UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Seismic analysis of pile foundations for bridges Thavaraj, Thuraisamy 2000

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2000-487288.pdf [ 14.95MB ]
Metadata
JSON: 831-1.0064051.json
JSON-LD: 831-1.0064051-ld.json
RDF/XML (Pretty): 831-1.0064051-rdf.xml
RDF/JSON: 831-1.0064051-rdf.json
Turtle: 831-1.0064051-turtle.txt
N-Triples: 831-1.0064051-rdf-ntriples.txt
Original Record: 831-1.0064051-source.json
Full Text
831-1.0064051-fulltext.txt
Citation
831-1.0064051.ris

Full Text

SEISMIC ANALYSIS OF PILE FOUNDATIONS FOR BRIDGES by T H U R A I S A M Y T H A V A R A J B. Sc. Eng.(Hons), University of Peradeniya, Sri Lanka, 1990 M . Eng., Asian Institute of Technology, Thailand, 1993 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE 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 We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A APRIL 2000 ©Thuraisamy Thavaraj, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Cw\) s The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT This thesis deals with the seismic response of bridges supported on pile foundations. A substantial portion of thesis is devoted in developing a better understanding of the seismic behavior of the pile foundations and their effects on the overall response of the bridge structure. Current methods for modeling the pile foundations are evaluated and some deficiencies are identified. New methods are developed that model better the seismic behavior of the pile foundations. Case studies based on the recorded response of a two span bridge under ambient vibration and strong earthquake shaking showed that the behaviour of the bridge foundations under strong shaking is highly nonlinear. A new lumped parameter model consisting a set of nonlinear springs and dashpots was developed to reproduce the nonlinear behaviour of the pile foundations during strong shaking. The nonlinear stiffnesses and damping ratios of the pile foundation are determined using an existing Quasi-3D finite element model. The lumped parameter model was incorporated in to a three dimensional stick model of the bridge superstructure to determine the seismic response of the bridge in the time domain. Seismic response analyses using the proposed model showed that the flexibility of the pile foundation and its nonlinear behaviour and the inertial interaction can significantly affect the behaviour of the bridge. A parametric study was conducted to see how different parameters affected the response of the bridge. One finding was that it was the relative stiffness of the foundation and the superstructure that controlled seismic response of the bridge. A new method was developed for determining the dynamic impedances of bridge abutments using simplified models of the abutments. Studies showed that the impedances of the abutments change significantly during strong shaking and that these changes should be taken into account. A finite element method was developed for the fully coupled seismic response analysis of pile foundations and superstructure in the time domain. This method treats the superstructure and ii foundation as two subsystems during the finite element solution process and hence requires less computer storage. The new method was used to evaluate the uncoupled analyses that are sometimes carried out in practice. Studies showed that the uncoupled analyses failed to give acceptable results. The Winkler model of a single pile is widely used in practice for the analysis of pile foundations. Therefore this method was evaluated in detail. A comprehensive program for the dynamic analysis of a single pile using a Winkler model was developed which included the currently available methods. This method was then used to evaluate the Winkler model and the use of p-y curves recommended by the American Petroleum Institute(API) by simulating a centrifuge test on a single pile under low level and strong shaking. Analyses using the API p-y curves gave poor estimates of the response of the pile under strong shaking. Studies showed that the extrapolation of the group behavior from the behavior of single pile using a group factor should consider both pile-to-pile interaction and the effects of superstructure-foundation interaction. The pile cap condition also needs to be modeled appropriately. An effective stress method was developed for the dynamic analysis of pile foundations in potentially liquefiable soils by incorporating a pore water pressure generation model into the Quasi-3D finite element method. The effective stress method was extensively verified using centrifuge test data on a single pile and (2x2) and (3x3) pile groups under low-level and strong shaking. A number of recommendations are made for future work to explore further some of the issues raised during the research study. i i i TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES xiv LIST OF FIGURES xv ACKNOWLEDGMENT xxv CHAPTER 1 INTRODUCTION 1 1.1 INTRODUCTION 2 1.2 OBJECTIVES OF THE THESIS 8 1.3 OUTLINE OF THE THESIS 9 CHAPTER 2 REVIEW OF METHODS ON THE DYNAMIC ANALYSIS OF PILES . 12 2.1 REVIEW OF METHODS ON THE D Y N A M I C RESPONSE OF A SINGLE PULE 13 2.1.1 Elastic Response 13 2.1.2 Non-Linear Response 15 2.2 REVIEW OF METHODS ON THE D Y N A M I C RESPONSE OF GROUP PILES 16 2.3 QUASI-3D FINITE E L E M E N T METHOD FOR THE D Y N A M I C ANALYSIS OF PILE FOUNDATIONS 17 2.3.1 Introduction 17 2.3.2 Dynamic Elastic Analysis of Pile Foundations Using Quasi-3D Finite Element Method in the Frequency Domain 18 2.3.3 Pile Head Impedances 19 2.3.4 Quasi-3D Finite Element Method for the Elastic Vertical Dynamic Analysis of Pile Foundations 24 iv 2.3.5 Rocking Impedances of Pile Groups 25 2.3.6 Quasi-3D Finite Element Method for the Dynamic Nonlinear Analysis of Pile Foundations in the Time Domain 27 2.3.7 Nonlinear Stiffness from PILE3D and PILIMP 29 2.3.8 Role of PTJLE3D and PILIMP in the Present Studies 29 CHAPTER 3 MODELING OF A SINGLE PILE USING WINKLER MODEL AND P-Y CURVES 31 3.1 REVIEW OF METHODS ON THE SINGLE PILE ANALYSIS USING WINKLER M O D E L A N D p-y CURVES 34 3.1.1 Static Behaviour 34 3.1.1.1 Linear Elastic Methods 34 3.1.1.2 Non-linear Methods 35 3.1.2 Dynamic Behaviour 36 3.1.2.1 Linear Elastic Methods 36 3.1.2.2 Nonlinear Methods 37 3.2 SINGLE PILE UNDER SEISMIC EXCITATION 39 3.2.1 Finite Element Formulation for Winkler Model 39 3.2.1.1 Element Stiffness Matrix 41 3.2.1.2 Element Mass Matrix 42 3.2.1.3 Element Damping Matrix 42 3.2.1.4 Element Load Vector 43 3.2.2 Equation of Motion 44 3.2.3 Solution Scheme 44 3.2.4 Near Field Behaviour 46 3.2.5 Free Field Behaviour 46 3.2.6 Equivalent Linear Approach Using Secant Stiffness 46 3.2.7 Incrementally Linear Elastic Approach Using Tangent Stiffness 47 3.2.8 Formulation of Global Mass, Stiffness and Damping Matrices 48 3.2.8.1 Global Mass Matrix 48 3.2.8.2 Global Stiffness Matrix 50 3.2.8.3 Damping Matrix 50 3.2.9 Computation of Correction Force Vector 52 3.2.10 Free Field Response 53 v 3.3 MODELING OF HORIZONTAL L A Y E R E D DEPOSIT 53 3.4 INTRODUCTION TO THE COMPUTER PROGRAMS PJEE-PY A N D FREEFLD 55 3.5 ANALYSIS USING SECANT A N D TANGENT STIFFNESS APPROACHES 56 3.5.1 Introduction 56 3.5.2 Model of the Pile and Soil 57 3.5.3 Near Field Behaviour 57 3.5.4 Input Base Acceleration 57 3.5.5 Response of Pile Under Earthquake Excitation 59 3.6 ANALYSIS OF CENTRIFUGE TEST OF SINGLE PILE 65 3.7 RESPONSE OF PILE UNDER M O D E R A T E L Y STRONG E A R T H Q U A K E EXCITATION 68 3.7.1 Introduction 68 3.7.2 Dynamic Analysis Using PILE3D 70 3.7.3 Dynamic Analysis of Centrifuge Test Using the Winkler Model and the p-y Curves 75 3.7.3.1 Model of the Soil-Pile-Structure System 75 3.7.3.2 Dynamic Response of Free Field 75 3.7.3.3 Dynamic Response of Single Pile using PILE-PY 75 3.7.3.4 Results of Analyses with kh=15000 kN/m 3 77 3.7.3.5 Results of Analyses with kh=2500 kN/m 3 80 3.7.3.6 Results of Analyses with k h H=E m a x 80 3.8 RESPONSE OF PILE UNDER L O W HARMONIC EXCITATION 86 3.9 ANALYSIS OF A SINGLE PILE E M B E D D E D IN C L A Y E Y SOIL USING WINKLER M O D E L A N D P-Y CURVES 89 3.9.1 Problem Description 89 3.9.2 PILE-PY Model of the Single Pile 89 3.9.3 PDLE3D Model of the Single Pile 91 3.9.4 Earthquake Input 91 3.9.5 Results of Analysis 91 3.10 DISCUSSION OF RESULTS 93 3.10.1 Single Pile in Sandy Soil 93 3.10.2 Single Pile in Clayey Soil 95 3.11 CONCLUSIONS 96 vi CHAPTER 4 MODELING OF BRIDGES FOR DYNAMIC ANALYSIS 98 4.1 REVIEW OF METHODS USED FOR THE D Y N A M I C ANALYSIS OF BRIDGES 100 4.1.1 Stick Model Studies 100 4.1.2 System Identification Studies 110 4.1.3 Single Bent Model Studies 112 4.1.4 Rigid Body Model Studies 115 4.1.5 Detailed Finite Element Model Studies - Coupled and Uncoupled Analysis 115 4.1.6 Code Based Models 119 4.2 C A S E STUDIES: D Y N A M I C BEHAVIOUR OF THE PAINTER STREET OVERPASS UNDER AMBIENT VIBRATION A N D E A R T H Q U A K E SHAKING 121 4.2.1 Details of the Painter Street Overpass 121 4.3 C A S E STUDY 1: D Y N A M I C BEHAVIOUR OF THE BRIDGE UNDER AMBIENT VIBRATION 124 4.3.1 Introduction 124 4.3.2 Mathematical Model of the Bridge 124 4.3.3 Stiffnesses of Springs 126 4.3.4 Calculation of Resultant Abutment Stiffness 126 4.3.5 Transformation of Stiffness 129 4.3.6 Resultant Column Foundation Stiffness 130 4.3.7 Modal Frequencies and Mode Shapes 134 4.3.7.1 Determination of Modal Frequencies and Mode Shapes 134 4.3.7.2 Comparison of Modal Frequencies and Mode Shapes 140 4.3.8 Fixed Base Model vs Flexible Base Model 141 4.4 C A S E STUDY 2: D Y N A M I C BEHAVIOUR OF THE BRIDGE UNDER E A R T H Q U A K E SHAKING 144 4.4.1 Rigid Body Model of the Painter Street Over Pass 144 4.4.2 Analytical Procedure 147 4.4.3 Analysis of Response of the Bridge under the Main Shock 149 4.4.3.1 Force-Deformation Behaviour of Abutments 149 4.4.3.2 Hysteresis Loops 149 4.4.3.3 Deformation Dependency of Stiffness 151 4.4.3.4 Time Variation of Stiffness 157 vii 4.4.4 Analysis of Response Using Power Spectral Density and the Transfer Functions 157 4.4.4.1 Introduction 157 4.4.4.2 Response in the Longitudinal Direction 160 4.4.4.3 Response in the Transverse Direction 160 4.5.3 Limitations of the Current Modeling of Bridge Foundations 165 4.6 CONCLUSIONS 168 CHAPTER 5 MODELING OF PILE FOUNDATION AS A LUMPED PARAMETER SYSTEM FOR NONLINEAR ANALYSIS 169 5.1 INTRODUCTION 170 5.2 INPUT MOTIONS . 170 5.3 NONLINEAR PILE CAP STIFFNESS A N D DAMPING 173 5.4 SEISMIC SUPERSTRUCTURE RESPONSE 173 5.5 N U M E R I C A L STUDIES OF NONLINEAR RESPONSE 174 5.5.1 Introduction 174 5.5.2 Lumped Parameter Model of a Single pile 174 5.5.2.1 Problem Description 174 5.5.2.2 Pile Head Impedances 178 5.5.2.3 Equivalent Lumped Parameter Model Response 180 5.5.3 Approximate Determination of Nonlinear Pile Head Impedance 181 5.5.4 The Effect of Superstructure Interaction 183 5.5.5 The Effect of Cross Translation-Rotational Impedance 187 5.6 CONCLUSIONS 187 CHAPTER 6 SEISMIC RESPONSE ANALYSIS OF BRIDGES 190 6.1 G E N E R A L COMPUTATIONAL M O D E L 191 6.1.1 Space Frame Member 191 6.1.1.1 Element Stiffness Matrix 192 6.1.1.2 Element Mass Matrix 194 6.1.2 Lumped parameter model of the Pile Foundation 194 6.1.3 Equations of Motion 195 viii 6.1.4 Formulation of Global Mass, Stiffness and Damping Matrices 196 6.1.4.1 Global Mass Matrix 196 6.1.4.2 Global Stiffness Matrix 196 6.1.4.3 Global Damping Matrix 196 6.1.5 Modes and Modal Frequencies 197 6.1.6 Dynamic Solution Approach 198 6.2 N U M E R I C A L STUDIES ON THE A A S H T O CODE BRIDGE 198 6.2.1 Introduction 198 6.2.2 Finite Element Model of the Bridge Structure 201 6.2.3 Nonlinear Stiffness and Damping of Pile Foundation Using PILE3D andPILIMP 201 6.2.4 Stiffness and Damping Based on " S H A K E " Moduli And Damping Ratio 204 6.2.5 Seismic Response of Bridge Under Transverse Earthquake Loading . . 