inelastic behaviour of the piers is discussed in Section 2.3.3.1. A 3500 kN gravity load was applied at the top of the piers to represent the weight of the bridges superstructure. 2.3.1 Beam Element Properties The beam element properties included the modulus of elasticity (E), the cross sectional area of the bridge column (A), mass density (p), the cross sectional moment of inertia (I) of the column, and the yield moment (M y). Typical values of E=25000 MPa and p=2400 kg/m were used for the modulus of elasticity and mass density of the concrete, respectively. The cross sectional area was that of a circular cross section with a diameter of 1.5 m. The moment of inertia of the cracked concrete and the yield moment of the pier were calculated for 1% longitudinal reinforcing steel as shown in Section 2.3.3.1. The inertial mass associated with the 3500 kN load from the bridge superstructure was included by assigning high mass density for the beam elements at the top of the piers. The center of mass at the top of the piers represented the centre of mass of the bridges superstructures. 21 2.3.2 Damping For the dynamic analyses of the bridge piers, the damping was assumed to be equivalent to 5% of the critical viscous damping of the first mode of vibration. This damping can be modeled in F L A C using Rayleigh damping. However, in F L A C , when stiff structural elements are modeled along with softer soil elements, applying Rayleigh damping results in a significant decrease of the time step required by the explicit time integration scheme of the program. Explicit time domain solutions are always stable but at the cost of long computational times due to small critical time steps that depend on the stiffness and the density of the material modeled. Thus, the time step is chosen by F L A C for numerical stability and cannot be controlled by the user. Using Rayleigh damping can result in impractically small time steps, causing the analysis runtime to become impractically long. For this reason, a non-Rayleigh damping option of F L A C , called "local damping", originally developed to increase the convergence rate of static solutions (Itasca 2005) was used. This damping, however, had to be calibrated for the dynamic analysis and had to be verified against results obtained from the analysis with viscous damping. To verify the non-Rayleigh damping of F L A C , the fixed-base bridge piers were modeled using both F L A C and the well-known structural analysis program SAP2000 (Computer and Structures 2003). Results obtained from the elastic dynamic analyses of the piers using both programs were compared for various input ground motions. Results demonstrated that the "local damping" represented an equivalent viscous damping of 5+0.5% of the critical damping. Figure 2.6 shows a comparison between the acceleration response spectra and the acceleration time histories of one of the bridge piers obtained from F L A C and SAP2000, when it was subjected to one the selected ground motions (see Chapter 4). As can be seen in this figure, the two responses match very well with minor differences. Finally, it must be mentioned that viscous damping is just a numerical convenience and it is not the real form of the physical damping of structural elements. Therefore the minor differences observed in Figure 2.6 are of no significant importance. 22 0.1 1 10 100 Frequency (Hz) 0.8 0 5 10 15 20 25 30 35 Time (s) Figure 2.6: Comparison of the elastic response of a pier obtained from F L A C with non-viscous damping and SAP2000 with 5% viscous damping T=0.6 s, T s i t e=1.0 s, Input Motion: Northridge Ci ty Terrace (Chapter 4) 2.3.3 Nonlinear Behaviour of the Bridge Piers Both material and geometric nonlinearity of the piers response were considered in this study. The material nonlinearity included the inelastic behaviour of the columns under bending moments (flexural yielding) and the geometric nonlinearity included the P-A effects. Material nonlinearity under shear (shear failure) was not considered. The assumption of flexural failure with no failure in shear is a reasonable assumption for especially the taller piers. Shorter piers are more likely to experience flexural and shear failures, however, this consideration was beyond the scope of this work. 23 2.3.3.1 Material Nonlinearity The inelastic response of the piers was accounted for through hysteretic behaviour of plastic hinges under bending moment. F L A C has options for modeling the plastic hinges of beam elements with elastoplastic response. It was observed, however, that under dynamic loading, the hysteretic behaviour of a plastic hinge modeled in F L A C was closer to a nonlinear-elastic response rather than an elastoplastic response. Therefore, modification of the program was necessary to ensure appropriate numerical modeling of the hysteretic behaviour of plastic hinges was achieved. The modifications were implemented through FLAC's programming language FISH. An elastoplastic hysteretic model, with and without stiffness degradation, was implemented. For the stiffness degrading hysteretic model, however, lack of robustness under dynamic loading was observed. Therefore, only the elastoplastic model was used in this study, although the stiffness degrading model is the preferred model for reinforced concrete material. Ruiz-Garcia and Miranda (2004) studied the effect of the stiffness degradation on the response of SDOF systems when subjected to ground motions recorded on soft soil (with no SSI) by comparing the results obtained from elastoplastic and stiffness-degrading systems and by plotting the mean ratios of the inelastic displacements obtained from the two models. Their results showed that the variation of response was mostly less than 30%. While the aforementioned variation of response is recognized here, it is emphasized that given the comparative nature of this research between the results obtained from the fixed-base and flexible-base piers (as it will be seen in the forthcoming chapters), and given the complexity of the performed nonlinear analyses, the dispersion of results caused by using an elastoplastic hysteretic model (instead of the preferred stiffness degrading model), will not play a significant role in the findings of this research. It should also be pointed out that there are still many uncertainties associated with modeling the inelastic behaviour of reinforced concrete columns, even when stiffness degrading models are used, and this is still a subject of research in structural engineering. 24 To estimate the yield moment and the cracked cross section properties o f the reinforced concrete columns, the commercial cross section analysis program X T R A C T (Imbsen 2 0 0 3 ) was used. X T R A C T is a program that can generate moment-curvature, axial load moment interaction, and moment-moment interaction responses for cross sections with nonlinear material model and with shapes that can be defined by the user. Two cross sections with about 1%, of longitudinal reinforcing steel were considered. One section was considered with 2 4 x 3 2 mm longitudinal reinforcing bars and the other section was considered with 2 4 x 3 0 mm bars. The section with less percentage o f reinforcement was used for the shorter piers with periods of 0 . 3 , 0.6 and 0.8 s to reduce the yield moment for the shorter piers with lower overturning moments. The yield strength of the reinforcing bars was 4 2 0 M P a . Figure 2 .7 shows an example moment-curvature curve obtained from the cross section analysis and its idealized bilinear curve (see Appendix A for more details). The cross section analysis resulted in the effective (cracked) cross section properties of the piers as well as their yield moments, their ultimate moments and their respective curvature ductility capacities. The cross section properties were then used to calculate the displacement ductility capacities of each prototype pier based on the method explained by Priestly et al. ( 1 9 9 6 ) , which is based on estimating the plastic hinge length of each pier. Table 2.1 lists a summary of the piers properties used in this study. 2 5 Table 2.1: Summary of the properties of the piers Fixed-Base Natural Period T (s) Column Diameter (m) Height of Pier (m) Yield Moment My (kN.m) Yield Deformation Ay (m) Displacement Ductility Capacity 0.3 1.5 3.50 5875 0.011 7.1 0.6 1.5 5.50 5875 0.027 6.0 0.8 1.5 6.65 5875 0.040 5.7 1.0 1.5 7.70 6350 0.057 5.3 1.5 1.5 10.05 6350 0.098 5.0 2.0 1.5 12.10 6350 0.142 4.8 As the final verification of the inelastic response of the bridge piers in F L A C , a comparison was made between the results of the analysis of a fixed-base pier obtained from F L A C and the results obtained from the nonlinear analysis of the corresponding SDOF system. The comparison showed a good match between the two results and verified the inelastic response of the pier with elastoplastic hysteretic behaviour in F L A C . 2.3.3.2 Geometric Nonlinearity The geometric nonlinearity or P-A effect was accounted for by running F L A C in the large deformation mode (Itasca 2005). Figure 2.8 shows a sample load-displacement curve of a pier with T=1.0 s when P-A effect was considered and shows the negative post-yield stiffness of the pier which is a typical result of P-A effect on columns. 1000.00 T= 1.0s 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Displacement (m) Figure 2.8: A sample load-displacement curve of a pier when P-A effects included 26 2.4 Modelling of Foundation The foundation consists of a 7.5x7.5x1.5 m pile cap and a 6x6 pile group with piles spaced at 1.25 m. The pile cap was modeled using plane strain elements having concrete properties. The piles were 0.3x0.3 m square precast reinforced concrete with 2.1% longitudinal reinforcing bars (4x25 mm bars). The piles were modeled using pile elements which are similar to beam elements with additional options that are used to simulate the pile-soil interface (see Section 2.5.1.2). The cracked cross sectional properties of the piles were used in the analysis. These properties were obtained through cross sectional analysis o f the piles using X T R A C T (Figure 2.9). 0.00 0.01 0.02 0.03 Curvature (1/m) 0.04 0.05 Figure 2.9: Moment-curvature curve obtained from the cross section analysis of the piles 2.5 Modeling of Soil-Foundation-Structure System In previous sections, modeling of individual components of the system was explained. In this section, the modeling of the system as a whole and the interaction of different components of the system with each other is explained. Figure 2.10 shows the geometry of the model. The connection of the pier to the pile cap was through relatively rigid beam elements that were connected to both the pile cap elements and the pier's beam elements. 27 between the pile cap and the soil medium and the connection between the piles and the soil medium. Another aspect of the system modeling is the approximations required to represent the system in a two dimensional plane strain model. This approximation is also explained in this section. Figure 2.10: Geometry of the F L A C model (top 30 m of soil) 2 8 2.5.1 Modeling of Soil-Foundation Interface 2.5.1.1 Pile Cap-Soil Interface The pile cap-soil interaction was modeled through interface elements made of a series of normal and shear springs that connect the opposing surfaces at the interacting nodes. Interface elements are used to model the slip and separation planes between the pile cap and the soil mass (Itasca 2005). Figure 2.11 shows a schematic description of the interface elements. The slider properties are obtained from the friction or, as in this case, the cohesion between the concrete and the soil which can be obtained from the soil's internal friction angle or cohesion respectively. The cohesion between concrete and soil was assumed to be equal to the cohesion of the soil (Ma and Deng 2000). Tensile strength was assumed to be zero, and the linear springs were assumed to be relatively stiff (normal stiffness kn = shear stiffness k s = 1*106 kPa). Interface elements are between all three faces of the pile cap and the soil. Gridpoint Zone w ks _ c _ r - v v ^ w r T Zone S = Slider T = Tensile Strength kn = Normal Stiffness L = Shear Stiffness Figure 2.11: Interface elements in F L A G (Itasca 2005) 29 2.5.1.2 Pile-Soil Interface The pile-soil interface was modeled using pile elements coupling springs as illustrated in Figure 2.12. The coupling springs are elastoplastic springs with effective stress dependent yield strength. The coupling springs can be illustrated by means of linear springs and sliders in series as shown in Figure 2.12. Shear sliders account for the slip of piles with respect to the soil and the normal sliders account for the piles pushing through the soil and the gap caused by that (Itasca 2005). The yield strength of the shear coupling springs depends on the cohesive strength in clayey soils. For sandy soils, it depends on stress-dependent frictional resistance along the pile-soil interface and can be determined from the friction angle of the soil. The limiting force of the normal spring simulates the three dimensional effect of the pile pushing through the soil (gapping accounted for) and is a function of the cohesive strength and the stress-dependent frictional resistance between the pile and the soil (Itasca 2005). In the absence of test results, the behaviour of the pushing pile can be evaluated by modeling sections of the pile at various elevations in the soil, the results of which can be used to calibrate the normal coupling springs as is demonstrated later in this section. Normal Coupling Spring Linear Spring Soil • Shear Coupling Spring Slider Pile Element Figure 2.12: Pile-soil coupling springs in F L A C 30 The shear yield strength criterion of the shear coupling springs are illustrated in Figure 2.13. The yield strength of the shear springs per unit length of pile is equivalent to the cohesive force between the pile and the soil, which is equal to the pile-soil cohesion multiplied by the perimeter of the pile, plus the friction force between the pile and the soil which is equal to the effective stress on the pile times the tangent of the pile-soil shear friction angle times the perimeter of the pile. These parameters can readily be determined by knowing the cohesion and the friction angle of soil along the pile. For the cohesive soil of this study, the pile-soil cohesion was taken equal to the cohesion of soil and the friction angle was zero. Note that since the cohesion of soil varies with depth, a different cohesion value was assigned for shear coupling springs of each segment of the pile elements in the model. The elastic stiffness of the springs were relatively high ( l x 10 6 kPa) so that the relative movement between the piles and the soil occur mostly when the piles slide. Shear Coupling Spring Yield Strength Pile-Soil Shear Cohesion x Perimeter Pile-Soil Shear Friction Angle Effective Stress x Perimeter Figure 2.13: Shear coupling springs yield criterion The yield strength of the normal coupling springs had similar criterion as that of the shear coupling springs as shown in Figure 2.14. However, determining the elastic stiffness and the yield strength of the normal coupling springs was not as straight forward as that of the 31 shear coupling springs, and additional numerical simulation was required for determining the values of the involving parameters, i.e. the elastic stiffness of the normal coupling springs, the normal coupling spring cohesion, and the normal coupling spring friction, so that the behaviour of the piles pushing through the soil could be appropriately modeled. This was performed by modeling cross sections of the pile in the horizontal plane of the soil as depicted in Figure 2.15. This Figure shows the cross section of one of the piles in a row of piles (spaced at 1.25 m). The left and the right boundaries of the model shown in this figure are restrained in the horizontal direction of the model because they are on the two axes of symmetry of the pile-soil system. The first axis of symmetry goes through the middle of the pile and the second one goes through the soil at half the distance between the piles. The connection of the pile to the soil was through interface elements with similar behaviour to those used for the pile cap (see Section 2.5.1.1). Since the cohesion of soil varies with depth, more than one model was necessary to simulate the behaviour of the pile at different depths. Note that the initial confining stresses of the soil in the numerical model were assigned to be equivalent to those at the depth being considered. Normal Coupling Spring Yield Strength Normal Coupling Spring Cohesion (normal cohesion) x Perimeter Normal Coupling Spring Friction Angle (normal friction angle) Effective Stress * Perimeter Figure 2.14: Normal coupling springs yield criterion 32 The pile was laterally pushed with slow velocity and its load-displacement was tracked until after it started to push through the soil without taking any more load. Note that the displacements were relative to the soil. Figure 2.15 shows the displacement of the pile with respect to the soil, as well as the vectors of the soil incremental displacements which show how the soil is being displaced as the pile pushes through the soil. An example load-displacement curve of the pile at the depth of 9.2 m is shown in Figure 2.16. The ultimate soil resistance per unit length of pile shown in Figure 2.16 is comparable with the reported values in the literature for soft clay (e.g. Reese and Van Impe 2001, API 1993). The stiffness of the normal coupling spring was taken as the secant stiffness obtained from the load-displacement curve as shown in Figure 2.16. The normal cohesion and the normal friction angle (of the normal coupling springs) are obtained from a set of two equations (with two unknowns) that describe the yield strengths at two different depths (yield strength per unit length of pile = normal cohesion + average effective stress on pile x tangent of the normal friction angle x perimeter of pile) (Itasca 2005). Here, the simulation was performed at three elevations. Two were used to obtain the two unknowns and the third was used to check the results obtained. Finally, the behaviour of the coupling springs thus calibrated was checked for piles modeled with pile elements under confining pressures corresponding to the depths of soil that were used to calibrate the springs. The pile was pushed laterally with slow velocity and the force-displacement of the normal springs were tracked and compared with that obtained from the analysis of the cross section of the pile in the horizontal plane. This was done to validate the model and to ensure that the calibrated springs could properly duplicate the pile-soil interaction as was observed in the simulations in the horizontal plane. Figure 2.17 shows this comparison for the normal coupling springs at a depth of 9.2 m. In this figure, a good agreement between the two load-displacement curves is observed. F L A C also takes into account the gapping between the piles and the soil, and the user can define what percentage of the gap formed by the piles is recoverable. In this work, no 33 recovery of the plastic deformation of the normal coupling springs was assumed in order to account for the full gap formation. Axes of Symmetry Figure 2.15: The displacement of the pile with respect to the soil and the vectors of the soil incremental displacements in the horizontal plane 3 4 14.00 12.00 10.00 GO 1 8.00 c £ 6.00 <D U 5 5 4.00 2.00 0.00 Secant Stiffness 0.000 0.005 0.010 0.015 Pile Displacement Relative to Soil (m) 0.020 gure 2.16: An example load-displacement curve of the pile at the depth of 9.2 a in £ c o 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 1 „ „ , . . , , ' „ , „ „ JLP ~ 1 f • - 1 / 1 j fl 9 •"• 1 1 1 I J i I f ' i U ! Pile Section in Horizontal Plane Normal Coupling Spring i ~ r f 1 0.000 0.005 0.010 0.015 Pile Displacement Relative to Soil (m) 0.020 Figure 2.17: Validation of the normal coupling springs at a depth of 9.2 m 35 2.5.2 Modeling of Structural Elements in Plane Strain Analysis Plane strain soil elements represent unit length in the out-of-plane direction of the model and therefore numerical modeling of three dimensional structures in plane strain requires approximations to account for the appropriate width of soil that interacts with each structural element. For piles with uniform spacing, while the effects of piles pushing through the soil are considered in calibrating the soil-pile interface coupling springs, the discrete effect of pile elements over this distance can be accounted for by linear scaling of material properties (Donovan et al. 1984, Itasca 2005, Potts and Zdravkovic 2001). Similar approximations can be made for the pier with the interacting soil to be assumed as wide as the pile cap or the total width of soil that the pile group interacts with. This is equivalent to modeling a row of piers uniformly spaced at, for instance, the width of the pile cap in the out-of-plane direction with rows of uniformly spaced piles underneath. By scaling the beam and pile element properties, the results of the simulation will represent the appropriate displacement response of the system and appropriate stresses in the soil, and only the forces in the structural elements will be scaled forces that can be corrected by using the same scale factor as that of the corresponding structural element. Therefore, if the properties of the pier and the piles and the pile-soil interface are scaled based on the width of the soil that they interact with, plausible representation of the soil-foundation-structure interaction in a plane strain model would be possible. Detailed explanation of the properties scaled can be found in FLAC's manual (Itasca 2005). 2.6 Sequence of Analysis The preparation of the numerical model for dynamic analysis included various stages of analysis for gravity loads to simulate the construction sequence. This must be done because of nonlinear stress-strain relationship of soil and because sudden application of all gravity loads causes unrealistic yielding of the soil. First, the constructed model of the soil layer was analyzed under its self weight to obtain the at rest soil stresses. In the second step, the foundation was constructed and the stresses were updated for the self weights. In the third stage, the bridge pier was constructed and the load on the bridge was applied incrementally. After addition of each increment of load, the stresses in the model 36 were updated again by running the analysis. The final updated model after application of all the gravity loads is the one used for the dynamic analysis. 2.7 System Physical Properties An advantage of fully coupled numerical models that include all the components of the soil-foundation-structure system is that they can be used to obtain the physical properties of the system that otherwise may not be easy to determine. These properties include the stiffness of the foundation, the stiffness of the pile-supported pier, and the initial natural period of the pile-supported pier (i.e. the elongated period of the pier). 2.7.1 Stiffness of Foundation Both translational and rotational stiffness of the foundation was obtained. For the translational stiffness, the rotational degree of freedom of the pile cap was restrained and the pile cap was pushed laterally by applying a lateral velocity at low amplitude ( l x l O 6 m/s). The force on the pile cap (base shear of the pier) and the pile cap translation (the displacement of the pile cap with respect to the free field) were tracked to obtain the translational stiffness of the foundation as illustrated in Figure 2.18. For the rotational stiffness, the rotational degree of freedom of the pile cap was released and its translational degree of freedom was constrained. The low velocity was applied to the top of pier resulting in overturning moment on the foundation. To obtain the rotational stiffness, the foundation moment (bending moment of the pier) and the rotation of the pile cap were tracked, as shown in Figure 2.19. The initial translational and rotational stiffness of the foundation can be taken as the tangent stiffnesses shown in Figures 2.18 and 2.19 respectively. The resulting initial translational stiffness is 3.0x105 kN/m and the resulting initial rotational stiffness is 1.75xl07 kN.m/rad 37 40000 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Rotation of Pile Cap (rad) Figure 2.19: Rotational stiffness of foundation 38 2.7.2 Stiffness of Pile-Supported Piers The stiffnesses of the pile-supported piers were estimated by laterally pushing the centre of the mass of each pier by applying low amplitude horizontal velocity (see Chapter 9 for more details). The base shear of each pier was plotted against its total (global) displacement as illustrated in Figures 2.20 to 2.25 which correspond to fixed-base periods of 0.3 to 2.0 s, respectively. The tangent stiffnesses obtained from these force-displacement diagrams are the initial stiffnesses of the flexible-base systems. For the sake of comparison, the force-displacement curves of the fixed-base piers are also plotted in Figures 2.20 to 2.25. The stiffnesses of the fixed-base and the flexible-base piers are listed in Table 2.2. The effect of the flexibility of the base of the piers can be seen in the figures and the table. As the piers become stiffer (higher pier-to-foundation stiffness ratio), the flexibility added by the foundation becomes more pronounced, which results in more deviation of the flexible-base stiffness from that of the fixed-base stiffness. Table 2.2: The Fixed-base and the flexible-base stiffnesses of the prototype bridge piers Fixed-base Natural Fixed-Base Flexible-Base Initial Period T Cracked Stiffness Cracked Stiffness (s) (kN/m) (kN/m) 0.3 153236 105000 0.6 39489 31790 0.8 22341 18800 1 14391 12450 1.5 6472 5615 2 3709 3225 39 Figure 2.20: Stiffness of the pile-supported pier with fixed-base period of T=0.3 s Figure 2.21: Stiffness of the pile-supported pier with fixed-base period of T=0.6 s 40 Figure 2.22: Stiffness of the pile-supported pier with fixed-base period of T=0.8 s 1000.00 0.00 0.02 0.04 0.06 0.08 0.10 Total Displacement (m) Figure 2.23: Stiffness of the pile-supported pier with fixed-base period of T=1.0 s 41 800.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Total Displacement (m) Figure 2.24: Stiffness of the pile-supported pier with fixed-base period of T=1.5 s Figure 2.25: Stiffness of the pile-supported pier with fixed-base period of T=2.0 s 42 2.7.3 Natural Period of the Pile-Supported Piers The natural period of the system is an important property of the system that indicates the period elongation of the pier due to the flexibility of its base. Since the system period depends on the amplitude of vibration due to nonlinearity of the soil, it would be more appropriate to estimate the elongated period of the vibration of the system at low amplitudes of motion. The natural periods thus obtained are regarded as the initial first mode natural period of the system (simply referred to as the system natural period). The system natural period was obtained by applying sinusoidal motions of various frequencies at the bottom of the soil layer and by estimating the system period by obtaining the transfer functions between the piers and the soil surface motions. The amplitude of the input motion was low to avoid the yielding of the piers. For each pier, this was carried out twice with different amplitudes of input harmonic motions and the average value was taken. Results are summarized in Table 2.3. Table 2.3: Natural periods of the pile-supported piers and the period elongations Natura Period Period Ratio Tsys/T Period Elongation (%) Fixed-Base T (s) Pile-Supported T s y s (s) 0.3 0.4 1.33 33 0.6 0.67 1.12 12 0.8 0.88 1.10 10 1 1.08 1.08 8 1.5 1.57 1.05 5 2 2.04 1.02 2 Table 2.3 lists the natural period of the fixed-base piers along with their corresponding elongated period due to SSI (T s y s). Researchers have previously shown that SSI depends on the pier-to-foundation stiffness ratio and have also shown that the stiffness ratio is correlated to the period ratio T s y s / T (e.g. Finn 2004). Therefore, the period ratio T s y s / T is also listed 2.2. The value of this ratio increases with increasing pier-to-foundation 43 stiffness ratio and therefore it is a good indicator of the flexibility of the base of the pile-supported bridge piers. The period and the period ratio o f the piers are used in this study as system variables, as a function of which the effects of SSI are described. 2.8 Summary In this Chapter, first the prototype systems used in this study were introduced. Then the construction of the numerical models representing the prototype systems was explained in detail for each component of the system and for the system as whole. The modeling of each component was verified where necessary and issues related to modeling of the system as a whole was discussed. The numerical models were used to estimate the systems physical properties. Estimation of physical properties demonstrated one of the applications of the direct methods of SSI analysis. This chapter dealt with verifying the numerical modeling at the component level. Next Chapter complements the verification of the modeling technique presented in this chapter at the system level. 44 3 ANALYSIS VERIFICATION STUDY: DYNAMIC ANALYSIS OF AN INSTRUMENTED PILE-SUPPORTED BRIDGE PIER In Chapter 2, the procedure of numerical modeling technique used in this study was presented and the modeling of various components of the system was verified. However, the overall system response still needs to be verified to validate the data obtained from the analyses of the prototype bridge pier systems. The verification of system response can be done by simulating the response of a system for which test results are available, or by simulating the response of a real system that has been subjected to a real earthquake and has recorded motions in response to that earthquake. Note that none of the two options provide a perfect means of calibrating or verifying numerical simulations and each method has its own advantages and limitations. Tests are usually performed under controlled environment with sufficient information about the system being tested. However, testing itself has limitations in representing real soil-foundation-structure systems which makes the test results less than perfect. As for real systems with recorded motions, the difficulties arise from the lack of complete information about the system. This includes the site soil, the foundation and the structural properties, as well as the input ground motion, which is a major source of uncertainty and can greatly influence the success of the simulation. In this research, a real bridge structure subjected to a real earthquake was chosen for the verification purposes. In order to make the case-history a suitable one for the purpose of this study, the bridge pier was chosen to have a simple configuration with similar characteristics to the prototype piers of this study. The case-history analysis of this bridge is presented in this chapter. 45 3.1 Structure and Instrumentation of the Bridge The bridge considered for this verification study is shown in Figure 3.1. It is an elevated section of the California Bay Area Rapid Transit (BART) system located just north of the Hayward Station (37.67° latitude and 122.086° longitude). This bridge was instrumented by the California Geological Survey (formerly the California Division of Mines and Geology) under its Strong Motion Instrumentation Program (CSMIP), and recorded strong motions during the 1989 Loma Prieta earthquake (Shakal et al. 1989). The significant effect of the SSI on seismic response of this bridge was previously investigated by Tseng et al. (1992) using the recorded strong motion. Figure 3.1: Photo of the Hayward BART Elevated Section 46 The pier modeled in this study is part of a three-span instrumented section of the bridge. This pier is labelled as "Bent 132" in Figure 3.2 which shows the geometry and the instrumentation of the bridge. This pier consists of a single reinforced concrete column with a hexagonal cross section having a 1.5 m (5') face-to-face dimension. The reinforcement of this column consists of two rings of Grade 60 reinforcing bars with a total of 44 #18 bars at its base (28 in its outer ring plus 16 bars in its inner ring). This pier supports simply supported decks consisting of concrete twin box girders. The foundation of this pier (see details in Figures 3.3 and 3.4) consists of a 5.5x5.5x1.7 m (18'xl8'x5.5') pile cap with 18 reinforced concrete piles with allowable load of 55 tonnes (60 tons). The inner piles are battered with a slope of 8:1 and the outer piles are battered with a slope of 4:1 and were driven approximately 15 m below the bottom of the pile cap. The as-built drawings of the foundation indicate two pile classes could be used for this foundation but it is not clear as to exactly which one was used. One of the pile classes is a tapered corrugated steel shell filled with concrete and the other pile class is a precast prestressed concrete square pile with 0.30 m (1') dimension. The properties of the latter pile class were used in this study. 47 H a y w a r d - B A R T E l e v a t e d S e c t i o n ( C S M I P S l a l i o n N o . 5 8 5 0 1 ) 111 B o n l 132 S E N S O R L O C A T I O N S , 7JL' 7 7 ' - f -6 ont 1 3 3 r - , , . Bant 134 elevation »58 k B . n t 13S 3 T 120 Tri, I O f *7 ' i 8 i t i J J To Hayward i—" IM. ^ J 1 i l l 1 i j j{—l *" t 1 BART S t a t i o n Deck Plan "1% 'I9 1 e e F i e l d S l r u c t u i e Reforence O r i o n l a t i o n : N = 3 1 0*" Note: l'=0.305 m i 13 C o n c r * t * Pad f i i i l E/W Sect ion Bent 132 C o n c r * t « P »d E/W Sect ion Bent 135 Figure 3.2: Geometry and instrumentation of the bridge (After Shakal et al. 1989). *8 P/LB PATTERN - £ PLAN SHOWN t-ht/ffrn riumb&r- denotes rht~ totat quantity of pi/«S r&gwr&cS pfr footing. Legend'. ~( fH* S a f t e r pile 4 6n / •fy* Batter piie 8 on I -fy Vertical pite Arrow indicates direction of batter Figure 3.3:. Foundation plan and pile pattern (per as-built drawings). Note: 1 '=0.305 m Bottom of footing Estimated, pi/et ftp aieyat/'orj 'Construetion Joint . r-<3F£kt/id /the QJ Figure 3.4: Cross section of the foundation (per as-built drawings) 49 3.2 Modeling and Verification Procedure Because the spans of the instrumented section of the bridge are straight and simply supported, it is reasonable to assume that the transverse and longitudinal vibrations of the pier are uncoupled and therefore they can be studied independently. Furthermore, Tseng et al. (1992) demonstrated that because of the continuous rails that are rigidly fastened to the girders, the coupling of response across the spans is significant in the longitudinal direction, but not significant in the transverse direction. Therefore, the transverse response of the pier 132 was selected for the purpose of this study. The modeling procedure follows that described in Chapter 2. The verification study was performed in three stages. In the first stage, after selection of the soil properties and the rock input motion, site response analysis was performed and the computed surface motion was checked against the measured surface motion. This was done mainly to verify the selected soil properties and bedrock motion. In the second stage, the dynamic response of the fixed-base structure obtained from F L A C was checked against that obtained from SAP2000 (Computer and Structures 2003) in order to verify the modeling of the pier and to calibrate the local damping of F L A C (see Chapter 2). In the third stage, the soil, the foundation and the structure were all included in the model. The calibration of the soil-foundation interface and the approximations required for the modeling of the structural members in a plane strain analysis were as explained in Chapter 2. Conclusions in regard to the success of the simulation are drawn by comparing the computed and the recorded responses of the pier. 3.3 Site Response Analysis 3.3.1 Soil Profile and Properties The available soil data was based on the original logs of boring at the bridge site which includes the soil classification, soil density and moisture content, and the blow counts to depths of up to 30 m. Undrained shear strength values in the upper 10 m of the soil were also available and the water table was at about 20 m below surface. This data shows that the site soil consists of sandy clay with sand and clayey sand layers in between. The 50 initial shear modulus of soil ( G m a x ) was estimated based on the available SPT data and was distributed proportional to the square root of the mean normal effective stress of soil within each layer (see Chapter 2). The mean normal effective stresses were obtained by performing static analysis and updating the stresses in soil before performing the dynamic analysis. The G m a x profile thus obtained is depicted in Figure 3.5. The depth of the bedrock is unknown and it was determined so that the calculated fundamental frequency of the soil layer matched that obtained from the recorded surface motion. Information regarding the site soil stiffness was also obtained from other available documents including Borcherdt (1994) and Stewart et al. (2001). The available data in these sources are in the form of mean shear wave velocity to the depth of 30 m (Vs30). Borcherdt (1994) and Stewart et al. (2001) report a Vs30 of 365 m/s and 276 m/s for this site, respectively. These values of Vs30 were compared to that estimated for the soil profile with the G m a x profile of Figure 3.5. Vs30 is calculated from the Equation 2.1. The Vs30 thus calculated for the soil layer with the G m a x profile of Figure 3.5 is 257 m/s, based on p = 2005 kg/m3 for the top sandy clay layer and p = 2038 kg/m3 for the other layers. This value of Vs30 is slightly lower than that reported by Stewart et al., but the difference is not significant. 51 Sandy Clay - Silty Sand Sandy Clay or Clayey Sand or Gravelly Sand 300 Figure 3.5: Distribution of G m a x The Mohr-Coulomb constitutive model was used for the soil. The bulk modulus was assumed to be three times the shear modulus (corresponding to a Poisson ratio of 0.35) for the static analysis. For the dynamic analysis, the bulk modulus was assumed to be equal to the shear modulus for the soil above the water table, and was assumed to be 50 times the shear modulus for the saturated soil below the water table. The strength parameters were obtained from either available shear strength of the soil or typical shear strength values based on the descriptions of the soil in the boring logs. An average cohesion of 100 kPa was considered for the top sandy clay layer, a friction angle of 9=34° was used for the silty sand layer, and an increasing-with-depth cohesion was assumed for the bottom sandy clay layer with a lower bound of 100 kPa. The distribution of cohesive strength of soil was assumed based on the equation proposed by Skempton (Craig 1987) as described by Equation 2.3 in Section 2.2.1.2. The plasticity index, PI, was assumed to be 15 for clay with low plasticity. It is noted that because of the lack of information about the bottom layer, the properties of sand and clay were used separately for sensitivity analyses to define the yield strength of soil, but only minor difference in response was observed partially due to minor yielding of soil in either cases. 52 The nonlinear hysteretic behaviour of the soil prior to yield was based on well known modulus reduction curves. For the sandy clay layers, the modulus reduction curve proposed by Sun et al. (1988) for clays with plasticity index of 10 to 20 was used. For the silty sand, the modulus reduction curve proposed by Seed and Idriss (1970) was used. Figure 3.6 shows these modulus reduction curves. Clay, PI=10-20 (Sun etal. 1988) c n « ^ /Qn^A ~~ i I J - : „ „ t m m L j a n u \ j r - c u a n u iu i i s& 0.0001 0.001 0.01 0.1 1 10 Shear Strain % Figure 3.6: Soil shear modulus reduction curves 3.3.2 Nonlinear Behaviour of Soil Elements in FLAC The modulus reduction curve and equivalent viscous damping exhibited by the soil element undergoing the cyclic loading are presented in Figures 3.7 and 3.8. Computation of these curves is explained in Section 2.2.1.3. Figure 3.7 shows a good match between the modulus reduction curves. Figure 3.8 shows that the soil element of F L A C was under-damped at low strains and was over damped at high strains, but overall was within the boundaries set by these curves at strains relevant to the recorded amplitudes of motion. To compensate for the low damping at small strains, Rayleigh stiffness damping was added to the numerical model. 53 Figure 3.7: Shear modulus reduction exhibited by a single element in F L A C Figure 3.8: Equivalent viscous damping exhibited by a single element in F L A C 54 3.3.3 Input Rock Motion The input rock motion is a major source of uncertainty and its selection is critical for a successful case history analysis. In this study, first, candidate input motions were selected from nearby surface motions recorded on firm soil or rock outcrop. Two closest rock outcrop sites were at the CSUH (California State University, Hayward) stadium grounds and the Hayward City Hall as listed in Table 3.1. The relative location and coordinates of these ground motion recording stations along with the coordinates of the site of the bridge and the epicentre of the Loma Prieta earthquake are shown in Figure 3.9. The Vs20 of these sites, also listed in Table 3.1, was obtained from Borcherdt (1994) and Stewart et al. (2001). Additional information on the site soil profile was obtained from Stewart and Stewart (1997) who summarized borehole data from various sources. Stewart and Stewart categorize both sites geology as rock. Table 3.2 shows the components of the recorded surface motions on each site. The most plausible input motion was chosen from these records as explained in the following sections. Table 3.1: Site's nearby rock outcrop motions Station Agency Station # Latitude (deg) Longitude (deg) v s 3 0 (Borcherdt 1994) m/s v s 3 0 (Stewart et al.2001) m/s Hayward C S U H Stadium Grounds CSMIP 58219 37.657 122.061 525 617 Hayward City Hall USGS 1129 37.679 122.082 N / A 745 Table 3.2: Components of the recorded ground motions Station Agency Station # Latitude Longitude Component's Clockwise Angle with respect to North (degree) Hayward C S U H Stadium Grounds CSMIP 58219 37.657 122.061 0 90 Hayward City Hall USGS 1129 37.679 122.082 64 334 55 N B Epicentre A Hayward BART Station B Hayward CSUH Stadium Grounds O Hayward City Hall 38 37.9 37.8 I- 37.7 37.6 37.5 37.4 37.3 37.2 37.1 37 122.2 122.1 122 121.9 Longitude (W) 121.8 Figure 3.9: Relative location and coordinates of the epicentre, the outcrops recording stations, and the bridge site free field recording station 3.3.4 Nonlinear Dynamic Analysis The site response analysis was performed using both input rock motions. Each ground motion of the Table 3.1 has two horizontal components as listed in Table 3.2 and the analysis was performed for all of these components. The resolved component of the CSUH Stadium Grounds record in the same direction as the recorded free field motion at the bridge site was also used. The computed surface response was then compared with the recorded surface motion and the correlation between them was studied. The correlation study was carried out in time and frequency domains. The purpose was to find the best input motion which results in a computed surface motion that is in best agreement with the recorded surface motion in time and frequency domains. The best input motion was found to be the Hayward City Hall 64° record (Table 3.2). It is noted that this record and the free field motion are relatively well aligned with only 24° angle between them. 56 3.3.5 Comparison of the Recorded and the Computed Surface Motions Figure 3.10 shows a comparison of the 2%-damped acceleration, velocity and displacement response spectra (SA, SV, and SD respectively) of the input motion, computed surface motion and the recorded surface motion when the Hayward City Hall 64° record was the input motion. This figure shows that the two surface responses are in good agreement at frequencies up to 3.5 Hz for the SA, SV and SD spectra; but at higher frequencies good agreement was not achieved for the SA spectrum, and to some extent for the SV spectrum. The spectral accelerations of Figure 3.10 show that the selected bedrock motion has peaks at 5.0 and 7.1 Hz, which are the frequencies at which the computed surface motion has peaks as well. This observation demonstrates that the higher vibration modes of the soil layer could greatly depend on the frequency content of the input ground motion. Further comparison of results in frequency domain can be made by obtaining the transfer functions between the input motion and the recorded surface motion, and between the input motion and the predicted surface motion. Figure 3.11 illustrates this comparison and shows that the simulation was capable of predicting the first two vibration frequencies of the site with natural frequencies of less than 3.5 Hz, but higher frequencies were not accurately captured. Figure 3.12 compares the computed and the recorded surface motions in time domain. This figure shows reasonable agreement between the two records with the Pearson correlation coefficients (Neter et al. 1996) of 0.41, 0.71, and 0.82 for acceleration, velocity and displacement response respectively. The good correlations between the velocities and between the displacements imply the appropriate selection of the input motion and site profile and properties. The low correlation of the accelerations is due to the lack of correlation between the high frequency components of the records. The predicted peak surface acceleration is about 25% less than that of the recorded surface motion, which can be explained by the lower energy of the predicted response at higher frequencies (between 3.5 and 10 Hz) as was observed in Figure 3.11. 57 0.75 Cn 0.25 1 10 Frequency (Hz) 100 C J 25 1 10 Frequency (Hz) 100 § 22.5 Frequency (Hz) Bedrock Motion (Hayward City Hall 64 deg.) • Recorded Surface Motion (Hayward BART Station FF 220 deg.) Computed Surface Motion ;ure 3.10: Comparison of the recorded and the computed response spectra 58 15 Between Input and Recorded Surface Motions Between Input and Computed Surface Motions 0' 0 5 10 15 20 25 Frequency (Hz) Figure 3.11: Comparison of transfer functions between input and recorded surface motions and between input and computed surface motions The observed discrepancies could arise from the uncertainties in the soil properties or from the uncertainties in the nature of the input rock motion, or could arise from the presence of significant surface waves in the recorded surface motion that cannot be accounted for in the presented analysis. Discrepancies could also result from the imperfections of the numerical representation of the system. However, the reasonable degree of agreement between the recorded and the computed surface motions demonstrates the validity of the model and the capability of the analysis to accurately capture the fundamental characteristics of the site. Since complete information about the site and the bedrock motion at the site are not available, no definitive conclusions can be drawn regarding the most important source of uncertainty causing the discrepancies at higher frequencies. The analyses results were more sensitive to the input ground motions suggesting that the discrepancies must, to a great degree, be related to the characteristics of the actual bedrock motion of the site that are not completely present in any of the available outcrop motions. It should be pointed out here that the strong dependency of the accuracy of the predicted site response on input rock motion characteristics has also been observed in the studies by other researchers, such as Kramer and Stewart (2004). Kramer and Stewart state that not knowing input rock motion characteristics as a-priori could introduce significant uncertainty to the results of ground response analysis which result from uncertainty and variability in input motion characteristics. 59 Displacement (cm) V e l o c i t y (cm/s) Accelerat ion (g) to —• N> o O «° bo n o 3 — s. to O 3 c —! 5-o o o 3 •a c — a. o —. o o o -i O , o Cu "5 O o -I fD oo TJ O co Ct 3 o O o 3 T3 C o a. o o a. on 2 o > H c -n — to to — ti 0= H 3 to O H 3 ' o 3.3.6 Comparison of FLA C and SHAKE SHAKE (Schnabel et al. 1972) is a program that has been widely used for site response analysis. Hence, the surface motion computed by F L A C was compared to that computed by SHAKE, as a final step in the verification process. The analysis was performed using SHAKE91 which is a modified version of the original program (Idriss and Sun, 1992). It is noted, however, that the fully nonlinear analysis of F L A C is superior to the equivalent linear analysis of SHAKE and the results can be expected to deviate especially at high soil strains. Nevertheless, at the level of strains observed in this case study, good agreement between the two analyses could be expected. The soil properties used in SHAKE are equivalent to that used in F L A C . The shear modulus curve used in the SHAKE analysis was that for clays with plasticity index of 10 to 20 (Sun et al. 1988), and the damping curve was that exhibited by F L A C as shown in Figure 3.8. The comparison of the surface acceleration, velocity and displacement response spectra obtained from the two programs is shown in Figure 3.13. It is observed that the surface motion obtained from SHAKE has slightly higher SA, SV, and SD amplitudes, indicating that the SHAKE model exhibited slightly less damping than the F L A C model. Also there is poor agreement of spectral accelerations in the vicinity of 7 Hz. These discrepancies are minor and an overall good agreement between the results is observed. This further verifies the numerical analysis performed with F L A C and its agreement with other well-established methods of site response analysis. 