206 6.2.6 Seismic Response of Bridge Under Longitudinal Earthquake Loading .217 6.2.7 Seismic Soil-Pile-Superstructure Interaction Analysis of Bridge 221 6.2.7.1 Mathematical Model Proposed by CalTrans for the Superstructure 221 6.2.7.2 Superstructure Model Parameters 222 6.2.7.3 Seismic Response of the Bridge When the Effect of Superstructure Interaction on the Foundation is included 222 6.3 PARAMETRIC STUDY 225 6.3.1 Introduction 225 6.3.2 Mathematical Model of the Bridge 225 6.3.3 Effect of Foundation Flexibility 231 6.3.3.1 Introduction 231 6.3.3.2 Foundation Stiffness Parameters 231 6.3.3.3 Superstructure and Pier Parameters 233 6.3.4 Effect of Different Parameters on Bridge Response 233 6.3.4.1 Effect of Nondimensional Stiffness Parameter, K P / K F L 233 6.3.4.2 Effect of Nondimensional Stiffness Parameter, K P * H 2 / K F L . . . 233 6.3.4.3 Effect of Nondimensional Parameter, H p /d 235 6.3.4.4 Effect of Nondimensional Stiffness Parameter, MVpd 3 235 6.3.4.5 Effect of Nondimensional Stiffness Parameter, K S A / K F L 235 6.3.4.6 Effect of Nondimensional Stiffness Parameter K S u / K F L 235 6.4 CONCLUSIONS 239 ix CHAPTER 7 STIFFNESS AND DAMPING OF BRIDGE ABUTMENTS 240 7.1 T R A N S V E R S E A N D V E R T I C A L STIFFNESS A N D D A M P I N G OF ABUTMENTS 242 7.2 LONGITUDINAL STIFFNESS A N D DAMPING OF ABUTMENTS 244 7.3 D Y N A M I C IMPEDANCES OF V E R T I C A L W A L L : ELASTIC RESPONSE 246 7.3.1 Modeling of Abutment Wall-soil System 246 7.3.2 Determination of Impedance of the Abutment Wall 250 7.3.3 Numerical Studies 251 7.3.3.1 Introduction 251 7.3.3.2 Finite Element Mesh 251 7.3.3.3 Wall and Soil Parameters 251 7.3.3.4 Impedances of the Vertical Wall 252 7.4 D Y N A M I C IMPEDANCES OF TRAPEZOIDAL A B U T M E N T S 252 7.4.1 Finite Element Model 252 7.4.2 Determination of Impedance 256 7.4.3 Elastic Impedances of Meloland Overpass Abutments 258 7.4.3.1 Introduction 258 7.4.3.2 Finite Element Model 258 7.4.3.3 Soil Properties 258 7.4.3.4 Elastic Impedances 259 7.5 NONLINEAR STIFFNESS A N D DAMPING A N D SEISMIC RESPONSE OF M E L O L A N D OVERPASS A B U T M E N T 261 7.5.1 Introduction 261 7.5.2 Method of Analysis 261 7.5.2.1 Time history response 261 7.5.2.2 Frequency and Stiffness and Damping Response 262 7.5.2.3 Finite Element Model, Model Parameters and the Earthquake Input 7.5.3 Results of Analysis 262 7.5.3.1 Acceleration Response 262 7.5.3.2 Stiffness and Damping Response 264 7.5.3.3 Frequency Response 264 7.6 CONCLUSIONS 267 x CHAPTER 8 SIMPLIFIED PILE-SUPERSTRUCTURE MODEL [SPSM] 270 8.1 E V A L U A T I O N OF SPSM 271 8.2 M O D E L PROPERTIES 273 8.2.1 Soil 273 8.2.2 Pile Foundations 273 8.2.3 Computational Models 275 8.2.4 Input Motion 275 8.3 RESULTS OF SPSM A N D F U L L GROUP A N A L Y S E S 275 8.4 EFFECTS OF PILE CAP ROTATION 280 8.5 CONCLUSIONS '. 283 CHAPTER 9 COUPLED ANALYSIS OF BRIDGE SUPERSTRUCTURE AND PILE FOUNDATION 284 9.1 INTRODUCTION 285 9.2 F U L L Y COUPLED SOLUTION A P P R O A C H 287 9.3 MODELING OF FOUNDATION SUBSYSTEM 290 9.4 MODELING OF SUPERSTRUCTURE S U B S Y S T E M 291 9.5 D Y N A M I C ANALYSIS PROCEDURE FOR A STRUCTURE WITH MULTI-SUPPORT EXCITATIONS 292 9.6 DETERMINATION OF SUPPORT REACTIONS 294 9.7 N U M E R I C A L STUDIES 295 9.7.1 Introduction 295 9.7.2 Problem Description 295 9.7.2.1 Superstructure 295 9.7.2.2 Soil Data 295 9.7.2.3 Pile Foundation Data 297 9.7.2.4 Earthquake Input 297 9.7.3 Modeling of Superstructure Subsystem 299 9.7.4 Modeling the Pile Foundation Subsystem 299 9.7.5 Soil-Pile-Superstructure Interaction Analyses 299 9.7.5.1 Introduction 299 9.7.5.2 Fully Coupled Analysis of Complete Structure Using PILE3D Program 300 xi 9.7.5.3 Iterative Coupled Analysis Using SOISTR Program 300 9.7.5.4 Uncoupled Analysis of Superstructure and the Pile Foundation Using SPFR-MS and PILE3D Program 300 9.7.5.5 Analysis of Pile Group Foundation Only Using the Program PILE3D 301 9.7.6 Results of Analysis 301 9.7.6.1 Iterative coupled approach vs Fully coupled approach 301 9.7.6.2 Uncoupled Approach vs Fully Coupled and Iterative Coupled Approach . . 304 9.7.6.3 Pile Foundation Alone vs Fully Coupled Analysis of Superstructure and Pile Foundation 312 9.7 CONCLUSIONS 312 EFFECTIVE STRESS ANALYSIS OF PILE FOUNDATIONS 317 10-1 INTRODUCTION 318 10.2 M-F-S POREWATER PRESSURE M O D E L 319 10.2.1 Introduction 319 10.2.2 Porewater Pressure under Undrained Conditions 319 10.2.3 Original Four Parameter Model for the Volumetric Stains under Drained Conditions 320 10.2.4 Parameters of the M-F-S Model 321 10.3 EFFECTIVE STRESS ANALYSIS OF PILE FOUNDATIONS 321 10.4 MODIFICATIONS OF SOIL PROPERTIES 322 10.5 D Y N A M I C ANALYSIS OF CENTRIFUGE TEST OF SINGLE PILES A N D PILE GROUPS 322 10.5.1 Centrifuge Tests 322 10.5.2 Effective Stress Dynamic Analysis of Single Pile 329 10.5.2.1 Introduction 329 10.5.2.2 Finite Element Model of the Single Pile-superstructure System 329 10.5.2.3 Soil and Pile Properties 329 10.5.2.5 Porewater Pressure Model Parameters 331 10.5.2.6 Earthquake Input Motion 331 10.5.3 Results of Single Pile Analysis 334 xii 10.5.3.1 Test I 334 10.5.3.2 Test K 338 10.5.4 Effective Stress Dynamic Analysis of (2x2) Group Pile 344 10.5.4.2 Finite Element Model of the (2x2) Group Pile-Superstructure System 344 10.5.5 Results of (2x2) Group Pile Analysis 344 10.5.5.1 Test I 344 10.5.5.2 Approximate Effective Stress Analysis of Test I 344 10.5.5.3 Total Stress Analysis of Test I 348 10.5.5.4 Linear Elastic Analysis of Test I 352 10.5.5.5 Test K 354 10.6 EFFECTIVE STRESS D Y N A M I C ANALYSIS OF (3X3) GROUP PILE . . . 354 10.6.1 Finite Element Model of the (3x3) Group Pile-superstructure System 354 10.6.2 Results of (3x3) Group Pile Analysis 358 10.6.2.1 Test I 358 10.6.2.2 Test K 358 10.7 CONCLUSIONS 362 CHAPTER 11 SUMMARY AND CONCLUSIONS 364 REFERENCES 376 APPENDICES 390 xiii LIST OF TABLES Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 4.9 Table 4.10 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8 Table 7.1 Table 8.1 Table 8.2 Table 8.3 Table 9.1 Table 9.2 Table 10.1 Table 10.2 Table 10.3 Comparison frequencies from ambient data and strong motion data Comparison modal frequencies between stick model and 11 model with fixed bases Comparison of modal frequencies between stick model and detail model List of strong motions recorded at painter street overpass Pile foundation stiffness Wall stiffness Footing stiffness Resultant stiffness of the abutments and the center pier foundation Computed modal frequencies and mode shapes Comparison of fixed base model and flexible base model frequencies Properties of bridge superstructure and the pier Summary of stiffnesses Summary of first transverse mode frequencies under different support conditions Summary of first longitudinal mode frequencies under different support conditions Nondimensional parameters chosen for the parametric study Foundation parameters chosen for the study The foundation stiffnesses calculated for the six different cases Superstructure and pier parameters Parameters, ejes and A used in the analysis Lateral stiffnesses of single pile and pile groups Group effect corresponding to initial elastic and minium lateral stiffnesses First mode frequency of the superstructure-pile foundation system Details of the simplified models of a bridge structure Summary of the four types of analyses x i v LIST OF FIGURES Fig. 2.1 Quasi-3D finite element model of the pile foundation Fig. 2.2 Pile head impedances Fig. 2.3 The mechanism of rocking in a pile group Fig. 2.4 Mechanical model used in PILE3D for the analysis of pile group with a rigid cap Fig. 2.5 Variation of lateral stiffness with time Fig. 3.1 Winkler model of the single pile for static analysis Fig. 3.2 A typical p-y curve Fig. 3.3 Winkler model of the single pile for the dynamic analysis Fig. 3.3 Winkler soil model for the dynamic analysis of a single pile under base excitation Fig. 3.5 A typical p-y curve suggested by API(1993) for sand Fig. 3.6 p-y curve for loading , unloading using Masing rule Fig. 3.7 p-y curve for first unloading and general reloading Fig. 3.8 A typical variation of shear modulus and damping ratio with shear strain Fig. 3.9 Finite element model for the analysis of free field soil Fig. 3.10 Soil-Pile model used in the analysis Fig. 3.11 Unit normalized input acceleration time history Fig. 3.12a Comparison of displacement at the superstructure mass Fig. 3.12b Comparison of acceleration at the superstructure mass Fig. 3.13a Comparison of displacement of pile at the surface Fig. 3.13b Comparison of acceleration of the pile at the surface Fig. 3.14a p-y hysteresis loops at 1-D depth Fig. 3.14b p-y hysteresis loops at 2-D depth Fig. 3.15 p and y time histories and p-y loops at 1-D depth Fig. 3.16 p and y time histories and p-y loops at 2-D depth Fig. 3.17a Comparison of displacement at the superstructure mass Fig. 3.17b Comparison of acceleration at the superstructure mass xv Fig. 3.18 Details of instrumented single pile used in the centrifuge test Fig. 3.19 Prototype model of the single pile test Fig. 3.20 Base input acceleration time history Fig. 3.21 Finite element model used in the PILE3D analysis Fig. 3.22 Variation of shear modulus and damping ratio of the sand with shear stain Fig. 3.23 Comparison of measured and computed acceleration time histories at the top of structure Fig. 3.24 Comparison of measured and computed displacement time histories at the top of structure Fig. 3.25 Comparison of measured and computed bending moment time histories at the soil surface Fig. 3.26 Comparison of measured and computed bending moment time histories at 3m depth Fig. 3.27 Comparison of maximum bending moment profiles Fig. 3.28 Comparison of measured and computed free field acceleration time histories Fig. 3.29 Comparison of measured and computed free field acceleration time histories Fig. 3.30 Comparison of measured and computed acceleration time histories at the top of structure Fig. 3.31 Comparison of measured and computed displacement time histories at the top of structure Fig. 3.32 Comparison of measured and computed bending moment time histories at the soil surface Fig. 3.33 Comparison of measured and computed bending moment time histories at 3m depth Fig. 3.34 Comparison of measured and computed acceleration time histories at the top of structure Fig. 3.35 Comparison of measured and computed displacement time histories at the top of structure Fig. 3.36 Comparison of measured and computed bending moment time histories at the soil surface Fig. 3.37 Comparison of measured and computed bending moment time histories at 3m depth Fig. 3.38a Computed p-y hysteresis loops at 0.25m depth Fig. 3.38b Computed p-y hysteresis loops at 0.75m depth xvi Fig. 3.39 Comparison of measured and computed acceleration time histories at the top of structure Fig. 3.40 Comparison of measured and computed displacement time histories at the top of structure Fig. 3.41 Comparison of measured and computed bending moment time histories at the soil surface Fig. 3.42 Comparison of measured and computed bending moment time histories at 3m depth Fig. 3.43 Comparison of maximum bending moment profiles Fig. 3.44 Comparison of maximum bending moment profiles Fig. 3.45 Model and the properties of soil and pile used in the analysis Fig. 3.46 Variation of shear modulus and damping ratio with shear strain Fig. 3.47 Input acceleration time history Fig. 4.1 Stick model with the boundary springs used by Wilson (1988) Fig. 4.2 Details of the Painter Street Overpass and the stick model with foundation springs used by Mc Callen and Romstad(1994) Fig. 4.3a Details of Meloland Overpass and the stick model with foundation springs used by Wilson and Tan(1990) Fig. 4.3b Finite element model of the Meloland Overpass Abutment used by Wilson and Tan(1990) Fig. 4.4 Bridge model used by Chen and Penzien (1977) Fig. 4.5 Single bent model of the Painter Street Overpass used by Makris et al., 1994) Fig. 4.6 Detail finite element model of Painter Street Overpass used by Mc Callen and Romstad(1994) (LL model) Fig. 4.7 Details of the Painter Street Overpass and the locations of strong motion instruments Fig. 4.8 Stick model of the Painter Street Overpass used in the modal analysis Fig. 4.9 The abutment of the Painter Street Overpass Fig. 4.10 Coordinate system used in the analysis Fig. 4.11 Measured shear modulus profile underneath the center pier Fig. 4.12 Computed first mode shape and the model frequency Fig. 4.13 Computed second mode shape and the model frequency Fig. 4.14 Computed third mode shape and the model frequency xvii Fig. 4.15 Computed fourth mode shape and the model frequency Fig. 4.16 Computed fifth mode shape and the model frequency Fig. 4.17 Computed sixth mode shape and the model frequency Fig. 4.18 Comparison of measured and computed first transverse model shapes Fig. 4.19 Comparison of measured and computed second transverse model shapes Fig. 4.20 Comparison of measured and computed first vertical model shapes Fig. 4.21 Comparison of measured and computed second vertical model shapes Fig. 4.22 Rigid body model of the Painter Street Overpass used in the analysis Fig. 4.23 Comparison of measured and linearly interpolated deformation of the center bent top Fig. 4.24 East abutment force and deformation time histories Fig. 4.25 West abutment force and deformation time histories Fig. 4.26 The east abutment force deformation loops for 0-15 sec during Main shock, 1992 Fig. 4.27 The west abutment force deformation loops for 0-15 sec during Main shock, 1992 Fig. 4.28 Determination of stiffness from force deformation loop Fig. 4.29 The stiffness deformation relationship for east abutment during Main shock, 1992 Fig. 4.30 The stiffness deformation relationship for west abutment during Main shock, 1992 Fig. 4.31 The stiffness deformation time histories for east abutment during Main shock, 1992 Fig. 4.32 The stiffness deformation time histories for west abutment during Main shock, 1992 Fig. 4.33 Comparison of unit normalized power spectral density of deck and free field motion in the longitudinal direction, Main shock, 1992 Fig. 4.34 Variation of amplitude of transfer function with frequency, Main shock, 1992 Fig. 4.35 Comparison of unit normalized power spectral density of deck and free field motion in the transverse direction, Main shock, 1992 Fig. 4.36 Variation of amplitude of transfer function with frequency, Main shock, 1992 Fig. 4.37 Comparison of unit normalized power spectral density of deck and free field motion in the longitudinal direction, Nov. 21, 1986, After shock Fig. 4.38 Variation of amplitude of transfer function with frequency, Main shock, 1992 Fig. 4.39 Comparison of unit normalized power spectral density of deck and free field motion in the transverse direction, Nov. 21, 1986, After shock Fig. 4.40 Variation of amplitude of transfer function with frequency, Main shock, 1992 Fig. 4.41 Typical variation of shear modulus and damping ratio with the shear stain xviii Fig. 5.1 Pile-Soil-Superstructure system Fig. 5.1 Lumped parameter model of pile-soil-superstructure system Fig. 5.2a Single pile with a lumped mass in a homogeneous media Fig. 5.2b Equivalent 2-DOF lumped parameter model Fig. 5.3 Input horizontal acceleration used in the Quasi-3D finite element analysis Fig. 5.4 The variation of shear modulus and damping ratio with shear strain Fig. 5.5 Finite element mesh used in the single pile analysis Fig. 5.6a Time histories of lateral, cross-coupling and rotational stiffnesses during strong shaking Fig. 5.6b Time histories of lateral, cross-coupling and rotational damping during strong shaking Fig. 5.7a Comparison of acceleration response of a single pile under strong shaking Fig. 5.7b Comparison of displacement response of a single pile under strong shaking Fig. 5.8a Comparison of stiffness time histories computed directly from PILE3D and by using the moduli and damping from S H A K E analysis Fig. 5.8b Comparison of damping time histories computed directly from PILE3D and by using the moduli and damping from S H A K E analysis Fig. 5.9a Comparison of acceleration response of a single pile under strong shaking Fig. 5.9b Comparison of displacement response of a single pile under strong shaking Fig. 5.10a The effect of inertial interaction on the lateral stiffness Fig. 5.10b The effect of inertial interaction on the cross-coupling and rotational stiffnesses Fig. 5.1 l a Comparison of acceleration response of a single pile under strong shaking Fig. 5.1 lb Comparison of displacement response of a single pile under strong shaking Fig. 6.1 A three dimensional space frame element Fig. 6.2 Details of the AASHTO(1983) bridge and the pile foundations Fig. 6.3a Input acceleration time history Fig. 6.3b Unit normalized power spectral density of input acceleration Fig. 6.4 Stick model of the bridge and the foundations springs and dashpots Fig. 6.5a Lateral and cross-coupling stiffness time histories for Am a x=0.5g Fig. 6.5b Rotational stiffness time histories for Am a x=0.5g Fig. 6.6a Shear modulus profiles xix Fig. 6.6b Damping ratio profiles from S H A K E Fig. 6.7 Bridge model used in the transverse vibration analysis Fig. 6.8 The effect of support conditions on the first mode frequency Fig. 6.9 The effect of support conditions on the first mode frequency Fig. 6.10a Comparison of acceleration time histories of the bridge deck Fig. 6.10b Comparison of displacement time histories of the bridge deck Fig. 6.1 l a Comparison of acceleration time histories of the bridge deck Fig. 6.1 lb Comparison of displacement time histories of the bridge deck Fig. 6.12a Comparison of acceleration time histories of the bridge deck Fig. 6.12b Comparison of displacement time histories of the bridge deck Fig. 6.13 Bridge model used in the transverse vibration analysis Fig. 6.14 The effect of support conditions on the first model frequency Fig. 6.15a Comparison of acceleration time histories of the bridge deck Fig. 6.15b Comparison of displacement time histories of the bridge deck Fig. 6.16 Model used in the pile-soil-superstructure interaction analysis using PILE3D Fig. 6.17 First transverse mode shape of the bridge Fig. 6.18 The effect of superstructure interaction on the lateral stiffness of the pile foundation Fig. 6.19 The effect of superstructure interaction on the first mode frequency Fig. 6.20a Comparison of acceleration time histories of the bridge deck Fig. 6.20b Comparison of displacement time histories of the bridge deck Fig. 6.21 A typical two span bridge Fig. 6.22 Model of the bridge used in the parametric study Fig. 6.23 Effect of nondimensional parameter, K S p / K F L on the period ratio Fig. 6.24 Effect of nondimensional parameter, K S p / K F L on the period ratio Fig. 6.25 Effect of nondimensional parameter, K S P H 2 / K F R on the period ratio Fig. 6.26 Effect of nondimensional parameter, H p /d on the period ratio Fig. 6.27 Effect of nondimensional parameter, M s / pd 3 on the period ratio Fig. 6.28 Effect of nondimensional parameter, K S A / K F L on the period ratio Fig. 6.29 Effect of nondimensional parameter, KSD/K?L on the period ratio Fig. 7.1 A typical two span bridge and the abutment embankment soil system Fig. 7.2 A typical abutment soil system xx Fig. 7.3 Plane strain finite element model of the Meloland Overpass used by Wilson and Tan(1990) Fig. 7.4 The abutment model used by Maragakis (1986) Fig. 7.5 The pressure diagrams behind a translating and rotating wall (Lam and Martin, 1986) Fig. 7.6 Type of problems considered in the vertical wall analysis Fig. 7.7 Finite element mesh used in the vertical wall analysis Fig. 7.8 Variation of nondimensional lateral, cross-coupling and rotational stiffness with dimensionless frequency Fig. 7.9 Variation of nondimensional lateral, cross-coupling and rotational damping with dimensionless frequency Fig.7.10 Finite element mesh used in the trapezoidal abutment analysis Fig. 7.11 Variation of lateral dynamic stiffness with frequency Fig. 7.12 Variation of vertical dynamic stiffness with frequency Fig. 7.13 Input base acceleration time history used in the analysis Fig. 7.14 Acceleration time history at the top of abutment Fig. 7.15 Variation of first transverse mode frequency with time Fig. 7.16 Variation of first vertical mode frequency with time Fig. 7.17 Variation of transverse stiffness with time Fig. 7.18 Variation of vertical stiffness with time Fig. 7.19 Variation of transverse damping with time Fig. 7.20 Variation of vertical damping with time Fig. 8.1 Simplified pile superstructure model (SPSM) Fig. 8.2 Simplified pile superstructure model (SPSM) and the properties of soil and the pile used in the analysis Fig. 8.3a Comparison of maximum bending moment profiles Fig. 8.3b Comparison of maximum shear force profiles Fig. 8.4a Comparison of maximum deflection profiles Fig. 8.4b Comparison of maximum rotation profiles Fig. 8.5a Comparison of maximum bending moment profiles xxi Fig. 8.5b Comparison of maximum shear force profiles Fig. 8.6a Comparison of maximum deflection profiles Fig. 8.6b Comparison of maximum rotation profiles Fig. 9.1 Schematic diagram of the soil-pile-superstructure interaction analysis Fig. 9.2 An iterative procedure for the coupled dynamic analysis of superstructure and pile foundation Fig. 9.3 Details of the superstructure and the pile foundation used in the analysis Fig. 9.4a Variation of Shear modulus and damping ratios with shear strain. Fig. 9.4b Maximum shear modulus profile Fig. 9.5a Comparison of pile cap acceleration time history Fig. 9.5b Comparison of pile cap displacement time history Fig. 9.6a Comparison of superstructure acceleration time history Fig. 9.6b Comparison of superstructure displacement time history Fig. 9.7a Comparison of maximum bending moment profiles Fig. 9.7b Comparison of maximum shear force profiles Fig. 9.8a Free field acceleration time history used as an input in the superstructure analysis Fig. 9.8b Comparison of dynamic shear force time histories Fig. 9.8c Comparison of unit normalized power spectral densities of the dynamic shear forces Fig. 9.9a Comparison of superstructure acceleration time history Fig. 9.9b Comparison of superstructure displacement time history Fig. 9.10a Comparison of pile cap acceleration time history Fig. 9.10b Comparison of pile cap displacement time history Fig. 9.11a Comparison of maximum bending moment profiles Fig. 9.11b Comparison of maximum shear force profiles Fig. 9.12a Comparison of pile cap acceleration time history Fig. 9.12b Comparison of pile cap displacement time history Fig. 9.13a Comparison of maximum bending moment profiles Fig. 9.13b Comparison of maximum shear force profiles Fig. 10.1 Layout single pile and (2x2) and (3x3) pile group models for the centrifuge test Fig. 10.2 Instrumented pile for single pile test xxii Fig. 10.3 Fig. 10.4 Fig. 10.5 Fig. 10.6 Fig. 10.7a Fig. 10.7 Fig. 10.8 Fig. 10.9 Fig. 10.10 Fig. 10.11 Fig. 10.12a Fig. 10.12b Fig. 10.13 Fig. 10.14 Fig. 10.15 Fig. 10.16 Fig. 10.17 Fig. 10.18a Fig. 10.18b Fig. 10.18c Fig. 10.19 Fig. 10.20 Fig. 10.21 Fig. 10.22 Fig. 10.23 Fig. 10.24 Fig. 10.25 Instrumented test piles and details of superstructure for (2x2) pile group Instrumented test piles and details of superstructure for (3x3) pile group Sketch of the