61 M 0.75 t / i 0.25 1 10 Frequency (Hz) 100 1 10 Frequency (Hz) 100 I 22.5 • FLAC SHAKE Frequency (Hz) 100 Figure 3.13: Comparison of spectral surface responses obtained from S H A K E and F L A C 62 3.3.7 Estimation of the Input Motion from the Surface Motion It is common in practice that when no outcrop motion is available, the input motion is estimated by deconvoluting the recorded surface motion down to the level of input motion by using, for instance, the program SHAKE. As pointed out in Section 3.3.5, a major source of discrepancy between the recorded and the computed surface motions is the input ground motion. Thus, in order to reduce the discrepancies associated with the input motion, the above-mentioned method was used to compute the input motion from the recorded surface motion. This was the final step in estimating an input motion that would result in a more plausible surface response at frequencies higher than 3.5 Hz. The ground motion at the bottom of the soil layer obtained from S H A K E was used as the input to the F L A G model and the surface motion was estimated. Figure 3.14 illustrates the site response when this computed input motion was used. As can be seen, except an increase of the spectral acceleration at 3.0 Hz, better agreement between the recorded and the computed surface motions was achieved in both frequency and time domains. The Pearson correlation between the recorded and the computed accelerations was increased from 0.40 to 0.78 and for the velocity was increased from 0.71 to 0.86, implying the improvement of the computed surface motion. 63 2% damping Recorded Surface Motion (Hayward BART Station FF 220 deg.) Computed Surface Motion i < 1 1 Kl 1 W , j , < . i . • I 1 0 5 10 15 20 25 30 35 40 Time (s) f 1 0 5 10 15 20 25 30 35 40 Time (s) Recorded Surface Motion (Hayward B A R T Station FF 220 deg.) Computed Surface Motion Figure 3.14: Surface motion computed by FLAC in response to the input motion generated by SHAKE 64 3.4 Analysis of the Fixed-Base Pier The fixed-base structure was analyzed with F L A C and SAP2000 to ensure the accuracy of the numerical model of the pier and to calibrate the non-Rayleigh damping used in the simulation as explained in Section 2.3.2. A stick model of the pier was developed by using beam elements that were connected to the grid through relatively rigid beam elements. Relatively rigid beam elements also represented the mass of the bridge deck at its centre of mass. Since concrete continues to gain strength with age and the actual modulus of elasticity of concrete structures exceeds the typical value corresponding to the 28-day compression strength by 20 to 50% (Priestly et al. 1996), the elastic modulus of concrete was taken 35% more than its typical 28-day value. For the recorded amplitude of motion, the gross cross section properties of the pier were used and the damping (presented by "local damping") was equivalent to 1.5% of the critical viscous damping. Plastic hinging at the pier's base and P-A effect were also considered in the analysis of the structure, but they did not play a role at the level of motions considered in this study. Using the above-mentioned properties, the fixed-base natural frequency of the structure was computed to be 2.35 Hz. 3.5 Analysis of Soil-Foundation-Structure Interaction Figure 3.15 depicts the model used for the soil-foundation-structure analysis. Non-reflecting boundaries were used for the soil, and the input motion was applied at the bottom of the soil mesh. The bridge pier was modeled as was explained in the previous section. The linear scaling of properties discussed in Section 2.5.2 was applied to account for the modeling of the three dimensional system in the two dimensional plane strain analysis. The pile cap was modeled using plane strain elements having concrete properties which interact with the surrounding soil through interface elements. No contact between the pile and soil was assumed at the bottom of the pile cap but the passive pressures on the sides were taken into account. The interface properties were estimated as described in Section 2.5.1. Piles were modeled with pile elements that interact with the surrounding soil through normal and shear coupling springs. The coupling springs were calibrated according to the 65 procedure described in Section 2.5.1.1 and the linear scaling of properties was applied. It is mentioned here that the analysis showed that the soil-structure-interaction of this bridge pier was dominated by the rocking of the foundation and showed no significant yielding in the normal springs at the level of the recorded motion. Figure 3.15: F L A C model (top 30 m of soil) of the instrumented bridge pier system 66 3.6 Response Ana l ys i s Figure 3.16 compares the acceleration response spectrum of the computed response of the bridge pier with that of the recorded response by sensor #12 shown in Figure 3.2. In general, results show a good prediction of the response by the simulation, especially at the fundamental frequency of 1.8 Hz and the site period of 1.0 Hz. This shows that while the fixed-base fundamental frequency of the pier is at 2.35 Hz, the fundamental frequency of pier-foundation-soil system is reduced by about 30% to 1.8 Hz due to the rocking of the foundation. This is an important observation and demonstrates the capability of the simulation to capture the primary effect of soil-structure interaction on the system's characteristics. Figure 3.16 also shows a minor peak in the recorded response at 3.7 Hz which was not captured by the simulation. This could be due to imperfections of the model, such as simplifications in the presentation of the deck, simplifications in modeling the complex pile configuration and the soil-pile interface, uncertain soil properties, or considering no coupling between the longitudinal and transverse response of the bridge. Although the actual behaviour of the bridge is certainly more complex than can be fully captured by such simulation, shortcomings may be reduced in light of more accurate information on the physical properties of the system and furthermore by employing three-dimensional simulations. 67 Figure 3.16: Recorded and computed acceleration response spectra of the pier 3.7 S u m m a r y This chapter presented a verification study on the numerical analysis methodology employed in this research by predicting the response of an instrumented bridge pier subjected to an actual earthquake. The analysis involved many uncertainties especially in regard to the input ground motion and the soil profile which were rationally dealt with. Good correlation between the computed and the recorded motions demonstrated the capability of the numerical simulation to capture the system's response and the effects that SSI had on the response of the bridge pier. The results of this chapter complemented the verification of the modeling technique presented in Chapter 2 at the system level. Therefore, the numerical models of the prototype piers were confidently used in the rest of this study to evaluate the effects of nonlinear SSI on the inelastic seismic response of bridge piers. Next chapter presents the dynamic analysis of the prototype bridge piers. 68 4 NONLINEAR DYNAMIC ANALYSIS OF THE PROTOTYPE BRIDGE PIERS This chapter presents the selection of input ground motions for the nonlinear dynamic analyses and describes various stages of dynamic analysis including site response analysis, analysis of the piers without SSI, and analysis of the system as a whole including SSI. The primary goal of this chapter is to examine the behaviour of the system rather qualitatively by observing the salient characteristics of the dynamic response of the system, without emphasizing on the statistics of response. Sample results are presented to gain insight into the contribution of various system components in the overall dynamic response of the system. 4.1 Selection of input Rock motions The ground motions used in this research were chosen to represent various levels of ground shaking with different time and frequency characteristics. They were selected from historic ground motions recorded on rock or very stiff soil. Al l the records were taken from the Pacific Earthquake Engineering Research (PEER) center's strong motion database (http://peer.berkeley.edu/smcat/). To avoid uncertainties introduced by the scaling of ground motions, the records were used as they are and were not scaled or further processed. The selected ground motions were all recorded on rock or stiff soil and were chosen from different seismically active regions. The ground motion parameter used for the selection purpose was the peak ground acceleration (PGA). Another criterion was limiting the corner frequency of the high-pass filter used in the processing of the records to 0.2 Hz. 69 The records were selected to represent moderate to severe ground shaking with a mean acceleration response spectrum that is comparable with the design response spectra proposed by the Canadian Highway Bridge Design Code (CAN/CSA-S6-00 2000) for Zonal Acceleration Ratio (A) of 0.2 and 0.3, representing, respectively, the seismic hazard of the cities of Vancouver and Victoria in the province of British Columbia. Table 4.1 shows a list of the 26 selected ground motions along with their magnitude, distance, the site class of the recording station, and the peak ground motions including the peak ground acceleration (PGA), the peak ground velocity (PGV), and the peak ground displacement (PGD). The listed records are sorted for PGA, ranging from 0.080g to 0.587g (g is the acceleration of gravity). Peak ground velocities are from 2.9 cm/s to 62.0 cm/s and peak ground displacements are from 0.2 cm to 51.8 cm. The selected ground motions are from earthquakes with magnitudes of 5.8 to 7.6 recorded at distances from 8.0 to 48.8 km. Site classifications of the recording stations are from three sources including the United States Geological Survey (USGS), Geomatrix Consultants, and Taiwan's Central Weather Bureau (CWB). USGS-A and USGS-B site classes are sites with vs30 > 750 m/s and 360 < vsi0 < 750 m/s respectively. Geomatrix A sites are those classified as "rock" and Geomatrix B sites are those classified as "shallow (stiff) soil". CWB 1 sites are described as "hard sites". Further information on the site classes can be found at http://peer.berkeley.edu/smcat/sites.html. Figure 4.1 illustrates the 5%-damped acceleration response spectra of the ground motions listed in Table 4.1 along with the mean, the maximum envelope, and the minimum envelope of the spectral accelerations. Figure 4.1 also shows the design spectra (Seismic Response Coefficient) from the Canadian Highway Bridge Design Code (CAN/CSA S6-2000) for the Soil Profile Type I (e.g. rock sites) and for the Zonal Accelerations (A) of 0.2 and 0.3. As can be seen in this figure, the mean spectrum is comparable to the design spectrum for A = 0.2. For the periods longer than 0.75 s, the amplitude of the mean spectrum is consistently lower than that of the design spectrum. This is attributed to the 70 conservative values of the design spectrum at longer periods which has been acknowledged in the literature (NCHRP Report 472, 2002). Figure 4.1: Acceleration response spectra of the input rock motions 71 Table 4.1: Input rock motions Earthquake Year Station Component Mw MI Ms D (km)* Site Class* PGA (g) PGV (cm/s) PGD (cm) DI D2 D3 Landers 1992 22161 Twentynine Palms 29P000 7.3 42.2 41.9 USGS A (Geomatrix A ) 0.080 3.7 2.3 Morgan Hill 1984 47379 Gilroy Array #1 G01320 6.2 16.2 USGS A (Geomatrix A ) 0.098 2.9 1.0 N. Palm Springs 1986 5224 Anza - Red Mountain ARM270 6 45.6 USGS A (Geomatrix A ) 0.104 5.2 0.6 Chi Chi, Taiwan 1999 HWA056 W 7.6 48.8 44.5 USGS A 0.107 11.7 17.6 Duzce, Turkey 1999 Mudumu MDR000 7.1 33.6 33.6 Geomatrix A 0.120 9.3 7.6 Friuli, Italy 1976 8022 San Rocco SRO270 6.1 6.1 5.7 12.7 Geomatrix A 0.134 7.6 2.0 Nahanni, Canada 1985 6099 Site 3 S3270 6.8 16 Geomatrix A 0.148 6.1 3.1 San Fernando 1971 127 Lake Hughes #9 L09021 6.6 23.5 20.2 USGS A (Geomatrix A ) 0.157 4.5 1.3 Northridge 1994 90017 LA - Wonderland Ave 185 deg 6.7 22.7 USGS A (Geomatrix A ) 0.172 11.8 2.8 Whittier Narrows 1987 24399 Mt Wilson - CIT Seis Sta A-MTW090 6 21.2 USGS A (Geomatrix A ) 0.186 4.6 0.2 Northridge 1995 127 Lake Hughes #9 L09090 6.7 26.8 28.9 USGS A (Geomatrix A ) 0.217 10.1 2.8 Kocaeli 1999 Gebze GBZ000 7.4 17 17 USGS A (Geomatrix A ) 0.244 50.3 42.7 Northridge 1994 90019 San Gabriel - E. Grand Ave. GRN270 6.7 41.7 39.5 USGS A (Geomatrix A ) 0.256 9.8 2.8 Chi Chi, Taiwan 2000 CHY029 W 7.6 15.3 15.3 CWB 1 (USGS B) 0.277 30.3 14.7 Northridge 1994 24592 LA - City Terrace LAC 180 6.7 37 35.4 Geomatrix A 0.316 14.1 2.4 Nahanni, Canada 1985 6098 Site 2 S2330 6.8 8 Geomatrix A 0.323 33.1 6.5 Chi Chi, Taiwan 1999 TCU089 TCU089-W 7.6 8.22 CWB 1 (USGS B) 0.333 30.9 18.5 Parkfield 1966 1438 Temblor pre-1969 TMB205 6.1 9.9 16.1 Geomatrix A (USGS B) 0.357 21.5 3.9 Coalinga 1983 1605 Skunk Hollow D-SKH270 5.8 12.2 Geomatrix A 0.375 16.4 6.2 Chi Chi, Taiwan 2000 TCU095 W 7.6 43.4 43.4 CWB 1 (USGS B) 0.378 62.0 51.8 Loma Prieta 1989 Gilroy Array 1 Odeg 6.9 11.2 10.5 USGS A (Geomatrix A ) 0.411 31.6 6.4 Mammoth Lakes 1980 54214 Long Valley dam (Upr L Abut) I-LUL000 6.3 15.5 Geomatrix A 0.430 23.6 7.5 Northridge 1995 90049 Pacific Palisades - Sunset Blvd SUN190 6.7 26.2 17.1 USGS B (Geomatrix B ) 0.469 31.0 5.3 Loma Prieta 1989 57217 Coyote Lake Dam (SW Abut) CYC285 6.9 21.8 Geomatrix A 0.484 39.7 15.2 Northridge 1994 24605 LA - Univ Hospital UNI005 6.7 34.6 32.8 Geomatrix A (USGS B) 0.493 31.1 2.4 Victoria, Mexico 1980 6604 Cerro Prieto VICT/CPE315 6.1 5.7 34.8 Geomatrix A (USGS B) 0.587 19.9 9.4 * DI: Closest to fault rupture, D2: Closest to surface projection of rupture, D3: Hypocentral + See http://peer.berkelev.edu/smcat/sites.html for further explanation on site classifications 4.2 Site Response Analysis Nonlinear dynamic analyses of the soil layers were performed by analyzing three soil columns corresponding to three prototype soil layers with three different heights. Given the 26 input ground motions and the 3 soil layers, the analyses resulted in 78 surface motions. Figure 4.2 illustrates the peak surface accelerations obtained from the analyses versus the peak accelerations of the input ground motions. This figure demonstrates how the input ground motions were amplified by the soil layers and shows the amplification is dependent on the amplitude of the input motion. The logarithmic fit shown in Figure 4.2 is for the sake of comparison with the curve proposed by Idriss (1990) shown in Figure 4.3. This is merely a qualitative comparison of the two curves and demonstrates that the amplification of ground motions predicted by the site response analyses of this study is comparable to that proposed by Idriss (1990). This verifies the general agreement of the analysis results with the available data in the literature. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Peak Input Motion Acceleration (g) Figure 4.2: Site acceleration amplification obtained from analysis 73 Acceleration on Rock Sites (g) Figure 4.3: Site acceleration amplification from Idriss (1990) Figure 4.4 depicts the 5%-damped acceleration response spectra of the free field surface motions obtained from the site response analyses along with the mean, the maximum envelope, and the minimum envelope of the spectral accelerations. Also shown in Figure 4.4 are the spectra (Seismic Response Coefficient) from the Canadian Highway Bridge Design Code (CAN/CSA S6-200 2000) for the soil profile Type IV (e.g. soft clays with depths of greater than 12 m) and for Zonal Accelerations (A) of 0.2 and 0.3. The mean spectral acceleration of the surface response is bound between the two design spectra for the periods of up to 1.0 s, but it is consistently lower for the periods of longer than 1.0 s. The lower values are attributed to the low values of the input motions at periods longer than 1.0 second as explained earlier. To better understand the dynamic response of the prototype soil layers, the mean spectral acceleration as well as the minimum and maximum envelopes of both the input and the computed free field surface motions are plotted together in Figure 4.5. It can be observed that the mean spectral response was amplified which is also the case for the minimum envelope. The maximum envelope, however, was not amplified over all periods which is due to the deamplification of motion resulting from the yielding of soil when subjected to very strong motions. 74 5% damping Period (s) Figure 4.4: Acceleration response spectra of the computed surface motions 0.5 1 1.5 2 2.5 Period (s) Figure 4.5: Comparison of the acceleration response spectra of the input and the surface motions 75 4.3 Ana l ys i s o f F i x e d - B a s e P ie rs The dynamic responses of the piers without SSI were used as a benchmark for evaluating the effects of SSI on the response of the pile-supported piers. The response of the piers without SSI was obtained by assuming that the base of the pier was fixed. The idealized piers with fixed base are referred to as the "fixed-base piers", indicating that SSI is not accounted for, as opposed to the "pile-supported piers" or the "flexible-base piers" for which SSI is included. The input motions for the analysis of the fixed-base piers were the surface motions obtained from the site response analyses explained in Section 4.2. The analysis of each fixed-base pier was performed in a step by step fashion beginning with linear elastic analysis, then nonlinear analysis without consideration of the P-A effects, and finally nonlinear dynamic analysis with the P-A effects. With 78 surface motions used as the input and with 6 prototype bridge piers, a total of 468 analyses were performed at each step of the analysis of the fixed-base piers (a total of 1404 analyses). The results of the analysis were examined to ensure the simulation was robust. In a number of cases, P-A effects resulted in excessive deformation of the pier due to excessive rotation of the plastic hinge beyond the capacity of the pier. These cases were considered as outliers and were excluded from the database of the seismic response of the system presented in Chapter 5. Figure 4.6 shows and example of the time-history of the displacement response at the centre of the mass of a fixed-base pier with natural period of 1.0 s located on a 40 m deep soil layer when subjected to the Taiwan Chi Chi TCU089 input motion (Table 4.1). This figure shows the response both with and without P-A effects and demonstrates the excessive deformation caused by P-A effects. Seismic demands obtained from the analysis of the fixed-base structures are presented in Chapter 5 with further discussions on the validity of the results. 76 Figure 4.6: Sample displacement response at the centre of mass of a fixed-based pier with and without P-A effects 4.4 Ana lys i s of So i l -Founda t ion -S t ruc tu re Systems The dynamic analysis of the system as a whole was carried out for the 3 soil layers, 6 bridge piers and 26 input ground motions. This resulted in a total of 468 nonlinear dynamic analyses of the soil-foundation-structure system. The analyses accounted for the nonlinearity of the soil, the nonlinearity of the structure (both material and geometric), and the nonlinearity of the soil-foundation interface. For the cases in which the pier yielded, the analysis was repeated with an elastic pier (no material nonlinearity). Similar to the analysis of the fixed-base piers, the cases in which P-A effects resulted in excessive deformation of the pier beyond its capacity were excluded from the database of the seismic response of the system. An example of the response of a pier and its foundation in time domain is presented in Figure 4.7. The absolute maximum values obtained from the response histories were used to collect the database of the seismic demands of the system. The statistics of the seismic demands thus obtained are presented in Chapter 5. 77 u 0_ o E o E D. 5 0.2 a " C I -0.2 Pier's Total Displacement Pier's Displacement with Respect to its Base i iv A A /\ A A / 15 20 25 Time (s) 0.005 0.0025 3 o o I -0.0025 o OH "0.005 Time (s) Figure 4.7: An example of the response of a bridge pier and its foundation in time domain (T=0.8 s, Tsjte=1.0 s, input motion: Nahanni Site 2) 78 In order to reduce the analysis runtime, only the top 30 m of the soil layers with depths of more than 30 m was included in the numerical model of the soil-foundation-structure systems. The input ground motions in these cases were the free field motions at the depth of 30 m obtained from the dynamic analyses of the soil columns as explained in Section 4.2. However, due to the nonlinearity of soil response, the surface responses thus obtained may show discrepancies from those obtained from the analyses of the soil columns. To ensure that these discrepancies were negligible, comparisons were made between the surface responses obtained from the two numerical models. Examples are shown in Figure 4.8 for a 40 m soil layer (TSj t e = 1.0 s) and in Figure 4.9 for a 100 m soil layer (T sj t e = 2.0 s) when the Northridge University Hospital input motion (Table 4.1) was used. These figures demonstrate that the discrepancies are minor and that they only exist at frequencies above 10.0 Hz in the spectral accelerations. Since the dominant response of the pile-supported bridge piers located on these soil layers are at frequencies below 10.0 Hz, the effect of the observed discrepancies on the system response is negligible. It is noted that the response spectra of Figures 4.8 and 4.9 are obtained for one of the strongest input ground motions used in this study with a PGA of 0.49g resulting in high degree of nonlinearity in the soil response. 79 1.25 0.75 0.25 Frequency (Hz) 112.5 <*> 37.5 h 1 10 Frequency (Hz) E 22.5 Full-Height (40 m) Layer 30 m Layer Frequency (Hz) Figure 4.8: Comparison of the surface spectral response of the 40 m soil layer obtained from the analysis of a 40 m soil column and a 30 m soil column 80 5% damping ""i Full Height (100 m) Layer 30 m Layer \ wV i f —-•-•J*'^ * /A' 0.1 1 10 100 Frequency (Hz) 0.1 1 10 100 Frequency (Hz) 0.1 1 10 100 Frequency (Hz) Figure 4.9: Comparison of the surface spectral response of the 100 m soil layer obtained from the analysis of a 100 m soil column and a 30 m soil column 81 4.5 Seismic B e h a v i o u r of the System To examine the behaviour of the system and its components, examples of its response spectra when subjected to the Nahanni-Site 2 input motion (Table 4.1) are presented in Figures 4.10 to 4.15. In these figures, the spectral acceleration, the spectral velocity and the spectral displacement of the transverse motion of the pile-supported piers and their foundations (on the pile cap) are plotted. Also shown are the free field surface response and the response of the corresponding fixed-base piers. These figures provide information about the dynamic behaviour of the soil-foundation-structure system and the effects of soil-structure interaction on the response of the foundation and the pier. In these figure, the following can be observed: • The responses of the piers with SSI are different and mostly (but not always) lower than that of the piers without SSI. As the structure becomes stiffer with lower natural period, SSI becomes more effective in modifying the response of the pier. In other words, SSI becomes more significant with increasing pier-to-foundation stiffness ratio and therefore plays a more important role in systems with higher period ratio T s y s / T (see chapter 2). This observation is further discussed in the forthcoming Chapters by studying the statistics of the seismic demands. • The foundation translational response is lower than or equal to the free field response. The reduction of response demonstrates the effects of SSI and becomes more significant for the piers with shorter natural period (i.e. higher period ratio T s y s /T) • It appears that regardless of the period of the piers, the foundation response is less than the surface response at frequencies greater that about 10 Hz. This can be explained by the kinematic interaction, which filters out the high frequency components of the ground motion (Mylonakis et al. 1997). The role of the kinematic interaction can further be explored by analyzing the system with a massless pier to exclude the inertial interaction. An example of the spectral 82 acceleration response of the foundation thus obtained is presented in Figure 4.16. The role of the kinematic interaction in reducing the response of the foundation with respect to the free field can clearly be observed in this figure. SSI does not always reduce the response of the pier. For instance, higher response of the pier without SSI can be observed for the pier with natural period of 0.8 s, depicted in Figure 4.12. Further discussion on this issue can be found in the following chapters. The spectral displacements can be higher for the piers with SSI compared to those without SSI. This indicates that the piers may experience greater total displacement demand due to the effects of SSI and the motion of the foundation in the soil. For piers with shorter natural periods where SSI is more effective, an increase of the spectral displacement at very low frequencies (less than 0.2 Hz) can be observed for the piers with SSI; but it cannot be observed in the response of the same piers without SSI. This is attributed to the nonlinearity of the soil-structure interaction and the yielding of the soil or the soil-foundation interface, which ultimately results in the movement of the foundation with respect to the free field. 