superstructure used in the (2x2) and (3x3) pile group tests and the model of superstructure used in the analysis. Finite element mesh for a single pile Variation of shear modulus and damping ratio with shear strain G m a x profile Input acceleration time history used in the centrifuge test of single pile and pile groups Comparison of porewater pressure time histories pass filtered Comparison of superstructure acceleration time histories xxiii Fig. 10.26 Fig. 10.27 Fig.10.28 Fig. 10.29 Fig. 10.30 Fig. 10.31 Fig. 10.32 Fig. 10.33 Fig. 10.34a Fig. 10.34b Fig. 10.35 Fig. 10.36a Fig. 10.36b Fig. 10.37a Fig. 10.37b Fig. 10.38a Fig. 10.39a Fig. 10.39b Fig. 10.39c Fig. A l . l Fig. A1.2 Fig. A2.1 Fig. A2.2 Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Comparison of Modulus of subgrade reaction as a function of relative density Parameters C,, C 2 and C 3 as a function of Angle of internal friction Variation of lateral, cross-coupling and rotational stiffness of a single pile with 8 Variation of 8 with E p / E s ratio xxiv ACKNOWLEDGMENT The author would like to express his sincere appreciation and gratitude to his thesis supervisor, Dr. W.D.L. Finn, for his advice, guidance and encouragement throughout this research and for his constant support and friendship. The author wishes to thank his supervisory committee members Dr. P. M . Byrne, Dr. C. E. Ventura, Dr. Y.P. Vaid, Dr. R.D. Campanella and Dr. M . K. Lee (BC Hydro) for their guidance. The author is grateful to Ms. K. Lamb, Graduate Secretary for her help and kind attention. The author is grateful to Dr. D. W. Wilson and Dr. R.W. Boulanger, University of California at Davis for providing the centrifuge test data for piles in liquefiable soils and for their help in the interpretation of the data. The data from the centrifuge tests carried out at the California Institute of Technology by Dr. B. Gohl is also gratefully acknowledged. The financial support provided by the University of British Columbia in the form of University Graduate Fellowship and Graduate Research Assistantship is gratefully acknowledged. The author wishes express his deepest gratitude to his mother, late father and his family members for their many sacrifices, strong support and encouragement throughout this research. The author would like to thank his colleagues and friends, particularly Sivathayalan, Premasiri and Ravichandran for providing him with a homely environment at U B C . xxv CHAPTER 1 Introduction 2 CHAPTER 1: INTRODUCTION 1.1 INTRODUCTION This thesis deals with one of the important and complex problems in soil-structure interaction, the seismic response of bridge foundations and their effects on the overall response of bridges. In the 1989 Loma Prieta earthquake and in the 1995 Kobe earthquake, many bridges suffered extensive damage and some failed due to the failure of the bridge foundations. These earthquakes demonstrated that the bridge foundations have an important role to play in the overall response of the bridge structure and the current design procedures to design the foundations and the bridge structures may be inadequate. The bridge piers and abutments are usually supported on pile foundations. The seismic soil structure interaction analysis involving the bridge superstructure and pile foundation is a complex problem. Coupled analysis of superstructure and pile foundation is not practically feasible due to limitations in computational capabilities such as the memory and time. To avoid the coupled modeling of the bridge superstructure and foundation, fixed base conditions are sometimes assumed in the design of bridges. Recognizing that such design may not be appropriate, many design codes such as CalTrans (California Department of Transportation), A AS HTO( American Association of State Highway and Transportation Officials) and ATC(Applied Technology Council) suggest that the flexibility of the foundation should be included in the in the superstructure model. This is usually done by modeling the foundations by means of springs and dashpots. However, these design codes do not prescribe how to model the foundation as a set of springs. Setting out clear guidelines for design requires a complete understanding of the dynamic behaviour of the pile foundations under earthquake loading and its effects on the bridge structural response. A substantial portion of this thesis is devoted in developing a better understanding of the seismic behaviour of the pile foundations and their effects on the structural response of the bridge. The seismic behaviour of the pile foundation is a difficult problem due to the soil-pile Chapter 1 : Introduction 3 interaction, pile-to-pile interaction, superstructure-foundation interaction and due to the non linear hysteretic behaviour of the soil. Many research efforts have solely focused on the analysis of pile foundations only. Novak (1991) gives an extensive review of the more widely accepted methods for for the analysis of pile foundations. Most of these methods can be categorized into two main groups. One is based on elastic continuum models and the other on the Winkler springs model. The elastic continuum models are suitable for studying the pile foundation under low excitations when the dynamic response is approximately elastic. Analytical solutions are limited to some special cases such as a single pile embedded in a homogeneous soil or in a layered soil. The behaviour of the pile group differs from that of a single pile due to the pile-to-pile interactions. As many of the available analytical solutions are for single pile only, group interaction factors are used to study the behavior of the pile groups. A fairly complete set of static interaction factors are available to study the behaviour of the pile group under static loading (Poulos, 1980). However, the dynamic interaction factors that are frequency dependent are more appropriate for studying the pile group under dynamic loading. Dynamic interaction factors have been presented by Kaynia and Kausel (1982) and Gazeta et al., (1991). However, they are available only for small pile groups with limited range of spacing and soil/stiffness ratio and are for elastic response. A Quasi-3D finite element method was developed to study the dynamic response of a single pile and pile groups (Wu, 1994, Finn and Wu, 1994, Wu and Finn, 1997). A full three dimensional analysis of pile foundations is not feasible for engineering practice at present because of the limitations in the computer storage and the substantial time needed for computations. Thus, the Quasi-3D finite element method was developed by relaxing some of the boundary conditions associated with a full 3D analysis. This method considers all the important effects in the seismic response of piles including the non-linear hysteretic behaviour of the soil, soil-pile interaction , inertial interaction, group effect and gapping and yielding. This method is not limited by the number of piles in the pile group and it treats the pile group as a group and the use of interaction factors does not arise. This method can also handle the variability of the soil properties is space. Numerous validation studies showed that this method is very accurate for excitation due to horizontally polarized shear waves propagating vertically. The Quasi-3D finite element method was implemented in the computer program PILE3D. Chapter 1 : Introduction 4 The Quasi-3D finite element method is chosen to analyze the pile foundations in this thesis and to determine the stiffnesses and damping of pile foundations. This method is also used in the evaluation of some of the approximate current methods of analysis of pile foundations and bridges used by sophisticated users such as CalTrans. The use of Winkler model has been very popular among practicing engineers and researchers to obtain the seismic response of a single pile. In the Winkler model, the soil surrounding the pile is replaced by a set of springs placed along the pile. These springs are modeled using a nonlinear load-deformation relationships known as p-y curves which relate the pressure to pile displacement. The use of American Petroleum Institute (API, 1993) recommended p-y curves is very common in the absence of any field data for the static nonlinear analysis of a single pile. Abghari and Chai (1995), Makris and Gazetas (1992), Gazetas and Dobry (1984), Kagawa and Kraft (1984), Matlock et al. (1979) have all used the Winkler model to obtain the dynamic response of a single pile. They placed a set of dashpots along the pile in addition to the springs to represent the damping in dynamic response. In seismic response analysis, the motions of the free field soil which is far away from the pile and unaffected by the presence of the pile is determined through a separate analysis and used as input motions in addition to the base excitations. The springs are usually modeled using the p-y curves recommended by API (1993) for cyclic loading. As this method is widely used, the effectiveness of the method and the use of API (1993) recommended p-y curves for dynamic loading is extensively studied here. For this purpose, a comprehensive method of analysis based on the Winkler model and finite element technique is developed. The effectiveness of this method is evaluated by centrifuge test data and by comparing the solutions with PILE3D solutions. The centrifuge test was carried out at the California Institute of Technology on a single pile in dry sand under low level and strong shaking (Gohl, 1991). In the seismic response analysis of bridges, various models are used to model the superstructure of the bridge, 2D and 3D stick models, single bent models and rigid body models. In all these models the bridge foundations and abutments are not explicitly modeled. Instead the stiffnesses of these components of the system are represented by a set of springs. Sometimes dashpots are also used in addition to represent the damping. The behaviour of the superstructures of the bridges under Chapter 1 : Introduction _ 5 dynamic loading are well understood and the superstructure models above are good enough to capture that behaviour. However, there is lack of knowledge on the dynamic response of the pile foundations and abutments and their effects on the overall response of the bridge structure. This has led to various ways of representing the pile foundations and abutments with springs and determining their stiffnesses. There is no well accepted way of modeling the foundation with springs and to determine their stiffnesses. Numerous studies that use the same superstructure model but different foundation models are evidence of this fact (Chen and Penzien, 1977, Gates and Smith, 1982, Wilson, 1988, Levine and Scott, 1989 Wilson and Tan, 1990, Mc Callen and Romstad, 1994, Werner etal., 1990). Lam and Martin(1986) carried out an extensive research study and presented guidelines to design the bridge foundations and abutments. They recommended that these guidelines could be used as supplement to the AASHTO(1983) seismic design guidelines. Their guide lines included procedures to determine the single and group pile stiffnesses and the stiffnesses of the abutments. In the determination of the single pile stiffness, the Winkler spring model was used. Methods were presented to determine both linear and nonlinear stiffness. In the nonlinear stiffness calculation, the load-deflection relationship of the pile is specified by p-y curves. A method was also presented to determine the group stiffnesses from the single pile stiffnesses. It can be noted here that the procedures recommended by Lam and Martin(1986) are essentially static and the