83 r n i —— Pier with SSI —— Foundation Translation — Free Field —— Pier without SSI A \ i ' i 0.1 1 10 100 Frequency (Hz) gure 4.10: Example of the response spectra of the soil-foundation-pier system (T=0.3 s (f=3.33 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 84 0.1 1 10 100 Frequency (Hz) 1 \ 1 — Pier with SSI — - Foundation Translation — Free Field • • Pier without SSI 0.1 1 10 100 Frequency (Hz) ;ure 4.11: Example of the response spectra of the soil-foundation-pier system (T=0.6 s (f=1.67 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 85 300 250 1 2 0 0 1 150 > | 100 a 50 0 0.1 Frequency (Hz) j \ \ 1 • Pier with SSI Foundation Translation Free Field Pier without SSI Frequency (Hz) 0.1 I 10 100 Frequency (Hz) gure 4.12: Example of the response spectra of the soil-foundation-pier system (T=0.8 s (f=1.25 Hz), T s i te-1.0 s, input motion: Nahanni Site 2) 86 Frequency (Hz) gure 4.13: Example of the response spectra of the soil-foundation-pier system (T=1.0 s (f=1.0 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 87 — Pier with SSI — Foundation Translation — Free Field — Pier without SSI 1 10 Frequency (Hz) 100 300 250 200 •3 150 100 I • Pier with SSI Foundation Translation Free Field Pier without SSI Frequency (Hz) 100 Pier with SSI Foundation Translation Free Field Pier without SSI 0.1 1 10 100 Frequency (Hz) gure 4.14: Example of the response spectra of the soil-foundation-pier system (T=1.5 s (f=0.67 Hz), Tsj,e=1.0 s, input motion: Nahanni Site 2) 8 8 200 • Pier with SSI Foundation Translation Free Field Pier without SSI Frequency (Hz) Frequency (Hz) S 40 •3 30 S 20 Pier with SSI Foundation Translation Free Field Pier without SSI 0.1 1 10 100 Frequency (Hz) ?ure 4.15: Example of the response spectra of the soil-foundation-pier system (T=2.0 s (f=0.50 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 89 Figure 4.16: Translation response of the foundation supporting a massless pier (kinematic interaction) 4.6 Summary The selection of the input ground motions were presented and various stages of dynamic analysis, including site response analysis, analysis of the fixed-base structures, and analysis of the system as a whole, were explained. The dynamic behaviour of the system was examined and the capability of the numerical model in representing the salient characteristics o f system response was demonstrated. Results were particularly presented to compare the response of the foundation with the free field surface motion and the response of the pier with SSI with the response of the pier without SSI, to demonstrate the effects of SSI on the response of the system. This chapter discussed the effects of SSI rather qualitatively by presenting examples of the responses of the system. Next chapter discusses the response of the system quantitatively by presenting the statistics of demands obtained from the analyses of the prototype bridge piers when subjected to the ground motions introduced in this chapter. 90 5 SEISMIC DEMANDS OF BRIDGE PIERS This chapter presents the statistics of the seismic demands extracted from the results of the nonlinear dynamic analyses discussed in Chapter 4. The presentation of results is in the form of cumulative frequency diagrams. Cumulative frequency diagrams illustrate what percentage of the data in a data set is smaller than or equal to any given value of the data in that data set. The presentation of results is not by merely the numerical descriptors of data, such as mean and standard deviation, because that will obscure useful information about the behaviour of the system that could otherwise be gained through observation of the distribution of data. The statistics of seismic demands, as presented in this chapter, directly provide insight into the role of SSI on the response of the piers. Further data processing is carried out in chapters 7 and 8 to better quantify this role. 5.1 Definition of Demand Parameters The demand parameters discussed in this dissertation are those relevant to displacement-based design of structures. The main demand parameters for the piers are the ductility and the total displacement demands. Other demand parameters are the strength reduction factor and the inelastic displacement ratio which relate the inelastic response of the piers to their corresponding response obtained from the analyses with elastic piers. The demand parameters of the foundation are the foundation translation and the foundation rotation. These parameters are explained in the following paragraphs. Note that the term "deformation" of the pier defines the displacement of the centre of mass at the top of the pier with respect to its base as explained in the following. Figure 5.1 shows the force-deformation relationship of a single-degree-of-freedom (SDOF) nonlinear system, by which the force-deformation relationship of the idealized 91 fixed-base bridge pier of this study, shown in Figure 5.2, can be described. In Figures 5.1 and 5.2, A y is the yield deformation which corresponds to the yield strength V y of the pier. The plastic deformation of the pier, corresponding to the plastic rotation 0 P , is denoted by A p (Equation 5.1). The maximum deformation of the pier with respect to its base, A c , is is the sum of the yield and plastic deformations (Equation 5.2). The corresponding ductility factor of the system, denoted by u., is defined by the Equation 5.3. A P = L - 9 P 1-1 = A, (5.1) (5.2) (5.3) V e and A e shown in Figure 5.1 are the force and the deformation of the corresponding elastic system, respectively. In other words, the base shear and the deformation of the pier obtained from an elastic analysis of the system are V e and A e , and the shear and the deformation obtained from an inelastic analysis are V y and A c. The relationship between the elastic and the inelastic demands is usually expressed in terms of the strength reduction factor, R, and the inelastic deformation ratio, C u (see Chapter 1 for background information). R and C u are used to estimate the strength and deformation demands of an inelastic system from their corresponding elastic demands obtained from the elastic analysis of the system. The strength reduction factor R is defined by Equation 5.4 and the inelastic deformation ratio C u is defined by Equation 5.5. In Equation 5.5, A e and A c are replaced by their corresponding values from the Equations 5.3 and 5.4, respectively, to obtain the relationship between the inelastic deformation ratio C u , the ductility factor \i and the strength reduction factor R. V y A y [ A„ R (5.4) (5.5) 92 The displacement parameters of the piers when the flexibility of their base is included are depicted in Figure 5.3. The flexible-base pier has two additional degrees of freedom defined by the foundation translation, denoted by A T , and by the foundation rotation, 0 R . Similar to the fixed-base pier, the ductility demand and the strength reduction factor of the flexible-base piers are defined by Equations 5.3 and 5.4 and by using the deformations of the piers with respect to their base as depicted in Figure 5.3. The total displacement of the flexible-base piers, however, is different than that of the fixed-base piers due to translation and rotation of the foundation. While the total displacement of the fixed-base piers is equal to A C , the total displacement of the flexible-base piers is described by Equation 5.6 in which A R is determined by Equation 5.7. A t o t = A y + A p + A R + A T (5-6) A R = L . e R (5.7) The aforementioned demand parameters were extracted from the displacements and rotations obtained from the dynamic analyses of the piers which were carried out with and without accounting for SSI, and with assuming both inelastic and elastic behaviour of the piers. 93 Force * A y A e A c Deformation Figure 5.1: Force-deformation relationship of an elastoplastic single-degree-of-freedom system A,0, = A c H A y A p I — H H Figure 5.2: Demand parameters of the idealized fixed-base bridge piers 94 AR A Y A P I — H H F i g u r e 5.3: D e m a n d parameters o f the idea l ized f lexible-base br idge piers 5.2 Seismic Demands of the Fixed-Base Piers A s prev ious ly expla ined, the seismic demands o f the fixed-base piers were used as a benchmark for evaluat ing the effects o f S S I on the response o f the pi le-supported piers. T h e analysis o f the f ixed-base piers were performed b y both assuming elastic behaviour o f the piers w i t h no plastic h i n g i n g ("elastic p iers") , and assuming inelastic behaviour o f the piers b y a l l o w i n g for y i e l d i n g o f the piers at the plastic hinges ("inelastic piers") . T h e analyses o f the inelast ic piers were performed w i t h and wi thout account ing for the P - A effects. C r a c k e d cross sect ional properties were used for a l l cases. T h e elastic deformations obtained f r o m the elastic analyses o f the piers and the piers y i e l d displacements l isted i n T a b l e 2.1 were used to compute the force reduct ion factors R f r o m the E q u a t i o n 5.4. T h e inelastic deformations o f the piers obtained f r o m the inelastic analyses were used to calculate the duct i l i ty factors p o f the piers f r o m E q u a t i o n 95 5.3. The inelastic displacement ratios C M are obtained from Equation 5.5 by using R and u thus calculated. The force reduction factors, ductility demands and inelastic deformation ratios are discussed first in order to validate the inelastic behaviour of the prototype piers in the absence o f SSI and before any further discussion in regard to the effects of SSI. Also discussed in this section is the effect of geometric nonlinearity (P-A effect) on the response of the fixed-base piers mainly to ensure that the results are meaningful and that the analysis is robust. 5.2.1 Strength Reduction Factors Elastic deformations at the centre o f the mass at the top o f the piers, A e , were obtained from the elastic analysis of the six prototype piers subjected to the 78 surface motions. Figure 5.4 shows the cumulative frequency distribution of R for the six prototype bridge piers, identified with their natural periods, and provides a means o f comparing the statistics of the strength reduction factors of the piers with different natural periods. This Figure demonstrates that the strength reduction factors are suitably distributed from values below 1, indicating no yielding, to values up to 4, indicating significant y ie ldingof the piers. It can be observed that the values of R for the piers with periods of 1.5 and 2.0 s are in general lower than that for the other piers. This is attributed to the low amplitude of the ground motions at these periods as shown in Figures 4.1 and 4.4. 96 0.0 1.0 2.0 3.0 4.0 5.0 6.0 R Figure 5.4: Distribution of the strength reduction factor R 5.2.2 Ductility Demands without P-A Effects Although the ductility demands with the P-A effect were ultimately used in this study to draw conclusions in regard to the role of SSI in the seismic response of the piers, the analysis of the fixed-base piers without the P-A effects were performed as an interim stage to study the effects of geometric nonlinearity on the response and to check the analysis results before adding more complexities to the numerical models. To obtain the ductility demands, inelastic displacements at the centre of mass at the top of the piers, A c , were obtained from the inelastic analysis of the six prototype piers subjected to the 78 surface motions. The yield displacements, A y , of the six piers are listed in Table 2.1. The ductility demands, p, were calculated using the Equation 5.3. Figure 5.5 shows the distribution of p for the six prototype bridge piers, identified with their fixed-base natural period. As expected, the distributions of the ductility demands show similar characteristics to that of the strength reduction factors and demonstrate that they can be up to 4.5. Note that the cases in which the piers did not reach to their yield point, are the cases with low level input ground motions. It is also reminded that if the 97 ductility demand of a pier exceeded its ductility capacity as listed in Table 2.1, the pier was considered to have collapsed in that case, therefore the corresponding data point was considered as an outlier and was excluded from the distributions. In the case of the fixed-base piers with no P-A effects, no collapse of the piers was observed. 0.8 i u c tu ST 0.6 > 3 0.4 U 0.2 0.0 0.0 2.0 3.0 M 4.0 1 fl* ' 'fi fhf I V -T=0.3 s -T=0.6 s T=0.8s IIJIL) -T=1.0s T=1.5s l/Jf / ! -T=2.0s - * " * 1 1 5.0 6.0 Figure 5.5: Distribution of the ductility demands without P-A effects 5.2.3 Ductility Demands with P-A Effects Figure 5 .6 depicts the distribution of the ductility demands when P-A effects are accounted for in the dynamic analysis of the fixed-base piers. This figure shows distribution of ductility demands with trends similar to that of the cases without consideration of the P-A effects. It can be observed that the maximum demands with P-A effects are greater than the cases without. In a few cases, P-A effects caused excessive deformation of the pier with excessive yielding beyond its ductility capacity, or caused dynamic instability of the system. The corresponding data points were considered as outliers and were excluded from the database of demands. Figure 5 .7 compares the ductility demands with and without P-A effects by plotting the distributions of the ratio of the ductility demand with P-A effects to that without (this is equivalent to the ratio of the 9 8 pier deformation with P-A to that without). It can be observed in this figure that the geometric nonlinearity is more pronounced for the taller piers with longer periods and that P-A effect can increase the demand by up to a factor of two. The cumulative distributions of Figure 5.7 also reveal that less than 15% of cases can be considered to have been affected by geometric nonlinearities with more than 20% increase of their ductility demands. This suggests that P-A effects did not play a significant role in changing the response of the fixed-base piers. However, it is pointed out that P-A effects can potentially play a more significant role in the response of the flexible-base structure due to the rotation of the foundation. Therefore, it must be accounted for in the demand estimation of both the fixed-base and the flexible-base piers so that the role of SSI can properly be investigated. 1.0 0.8 >. u c I 0.6 u > | 0.4 3 3 u 0.2 1 0.0 0.0 1.0 2.0 3.0 1' 4.0 Sf" ' J^!'' ' _uir--T=0.3 s I lJ -T=0.6 s If wf* §J -T=0.8s -T=1.0s ... -fMfJ^- • • -T=1.5s [iSi J \ -T=2.0s -U-sJL 1 ; , 5.0 6.0 Figure 5.6: Distribution of the ductility demands with P-A effect 99 Figure 5.7: Comparison of the ductility demands with P-A effect to those without 5.2.4 Comparison of Ductility and Strength Reduction Factors The relationship between the ductility demands and the strength reduction factors are studied through inelastic deformation ratios Cu. (Equation 5.5). This comparison provides an understanding of the inelastic behaviour of the fixed-base piers before inclusion of SSI in the analysis. The cumulative frequency distributions of the inelastic deformation ratios without P-A effects are plotted in Figure 5.8. Note that the data points with no yielding of the pier are excluded from the plots of this figure. This figure depicts the behaviour of the pier with different natural periods and shows that the ductility and the strength reduction factors are not equal (i.e. C ^ l ) in more than half of the cases, although the median C^is about 1 (the equal displacement rule). The difference between R and p. becomes more significant with decreasing natural period of the structure and with increasing R as shown in Figure 5.10. This was the expected behaviour based on the available R-u-T relations obtained from analyzing SDOF systems (e.g. Chopra and Chintanapakdee 2004, Ruiz-Garcia and Miranda 2004, FEMA-440 2005). Similar results were obtained when P-A effect was considered. The inelastic deformation ratios with the P-A effect are plotted in Figure 5.9. 100 Figure 5.8: Distribution of the inelastic deformation ratios without P-A effect Figure 5.9: Distribution of the inelastic deformation ratios with P-A effect 101 1.0 1.5 2.0 2.5 3.0 3.5 4.0 R Figure 5.10: R-u relationship of the fixed-base piers without P-A effects 5.3 Seismic D e m a n d s of P i l e -Suppo r ted P ie rs In this section, the seismic demands of the pile-supported bridge piers obtained from nonlinear dynamic analyses of the soil-foundation-structure system are presented. Similar to the fixed-base piers, the analyses of the pile-supported piers were performed by both assuming elastic behaviour of the piers with no plastic hinging ("elastic piers"), and assuming inelastic behaviour of the piers by allowing for the yielding of the piers at the plastic hinges ("inelastic piers"). All analyses of the pile-supported piers included the P-A effects and they were not repeated without P-A effects because of the time-consuming nature of the analyses including SSI. Cracked cross sectional properties were used for all cases. 102 5.3.1 Strength Reduction Factors The strength reduction factors are obtained from the analysis of the soil-foundation-structure system with elastic bridge piers for the 6 prototype piers, 3 soil layers and 26 input rock motions. To calculate R for the flexible-base piers, A e is measured with respect to the base of the pier and does not include the deformations caused by the foundation. The same applies to A y and therefore the values of A y are the same as of those used for the fixed-base piers listed in Table 2.1. Figure 5.11 illustrates the distribution of R for the six prototype bridge piers, identified with their fixed-base natural periods. It can be observed in this figure that the strength reduction factors are in general lower than those of the corresponding fixed-base piers (Figure 5.4), indicating the lower strength demands of the piers when SSI is included. It also appears that the reduction of R is more significant for the piers with shorter period. None of the piers with the fixed-base period of 0.3 seconds had an R value of greater than 1.0, indicating that these piers did not deflect beyond their elastic limit as a result of the dissipation of energy by SSI. Figure 5.11: Distribution of the strength reduction factors 103 5.3.2 Ductility Demands Figure 5.12 shows the distributions of the piers ductility demands as defined by Equation 5.2. Note that the ductility demands are the local ductility demands which are related to the inelastic deformations of the piers with respect to their base (Figure 5.3). In a few cases, P-A effects caused excessive deformation of the pier with excessive yielding beyond its ductility capacity, or caused dynamic instability of the system. The corresponding data points were considered as outliers and were excluded from the database of demands. The distributions of Figure 5.12 show that ductility demands can be greater than the strength reduction factors shown in Figure 5.11. Note, however, that since the pier with the period of 0.3 seconds did not yield in any of the cases, there is no difference between R and u of this pier and they both are less than one. As was previously explained, this is attributed to the energy dissipation by SSI which is significant for this pier. The relationship between the ductility demands and the strength reduction factors of the pile-supported piers is discussed in the next section. Figure 5.12 also shows that the distributions of the ductility demands vary from those of the corresponding fixed-base piers depicted in Figure 5.6. Detailed discussions on the effects of SSI on the ductility demands are in Chapter 7. 104 5.3.3 Comparison of Ductility and Strength Reduction Factors Figure 5.13 depicts the inelastic deformation ratio C u of the pile supported bridge piers obtained from Equation 5.5. This figure shows that the inelastic deformation ratios of the pile-supported piers are greater than that of the fixed-base piers (shown in Figure 5.9), which implies that the p/R ratios of the pile-supported piers are greater than those of the fixed-base piers. Therefore, the deviation between p and R of the pile-supported piers is greater than that of the fixed-base piers. This implies that the R-p-T relationships when SSI is included are different than R-p-T relationships when SSI is not included. The importance of this observation is that if the SSI analysis is carried out with the elastic piers, and if the inelastic deformations of the piers are estimated by using the R-p-T relationships obtained from analyzing SDOF systems (e.g equal displacement rule), then the inelastic response of the structure may be underestimated. This is an important observation that also has implications for the seismic demand estimation of foundations, and requires further investigation. Further discussions on this issue are presented in Sections 5.3.4, 5.3.6 and 6.5. 0.0 -t i — I ! 1 ! 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Column Inelastic Deformation Ratio Figure 5.13: Distribution of the inelastic deformation ratios of the pile-supported structure 105 5.3.4 Effects of Piers Modeling Assumption on the Ductility Demands As it was explained in Chapter 1, in practice, the structural analysis is normally performed by analyzing the elastic system to obtain the elastic displacements, and the corresponding inelastic displacements are obtained by employing inelastic deformation ratios obtained from parametric studies of SDOF systems. It was also shown in the previous section that the R-p-T relationships of the flexible-base piers are different than R-p-T relationships of the fixed-base piers. Since the fixed-base piers represent SDOF systems, the comparison between R-p-T relationships of the fixed-base piers with those of the flexible-base piers can demonstrate how accurately the inelastic displacements of the pile-supported piers can be estimated from their corresponding elastic displacements by using inelastic deformation ratios. The R-p-T relationships of the pile-supported pier are compared to those of the fixed-base piers. The comparison is made by plotting the inelastic deformation ratios, C u = p/R, versus the force reduction factor R at each period of the pier. This is done for both fixed-base and the flexible-base piers and is shown in Figures 5.14 to 5.18. Note that only the results obtained from the cases in which the piers yielded are used and therefore there is no plot for the piers with the period of 0.3 s because no yielding was observed in this case. Each figure shows the C u -R scatter plot, along with its associated trend line, for both the fixed-base and the flexible-base piers. For the fixed-base piers with periods of 0.8 s or greater, the trend lines show that for any given R of the piers, the average inelastic deformation ratio is very close to 1.0. This implies the validity of the equal displacement rule for these piers which could be expected. For the piers with period of 0.6 s, the inelastic deformation ratio increases with increasing R which is the expected behaviour for acceleration sensitive structures with shorter period (e.g. Chopra and Chintanapakdee 2004, Ruiz-Garcia and Miranda 2004). Therefore, the behaviour of the fixed-base piers is consistent with what was expected from the published literature in regard to the behaviour of SDOF systems. 106 Figures 5.14 to 5.18 illustrate different behaviour of the pile-supported piers and show relatively sharp increases of the inelastic deformation ratios with increasing strength reduction factor of the piers with periods of up to 1.5 s. The deviation of the inelastic deformation ratios of the pile-supported piers from that of the fixed-base piers is dependent on the natural period of the piers and increases with decreasing period. Therefore, the inelastic deformations of the pile-supported piers would be underestimated if the inelastic deformation ratios obtained from SDOF systems, represented here by the fixed-base piers, are used to modify the elastic deformations. This implies that the commonly used equal displacement rule may underestimate the ductility demands of pile-supported piers. Different R-u-T relationships of the fixed-base and flexible-base structures have also been observed by other researchers such as Aviles and Perez-Rocha (2005). This behaviour can be explained by the fact that SSI becomes less effective with decreasing pier-to-foundation stiffness ratio (i.e. increasing period), as was observed in Chapter 4. When the pier is yielding, the pier-to-foundation stiffness ratio is decreased, and that results in the reduced effectiveness of SSI in dissipating the energy. Thus, in the analysis with the inelastic pier, SSI is less effective in reducing the response. This behaviour results in inelastic displacements that are higher than the elastic ones. Obviously, this phenomenon is not represented by the fixed-base piers and therefore the inelastic deformation ratios obtained from the analysis of the fixed-base piers (SDOF systems) cannot represent those of the pile-supported piers. 