complexities due to the dynamic group effect and the nonlinear hysteretic behaviour of the soil were simplified to guide the practicing design engineers. The modeling of the bridge foundations is complicated by its different behaviour under low level excitations and strong shaking. Therefore, it is important to understand the behaviour of the bridge under these two levels of shaking. Painter Street Overpass is a two span bridge located in Rio Dell, California. For this bridge, both ambient vibration records and strong motion records are available (Gates and Smith, 1982, Ventura et al, 1994, Goel and Chopra, 1994). These two sets of records are used in two separate case studies to study the behaviour of the bridge abutments and pile foundations under low level and strong shaking. The findings from theses two case studies and from several other past studies are then used to identify the limitations in the current modeling of Chapter 1 : Introduction 6 foundations for seismic response analysis. The case studies have also led to the development of a new strategy that overcomes many of these limitations. The main limitation in all the current models is their inability to capture explicitly the nonlinear behaviour of the pile foundations and abutments during strong shaking. As a first step in the new strategy, a method is developed to model the pile foundations as a lumped parameter system. In the lumped parameter model, the pile foundation is represented by a set of nonlinear springs and dashpots. The spring damper model, though of a reduced order compared to the actual 3D nature of pile foundation, is expected to replicate the nonlinear behaviour ofthe pile foundations. This is achieved by calculating the stiffness and damping from a nonlinear Quasi-3D finite element analysis and then using it in the lumped parameter model. In the course of the study various factors affecting the stiffness, damping and the dynamic response of the pile foundations are studied in detail. The second step is to develop an analytical procedure that captures the nonlinear response of the bridge structure caused by the localised nonlinear behaviour of the foundations under earthquake loading. This is achieved by incorporating the lumped parameter model of the pile foundations into a 3D stick model of the bridge structure. This model is then used to investigate the importance of including the foundation in the mathematical model of the bridge and the overall effects of the foundations on the response of the bridge. Various options to calculate the foundation stiffness are evaluated. A parametric study is then conducted to investigate the relative importance of various superstructure and foundation stiffnesses on the dynamic response of the bridge. The bridge decks are usually supported at their ends on abutments. The support conditions play an important role in the seismic response of the bridges. The support conditions at the abutments are commonly assumed as either roller supported or a hinge. These conditions are not appropriate if the abutments are an integral part of the bridge, such as short stiff bridges that are supported on monolithic abutments. Thus, a method is described in this thesis to determine the dynamic stiffness and damping of abutments under elastic conditions. Under earthquake shaking, the stiffness and damping of the abutments are affected by the nonlinear hysteretic behaviour of the soil. A method is presented in this thesis to determine the nonlinear stiffness and the damping of the abutments. Chapter 1 : Introduction 7 This method is similar to the method that is used to determine the nonlinear stiffness and the damping of the pile foundations. The new method is then used to investigated the stiffness and damping characteristic of the abutments of the Meloland Road Overpass under strong shaking. California Department of Transportation (CalTrans) is the leading institution in the United States involved in the design and retrofitting of bridges in seismically active areas. CalTrans currently practices a simplified pile-superstructure model and nonlinear p-y curves to study the dynamic behaviour of bridge foundations under strong shaking. The effectiveness of this simplified pile-superstructure model is evaluated in detail using P1LE3D. A fully coupled analysis of superstructure and foundations is the ideal way to obtain the seismic response of a bridge structure. However, it is not yet feasible in engineering practice because it is very time consuming and impractical due to the limitations in the computational capabilities currently available on personal computers. Therefore, various techniques are used to approximate the behaviour of the foundations while analysing the superstructure and vice versa while analysing the foundations. In these analyses, the superstructure and foundations are not fully coupled but the coupled response is approximated. Sometimes uncoupled analysis of superstructure and foundations are also carried out. An iteratively coupled method of analysis is developed in this thesis for determining the coupled response of bridge superstructure and pile foundations. The iterative coupled analysis required the development of a method for response analysis of superstructure that allowed input at multiple supports and a modification of the PILE3D analysis to allow the input of dynamic forces and moments at the pile head. The linking of these two procedures with routine superstructure analysis allowed iterative coupled analysis of bridge and the foundations. This iterative coupled analysis is used directly in evaluating the adequacy of simpler more approximate methods. Its advantage over PILE3D is that it can accommodate much larger structures because the iterative process drastically reduces the memory requirements. Liquefaction has been one of the key causes of damage and failure of pile foundations in major earthquakes. Liquefaction in granular soils occurs due to the development of porewater pressure under undrained conditions. The increase in porewater pressure reduces the effective stresses in the Chapter 1 : Introduction 8 soils. As the effective stresses rather than total stresses control the deformations in soils, an effective stress analysis of pile foundations is required to obtain the seismic response of pile foundations in potentially liquefiable sites. A new effective stress method is described in this thesis for the analysis of pile foundations. The new effective stress method is developed by incorporating the Martin-Finn-Seed (Martin et al., 1975) porewater pressure generation model into the Quasi-3D finite element method described previously which is a total stress method. The new effective stress method is extensively verified using centrifuge test data on a single pile and (2x2) and (3x3) pile groups. The centrifuge tests were carried out on a potentially liquefiable soils at the University of California, Davis under low-level and strong shaking (Wilson et al., 1997). 1.2 OBJECTIVES OF THE THESIS The basic objectives of this thesis are to provide a more complete and fundamental understanding of the seismic behavior of pile foundations under earthquake loading and on this basis develop methods that advance the capability of analyzing the seismic behavior of bridges on pile foundations. The detailed steps in achieving these objectives are outlined in the following chapter summaries; a general overview of the main steps are as follows. # Identify the shortcomings of current methods for modeling pile foundations for bridges through a review of case histories and the detailed analyses of a particular bridge for which extensive data are available for conditions of ambient vibration and strong earthquake shaking. Also the suitability of the widely used p-y curves to represent the interaction of soil and pile will be evaluated. # Development of a relatively simple new method overcoming the shortcomings in current methods. In this method, the pile foundation is modeled as a lumped parameter system consisting a set of nonlinear springs and dashpots for which time histories of stiffness and damping under strong shaking are calculated using the Quasi-3D finite element nonlinear Chapter 1 : Introduction 9 analysis. The inertial interaction between superstructure and foundation is modeled approximately. • Development of a more sophisticated method for the coupled response of the bridge superstructure and pile foundations which can model the inertia effects of the superstructure more accurately. The new method does not require the representation of pile foundation as a set of springs and dashpots. A l l the methods of analysis described so far do not take into account any excess porewater pressures due to the seismic shaking. The Quasi-3D finite element method described earlier is modified to include seismic porewater pressures by incorporating an effective stress constitutive relationship for the material and a porewater pressure generation model. The new analysis method can take into account the loss of stiffness and strength due to the development of seismic porewater pressures. 1.3 OUTLINE OF THE THESIS Chapter 2 Chapter 2 gives a review of the existing methods for determining the dynamic response and dynamic impedances of pile foundations. Chapter 3 Chapter 3 is devoted in studying the dynamic behavior of a single pile using Winkler model and p-y curves. A comprehensive method is developed for the dynamic nonlinear analysis of single pile by incorporating the current procedures and it is then used to evaluate the effectiveness of the Winkler model and the API(1993) recommended p-y curves. In the evaluation study, data from a centrifuge test on a single pile are used. In this chapter, a computer program FREEFLD for the nonlinear dynamic analysis of horizontally layered soil and a computer program PILE-PY for nonlinear dynamic analysis of a single pile using Winkler model and p-y curves are introduced. Chapter 1 : Introduction 10 Chapter 4 Chapter 4 provides an extensive review of the currently used bridge models and analytical procedures. Two case studies on the dynamic behaviour of the Painter Street Overpass under ambient vibration and strong shaking are described in detail and the limitations in the current procedures are identified. Chapter 5 Chapter 5 describes a lumped parameter model for the nonlinear soil-pile-structure interaction analysis using a lumped parameter system. Various factors affecting the impedance of the pile foundations are studied in detail and approximate procedures to calculate the impedances are also evaluated. In this chapter a computer program LUMPILE is introduced for nonlinear dynamic analysis of a 2-DOF lumped parameter system in the time domain. Chapter 6 Chapter 6 presents the development of a new three dimensional model of a bridge and a nonlinear analytical procedure. The model consists of a three dimensional space frame for the superstructure and nonlinear springs and dashpots for the foundation. A seismic response study on a three span bridge founded on pile foundations is described and the role of the flexibility of the foundation on the overall response of the bridge structure is investigated. This chapter also presents a parametric study on a simplified single bent model of bridges. A computer programs BRIDGE-NL for the nonlinear 3D analysis of a bridge superstructure and pile foundations is introduced in this chapter. Chapter 7 Chapter 7 describes a new method for determining the dynamic impedance of bridge abutments. A seismic response study on the Meloland Overpass abutment and the stiffness and damping characteristics of abutments under strong