107 T = 0.6 s Figure 5.14: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 0.6 s Figure 5.15: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 0.8 s 108 Figure 5.16: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 1.0 s Figure 5.17: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 1.5 s 109 T = 2.0 s 3.0 • Pile-Supported Pier • • Fixed-Base Pier 2.0 ^ • 0.0 0.0 1.0 2.0 3.0 R Figure 5.18: Comparison of inelastic deformation ratios for the fixed-base and flexible-base piers; T = 2.0 s 5.3.5 Total Displacem ent Dent ands Total displacement of bridge piers is another important demand parameter used in the design of bridge components such as deck support length, restrainers, and expansion joints. Therefore accurate estimation of total displacements when SSI included can have important design implications for bridge structures. Figure 5.19 illustrates the distribution of the total displacements of the piers normalized with respect to their heights. It is more informative, however, to separate the portion of the total displacements induced by the deformation of the piers with respect to their base from the portions induced by the rotation and translation of foundation. Figure 5.20 shows the distribution of the normalized deformation of the piers with respect to their base, called piers local drift. Comparison of Figures 5.19 and 5.20 shows a clear difference between the two distributions and demonstrates that while the normalized total displacements of the piers are comparable for all natural periods, the local drifts of the piers are not, and the stiffer piers with shorter natural periods experience less local drift 110 than the piers with longer period. This can further be investigated by obtaining the piers portion of the total displacements by calculating the ratio of the deformation of the pier with respect to its base to the total displacement of the pier. Figure 5.21 depicts the distribution of the piers portion of the total displacements. This figure demonstrates that the deformation of the piers account for the decreasing portion of the total displacements with decreasing natural period of the piers. Consistent with previous observations, this implies that SSI plays a more significant role in the response of the piers with shorter period and demonstrates the importance of displacements caused by the motion of the foundation. Further to the aforementioned observations, a direct comparison of total displacements of the pile-supported piers with the displacements of the fixed-base piers can demonstrate the effects of SSI on the total displacement of pile supported piers. This comparison is presented in Chapter 7 and the seismic demands of the foundation are discussed in Chapter 6. Figure 5.19: Distribution of the total displacement of the piers normalized with respect to their height 111 1.0 0.8 i >> o c u I 0.6 u_ u > I 0.4 3 £ 3 o 0.2 0.0 1 I 7 T=0.3 s ; — — T=0.6 s " 1 / T=0.8s I T"= 10s j T=1.5s T=2.0s -1 i 1 1 0.0 1.0 2.0 3.0 4.0 Pier Local Drift (% of Height) 5.0 6.0 Figure 5.20: Distribution of the local drift of the piers 0.8 0.4 0.6 8 1.0 1.2 Pier Portion of the Total Displacement 1.4 Figure 5.21: Distribution of the piers portion of the total displacements 112 5.3.6 Effects of Piers Modeling Assumption on the Total Displacements It was observed in Section 5.3.4 that SSI analyses with elastic piers may underestimate piers inelastic deformations and ductility demands. In this section, the difference that the modeling assumption of the piers makes in the total displacements is investigated. Figure 5.22 shows the distributions of the ratio of the total displacements obtained from the SSI analyses using the elastic piers to those obtained from the SSI analyses using the inelastic piers. These distributions show that the total displacements with the elastic piers can be greater or less than those of the inelastic piers, although there is more tendency towards decreasing the total displacements. Therefore, it is more likely for the SSI analyses with the elastic piers to underestimate the total displacement demands. The results presented here are not fully conclusive and further research is required to better understand the role of the piers inelastic behaviour in total displacement demand estimations. 1 0 4 E - A T=0.6 s - - -T=0.8s •j\ T=1.0s j J T=1.5s T=2.0s 0.0 0.5 1.0 1.5 2.0 Ratio of Total Displacements Figure 5.22: Distribution of the ratio of total displacements (SSI with elastic piers to SSI with inelastic piers) 113 5.4 S u m m a r y In this chapter, the demand parameters were introduced and the statistics of the seismic demands of the piers were presented. It was observed that: • Suitably distributed database of demands was obtained. • The relationship between the elastic and the inelastic demands of the fixed-base piers are in general agreement with what would be expected from the results of past research on the behaviour of SDOF systems. • The ductility and the strength demands of the piers with SSI were different than that without SSI. • The difference between the demands of the piers with SSI and the demands of the piers without SSI was more significant for the piers with shorter natural period. • The relation between the force reduction factors R and the ductility demands p when SSI included was different than that when SSI was not included. It was observed that for a given R, ductility demands were higher for the system with SSI. The difference was more pronounced for the piers with shorter period where SSI is more effective. This denotes that the inelastic deformations of the pile-supported piers would be underestimated if the inelastic deformation ratios obtained from SDOF systems are used to estimate the inelastic deformations of the pile-supported piers from their elastic deformations. Hence, the commonly used equal displacement rule may underestimate the ductility demands of pile-supported piers. • The study of the total displacements of the pile-supported piers revealed that the deformation of the piers accounts for a decreasing portion of the total displacements with decreasing natural period of the piers. This demonstrates the 114 importance of the displacements caused by the translation and the rotation of the foundation and implies that SSI is more effective for the piers with shorter period. It was observed that the total displacements obtained from the SSI analyses with elastic piers are more likely to be less that those obtained from SSI analyses with inelastic piers. This means that SSI analysis with elastic piers may underestimate the total displacements. However the results presented here were not fully conclusive and further research is required to better understand the role of piers inelastic behaviour in total displacement demand estimations. 115 6 SEISMIC DEMANDS OF FOUNDATIONS This section presents the seismic demands of the pile foundations. Foundation demands include the rotation and the translation of foundation, as well as the demands of the structural components of the foundation. The primary objective of this chapter is to study the seismic demands of the foundations and their relation to the demands of the piers in order to better understand the role of the foundation response in the overall system response. Detailed investigation in regard to foundation design is beyond the scope of this work. 6.1 Translation of Foundation Translation of foundation, shown by A T in Figure 5.3, is defined as the horizontal movement of the pile cap with respect to the free field. The translation of foundation is measured on the surface of the pile cap at the connection point of the pier. The distributions of peak translations are plotted in Figure 6.1, where each foundation is identified by the fixed-base natural period of the pier that it supports. In this figure, translations of up to 0.08 m can be observed, which denote the importance of SSI. While Figure 6.1 shows the magnitudes of the translation of foundations, it is more informative to convert these translations into the resulting displacement of the pier expressed as the fraction of the total displacement of the pier. Resulting distributions are plotted in Figure 6.2 which displays the effects of SSI on the response of piers through the translation demand imposed on the foundation. It can be observed that the contribution of the foundation translation in total displacement of the piers is different for different piers with different natural periods. The distributions of Figure 6.2 show that the foundation translation contributes significantly in displacing the stiffer piers with shorter periods, and this significance decreases with increasing period of the pier. Therefore, the role of 116 SSI through translation of foundation increases as the pier-to-foundation stiffness increases. Figure 6.2: Contribution of the foundation translation to the total displacements of the piers 117 6.2 Ro ta t i on o f F o u n d a t i o n The rotation of the foundation is measured by the rotation of the pile cap. Figure 6.3 shows the distributions of the peak rotation of foundations. This figure shows that the distribution of the rotation of foundation does not vary with the natural period of the piers as much as it does for the translation of foundation. Similar to the translations, the rotations are converted into the resulting displacements of the piers and are expressed as the fraction of the total displacements of piers. Resulting distributions are plotted in Figure 6.4 which displays the effects of SSI on the response of piers through the rotation demand imposed on the foundation. It can be observed that while the contribution of the foundation rotation in total displacement of the piers is different for different piers and increases with decreasing natural periods, this difference is not as pronounced as is for the translation of foundation. Figure 6.4 shows that the rotation of foundation can contribute to up to 40% of the total displacement of the piers and denotes the significance of SSI through the rotation of foundation. Figure 6.3: Distribution of peak rotation of foundation 118 Figure 6.4: Contribution of the foundation rotation to the total displacements of the piers 6.3 Comparison of the Translation and the Rotation of Foundations To better understand the relative role of the translation and the rotation of foundation on the total displacement response of each pier, and its relation to the natural period of the pier, the distributions of Figures 6.2 and 6.4 are rearranged in separate plots for each pier as depicted in Figure 6.5. This figure reveals that for the piers with shorter periods, SSI affects the total displacement of the piers primarily through the translation of foundation rather than the rotation. However, with increasing period of the pier, the contribution of the translation becomes less significant while the contribution of the rotation remains relatively unchanged, resulting in slightly more contribution of rotation for piers with longer periods. Due to the less effectiveness of SSI for the piers with longer periods, less contribution of both translation and rotation may be expected for these piers, however, the reduction of the role of rotation is not significant due to the taller structure of the piers and the resulting higher overturning moments on the foundation. 119 In summary, Figure 6.5 shows that for the shorter piers, the interaction is more in the form of translation of foundation rather than the rotation of foundation. For longer period piers, rotation is the more dominant form of SSI. This difference of behaviour can have design implications for the piles. For instance, lateral motion of piles induces bending moment in the piles and in the connection of the piles to the pile cap, while rotation causes axial forces in the piles and pull-out forces in the pile to pile cap connections. This results in different seismic demands of the piles. 120 T = 0.3 s T = 0.6s 1.0 u 0.8 C o e 0.6 u 1 0.4 E O 0.2 0.0 1 ; ] f / f ! 1 / " / ; / j •^ —Translation Rotation 0.0 0.2 0.4 0.6 0.1 Fraction of Total Displacement 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of Total Displacement ().() 0.2 0.4 0.6 0.8 Fraction of Total Displacement 1.0 g0.8 U I" 0.6 u-u 1 0.4 S 0.2 0.0 0.0 0.2 T= 1.0 s . . . . . i / J j - - - j - — ~ - -__f _._] J 1__ —» Translation — Rotation 0.4 0.6 o.: l.o Fraction of Total Displacement T= 1.5 s T= 2.0 s 1.0 £0.8 U 3 c r e o.6 I 0.4 0.2 4-0.0 f \ r . . . U I — Translation Rotation 1 -j 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of Total Displacement — Translation • Rotation 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of Total Displacement Figure 6.5: Comparison of the contribution of the foundation translation and rotation to the total displacements of the piers 121 6.4 Demand of Piles The seismic demand of the piles, including the deformations, the bending moments, and axial and shear forces were directly obtained from the dynamic analyses. The piles were intended to remain elastic and therefore they were solely screened to verify that they remained elastic. Further study of the structural response of the piles is beyond the scope of this dissertation. However, the numerical results of the dynamic analyses can be used in future to expand this study to include detailed investigation of the demands of piles. 6.5 Effects of Piers Modeling Assumption on the Demands of Foundations It was shown in Chapter 5 that performing SSI analysis with elastic piers could underestimate the inelastic deformation of the piers. This hints the possibility of the overestimation of the demands of the foundation. This possibility is investigated here by determining the ratio of the demands when elastic piers were used in the analyses to the demands when piers where modelled with inelastic material. This demand ratio was obtained for both the translation and the rotation of foundation. Figure 6.6 and Figure 6.7 show the distribution of the aforementioned ratios (note that the wiggled shape of the curve for T=2.0 has no specific meaning since the curve simply connects the data points) and Table 6.1 lists the mean values of the ratios with their respective standard deviations. Figure 6.6 shows that the predicted translation of foundation is greater when elastic structure is used in the numerical model. The translation demands obtained from the SSI analyses with elastic piers are on average up to 45% greater than the translations obtained from the analyses with inelastic piers. The average rotation demands are up to 25% greater. Higher predicted demands from the analyses with elastic piers result from higher forces transferred by the elastic piers to the foundation. In other words, in the analysis with inelastic piers, there is a maximum limit of the forces that can be transferred by the piers to the foundation, beyond which the piers begin to yield, and therefore the yielding pier induces less forces on the foundation compared to its corresponding elastic pier. 122 These results signify the importance of proper modeling of the structural elements for seismic demand estimation of foundations and have important implications in design and retrofit of pile foundations as the overestimated demands can add to the cost of the construction of the foundation. It is common to evaluate the seismic demand of foundations by using elastic structure in the analysis and therefore the preceding observations suggest that proper material modeling of structures in SSI analysis can result in savings. Figure 6.6: Ratio of foundation translation demands estimated from analyses with elastic piers to those estimated with inelastic piers 123 o.o -\ 1 i 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ratio of Rotations Figure 6.7: Ratio of foundation rotation demands estimated from analyses with elastic piers to those estimated with inelastic piers Table 6.1: Mean ratio of foundation demands estimated from analyses with elastic piers to those estimated with inelastic piers Translation Rotation Period Mean Standard Deviation Mean Standard Deviation 0.3 N/A* N/A N/A N/A 0.6 1.04 0.08 1.00 0.07 0.8 1.22 0.23 1.14 0.20 1 1.12 0.18 1.08 0.15 1.5 1.11 0.11 1.15 0.20 2 1.44 0.52 1.24 0.28 * No yie ding of the pile-supported pier was observec 124 6.6 S u m m a r y In this chapter the seismic demands of the foundations were studied by obtaining the translations and the rotations of the foundations and by determining their contribution into the total displacement demands of the piers. Also studied were the effects of the inelastic behaviour of the piers on the response of the foundation and how the lack of its direct consideration in SSI analysis would affect the estimated demands. The following was observed: • Translation of foundation contributes significantly in displacing the stiffer piers with shorter natural periods, but it becomes less significant with increasing period of the piers. Therefore, the role of SSI through translation of foundation increases as the pier-to-foundation stiffness increases. • The contribution of the foundation rotation in the total displacements of the piers increases with decreasing natural periods, however, the difference is not as pronounced as is for the translation of foundation. • For shorter piers with shorter periods, the interaction is more in the form of translation of foundation rather than the rotation of foundation. For taller piers with longer periods, rotation is the more dominant form of SSI. • SSI analysis with an elastic structure may overestimate the foundation demands. This signifies the importance of the proper modeling of the inelastic behaviour of structural elements in SSI analysis for seismic demand estimation of foundations and the potential for reducing the cost of the construction of the foundation. 125 7 EFFECTS OF SOIL-STRUCTURE INTERACTION ON THE SEISMIC DEMANDS OF PIERS In Chapters 4 to 6, nonlinear dynamic analyses and estimated seismic demands of the piers and the foundations were presented. The demands of piers were studied with and without consideration of SSI, by obtaining the response of both the pile-supported piers and their corresponding fixed-base piers. Effects of SSI on the behaviour of the system were explored and the contribution of foundation response in the overall response of the piers was demonstrated. In this chapter the relevance of including SSI in seismic demand estimation of pile supported bridge piers is further discussed by comparing the demands of the flexible-base systems to those of the corresponding fixed-base systems when SSI was ignored. This chapter aims at demonstrating the effects of SSI in a format useful for structural design to effectively demonstrate when SSI must be accounted for, and what would be its effects on the response of the piers. Ductility and total displacement demands of the piers are examined and it is shown how the response without SSI is different than the response when the effects of SSI are accounted for. The significance of SSI is discussed and conclusions are made on the relevance of SSI in seismic design of pile-supported bridge piers. The comparisons are made through various response ratios which are introduced and explained in the following sections. 126 7.1 Ef fects of So i l -S t ruc tu re In teract ion on Duc t i l i t y D e m a n d s To demonstrate how the ductility demands of the pier with SSI is different than that without SSI, the ductility demands of the pile-supported piers are compared to the ductility demands of the fixed-base piers. Note that results presented here are for all cases, whether the piers yielded or not. The comparison is made by calculating the ratio of the pile-supported piers ductility demand, where SSI included, to the ductility demand of the corresponding fixed-base piers, where SSI was not included. This ratio of ductility demands is called pier ductility demand ratio and is denoted by DDR as shown in Equation 7.1. DDR = ^ Pile-Supported Fixed—Base DDR is an indicator of whether SSI increases or decreases the ductility demand of piers. If DDR is less than 1.0, SSI is decreasing the ductility demand and therefore it has beneficial effects. If DDR is greater than 1.0, then SSI is increasing the ductility demand, meaning that ignoring SSI is an unconservative assumption. The importance of this consideration is that SSI in general is assumed to reduce the ductility demands of structures and therefore ignoring it is usually assumed to be a conservative assumption. Figure 7.1 plots the distributions of pier ductility demand ratios, DDR, for the six prototype bridge piers identified by their fixed-base natural periods. The mean values of DDR at each period, corresponding to each distribution in Figure 7.1, are presented in Figure 7.2 along with a curve fit showing the trend of the data (see Chapter 8 for regression analysis). The following is observed: • The ductility demands of the piers with shorter periods are more affected by SSI. As previously observed, the effects of SSI increases with decreasing natural period of piers. • In most cases, SSI reduces the ductility demand of the piers (DDR < 1.0), however, this is not true for all cases and increased ductility demands are also 127 observed. The number of cases with increased ductility demand is more for the piers with longer periods. This indicates that while SSI becomes less effective with increasing period of the system, it is more likely to increase the ductility demand of the piers. Mean values of DDR suggest that SSI always affects the response of the piers by decreasing the ductility demands, while the distributions of DDR show the possibility of increased ductility demands as well. Therefore relying on the mean values of DDR can be misleading in regard to the effects of SSI. Proper observation of the data requires accounting for the dispersion of DDR and results obtained should be studied probabilistically so that accurate conclusions in regard to the effects of SSI could be drawn. Figure 7.1: Distributions of pier ductility demand ratios (DDR) 128 1.0 0.2 4 0.0 -I . 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 Fixed-Base Period (s) Figure 7.2: Mean values of pier ductility demand ratios (DDR) 7.2 Effects of Soil-Structure Interaction on Total Displacement Demands The effects of SSI on the total displacement demands are studied in a similar fashion to that of the ductility demands by obtaining the ratio of piers total displacement demand when SSI was included (pile-supported pier) to the corresponding displacement demand when SSI was not included (fixed-base piers). This ratio of total displacement demands is called pier total displacement demand ratio and is denoted by TDR as shown in Equation 7.2. ™ „ ( A t o t ) Pile-Supported . 1L>K=—— ( A t o t ) Fixed-Base TDR is an indicator of whether SSI increases or decreases the total displacement of piers. If TDR is less than 1.0, SSI is decreasing the total displacement. If TDR is greater than 129 1.0, then SSI is adding to the total displacements imposed on the pier. Quantifying the additional displacements caused by SSI has important design implications for structures such as bridges with displacement-sensitive components which can be affected by additional displacements imposed by SSI. Figure 7.3 illustrates the distributions of pier total displacement demand ratios, TDR, for the six prototype bridge piers. The mean values of TDR at each period, corresponding to each distribution in Figure 7.3, are presented in Figure 7.4 along with a curve fit showing the trend of the data (see Chapter 8 for regression analysis). The following is observed: • In most cases SSI increases the total displacement of the piers (TDR > 1.0), however, this is not true for all cases and decreased total displacements are also observed. • The total displacements of piers with different periods are affected by SSI more or less equally for periods more that 0.6 seconds. SSI becomes exceedingly effective in increasing the total displacements of the piers with periods less than 0.6 seconds. • Mean values of TDR suggest that SSI always affects the response of the piers by increasing the ductility demands, while the distributions of TDR suggest the possibility of decreased total displacement demands as well. Therefore relying on the mean values of TDR can be misleading in regard to the effects of SSI on total displacement demands. Proper observation of the data requires accounting for the dispersion of TDR, and results obtained should be studied probabilistically so that accurate conclusions in regard to the effects of SSI could be drawn. 