shaking are also presented in this chapter. A computer program ABUT-STD for determining the dynamic impedances of abutments and a computer program A B U T - N L for the nonlinear dynamic analysis of the abutments in the time domain are introduced in this chapter. Chapter 1 : Introduction 11 Chapter 8 Chapter 8 describes the detailed evaluation of a simplified pile-superstructure model of a bridge using PILE3D. Chapter 9 Chapter 9 presents a new approach for the coupled analysis of bridge superstructure and pile foundation. As a part of this new approach, a method is developed to analyze the superstructure under multi support excitations and to calculate the support reactions. PILE3D was extended to handle dynamic forces and moments in addition to the base excitations. A detailed numerical study is presented to verify the new approach and to investigate the effectiveness of the various uncoupled analysis of superstructure and foundations. A computer program SPFR-MS for the dynamic analyses of a bridge superstructure in the time domain under multi support excitations and for determination of support forces and a computer program SOISTR for the coupled analysis of bridge superstructure and pile foundations in the time domain are introduced in this chapter. Chapter 10 Chapter 10 presents a new method for the effective stress dynamic analysis of pile foundations. The new method is verified by simulating the centrifuge tests that were carried on a single pile and pile group under low level and strong shaking in potentially liquefiable soils. A computer program PILE3D-E is introduced in this chapter for the nonlinear effective stress analysis of pile foundations. Chapter 11 Chapter 11 summarizes the developments described in earlier chapters and presents the conclusions arising from the various studies. 12 CHAPTER 2 Review of Methods on the Dynamic Analysis of Piles 13 CHAPTER 2: REVIEW OF METHODS ON THE DYNAMIC ANALYSIS OF PILES 2.1 REVIEW OF METHODS ON THE DYNAMIC RESPONSE OF A SINGLE PILE 2.1.1 Elastic Response A number of analytical and numerical procedures have been developed to study the dynamic response of a single pile. Most of the approaches are based on two distinctly different types of models namely elastic continuum models and lumped mass-spring-dashpot models. An elastic continuum model in which the pile and the soil are fully coupled in a unified system is more appropriate for analyzing the pile under very low level shaking where soil behaves more or less elastically. The lumped parameter model is appropriate for strong motion analysis because of its ability to model the non-linear behaviour of the soil under strong shaking. In the lumped parameter model, the response of the pile is separated from the surrounding soil medium and the contribution of the soil medium to the dynamic response of the pile is represented through empirically or analytically derived Winkler springs and dashpots. Soil-pile interactions are very difficult to model using continuum mechanics. Accurate solutions are very difficult to obtain, even for the ideal assumptions of linear elastic or visco- elastic, homogeneous soil with the pile being welded to the soil. Therefore, approximate solutions have been developed. Tajimi(1966) solution of the horizontal vibration of an end bearing pile in a homogeneous layer was the first of its kind. His solutions neglected the vertical component of the motions. Since 1970's a significant advance has been made in solving the problem of an elastic beam vibrating in a homogeneous or non-homogeneous elastic isotropic medium subjected to dynamic pile head loadings by Novak (1974), Nogami and Novak (1977), Novak and Aboul-Ella (1978), Novak et al. (1978) and Novak and Sheta (1980, 1982). Novak (1974) identified the dimensionless parameters of the problem and gave a number of Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 14 design charts and formulae for the dynamic stiffness and damping of piles. Material damping was later included in the closed form expressions for soil reactions in Novak et al.(1978). A more rigorous solution similar to that of Tajimi(1966) was formulated for the horizontal response by Novak and Nogami(1977). These approximate solutions provided the basic insight into the behaviour of the soil pile system. Novak's formulation is focused on the pile head impedances that have great influences on the pile supported structures. The pile head impedances are defined as the ratio between the complex valued displacement (or rotational) response at the pile head and the harmonic forces (or moments) applied at the pile head. The pile head impedance, K can be expressed as K=k+ic in which the real part, k represents the stiffness of the pile head and the imaginary part, c represents the damping. The damping, c consists of both material or hysteretic damping and the radiation damping. The material damping represents the energy losses due to the hysteretic behaviour of the soil. It is usually small during low level shaking and is high during strong shaking. The radiation damping which represents the energy losses away from the pile is dominant under low level shaking. Gazetas and Dobry(1984) proposed closed form expressions for computing the radiation damping and these expressions are found to be in good agreement with those developed by Novak et al. (1978) and Rosset and Angelidas(1980). Gazetas and Dobry's expressions which are are frequency and depth dependent are used in this thesis. Novak (1974) developed pile head impedances for a homogeneous elastic medium using the plane strain approach. Nogami and Novak (1977) used plane displacement soil reactions in developing the pile head impedances. The latter have compared their solutions with the three dimensional finite element solutions of Kuhlmeyer (1979) and found that the results are similar. Novak and Aboul-Ella (1977) extended the plane strain solution approach to include layered media and incorporated the analysis in the computer program P R A Y . This code was later used by Novak and El-Sharnouby(1983) to generate design charts and tables for parabolic and homogeneous soil profiles. The PILAY program is extensively used in the elastic analysis of pile under low amplitude vibrations. Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 15 The pile head impedance generally depends on the frequency. Novak's plane strain approach gives very good results at high frequencies typical of machine foundations but does not give realistic results at typical earthquake ground motion frequencies. For the lower frequencies the solutions proposed by Kaynia and Kausel (1982) and Wu and Finn (1997a) are appropriate. They demonstrated that, at lower frequencies, the stiffness of the pile remains approximately constant at its static value. However, the damping increases with the increases in frequency. 2.1.2 Non-Linear Response The pile head impedances derived using elastic theories are appropriate for low level shaking during which small strains are developed in the soil and the pile. During strong shaking, the pile head impedances depend on the level of shaking due to the nonlinear response of soil at high strain, pile separation (gapping) slippage and friction. It is very difficult to incorporate these nonlinear effects into the continuum models. Thus most often lumped mass-spring-dashpot models have been used when nonlinear analysis is required. In these lumped parameter models, the interaction between the pile and the soil in the near field is usually accounted for by the use of nonlinear springs and dashpots placed along the length of the pile. The behavior of these springs and dashpots are modeled using a curve called the p-y curve. This model was dealt with in detail in Chapter 3. Nonlinear pile stiffnesses were investigated by Angelidas and Rosset (1981) using the finite element theory. Even neglecting the slippage and gapping, they demonstrated that a dramatic reduction in the horizontal stiffness occurs due to strain softening. The equivalent linear method (Seed and Idriss, 1967) was used in this investigation to model the nonlinear behaviour of the soil. The nonlinear lumped parameter models are appropriate for single pile only. The group behaviour that may differ from the behaviour of a single pile cannot directly be obtained form this type of lumped parameter models. Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 16 2.2 REVIEW OF METHODS ON THE DYNAMIC RESPONSE OF GROUP PILES In the seismic design of structures such as buildings and bridges, springs are used often to represent the action of the pile foundations. The springs constants are deduced from the pile group impedances. The pile group impedance may not simply be a sum of the impedances of the single piles in the pile group. It may differ significantly due to dynamic pile-soil-pile interaction. This pile-soil-pile interaction effect is usually assessed through the use of interaction factors. The concept of interaction factors was first proposed for static loading (Poulos, 1971). A fairly complete set of static interaction factors has already been developed. (Poulos, 1971, Butterfield and Banerjee, 1971, Poulos and Davies, 1980). The pile group stiffness can be estimated using these interaction factors and it is smaller than the sum of individual pile stiffnesses. The dynamic interaction factor approach has been proposed by Kaynia and Kausel (1982) as an extension to the widely used static interaction factor approach. A set of dynamic interaction factors is available for floating piles in homogeneous soil and for a limited selection of parameters such as the number of piles in the pile group, spacing of piles and the pile-soil stiffness ratio in Kaynia and Kausel (1982) and for vertical vibration in linearly non-homogeneous soil in Banerjee (1987). E l Marsafawi et al. (1992a, 1992b) presented approximate procedures for estimating the dynamic interaction factors. A comprehensive and easy to use dimensionless graphs of complex valued dynamic interaction factors versus frequency were developed by Gazetas et al. (1992) for vertical, horizontal and rocking harmonic excitation at the pile head. These readily applicable graphs were prepared using the rigorous analytical and numerical formulations developed by Kaynia and Kausel (1982) for the dynamic analysis of a pile group in a layered half space. In addition, the simplified analytical method of Dobry and Gazetas (1988) and Gazetas and Makris (1991) were also used. Closed form expressions for dynamic interaction factors derived from simplified wave interference theory after calibration with numerical solutions are also available. (Gazetas, 1991). The computer program D Y N A 3 (Novak et al., 1990) is widely used to calculate the group Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 17 stiffness and damping of a pile group in non-homogeneous media. Novak's plane strain approach is used in this program to calculate the individual pile stiffness and damping. The group pile stiffness and damping is then estimated using the interaction factors. The interaction factors used in the D Y N A 3 program are a combination of static interaction factors proposed by Poulos and Davies (1980) for vertical loading and El Sharnouby and Novak (1985) for horizontal loading and the dynamic interaction factors by Kaynia and Kausel (1982). The group stiffness and damping calculated by the D Y N A 3 program are appropriate for low level shaking only. The motion of the pile cap may differ from the motions of the free field surface due to the kinematic interaction between the pile and the soil. The pile cap motions may include a rotational component in addition to the translational component. The effect of kinematic interaction on the pile head motion has extensively been studied by Gazetas (1984), Fan et al. (1991) and Wu (1994). 