130 Figure 7.3: Distributions of pier total displacement demand ratios (TDR) 2.0 1.5 Q H u 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 Fixed-Base Period (s) 2.5 Figure 7.4: Mean values of pier total displacement demand ratios (TDR) 131 7.3 Comparison of the Effects of Soil-Structure Interaction on Ductility and Total Displacement Demands To better observe the effects of SSI on the response of the piers, the mean values of DDR and TDR are plotted together in Figure 7.5. This figure demonstrates that SSI decreases ductility demands at the expense of increasing total displacement demands, although it is emphasized again that relying on the mean values as presented in Figure 7.5 can be misleading in regard to the effects of SSI. c o 2.0 1.8 1.6 1.4 1.2 1.0 4, 0.8 0.6 0.4 0.2 0.0 0 Ductility Demand Ratio (DDR) Total Displacement Ratio (TDR) 0 0.5 1.0 1.5 Fixed-Base Period (s) 2.0 2.5 Figure 7.5: Mean response ratios (DDR and TDR) It is also informative to relate DDR and TDR to an overall system parameter rather than merely the fixed-base period of piers. An important system parameter in regard to the effects of SSI is the stiffness of the pier relative to the stiffness of the foundation, described by the pier-to-foundation stiffness ratio. As was observed in Chapter 2, and has been shown by other researchers (e.g. Finn 2004b), the period elongation of the system due to SSI is correlated to the pier-to-foundation stiffness ratio. Therefore, the ratio of the system initial period to the period of the fixed-base pier is a convenient system parameter 132 that can be employed. Greater system-to-fixed-base period ratio (T s y s /T) indicates greater period elongation and more flexibility at the base of the pier due to higher pier-to-foundation stiffness ratio. Since piers with shorter periods are stiffer, the period elongation increases with decreasing natural period of the fixed-base pier (see Chapter 2). Obviously one can expect to observe greater effectiveness of SSI with greater flexibility of the base, i.e. with greater period elongation. Figure 7.6 depicts the plots of the mean DDR and the mean TDR versus the dimensionless period ratio, T s y s / T . An interesting observation in Figure 7.6 is that the relationship between the demand ratios and the period ratio T s y s / T of the system is rather linear (although it should be recognized that the selection of the prototype systems based on the piers fixed-base natural period has resulted in the lack of data points between T s y s / T of 1.5 and 3.0). An advantage of the presentation of results as a function of T s y s / T is that the results obtained from the analysis of the prototype systems of this study can be compared to any similar study with different prototype systems but with similar T s y s / T ratios. If the comparisons show that DDR and TDR thus obtained can be applicable to different soil-foundation-structure systems based on their T s y s / T value, then DDR and TDR curves can be used to modify the response obtained without SSI to estimate the demands including SSI. It is reminded, however, that using the mean values of DDR and TDR without consideration of the scatter of results may obscure some aspects of the system response. Therefore, a probabilistic approach is required so that DDR and TDR can be reliably used to estimate the effects of SSI on the response of bridge piers. The probabilistic assessment of the effects of SSI and further discussion on the application of DDR and TDR for performance-based estimation of the effects of SSI is presented in Chapter 8. 133 o ca OS 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 * i • Ductility Demand Ratio (DDR) A Total Displacement Ratio (TDR) ; ; i i i i 1 1.00 1.10 1.20 Tsys/T 1.30 1.40 Figure 7.6: Mean demand ratios (DDR and TDR) as functions of the period ratio T s y s / T 7.4 S u m m a r y Effects of SSI on the behaviour of the system were explored by presenting the pier ductility demand and the pier total displacement ratios (DDR and TDR). It was observed that: • The effects of SSI on the ductility and the total displacement demands is increased with decreasing period of the piers, or with increasing period ratio of the system (T s y s /T). • In most cases, SSI reduces the ductility demand of the piers (DDR < 1.0), however, this is not true for all cases and increased ductility demands were also observed. It is more likely that piers with longer periods (lower T s y s /T) experience greater ductility demands due to SSI • The reduction of the ductility demands is at the expense of increasing total displacement demands. 134 Relying on the mean values of DDR and TDR can be misleading in regard to the effects of SSI on the seismic demands of the system. Proper observation of the data requires accounting for its dispersion. Therefore, results obtained should be studied probabilistically so that accurate conclusions in regard to the effects of SSI could be drawn. 135 8 PROBABILISTIC ASSESSMENT OF THE EFFECTS OF SOIL-STRUCTURE INTERACTION It was explained in Chapter 7 that due to the dispersions of DDR and TDR, relying merely on their mean values can be misleading in regard to estimating the effects of SSI on the ductility and total displacement demands of the bridge piers. Therefore, results obtained should be studied probabilistically to properly account for the uncertainties of DDR and TDR, so that they can be used for seismic demand estimations within the framework of performance-based design. The source of uncertainties for DDR and TDR are the input earthquake ground motions used in seismic demand estimations of the bridge piers. The input motions are random in nature and hence are DDR and TDR. Besides the input motions, the system parameters also have uncertain natures and should be treated as random variables. Therefore, proper probabilistic assessment of DDR and TDR should also account for the randomness of the system parameters to which DDR and TDR are dependent. In this regard, one of the objectives of the probabilistic assessments presented in this chapter is to account for the uncertainties of DDR and TDR by estimating the probabilities that they are greater than predetermined target values. Target values are chosen based on the performance objectives. For instance, the probability of DDR>1.0 defines the probability of SSI increasing the ductility demand of the pier, which is the probability of ignoring SSI being an unconservative assumption. Moreover, with the evolution of performance-base design methodologies, probabilistic estimate of the performance of structures under earthquake loadings is needed to be accounted for explicitly as an integrated part of structural design so that the system can be designed and optimized to meet various limit states with different reliabilities tailored 136 to the needs of the project. Hence, another objective of the probabilistic assessments presented in this chapter is to estimate the effects of SSI on the performance of the piers with given target reliabilities. For instance, TDR can be used to estimate the total displacement of a bridge pier based on the total displacement of that pier without SSI. This can be done by multiplying the total displacement of the fixed-base pier by TDR. The mean value of TDR can be used for this purpose; however, there would be about 50% chance that TDR is greater than the mean TDR which may represent too much uncertainty. To find TDR with higher level of confidence, it can be estimated for a target reliability that reflects the corresponding desired level of confidence. For instance, if r is found such that the probability of TDR<r is p, then TDR=r can be used to estimate the effects of SSI on the total displacement of the pier with the reliability of p. Note that the reliability analysis is performed by estimating the probability of not meeting the performance objective, i.e. the probability of TDR>r (in other words, if the probability of TDR>r is P, then P is the probability that total displacement is greater than that predicted by TDR). In the following sections, first the methodology used for reliability analysis is explained. Then results of the reliability analyses are presented towards the stated objectives of this chapter, i.e. probabilistic data processing, and application for performance design. 137 8.1 Methodology The performance function (or limit state function) for reliability analysis is in the general form of Equation 8.1, where G is the performance function, C is the capacity and D is the demand, x is the vector of all random variables, and xc and xd are vectors of random variables associated with D and C respectively. Reliability analyses are performed to estimate the probability of non-performance, which is the probability of G(x) <0. G(x) = C(xc)-L\x(l) (8.1) In this study, the demand parameters (D) of interest are DDR and TDR which were estimated in Chapter 6 as functions of T (i.e. x^ = T) and T s y s / T (i.e. Xd = T s y s /T) . It must be recognized that DDR and TDR are not solely dependent on the natural periods and they also depend on other system physical properties, as well as input ground motions. Particularly, the response at each T is dependent on the 76 ground motions at the bridge site which are the combination of the 26 input rock motions passing through 3 different soil layers. Such representation of results implies that instead of representing the effects of various ground motions and sites individually, the group effect of all of them is represented for a specific choice of system variables, i.e. natural period, which is chosen based on the objectives of this work. More on grouped sets of data can be found in Foschi (2005). The capacity parameter C in this work is not dependent on any random variable and is a limit value that is set for D (e.g. r in the previous section) based on the objectives of the reliability analysis. If the limit value of D is denoted by A i m , (i-e. C = A i m ) , m e performance function can be expressed by G(xd) = DUm - D(xd) and the reliability analysis would be the estimation of the probability of D ] i m - D(xd) < 0. Reliability analysis can be carried out using different methods such as the first order reliability method (FORM), the second order reliability method (SORM) and the Montecarlo simulations using variance reduction techniques. Reliability analysis using FORM or SORM requires estimation of the limit-state surface G and structural responses D as functions of the intervening random variables x,t. However, these functions which 138 define the input-output relationship of the system cannot always be defined explicitly and they should be approximated through fitting response surfaces to discrete values of response obtained for specific combinations of the intervening random variables. For simulation techniques such as Montecarlo simulation, the discrete values of response can be directly used, however due to low probabilities of failure of structures, simulations may necessitate a large number of performance function evaluation which involves the computationally demanding task of nonlinear dynamic analysis for structural systems subjected to earthquake ground motions (Zhang and Foschi 2004). Therefore, response surfaces will be required to make the simulations efficient. Note that in this work, the responses are obtained only at six discrete values of the piers natural periods and therefore response surfaces are required to describe DDR and TDR as continuous functions of T over the range of T that is being studied. The response surfaces can be obtained through fitting regression models to the discrete values of response. For nonlinear systems, this requires nonlinear regression models which cannot always provide a good fit to the available data, especially for highly nonlinear responses and for systems with multiple inputs. Alternatively, the response surfaces can be approximated by artificial neural networks and by training the network using the available discrete data to represent the complex input-output relationships between the input random variables and the response of the system. Neural networks can provide robust representations of complex systems with many input variables that otherwise cannot be represented by nonlinear regression models. Examples of the application of neural networks for reliability analysis can be found in Foschi (2005), Zhang and Foschi (2004) and Goh and Kulhawy (2003). To perform the reliability analysis, the distributions of demands must be verified as the first step to ensure that they can be represented by known probability distributions. The distributions of DDR and TDR, presented in Figures 7.1 and 7.3, show that they can be represented by lognormal distributions. If the mean of the demand parameter D (either DDR or TDR) is shown by D, with the corresponding standard deviation of a, then D 139 can be calculated from Equation 8.2, which assumes a lognormal distribution for D. R in this equation is a standard normal variable (Foschi 2005). DfT) = D(T) (8.2) ^l+(cXT)/D(T)) : D can readily be estimated from Equation 8.2 for the six discrete values of T by knowing D and cr at each T, or it can be estimated continuously for a range of T, if response surfaces of D and cr are available over that range of T. Reliability analyses are thus carried out by obtaining response surfaces for D and o by using well-known nonlinear regression models and by using neural networks. The reliability analyses were performed by using the general reliability analysis program R E L A N (Foschi et al 2000) by performing Montecarlo simulations using response surfaces. The estimation of response surfaces is explained in the following section. 8.2 Response Surfaces The distributions of DDR and TDR are presented in Figure 7.1 and Figure 7.3, respectively, and Table 8.1 summarizes their means and standard deviations. DDR and TDR are expressed as functions of both the fixed-base period T and the period ratio T s y s / T , as shown in Chapter 7. Hence, response surfaces of means and standard deviations are obtained as functions of both T and T s y s / T (application will be shown later). Response surfaces are obtained using regression models and artificial neural networks. These methods are explained in the following. 140 Table 8.1: Summary of the statistics of results Fixed-Base Natural System Natural Period Tsys (s) Tsys/T Ductility Demand Ratio (DDR) Total Displacement Ratio (TDR) Period T(s) Mean Standard Deviation Mean Standard Deviation 0.3 0.4 1.33 0.39 0.12 1.50 0.66 0.6 0.67 1.12 0.64 0.19 1.08 0.28 0.8 0.88 1.10 0.83 0.20 1.10 0.33 1 1.08 1.08 0.87 0.19 1.06 0.29 1.5 1.57 1.05 0.93 0.16 1.05 0.16 2 2.04 1.02 0.94 0.16 1.05 0.20 8.2.1 Regression Models Various regression models, including linear and nonlinear models, were considered. The selection of the model depends on the problem at hand and the degree and the type of the nonlinearity presented in the data. The models considered were polynomial, exponential, power curve, and logarithmic regression models. The estimation of the parameters of these regression models was carried out by the least squares method which involves minimizing the sum of the squares of the errors performed by direct numerical search procedures (Neter et al. 1996). Unlike linear models, since it is usually not possible to find analytical expressions for the least squares of nonlinear regression models, the estimation of nonlinear regression model parameters requires intensive computations that can be carried out by available standard programs such as Mathcad (Mathsoft 2004) and M A T L A B (Mathworks 2004). For each case, various models were tested and the one with the smallest least square error was chosen. Figure 8.1 shows the mean values of DDR and TDR as functions of T, along with their associated nonlinear regression model. Figure 8.2 shows the corresponding standard deviations and their nonlinear regression fit. The regression models as a function of T s y s / T were rather linear as can be seen in Figures 8.3 and 8.4. 141 2.0 0.2 -0.0 -0.0 0.5 1.0 1.5 2.0 2.5 Fixed-Base Period T (s) Figure 8.1: Nonlinear regression fit to mean response ratios (DDR and TDR) as functions of the piers fixed-base period T Figure 8.2: Nonlinear regression fit to standard deviations of response ratios (DDR and TDR) as functions of the piers fixed-base period T 142 Figure 8.3: Linear regression fit to mean response ratios (DDR and TDR) as functions of the period ratio T s y s / T Figure 8.4: Linear regression fit to standard deviations of response ratios (DDR and TDR) as functions of the period ratio T s y s / T 143 8.2.2 A rtificial Neural Networks The development of artificial neural networks was inspired by the central nervous system with the goal of learning from experience. Artificial neural networks are self-learning mechanisms composed of simple interconnected processing elements, or neurons, which operate in parallel and are trained to map a specific input to a specific target output. Therefore, neural networks are capable of capturing the complex input-output relationships, making them suitable for simulating the behaviour of complex nonlinear systems. Among various neural network architectures, the feed-forward back-propagation network is the most popular one which includes an input layer, one or several hidden layers, and an output layer. For each link between neurons, there is an associated weight that is adjusted during the training of the network to achieve the optimum performance of the network. There is also a bias and a transfer function associated with each neuron. When operating neural networks, each neuron calculates the weighted sum of the inputs from the preceding layer. The results are then passed on to a transfer function and the resulting output becomes the input for the neurons of the next layer until the output layer is reached. During training, the number of neurons and the corresponding weights are adjusted to minimize the square of the prediction error or to make the error less than a specific value (Zhang and Foschi 2004, Patternson 1997). Feed-forward back-propagation networks were designed and trained for both mean and standard deviations using both R E L A N and M A T L A B . R E L A N directly uses the neural network trained by its built-in algorithm to perform the reliability analysis, the results of which are presented in the following section. In training the neural network, the over-fitting of the data must be avoided as it reduces the generality of the network. Thus, it is important to validate the network using a set of data that was not used for the training of the network. It must also be noted that when using a trained neural network to simulate D and a as functions of input random variables (i.e. T and T s y s /T) , only values of input variables that are within the bounds of these variables used for the training of the neural network must be used. This is because the network does not provide reliable estimates when it is used to simulate the response for input values that are outside the training 144 range. Detailed explanation of the implementation of NN in reliability analysis can be found in Foschi (2005). 8.3 Probabilistic Data Processing It was shown that relying on the mean values of DDR and TDR without consideration of their dispersions can be misleading in regard to the effects of SSI on the seismic demands of the system. Therefore, an objective of the probabilistic assessments presented in this chapter is to describe DDR and TDR probabilistically in order to account for their uncertainties, so that accurate conclusions in regard to the effects of SSI could be drawn. As explained in Chapter 1, of particular interest is to explore the possibility of SSI increasing the ductility demand of the piers. This possibility depends on the characteristics of the input rock motions and the site soil profile and properties. It is pointed out that a simple approach to this problem is to directly using the cumulative frequency diagrams. For instance, Figure 7.1 shows that about 30% of piers with fixed-base period of 2.0 seconds have DDR greater than 1.0, and so one can tell that there is about 30% probability of DDR>1 for this pier, meaning that there is 30% chance of SSI increasing the ductility demand of the pier with T=2.0 s. However, reliability analysis as explained here has the advantage of accounting for the uncertainties of the system parameters, such as T, that were treated deterministically in the dynamic analysis. It also has the advantage of providing probabilities over a range of periods rather than a limited number of discrete values of T. Moreover, the reliability analyses can be used for the performance-based assessment of DDR and TDR as is discussed in the next section. To answer the question about the circumstances under which SSI may increase the demands, either ductility demand or total displacement demand, the probability of DDR>1.0 and the probability of TDR>1.0 were estimated. The variation of T was assumed to be 10%. The variation of T accounts for the combined uncertainties of the mass and the stiffness of the system. Reliability analyses were performed using both regression models and neural networks. Results are presented in Figure 8.5 for DDR and in Figure 8.6 for TDR. Note that a comparison of the results obtained from regression models and neural networks is made here as a verification of the reliability analysis 145 methodology using neural networks. Although using neural networks was not necessary here, verifying the application of NN will ensure that the presented approach to the data processing can be employed for future works with more number of input variables and possibly more complex response surfaces that cannot be represented by conventional nonlinear regression models. The probability of DDR>1 represents the probability of SSI increasing the ductility demand of the bridge pier and the probability of TDR>1 represents the probability of SSI increasing the total displacement demand of the pier. Figure 8.5 shows that the probability of SSI increasing the ductility demand of the piers increases with increasing period of the structure. Since the overall effects of SSI on the response of the piers decreases as the natural period increases (the pier-to-foundation stiffness ratio decreases), it is more likely that the scatter of DDR results in values of DDR greater than 1.0. Figure 8.5 also shows more than 25% probability of DDR>1 for piers with periods of greater than 1.0 second when subjected to the wide range of ground motions used in this study. This is a considerable probability and emphasizes that SSI should not be ignored as a conservative assumption in the design of the pile-supported bridge piers. Figure 8.6 presents the probability of SSI increasing the total displacement demand of the piers. It can be seen that this probability is higher for shorter stiffer piers with lower natural periods for which the effects of SSI is more pronounced. It can also be observed that there is at least about 60% chance of larger total displacement demands of the pile-supported bridge piers compared to the fixed-base piers which confirms the necessity of attention to the design of displacement sensitive bridge components when SSI is involved. The comparison of the reliabilities obtained using nonlinear regressions and neural networks shows that results obtained are in reasonable agreement with minor differences that stem in the errors presented in each of the response surfaces. The comparison of the reliability estimates using neural networks with that obtained from nonlinear regression models demonstrates the successful application of neural networks for reliability analyses 146 of this nonlinear system. This case study employed simple neural network representation of small data sets of means and standard deviations so that the results obtained could be compared with that obtained by utilizing well-known response surface estimation methods. However, as previously pointed out, the application of neural networks in the context of the presented methodology for reliability analysis can be expanded to systems with larger number of input random variables and with larger data sets for which nonlinear regression models cannot provide a robust description of the response surface. o 4 0 0 Nonlinear Regression Models 0.0 0.5 1.0 1.5 2.0 2.5 Fixed-Base Period (s) Figure 8.5: Probability of DDR>1 as a function of the piers fixed-base period T 147 : i ' I 1 1 ! I ; ; Nonlinear Regression M o d e l Neural Network i i 0.0 0.5 1.0 1.5 2.0 2.5 Fixed-Base Period (s) Figure 8.6: Probability of TDR>1 as a function of the piers fixed-base period T The probabilities of DDR>1 and TDR>1 as functions of the period ratio T s y s / T are presented in Figures 8.7 and 8.8. The variation of T s y s / T was assumed to be 5% which in fact, can account for uncertainties involved in estimating the period elongation of the piers caused by the flexibility of their bases. Figure 8.7 shows that the probability that ductility demands when SSI included are greater than the ductility demands when SSI is not included increases with decreasing period elongation (T s y s /T), i.e. with decreasing effectiveness of SSI. The increase in ductility demand is more than about 25% for systems with period elongation of less than 5% (Tsys/T<1.05). Figure 8.8 illustrates that the probability of SSI increasing the total displacement demands increases with increasing period elongation (T s y s/T), i.e. with increasing effectiveness of SSI. Overall, the observation for the simple prototype bridge piers of this research is that for piers with less flexibility at the base, the concern in regard to ignoring SSI is related to the structure of the pier and the estimation of the ductility that must be provided for them. For piers with more flexible base, the concern is related to the displacement sensitive bridge components that are supported by the piers and not the piers themselves. 148 0.9 1.0 1.3 Tsys /T Figure 8.7: Probability of DDR>1 as a function of the period ratio T s y s / T Figure 8.8: Probability of TDR>1 as a function of the period ratio T s y s / T 149 8.4 Performance-Based Assessment of the Effects of Soil-Structure Interaction on the Seismic Demands of the Piers Results presented in the previous section were concerned with the probability of SSI increasing the demands of the piers regardless of the magnitude of the amplification of demands. Moreover, probabilities can be estimated for SSI decreasing the demand and not just increasing the demand. Hence, reliability analyses can be carried out for various performance objectives, defined by the values chosen for D l i m as discussed in Section 8.1. The curves of Figures 8.9 and 8.10 were obtained for Dnm= r, where r is from 0.8 to 1.5, and show the probability of DDR>r and TDR>r, respectively, as functions of T s y s / T . It can be observed, for instance in Figure 8.9, that for a pier with T s y s / T of 1.05, while the probability of DDR>1.0 is about 30%, it is less than 5% for DDR>1.3 which is a much lower probability. Therefore, if the column has 30% ductility reserve, then one can say that ignoring SSI will not pose a significant risk to the pier (less than 5%). As another example, for a system with T s y s / T of 1.20, the probability of DDR>0.9 is less than 10%, which means that the ductility demand can be reduced by 10% with more than 90% confidence. Such reduction of demand can become a source of saving especially in the retrofit of bridge piers. It is reminded that the probabilities here are not total probabilities and they are conditional on the occurrence of all the ground motions used in this study. Calculation of total probabilities requires estimation of the probability of occurrence of the input ground motions which is not within the scope of this work. 150 151 While the curves of Figures 8.9 and 8.10 are very informative, it is more convenient for design purposes to rearrange these curves for the purpose of estimating the effects of SSI on the performance of the piers with given target reliabilities. Figures 8.11 and 8.12 show the probabilities of DDR>r and TDR>r as functions of r for bridge piers with T s y s / T from 1.02 to 1.3. The advantage of such presentation of the results of reliability analyses is that for a specific pier with known T s y s / T , the value of r can be found for the desired level of confidence. For example, the curves of Figure 8.12 can be used to obtain the value of r for a bridge pier with T s y s / T of 1.06 with a corresponding confidence of 80%. As shown in this figure, the value of r associated with 20% probability of TDR>r obtained from the curve for Tsys/T=1.06 is 1.23. This means that if the total displacement of the bridge pier is obtained from the analysis of the corresponding fixed-base pier, then the effects of SSI can be accounted for by multiplying this fixed-base displacement by r=1.23 to obtain the total displacement of the pier with 80% confidence. 152 Figure 8.12: Probability of TDR>r as a function of r for T s y s / T of 1.02 to 1.30 To better demonstrate the proposed practical application of the above SSI modification curves, a bridge example, shown in Figure 8.13, is considered. This bridge has simply supported spans with soil-foundation-pier systems similar to those of this study. It is assumed that the input motions used in this study describe the seismic hazard at the site of the bridge. To design the supports of the deck, the support length must be estimated which is a parameter sensitive to the relative displacement of the piers with respect to each other. The relative displacement of the piers with respect to each other, on the other hand, is described by the total displacements of the two piers which are demand parameters that get affected by SSI. Therefore the effects of SSI on the total displacements of the two piers of the bridge must be estimated. For a simple bridge like this bridge, however, performing SSI analysis is likely not justifiable due to practical constraints and therefore a simplified method can be of great help. This task can be quickly performed with the availability of curves such as those of Figure 8.12 which can be used to obtain SSI modification factors to modify the demands obtained from the analysis of the fixed-base piers. The level of confidence in the modification factor can be chosen by the designer. For instance a confidence level of 80% results in r = 1.23 for the 153 pier with T s y s / T = 1.06 and in r = 1.62 for the pier with T s y s / T = 1.2. Having obtained the values of r for the two piers, the total displacement of the piers with SSI can be calculated from the displacements of the fixed-base piers. Simply Supported Span 1 * Figure 8.13: A bridge example with simply supported spans and pile-supported piers on soft soil The curves of Figures 8.11, called here "performance-based SSI assessment diagrams for ductility demand" and the curves of Figure 8.12, called here "performance-based SSI assessment diagrams for total displacement demand", have a number of interesting features that make them appealing for performance-based assessment of the effects of SSI on the response of simple bridges: • One feature is the explicit consideration of the dispersions of the database of demands that was used to generate these curves. Rather than crudely using mean or median values with unknown levels of reliability, these curves provide the flexibility of choosing the level of confidence in the estimated modification factors. Therefore, the designer can either use a uniform level of reliability for all performance objectives or can use different levels of reliability for different performance objectives tailored for a specific project. 154 These curves account for the uncertainties of the system parameters, such as the natural period, that are used to describe the system. The variability of the system parameters must be decided upon prior to generating the curves. Various families of curves can be constructed to account for various levels of uncertainty in the system parameters so the designer can have the flexibility of choosing the level of uncertainty in the estimated system parameters. In the examples presented here, only system periods, with and without SSI, were considered to construct the curves. However, this can be extended to other system parameters, or combination of system parameters, when constructing the curves. Constructing the curves as functions of T s y s / T can expand the domain of the applicability of these types of curves for various combinations of piers, foundations and soils that are different from those used to generate the curves, but have similar T s y s / T characteristics. This possibility must be investigated by performing nonlinear dynamic analyses of various pile-supported bridge pier systems on soft soils. If similar behaviour of systems with similar T s y s / T is observed, then such curves become even more appealing for implementation in performance-based design codes for typical highway bridges. Simplified reliable estimation of T s y s , however, remains a challenge which must be addressed if such curves are to be implemented with a design code format. If the input ground motions used to construct the curves are all selected to represent a specific level of seismic hazard, then families of curves can be constructed for various levels of seismic hazard. Merits of such selection of ground motions as opposed to the selection criteria used here must be investigated. If the effects of SSI on the response of the piers are not strongly dependent on the amplitude of the input ground motions, then separating these curves for various levels of seismic hazard is not necessary. A preliminary observation of results obtained in this study suggests that DDR and TDR are to 155 some degree dependent on the amplitude of ground motions. Figures 8.14 to 8.16 show the relationships between DDR and spectral acceleration, spectral velocity and spectral displacement of the surface motions at the fixed-base period of the piers and illustrate that SSI becomes more effective for stronger ground motions. Further discussion on this topic is beyond the scope of this dissertation and is recommended for future research. 156 T = 0.3 s 0.0 0.5 1.0 SA(0.3)(g) T = 0.6s 0.0 0.5 1.0 SA(0.6)(g) 1.5 1.0 ai Q Q 0.5 H 0.0 0.0 T = 0.8 s 0.5 1.0 SA(0.8)(g) 1.5 Figure 8.14: Correlation between DDR and site surface spectral acceleration at the natural period of the fixed-base pier 157 T = 0.3 s 0.0 10.0 20.0 30.0 40.0 50.0 60.0 SV(0.3) (cm/s) T = 0.6 s 0.0 20.0 40.0 60.0 80.0 100.0 120.0 SV(0.6) (cm/s) T = 0.8s 1.5 1.0 4 a a 0.5 0.0 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 SV(0.8) (cm/s) Figure 8.15: Correlation between DDR and site surface spectral velocity at the natural period of the fixed-base pier 158 T = 0.3s T = 0.6s 4.0 6.0 8.0 SD(0.6) (cm) 10.0 12.0 1.5 1.0 a a 0.5 0.0 0.0 T = 0.8s 5.0 10.0 15.0 SD(0.8) (cm) 20.0 gme 8.16: Correlation between D D R and site surface spectral displacement at the natural period of the fixed-base pier 159 8.5 Summary The data obtained from the nonlinear dynamic analyses were processed probabilistically to explicitly account for the dispersion of results and the uncertainties involved in the estimated system parameters, so that accurate conclusions in regard to the effects of SSI on the response o f the bridge pier could be drawn. In summary, the following was observed: • The probability of SSI increasing the ductility demand of the piers increases with increasing period of the piers or with decreasing period ratio (T s y s /T) of the system. The increase in ductility demand is more that about 25% for systems with period elongation of less than 5% (T s y s/T<1.05) or for piers with periods of greater than 1.0 second. This is a considerable probability and emphasized that SSI should not always be ignored as a conservative assumption in the design of the pile-supported bridge piers. • The probability of SSI increasing the total displacement demand of the piers is higher for piers with lower natural period or with higher system period ratio (T s y s /T) . There is at least about 60% chance of larger total displacement demands of the pile-supported bridge piers compared to their corresponding fixed-base piers which confirms the necessity of attention to the design of displacement sensitive bridge components when SSI is involved. Results of the reliability analyses were then presented in a format useful for performance-based assessment of the effects of SSI on the response of pile-supported bridge piers by using their corresponding fixed-base response. The application of the proposed methodology was demonstrated by a simple example and the appealing features of the proposed methodology for performance-based design of bridges were discussed and suggestions were made for future research to explore the implementation of the proposed methodology. 160 9 EVALUATION OF NONLINEAR STATIC PUSHOVER ANALYSIS FOR DEMAND ESTIMATION INVOLVING SOIL-STRUCTURE INTERACTION Nonlinear static pushover analysis has received much attention in recent years for seismic demand estimations of structures. The idea behind pushover analysis is to determine the inelastic deformations and the inelastic forces of various components of a system by pushing the structure monotonically at a reference point, e.g. the centre of mass of the bridge superstructure, to the expected total (global) displacement of that reference point under earthquake loading (target displacement). This infers that the target displacement must be determined before performing the pushover analysis. This is normally done by performing linear elastic analysis of the structure to obtain the elastic displacement and then to estimate the inelastic displacement by using equal displacement rule, or other modification factors, or any other approximate method (e.g. FEMA-356 2000, ATC-49 2003). Assuming that correct target displacements are estimated, pushover analysis provides an exact representation of the response of single degree of freedom (SDOF) systems, such as the bridge piers of this study with fixed base (without SSI). However, for multi-degree of freedom systems, such as multi-story frame structures, or bridge piers with flexible base, there might be shortcomings associated with this method, and the pushover prediction of the inelastic forces and the inelastic deformations of systems components might not match those obtained from nonlinear dynamic analysis. This has been the subject of research for building structures (with many degrees of freedom) but not much for bridge piers with simpler structural form that can be represented by SDOF systems. The simple 161 structural form of bridge piers eliminates the concerns in regard to the shortcomings of pushover analysis for this type of structures. However, adding the flexibility of the base of the piers (due to SSI) makes the flexible-base pier a more complicated system. Hence, the objective of this chapter is to evaluate the accuracy of the pushover analysis by comparing the pushover prediction of demands with those obtained from the nonlinear dynamic analyses. Note that the bridge piers of this study have the simplest form and are SDOF systems when their base is fixed. Therefore, since pushover analysis represents the exact behaviour of the fixed-base piers, any difference between the results obtained from the analyses of the flexible-base piers is attributed to the effects of SSI on the response of the system. 9.1 Base Shear Demands To perform the pushover analyses, the same numerical models of the pile-supported bridge piers that were used to perform the nonlinear dynamic analyses were used. Very low amplitude constant velocities (le-6 m/s) were applied horizontally at the centre of mass at the top of the piers, as shown in Figure 9.1, and the resulting base shears and total displacements of the piers were tracked. The force-displacement curves thus obtained from the pushover analyses were plotted along with the demands obtained from the nonlinear dynamic analyses. Resulting plots are shown in Figures 9.2 to 9.7 for the piers with fixed-base periods of 0.3 s to 2.0 s. Note that the displacements in these figures are the total displacements (global displacements) of the piers as shown in Figure 5.3 and they are normalized with respect to the height of the piers. Figure 9.2 shows that the shear forces predicted by the pushover analysis are different than those obtained from the nonlinear dynamic analyses of the pier with T = 0.3 s. It can be observed in this figure that for target displacements of about 0.4% of the piers height and above, the shear forces predicted by the dynamic analyses are lower than that of the pushover curve. This is because larger deformations are occurring in the soil when the system in analysed dynamically as apposed to when it is analysed statically by pushover analysis. Hence, the pushover analysis predicts larger contribution of the deformation of the pier in the total displacements compared to that predicted by the dynamic analysis, 162 which results in higher predicted shear forces for a given target displacement. Same difference can be observed for the piers with T = 0.6 s (Figure 9.3), but with less deviation of the results obtained from pushover and dynamic analyses. A s can be observed in Figures 9.4 to 9.7, for piers with periods greater than 0.6 s, plausible predictions are made by the pushover analyses. Thus, it is concluded that as SSI becomes more effective with decreasing natural period of the pier, the predictions of pushover becomes less accurate as it cannot properly account for the dynamic behaviour of the soil when the interaction is significant. Note that the observations here are consistent with the observations in previous chapters in regard to the displacements o f the system and the increasing contribution of foundation translation in the response o f the piers with decreasing natural periods. Applied Velocity = le-6 m/s • • Plastic Hinge \ . Figure 9.1: Numerical model for pushover analysis 163 T = 0.3 s 2000.00 1500.00 1 ™ 1000.00 CQ 500.00 0.00 j , t 1 / • • ' • • 1 llfc^ ^  * * ^ <J^*I^I "" ~% -Mr • Nonlinear Dynamic Analysis / Pushover Analysis 1 1 1 0.0 0.5 1.0 1.5 2.0 Normalized Total Displacement (% of Height) Figure 9.2: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.3 s T = 0.6 s 2000.00 _ 1500.00 2 55 1000.00 CQ 500.00 • Nonlinear Dynamic Analysis """"Pushover Analysis 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Normalized Total Displacement (% of Height) Figure 9.3: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.6 s 164 T = 0.8 s 2000.00 1500.00 £ 1000.00 500.00 0.00 • Nonlinear Dynamic Analysis ——Pushover Analysis 0.0 0.5 1.0 1.5 2.0 2.5 Normalized Total Displacement (% of Height) 3.0 Figure 9.4: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.8 s T= 1.0 s 2000.00 1500.00 5 1000.00 CQ 500.00 0.00 • Non linear Dynamic Analysis Pushover Analysis L • .» • % . 1 -0.0 0.5 1.0 1.5 2.0 2.5 3.0 Normalized Total Displacement (% of Height) Figure 9.5: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 1.0 s 165 T = 1.5 s 2000.00 1500.00 55 1000.00 500.00 H 0.00 • Nonlinear Dynamic Analysis Pushover Analysis 0.0 0.5 1.0 1.5 2.0 2.5 Normalized Total Displacement (% of Height) 3.0 Figure 9.6: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 1.5 s _ 1500.00 <*> 1000.00 • Nonlinear Dynamic Analysis Pushover Analysis I 1 i 1 -. r y * * ^ * * • i - ' • ' 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Normalized Total Displacement (% of Height) Figure 9.7: Comparison of force-displacement relationships with SSI obtained from pushover and nonlinear dynamic analyses of the pier with T = 2.0 s 166 9.2 Global Ductility Demands Global ductility demand is another demand parameter used to measure the inelastic response of bridge piers. Global ductility demand is similar to local ductility demand except that instead of using the displacements measured with respect to the base of the pier, the total displacements (global displacements) are used to calculate global ductility demand, as shown in Equation 9.1 (refer to Figure 5.3 for the parameters) U global A tot A s / S A c +(A T + A R ) A y +(A T + A R ) (9.1) Estimation of the global ductility demand of a system requires estimation of the total displacement demands (global displacement) A t o t of the system as well as the system yield displacement (global yield displacement) A y y s which could be estimated by performing pushover analysis. Note that this is another application of pushover analysis which is the estimation of the capacity of structural systems. For capacity estimation purposes, pushover analysis is performed by pushing the structure to large nonlinear deformations, and by plotting the resulting force-displacement curve. This curve is then idealized into a bilinear or multi-linear curve from which the global yield displacement of the system is extracted. Assuming that total displacement demands are accurately predicted, the objective here is to examine the accuracy of pushover analysis in predicting the global ductility demands. For this purpose, the total displacements obtained from nonlinear dynamic analyses are used with the system yield displacements obtained from pushover analysis. Resulting global ductility demands are then used to calculate the ratio of the global ductility demands to their corresponding local ductility demands. The mean values of these global-to-local-ductility-ratios are plotted in Figure 9.8. This figure shows that the predicted global ductility demands were higher than their corresponding local ductility demands, especially for the piers with periods of less than 1.0 s. This is an important observation because global ductility demands are expected to be less than local ductility demands (for 167 local ductility demands greater than 1.0) since they are computed by adding (A T + A R ) to both numerator and denominator of the equation for the local ductility demand (Miocai = y " = A c / A y ) . O I I I '5 1.6 " - - - - j - - - -§ 0.2 0.0 -i 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 Fixed-Base Period (s) Figure 9.8: Mean ratios of the global ductility demands to the local ductility demands The reason for the overestimation of ductility by global ductility factor is because it employs the total displacement of the system which includes the rigid body motion of the pier due to translation and rotation of foundation as explained in the previous paragraph. Thus, since the total displacement of the system is not necessarily accurate representative of the element deformation, significant discrepancies between global and local ductility demands can be observed when SSI is significant. To further explain this reasoning, the pier deformations (local displacement) are plotted against total (global) displacements of the system, both obtained from the dynamic analyses. This plot is shown in Figure 9.9 for the piers with T = 0.3 s and in Figure 9.10 for the piers with T = 0.6 s. The onset of the yielding of the piers and the onset of the yielding of the soil-foundation-pier systems, as predicted by the pushover analysis, are also shown in these figures. The onset of the 168 yielding of the pier is identified by pi o c ai > 1 and the onset of the yielding of the soil-foundation-pier system is identified by u gi 0bai > 1- Figure 9.9 clearly shows that pushover analysis may predict yielding of the pier while dynamic analysis predicts no yielding of the pier, even at large total displacements. This is the reason why global ductility factor, in this case, indicates yielding while local ductility factor indicates no yielding of the pier. Therefore in this case, using global ductility demand can be a misleading indicator of the inelastic response of the pier. Figures 9.9 and 9.10 show the cases for which global ductility demand is greater than 1 while the local ductility demand is still less than 1 with no yielding of the pier. Results presented here denote that global ductility demands should not be used when SSI is significant. In general, Figures 9.10 and 9.11 demonstrate the shortcoming of the pushover analysis in accurate prediction of the relative contribution of piers deformations in the total displacements of the soil-foundation-pier systems when SSI is significant. T = 0.3s 0.0 0.5 1.0 1.5 2.0 Total Displacement (% of Height) Figure 9.9: Comparison of the pier deformations with total displacements obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.3 s 169 T = 0.6s 2.00 o 1.50 0.