2.3 QUASI-3D FINITE ELEMENT METHOD FOR THE DYNAMIC ANALYSIS OF PILE FOUNDATIONS 2.3.1 Introduction A Quasi-3D finite element method was developed for the elastic and nonlinear dynamic analysis of pile foundations and implemented in a computer program called PILE3D (Wu, 1994, Finn and Wu, 1994, Wu and Finn, 1997a and 1997b). In this method, a simplified quasi-3D wave equation was used to describe the dynamic motion of the soil under horizontal shaking. The coupled equations of motion between the pile and soil were solved by using the finite element method. The PILE3D considered all the important factors affecting the pile group behavior under earthquake loading except seismically induced porewater pressures. The factors that were taken into account include: Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 18 • The nonlinear hysteretic behaviour of the soil • Soil-pile interaction or kinematic interaction • Superstructure or inertial interaction • Pile-soil-pile interaction or the group effect • Yielding and gapping • Frequency dependency • Spatial variability of the soil The method for the elastic analysis was verified (Wu, 1994, Finn and Wu, 1994, Wu and Finn, 1997a) against the elastic solutions proposed by Kaynia and Kausel(1982) and Novak(1990) and using the data from a full scale vibration test on an expanded base concrete pile and on a 6-pile group supporting a large transformer (Sy, 1992 and Sy and Siu, 1992). The nonlinear method was verified (Wu, 1994, Finn and Wu, 1994, Wu and Finn, 1997b) using the centrifuge test data on a single and group piles (Gohl, 1991). The use of quasi-3D method greatly reduced the memory and time requirements and the analysis of pile foundations involving large pile groups using the PILE3D program became computationally feasible. 2.3.2 Dynamic Elastic Analysis of Pile Foundations Using Quasi-3D Finite Element Method in the Frequency Domain Under vertically propagating shear waves, the foundations soils undergo mainly shear deformations in the X O Y plane, except in the area near the pile where extensive compression deformations develops in the directions of shaking. The compressive deformations also develop shearing deformations in the Y O Z plane, as shown in the Fig. 2.1. In the light of these observations, assumptions were made that the dynamic motions are governed by shear waves in the X O Y and Y O Z planes, and compression waves in the shaking direction, Y . Deformations in the vertical direction and normal to the direction of shaking are neglected. Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 19 8 Node Solid Element (soil) V ^ d y ^ d y ) V „ ( d v , / d y ) 2 Node Beam Element (Pile) Fig. 2.1 The Quasi-3D Finite Element Model of the Pile Foundation Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 20 Let v represents the displacement of soil in the direction of shaking. The compressional force d2v per unit volume is 0 G * . The shear forces per unit volume in the X O Y and Y O Z planes are dy2 * d2v » 8 2v G and G Respectively. The two shear waves propagate in the Z direction and X direction dz2 dx2 d2v respectively. The inertial force per unit volume is p s G * . Applying dynamic force equilibrium dt2 in the Y direction, the governing equation describing the free vibration of the soil continuum is written as, G ^ + 0 G ^ + G ^ = p ^ (2 1) dx2 dy2 dz2 sdt2 ( Z A ) where p s is the mass density of the soil, and G* is the complex shear modulus. The complex shear modulus G* is expressed as G*=G(l+i 2X) in which G is the shear modulus of the soil and X is the strain dependent damping ratio of the soil. The parameter 6 is given by 0= 2/(l-u) where u is the Poisson's ratio of the soil. Free horizontal displacements were allowed at the lateral boundaries of the finite element model. Piles are modeled using ordinary Euler beam theory. Pile bending occurs only in the Y O Z plane. d4v d2v V P ^ * PPAP^7 < " ) where E p is the Young's modulus of pile, L, is the second moment of area, p p is the mass density of pile and A p is sectional area of the pile. Dynamic soil-pile-structural interaction is maintained by enforcing displacement compatibility between the pile and the soils. An appropriate finite element procedure is employed that couples the motion of the soil and the pile and enforces displacement compatibility between Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 21 them. An 8 node brick element is used to represent the soil element and a 2 node beam element shown in Fig. 2.1 is used to simulate the pile. The global governing dynamic equilibrium equation in matrix form can be expressed as [M*]{v} + [C*]{v} + [K*]{v} = |P(t)} (2.3) in which [M*],[C*] and [K*] are the mass, damping and stiffness matrices of the soil pile system vibrating in the horizontal direction. P(t) is the external load vector and v,v and v are the nodal acceleration, velocity and displacement. 2.3.3 Pile Head Impedances The impedances Ky are defined as the complex amplitudes of harmonic forces (or moments) that have to be applied at the pile head in order to generate a harmonic motion with a unit amplitude in the specified direction(Novak, 1991). The concept of translational and rotational impedances is illustrated in Fig. 2.2. The translational, the cross-coupling and the rotational impedances of the pile head are represented by K v v , K v 9 and K e e respectively. The translational, cross-coupling and rotational impedances are defined as follows. Translational Impedance- K v v : The complex valued pile head harmonic shear force required to generate unit harmonic lateral displacement at the pile head while the pile head rotation is fixed. Cross-Coupling Impedance-Kv6: The complex valued pile head harmonic moment generated by the unit harmonic lateral displacement at the pile head while the pile head rotation is fixed. Rotational Impedance-Kee: The complex valued pile head harmonic moment required to generate unit harmonic rotation at the pile head while the pile head rotation is fixed. Since the pile head impedances K v v , K v 9 and K e e are complex valued they are expressed by their real and imaginary parts as Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 22 Fig. 2.2 Pile Head Impedances Chapter 2 : Review of Methods on the Dynamic Analysis of Piles K.j = k i r i C . i o r K y k u , i ( 0 c i i 23 (2.4) in which ky and Cy are the real and imaginary parts of the complex impedances respectively. cij=Cij/co is the coefficient of equivalent viscous damping and (0 is the circular frequency of the applied load. k ; j and C y are usually referred as the stiffness and damping at the pile head. Pile head impedances are evaluated as functions of excitation frequency by subjecting the system to the harmonic loads. If P(t)=P0 e l u t and the steady state displacement vector v=v0 e10>t, the Eq. 2. 3 takes the form [ K l g l o b a l K l = ( P o ) (2.5) where [K] g l o b a l is a complex valued matrix given by [ K W = [K *] +i(o[C *] - to 2[M *] (2.6) By definition, the lateral and cross-coupling stiffness can be obtained by applying a unit lateral displacement at the pile head under the condition of zero pile head rotation. Then the Eq. 2.5 becomes V < 0 • = - K e " 0 where v r is the displacements of the nodes other than pile head. Eliminating the row corresponding to zero rotation and dividing by K v v , the Eq. 2.7 is re written as 1 K . 1 0 (2.8) K. Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 24 The lateral pile head impedance, K v v can be obtained by calculating displacement, v p caused by a unit lateral shear force at the pile head of fixed head pile. The lateral pile head impedance is expressed as K w = -7 (2.9) v p The moment at the pile head, M p corresponding to v p can also be easily calculated. Hence, the cross-coupling impedance, K v 6 is expressed as M p K v 6 = — (2.10) V p Similarly, by applying a unit moment at the pile head and calculating the pile head rotation, 6P under the condition of zero pile head deflection, the rotational impedance, K e e can be calculated and is expressed as Kee = Jp (2-11) 2.3.4 Quasi-3D Finite Element Method for the Elastic Vertical Dynamic Analysis of Pile Foundations Under vertically propagating compression waves, the soil mainly undergoes compressive deformations in the vertical direction. In the two horizontal directions, shearing deformations are generated. Although compression occur in the two horizontal directions, assumptions are made that the normal stresses in the two horizontal directions due to vertical excitations are small and can be ignored. Therefore the dynamic motions of the soil are governed by the compression wave in the vertical direction and the shear wave propagating in the two horizontal directions X and Y . Analogous to the governing equation in the horizontal direction, the quasi-3D wave equation of soil in the vertical direction is given by Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 25 G , ^ w + Q . ^ y v + Q , aV = ^ w ax2 a y 2 z az2 s at2 ( } where w is the soil displacement in the vertical direction. The parameter 0Z is given by 0z=2(l+u) based on equilibrium of the model in the vertical direction. The equation of motion of pile can be expressed as. ~ . a2w . a2w E A = p A (o ]3) P VA5) The global dynamic equilibrium equations are written in the matrix form as [ M ; H W } + [c;]{wj + [ K ; ] { W } = (p z(t)i (2.14) in which [ M Z * ] , [ C Z * ] and [ K J * are the mass, damping and stiffness matrices of the soil-pile system vibrating in the vertical direction. P z ( t ) is the external load in the vertical direction and w,w and wand the relative acceleration, velocity and displacement in the vertical direction. 2.3.5 Rocking Impedances of Pile Groups The rocking impedance of pile group is a measure of the complex resistance to rotation of pile cap due only to the resistance of each pile in the group to vertical displacements. The rocking impedance of a pile group is defined as the summation of the moments of the axial forces around the center of rotation of the pile cap for a harmonic rotation with unit amplitude at the pile cap. This definition is quantitatively expressed as where ri is the distance between the center of rotation and the pile head centers, and F ; is the amplitude of axial forces at the pile heads, as shown in Fig. 2.3. Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 26 Fig. 2.2 Pile Head Impedances Center of Rotation Pile Cap Pile-r. Soil Rigid Base Fig. 2.3 Rocking Mechanism in a Pile Group Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 27 In the analysis the pile cap is assumed rigid. For a unit rotation of the pile cap, the vertical displacements W j P at all pile heads are determined according to their distances from the center of rotation rr For a given combination of vertical displacements W j P at the pile heads, the corresponding axial forces Fi at these pile heads are determined from Eq. 2.14. The rocking impedance of a pile group is then determined using Eq. 2.15. The method of determination of pile head impedances was implemented in a computer program called PILIMP(Wu and Finn, 1997a) 2.3.6 Quasi-3D Finite Element Method for the Dynamic Nonlinear Analysis of Pile Foundations in the Time Domain The method proposed by Wu (1994) for the nonlinear analysis is an extension of the method for the elastic analysis described previously. Adjustments were made to the formulation for the elastic analysis to accommodate the time domain analysis. Similar to Eq. 2.1 , the dynamic governing equation under free vibration of the soil continuum is expressed as „ d2v fl„ a2v _ a2v a2v G + 6G + G = p (o 16s* dx2 dy2 dz2 sdt2 { } where G is the shear modulus of the soil. The hysteretic damping was represented using equivalent viscous damping and it was assumed to be of Rayleigh type. The piles were modeled in the same way as for the elastic analysis. The global dynamic equilibrium equation of motion under horizontal shaking is written in the matrix form as [M]{v} + [C]{v} + [K]{v} = [M]{I}vo(t) (17) Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 28 in which v0(t) is the base acceleration, {1} is a unit column vector, and v,v and v are the relative nodal acceleration, velocity and displacement respectively. [M],[C] and [K] are mass, damping and stiffness matrices respectively. Direct step by step integration using the Wilson- method was employed to solve the equation of motion. The nonlinear hysteretic behaviour of the soil is modeled using a variation of the iterative equivalent linear method used in the S H A K E program.