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Total Displacement (% of Height) Figure 9.10: Comparison of the pier deformations with total displacements obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.6 s 9.3 Summary Nonlinear static pushover analysis is employed for both, demand and capacity estimation of structural systems. For demand estimation of a system, pushover analysis is performed to determine the inelastic deformations and the inelastic forces of various components of the system by pushing the structure monotonically to an already-estimated target displacement. For capacity estimation, pushover analysis is performed to push the structure to large nonlinear deformations so that the global capacity parameters of the system, such as the global yield displacement of the system, can be determined. Predictions of demand and capacity by pushover analysis are accurate when the system can reliably be replaced by an SDOF system, otherwise, results obtained from pushover analysis might not be reliable. Since pile-supported bridge piers are not well represented with an SDOF system when SSI is significant, the objective of this chapter was to investigate the accuracy of pushover analysis for demand estimation by comparing demands obtained from nonlinear dynamic analysis and pushover analysis. Since capacities estimated by pushover are used in global ductility demand estimations, the 170 accuracy of capacity estimation by pushover analysis was also studies. The following is observed: • Pushover analysis cannot properly capture the dynamic soil-structure interaction when the interaction is significant and therefore makes inaccurate prediction of the relative contribution of piers deformation in the total displacements of the systems with high SSI. Pushover analysis predicts greater contribution of the deformation of the pier in the total displacements compared to that predicted by the dynamic analysis. This results in discrepancies between both demands and capacities predicted by pushover analysis and those predicted by nonlinear dynamic analyses • When SSI is significant (e.g. T s y s / T > 1.15), the capacity of the system is governed by yielding in soil when dynamic analyses are performed, but this is not the case when pushover analysis is performed and the capacity of the system is governed by yielding of the pier. • Pushover analysis overestimates the base shear of the piers when SSI is significant (e.g. systems with T s y s / T > 1.15). The predictions of pushover become less accurate as SSI becomes more effective with decreasing natural period of the piers (increasing T s y s /T) . The reason is that pushover cannot properly capture the dynamic soil-structure interaction when the interaction is significant (explained above). • Global ductility factor overestimates the ductility demand of piers when SSI is significant. This is attributed to the significant rigid body motion of the pier due to SSI and the resulting total displacements that are not representative of the deformation of the pier. 171 10 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE R E S E A R C H This dissertation presented a study on the effects of nonlinear soil-structure interaction on the inelastic seismic response of pile-supported bridge piers by performing nonlinear dynamic analyses of prototype pile-supported bridge pier systems located on soft soil. The motivation behind this research was to address practical issues related to SSI with the goal of providing a link between the geotechnical aspects of SSI and the practice of structural engineering with a performance-based design approach. This research was performed in four phases. The first phase was to explore an efficient implementation and practical application of the direct methods of SSI analysis with a system approach that includes proper mathematical representation of the soil, foundation, structure, and the soil-foundation interface and that accounts for the nonlinearities and radiation damping of the system. The numerical models were made with the objective of minimizing the required simplifications within practical limitations and without bias towards simplifying either the soil, or the foundation or the structure subsystems, as it might be observed in practice (Chapters 2,3, and 4). In this phase, the following conclusions were made: • Direct methods of SSI analysis can be successfully implemented to estimate the demands of various components of SSI systems and to observe their relationship with respect to each other so that a complete picture of the seismic response of the system is provided. 172 • At present time, SSI analysis with direct methods is still a challenging task that requires addressing various issues in regard to modelling the nonlinear behaviour of soil, structure, and soil-foundation interface. There are imperfections and uncertainties associated with any method of analyzing SSI and the direct methods are no exceptions. It is emphasized, however, that compared to other methods, such as p-y springs methods that employ complex spring elements to account for a combination of various system properties, direct methods use basic physical properties of the system and provide a more transparent means of simulating SSI. More complicated system effects, such as pile group effect, are accounted for through the simulation itself and therefore, using direct methods are not necessarily the most complex approach to the problem. The drawback of using direct methods is the long required runtime which with the ever increasing speed of personal computers continually becomes of lesser importance. The problem of costly runtime can be dealt with by employing techniques of efficient modeling as demonstrated in this study. The large number of analyses carried out in this study remarks the relative efficiency and the robustness of the simulations. • Some of the challenges involved in the numerical simulations were associated with the modeling of nonlinear behaviour of structural elements. There is a need for integrating advanced modeling techniques of both structural and soil elements into commercial programs for more efficient and transparent application of these programs by both structural and geotechnical engineers. The second phase of this research was to evaluate the effects of nonlinear SSI on the inelastic seismic response of pile-supported bridge pier systems by employing the numerical models developed in the first phase to construct response databases with and without accounting for SSI. The evaluation was carried out by. studying the general behaviour of the system and its components, i.e. the soil, the foundation and the piers, in response to earthquake loadings (Chapters 4 and 5), and by studying the differences in the response of the piers with and without inclusion of SSI (Chapters 5, 6 and 7). Results 173 were classified and presented as functions of the system natural periods, a format useful in structural engineering. The following conclusions are made: • The role of SSI in the response of the pile-supported bridge piers becomes more significant as the natural period of the pier decreases. This results from the higher pier-to-foundation stiffness ratio of the piers with shorter natural periods. The pier-to-foundation stiffness ratio can be measured by the ratio of the piers elongated period (system's natural period) to the fixed-base natural period of the pier, T s y s / T . Hence, it can be said that the role of SSI in the response of the pile-supported bridge piers becomes more effective as the T s y s / T period ratio increases. • In general, SSI may affect the response of the piers favourably by reducing the ductility demand of the piers and may affect the response adversely by increasing the total displacement demand of the piers. The reduction of the ductility demands is typically at the expense of increasing the total displacement demands. SSI may also decrease the total displacement demands, but that is not a dominant behaviour and can be ignored as a conservative assumption. SSI may also affect the response adversely by increasing the ductility demands of the piers. However, increasing the ductility demands, even though it is not the dominant effect of SSI, cannot be ignored as a conservative assumption and its possibility must be accounted for probabilistically (see conclusions of the third phase). • The deformation of the piers accounts for the decreasing portion of the total displacements of the system with decreasing natural period of the piers. This implies the increasing importance of the foundation response with decreasing natural period of the piers (i.e. increasing T s y s / T period ratio). • For shorter piers with shorter natural periods (i.e. with higher T s y s / T period ratios), the interaction is more in the form of the translation of foundation rather than the rotation of foundation. For taller piers with longer natural periods (i.e. lower T s y s / T period ratios), rotation is the more dominant form of SSI. 174 The effects of the modeling assumption of the piers on the estimated demands of the pile-supported bridge pier system were also studied in the second phase of this research and it was concluded that • The relationship between the force reduction factor R and the ductility demands p when SSI was included was different than that when SSI was not included and showed that for a given R, ductility demands were higher when SSI was included. The difference was more pronounced for piers with shorter period where SSI was more effective. This implies that the inelastic deformation of the pile-supported piers would be underestimated if the inelastic deformation ratios obtained from SDOF systems are used to obtain the demands from elastic analyses of the piers. This, for instance, means that the commonly used equal displacement rule may underestimate the ductility demands of pile-supported bridge piers. • SSI analysis with elastic pier may overestimate foundation demands. This signifies the importance of proper modeling of the inelastic behaviour of the structural elements in SSI analysis for seismic demand estimation of foundations and potential for reducing the cost of the construction of foundations. • The total displacements obtained from the SSI analysis with elastic piers can be greater or less than the total displacements obtained from the SSI analysis with inelastic piers. However, there is more tendency towards greater total displacements obtained from the inelastic piers. Therefore, it is more likely for the SSI analysis with elastic piers to underestimate the total displacement demands. The results presented here are not fully conclusive and further research is required to clarify the role of piers inelastic behaviour in total displacement demand estimations. The third phase of this study involved performing reliability analyses to provide a probabilistic picture of the effects of SSI on the seismic demands of bridge piers by including the uncertainties of the system parameters and by accounting for the 175 dispersions introduced by the variability of input ground motions and soil conditions so that clear conclusions in regard to the effects of SSI on the response of the piers can be made (Chapter 8). It was observed that • The probability of piers experiencing greater ductility demands when SSI is included increases with the increasing natural period of the piers (i.e. with decreasing T s y s /T) . For the prototype systems and the input ground motions of this study, and by assuming 10% variation for the natural period of the piers (T), the probability of an increase in the ductility demand is more than 25% for piers with natural periods greater than 1.0 s (systems with Tsys/T<1.08 as shown in Table 2.3). Similarly, by assuming 5% variation for T s y s / T , the probability of an increase in the ductility demand is more than 25% for systems with Tsys/T<1.05 (i.e. natural periods greater than 1.0 s). This is a considerable probability and confirms that SSI cannot always be ignored as a conservative assumption in the design of the pile-supported bridge piers. • The probability of SSI increasing the total displacement demand of the piers is significant in general and increases with decreasing natural periods of the system (i.e. with increasing T s y s /T) . This confirms the necessity of attention to the design of displacement sensitive bridge components when SSI is involved. Furthermore in the third phase of this study (Chapter 8), a method for approximate estimation of the effects of SSI on the ductility and the total displacement demands based on the response statistics was proposed and its practical implementation in a performance-base design guideline was explored. The idea of the proposed method is to use the statistics of demands obtained from the analyses of prototype systems to calculate the effects of SSI on similar systems by modifying their corresponding fixed-base demands. SSI modification factors, which are functions of the fixed-base elastic natural period of the piers, can be used to estimate the inelastic displacements of the SSI system from the inelastic displacement of its corresponding fixed-base system in a similar 176 fashion to estimating the inelastic displacements of structures from their corresponding elastic displacements by using inelastic deformation ratios. While the concept behind the SSI modification factors is the same as the inelastic deformation ratios, the domain of applicability of the SSI modification factors obtained from each statistical study could be very limited due to large variations in soil properties and the many possible combinations of pier and foundation properties which could ultimately compromise the practicality of such modification factors. However, a common ground for various databases may be found i f the results are expressed as functions of the period ratio T s y s /T. This ratio represents the overall system property defined by the stiffness of the pier and the stiffness of the foundation which itself depends on the stiffness and the configuration of the structural components of the foundation and the soil stiffness. This possibility needs to be investigated and i f this common ground for the statistics of the seismic demands of pile-supported bridge piers is proved to exist, then SSI modification factors can be implemented in a design guideline for simple typical bridges as functions of T s y s /T. A challenge in this case remains to be the accurate estimation of T s y s without a complicated numerical model and that could be the focus of another line of research for SSI of bridge piers. The concept of SSI modification factors proposed here differs from that of the inelastic deformation ratios in the literature in its explicit consideration o f the scatters observed in the statistics of the response of the prototype systems. This implies that instead of presenting the SSI modification factors by merely the mean (or median) of the statistics obtained, they are presented by their values associated with various percentiles of their probability distributions. The proposed SSI modification factors also differ from the inelastic deformation ratios in their explicit consideration of the uncertainties of the system properties to which they are dependent to. This means that the uncertainties in the natural periods of the system can be explicitly accounted for in the selection of SSI modification factors. This feature can especially include the uncertainties involved in T s y s and account for approximations involved in its calculation. The joint consideration of the scatter in the statistics of response (caused by the variation of ground motions and soil 1 7 7 layers) and the uncertainties of the system parameters was made by performing reliability analyses and thus SSI modification factors were estimated for various performance criteria with several levels of confidence as functions of the system period ratios T s y s / T with given uncertainty. Results were presented by SSI modification curves from which SSI modification factors can be obtained after deciding on the performance objective and the level of confidence required in the predicted performance. Given the prescriptive background of design codes and the current need for performance-based and yet prescriptive codes, the selection of the confidence level could be regarded as a subjective matter and therefore guidelines can be provided for the selection of the level of confidence. Although this represents yet another prescriptive procedure, it can provide designers with more flexibility to meet specific needs of specific projects. The merits and limitations of such procedures need to be explored by researchers and practicing engineers as well. If the input ground motions used to construct the SSI modification curves are all selected to represent a specific level of seismic hazard, then families of curves can be constructed for various levels of seismic hazard. Merits of such selection of ground motions, as opposed to the selection of ground motions with various amplitudes as was done in this study, must be investigated. The preliminary observation of the results of this work demonstrated that the SSI modification factors are to some degree dependent on the amplitude (intensity) of ground motions. Further research is needed to verify this and to investigate the effects of ground motions parameters on the SSI modification factors. The fourth and last phase of this research included an investigation on the nonlinear static pushover analysis in providing accurate estimates of seismic demands (and capacity) of the pile-supported piers including SSI (chapter 9). It was concluded that • Pushover analysis cannot properly capture the dynamic soil-structure interaction when the interaction is significant and therefore makes inaccurate prediction of the relative contribution of the piers deformation in the total displacements of 178 systems with high SSI. Pushover analysis predicts greater contribution of the deformation of the pier in the total displacements compared to that predicted by the dynamic analysis. This results in discrepancies between the demands and the capacities predicted by the pushover analyses and the demands and the capacities predicted by the nonlinear dynamic analyses. When SSI is significant (e.g. T s y s / T > 1.15), the capacity of the system is governed by yielding in soil when dynamic analyses are performed, but this is not the case when pushover analysis is performed and the capacity of the system is governed by yielding of the pier. 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Dynamic Nonlinear Analysis of Pile Foundation Using Finite Element Method in the Frequency Domain, Canadian Geotechnical Journal, Vol. 34, No. l,pp 34-43. Zhang, J. and Foschi, R.O., (2004). Performance-Based Design and Seismic Reliability Analysis Using Designed Experiments and Neural Networks, Journal of Probabilistic Engineering Mechanics, Vol. 19, pp 259-267 185 APPENDIX A CROSS SECTIONAL PROPERTIES OF T H E PIERS The material presented in this appendix is taken from the outputs of the cross section analysis program X T R A C T (Imbsen 2003). It includes the cross section geometry and the material properties of the piers and the results of the moment-curvature analysis performed by the program. Cross Section of Piers with Periods of 0.3,0.6, and 0.8 s Section Details: XCentroid: -.1364E-10 m Y Centroid: .44S9E-16 m Section Area: 1.762 mA2 I gross about X: .2503 m'M I gross about Y: .2503 m'M Reinforcing Bar Area: 16.97E-3 m"2 Perc ent Longitudinal Ste el: .9626 % Overall Width: 1.497 m Overall Height: 1.500 m Number of Fibers: 518 Number of Bars: 24 Number of Materials: 3 Material Types and Names: UnconfinedConcrete: H Unconfinedl Strain Hardening Steel: I Steell Confined Concrete: H Confined7 A . • • . > & r *i ; .1 m IS* V a • 7 B | I.- '• ,§f ...... V ••# • / I • • • ' • Comments: Section Type: Circular Column Type of Reinforcing: Spiral Reinforcing Transverse ReinforcingBarSize: 12 mm Spacing of Transverse Steel: 70.00 mm Outside Diameter: 1500 mm Cover Thickness: 50.00 mm Number of Longitudinal Bars: 24 Longitudinal Bar Size: 30 mm Cover Concrete: Unconfinedl Column Core Concrete: Confined7 Longitudinal Steel: Steell 186 Section Detai ls : X Centroid: Y Centroid: Section Area: L o a d i n g Detai ls : Constant Load - P: Incrementing Lo ads: Number of Points: Analysis Strategy: Analys is Resul ts : Failing Material: Failure Strain: Curvature at Initial Load: Curvature at First Yield: Ultimate Curvature: Moment at First Yield: Ultimate Moment: Centroid Strain at Yield: Centroid Strain at Ultimate: NA. at First Yield: N.A. at Ultimate: Energy per Length: Effective Yield Curvature: Effective Yield Moment: Over Strength F actor: EI Effective: Yield EI Effective: Bilinear Harding Slope: Curvature Ductility: Comments: User Comments -.1364E-10 m .4489E-16 m 1.762 mn2 3500 kN MyyOnly 30 Displacement Control Confined? 11.31E-3 Compression .2875E-16 1/m 2.268E-3 1/m 40.52E-3 1/m 4661 kN-m 6622 kN-m .571 IE-3 Tension 16.40E-3 Tension .2518 m .4049 m 243.7 kN 2.860E-3 1/m 5876 kN-m 1.127 2.05E+9 N-mA2 1.98E+7 N-mA2 .9639 % 14.17 A / S I &'• B i l l i 1 mw \ Moments about the Y-Axis - kN-m 70001 0.00 0.01 0.02 0.03 0.04 Curvatures about the Y-Axis - 1/m 0.05 Moment Curvature Relation Moment Curvature Bilinearization 1 8 7 Cross Sect ion of P ie rs w i th Per iods of 0.3, 0.6, a n d 0.8 s Section Details: XCentroid: -.1539E-10 m YCentroid: .4135M6 m Section Area: 1.762 mA2 I gross about X: .2508 m'M I gross about Y: .2508 m'M Reinforcing Bar Area: 19.30E.3 mA2 Perc ent Longitudinal Ste el: 1.095 % Overall Width: 1.497 m Overall Height: 1.500 m Number of Fibers: 518 Number of Bars: 24 Number of Matenals: 3 Material Types and Names: UnconfinedConcrete: I Unconfinedl Strain Hardening Steel: I Steell Confined Concrete: •Confined? a*1*' Comments: Section Type: Circular Column Type of Reinforcing: Spiral Reinforcing Transverse Reinforcing Bar Size: 12 mm Spacing of Transverse Steel: 70.00 mm Outside Diameter: 1500 mm Cover Thickness: 50.00 mm Number of Longitudinal Bars: 24 Longitudinal Bar Size: 32 mm Cover Concrete: Unconfinedl Column Core Concrete: Confined7 Longitudinal Steel: Steell 188 Section Details: X Centroid: Y Centroid: Section Area: Loading I it-tails: Constant Load - P: Incrementing Lo ads: Number of Points: Analysis Strategy: Analysis Results: Failing Material: Failure Strain: Curvature at Initial Load: Curvature at First Yield: Ultimate Curvature: Moment at First Yield: Ultimate Moment: Centroid Strain at Yield: Centroid Strain at Ultimate: NA. at First Yield: NA. at Ultimate: Energy per Length: Effective Yield Curvature: Effective Yield Moment: Over Strength Factor EI Effective: Yield EI Effective: FJiline ar H arding Slope: Curvature Ductility: Comments: User Comments -.1539E-10 m .4135E-16 m 1.762 mA2 3500 kN Myy Only 30 Displacement Control Confine d7 11.31E-3 Compression .3135E-16 1/m 2.287E-3 1/m 39.43E-3 1/m 5014 kN-m 7175 kN-m 5584E-3 Tension 15.66E-3 Tension .2442 m .3971 m 256.2 kN 2.895E-3 1/m 6348 kN-m 1.130 2.19E+9 N-mA2 2.26E+7 N-mA2 1.033 % 13.62 Moments about the Y-Axis - kN-m 8000' 7000' 6000' 5000' 4000' 3000' 2000' 1000-U " . . . . . . . 0.00 0.01 0.02 0.03 0.04 Curvatures about the Y-Axis - 1/m Moment Curvature Relation Moment Curvature FJilinearization 189 Material Properties Used in the Computation of Moment-Curvature Relations Unconfined Concrete Input Parameters: Tension Strength: 28 Day Strength: Post Crushing Strength: Tension Strain Capacity: Sp ailing Strain: Crushing Strain: Elastic Modulus: Secant Modulus: Model Details: -3000 kPa 30.00E+3 kPa 2.50E+7 kPa 2176 kPa For Strain-£< 2-£ t fc-0 f c * r For Strain- s< 0 fc=£-Ec For Strain-£<£ c u fc = r _ 1 + x r (s-£ ) F o r S k a m - £ < £ s p fc = f c u + (f c p - f c u ) . . ^ ( £ s p - £ c u ) Ec Ec - E , C C £ = Concrete Strain fc» Concrete Stress Ec = Elastic Modulus E s e c = Secant Modulus £ j = Tension Strain Capacity s c u = Ultimate Concrete Strain s c c = Strain at Peak Stress = .002 £ = Spalling Strain r f c = 28 Day Compressive Strength f c u = Stress at £ m f = Post Spalling Strength cp stress -kPa 30000' 20000 10000 ' ' V -0.001 0.001 0.002 0.003 0.004 0.005 -10000-strain Material Color States: I Tension strain after tension capacity I Tension strain before tension capacity B Initial state fl Compression before crushing strain I Compression before end of spalling D Compression after spalling Reference: Mander, J.B., Priestley, M. J. N., "Observed Stress-Strain Behavior of Confined Concrete", Journal of Structural Engineering. ASCE, Vol. 114, No. 8, August 1988, pp. 1827-1849 Comments: User Comments 1 9 0 Confined Concrete Input Parameters: Tension Strength: 28 Day Strength: Confined Concrete Strength: Tension Strain Capacity: Strain at Peak Stress: Crushing Strain: Elastic Modulus: Secant Modulus: Model Details: « c c = 0 0 2 E c 1 + 5 - | — - 1 f cc c Ec - E , 0 kPa 30.00E+3 kPa 36.56E+3 kPa 0 Tension 4.187E-3 11.31E-3 Compression 2.59E+7 kPa 1267 kPa For Strain- s< 2st fe- 0 F or Strain- £<0 te- z-Ec For Strain- s < s c u fc = fee" r - 1 + xr stress - kPa 40000T 0.000 0.002 0.004 0.006 0.008 0.010 0.012 strain Material Color States: [3 Tension strain after tension capacity [D Tension strain before tension capacity D Initial state ES Compression before crushing strain Reference: Mander, J.B., Priestley, M. J. N., "ObservedStress-Strain Behavior of Confined Concrete", Journal of Structural Engineering, ASCE, Vol. 114, No. 8, August 1988, pp. 1827-1849 Comments: Use-rCorrtmentc £ = Concrete Strain fc » Concrete Stress Ec = Elastic Modulus £ t = Tension Strain Capacity £ c u » Ultimate Concrete Strain £ c c = Strain atPeakStress f = 28 Day Compressive Strength f c c = Confined Concrete Strength 191 Steel Input Parameters: Yield Stress: Fracture Stress: Yield Strain: Strain at Strain Hardening: Failure Strain: Elastic Modulus: A dditional Information: Model Details: 420.0E+3 kPa 655.0E+3 kPa 2.100E-3 11.50E-3 .1200 2.00E+8 kPa Symetric Tension and Comp. For Strain - £< s^ For Strain - £< £ , fs = Es fs = f . sh *" * y ForSt ra in-£< £ s u fs = f u - (f u - f y ) • sh £ = Steel Strain fs = Steel Stress f = Yield Stress f,, = Fracture Stress S = Yield Strain £ s k = Strain at Strain Hardening £ s u = Failure Strain E = Elastic Modulus stress - kPa 700000 600000 500000 400000 300000 200000 100000 192 

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