(Schanabel et al., 1972). In S H A K E , constant values of shear modulus and damping ratio are assigned to each soil layer for the entire duration of the shaking compatible with the effective strain in each layer during the previous iteration. The analyses are repeated until the specified tolerances are met. The effective strain is defined as the 65% of the maximum shear strain experienced by the soil element during an iteration. To approximate better the nonlinear behaviour of soil under strong shaking, in PILE3D compatibility between the secant shear modulus and damping ratio and shear strain can be enforced at each time step during the integration of the equation of motion. This ensures the time histories of moduli and damping ratios in each soil element are followed during the analysis. In PILE3D, instead of updating the shear moduli and damping ratios at each time step, one can also select longer time intervals for updating to reduce the computational time if error limits are not exceeded. Additional features such as tension cutoff and yielding were also incorporated to simulate the possible gapping between the soil and pile near the soil surface and yielding in the near field. The response of the pile group to seismic excitation by shear waves propagating vertically is analyzed using the model in Fig. 2.4. The rocking stiffness of the pile cap which is primarily due to the vertical resistance of the piles is represented by a rotational spring. The spring is updated at selected time intervals during the seismic response analysis. This done by carrying out the analysis in the vertical mode using the current soil properties at that time. The new rocking stiffness is returned to the program operating in the horizontal mode and the analysis is continued alternating between rocking and horizontal shaking to the end of the earthquake. Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 2.3.7 Nonlinear Stiffness from PILE3D and PILIMP 29 During strong shaking, the pile head impedances do not remain constant but vary due to the nonlinear behaviour of the soil. PILE3D and PILIMP program can be used to determine these nonlinear pile head impedances as time histories. In this case, the pile foundation is first analyzed using the PILE3D program in the time domain and during this analysis, the shear modulus and damping ratio of soil elements are traced as time histories. These time histories are then used in PILIMP analysis to determine the time variation of stiffness and damping of the pile foundations. The method to determine the time histories of stiffness and damping are described in detail in Chapter 6. A centrifuge test on a single pile under a peak base acceleration of 0.16g was simulated using the PILE3D program by Wu and Finn(1997b). The variation of lateral stiffness during the shaking obtained by Wu and Finn(1997b) using PILE3D is shown in Fig. 2.5. This figure shows that the lateral stiffness reduces from its initial value of 145 MN/m to 20 MN/m during the periods of strong shaking and it increases after the level of shaking has reduced. The determination of nonlinear stiffness and damping of pile foundations as time histories is an important feature in PILE3D. 2.3.8 Role of PILE3D and PILIMP in the Present Studies The PILE3D and PILIMP are introduced in detail here as they are extensively used in this thesis. They are used as a bench mark analysis in the evaluation of various alternative and approximate methods for the analysis of pile foundations. Various bridge foundation models are also developed using the PILE3D and PILIMP. PILE3D was also modified and used in the development of a coupled bridge superstructure-foundation model. The total stress analysis method implemented in PILE3D was used as the basis in the development of a method for the effective stress analysis of pile foundations. Chapter 2 : Review of Methods on the Dynamic Analysis of Piles 30 Fig. 2.4 Mechanical Model Used in PILE3D for the Analysis of Pile Group with a Rigid Pile Cap 200 Fig. 2.5 Variation of Lateral Stiffness with Time CHAPTER 3 Modeling of a Single Pile Using Winkler Model and p-y Curves 32 CHAPTER 3 : MODELING OF A SINGLE PILE USING WINKLER MODEL AND P-Y CURVES The nonlinear response of a single pile embedded in a continuum is a three dimensional problem. However, a full 3D analysis using finite element method is not practically feasible due to the limitations in the computer storage and the substantial computer time involved. In practice an approximate analysis is used based on the Winkler model of the soil. In the Winkler model, the soil surrounding the pile is replaced by a set of springs placed along the length of the pile. The springs represent the stiffness of the surrounding soil. In linear elastic analysis, constant values of stiffness are assigned to the springs. In nonlinear analysis , the springs are expected to follow a predefined load-displacement relationship known as the p-y curves. A typical Winkler model is shown in Fig. 3.1 and a typical p-y curve is shown in Fig. 3.2. The Winkler model is relatively simple and it requires much less computational effort and cost compared to the 3D finite element model. This model has been very popular among the practicing engineers and researchers to obtain the static nonlinear response of a single pile. The Winkler model may also be used to obtain the dynamic response of a single pile. In this case, a set of dashpots are placed along the pile in addition to the springs to model the damping of the soil(Fig. 3.3). The nonlinear behaviour of the soil is modeled using the nonlinear p-y curves for cyclic loading. The effectiveness of the Winkler model in capturing the seismic response is evaluated herein. First, the theories behind the Winkler model are reviewed in detail. A method of analysis based on the Winkler model and the finite element technique is developed to obtain the nonlinear seismic response of a single pile. The new method incorporates various currently available models. This method is then used to evaluate its effectiveness of the Winkler model in capturing the nonlinear seismic behaviour of single pile. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 33 P i l e h ^ A / V V V — Winkler Springs ^ A A A A r -HAAMr -vwvv Fig. 3.1 The Winkler Model of a Single Pile for Static Analysis Fig. 3.2 A Typical p-y Curve Pile Winkler Springs and Dashpots Fig. 3.3 The Winkler Model of a Single Pile for Dynamic Analysis Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 34 3.1 REVIEW OF METHODS ON THE SINGLE PILE ANALYSIS USING WINKLER MODEL AND p-y CURVES 3.1.1 Static Behaviour 3.1.1.1 Linear Elastic Methods Closed form solutions to the problem of a single pile with linear springs are available for various distributions of soil stiffness with depth. These distributions include uniform, linear and parabolic. The linear springs stiffnesses are normally expressed in terms of the modulus of subgrade reaction as follows. k s = k z s (3.1) in which k s is the linear spring stiffness, k z is the modulus of subgrade reaction and s is the spacing between the springs. Modulus of Subgrade Reaction : The modulus of subgrade reaction is not a fundamental property of the soil. Several empirical expressions are available for the modulus of subgrade reaction for different types of soils. These correlations are usually obtained by fitting the closed form solutions to experimental results. Palmer and Thompson(1948) suggested that the distribution of modulus of subgrade reaction with depth is of the form k z=kL(z/L)n where k z is the modulus at depth z. k L is the known modulus at depth L and n is an empirical constant which is equal to or greater than zero. The most common assumptions are that the modulus is constant for clay (n=0) and linearly varies with the depth for granular soils (n=l). Terzaghi (1955) also suggested a similar form of equation for sand i.e. kz=k(z/d) where k z is the modulus of subgrade reaction at depth z, k is the modulus of subgrade reaction that depends only on the type and density of soil and d is the diameter of the pile. Reese et al.(1974) also agreed with the Terzaghi's distribution of modulus for sand. However, the values of k suggested by Terzaghi were much smaller than those recommended by Reese et al.(1974). It can be noted here that the values recommended by Reese et al.(1974) Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 35 correspond to the initial tangent stiffness of the load displacement curve and the values recommended by Terzaghi(1955) corresponds to the secant stiffness of the load displacement curve at typical design load levels. Various factors affecting the modulus of subgrade reaction include the pile load level, pile diameter and relative pile-soil stiffness. The load versus displacement or the load versus rotation relationship for laterally loaded piles are nonlinear due to the nonlinear nature of the soil. The subgrade reaction approach based on the linear theory can only be expected to give an approximate solution only. A more logical approach is to use nonlinear springs instead of linear springs to model the soil-pile interaction. 3.1.1.2 Non-linear Methods The nonlinear behaviour of a single pile under a static loading can be modeled using the nonlinear springs. The load deformation characteristics of these nonlinear springs are usually assumed to follow a specified curve called the p-y curve. The concept of p-y curves was first proposed by McClelland and Focht (1958). Several methods have been developed since for constructing the p-y curves for various soils. These methods are based on semi-empirical, insitu and finite element methods. The method proposed by Reese et a l . (1974) to construct the p-y curve was widely used and incorporated originally in the design guide of American Petroleum Institute (API). This procedure was initially based on the back analysis of full scale pile load tests on sand at Mustang Island in Texas (Cox et. al., 1974, Reese, L.C. , 1977 and Reese et al., 1974). The Reese et al.(1974) p-y curve for sand consist of three straight lines and a parabola. Initial portion of the curve is a straight line representing the elastic behaviour and the final portion of the curve is also a straight line representing the ultimate resistance. The intermediate portion consists of a parabola and a sloping line and these two are selected empirically to fit the experimentally derived p-y curve. The initial slope of the curve is assumed to vary linearly with the depth and can be expressed as k=kh z where k h is the initial slope of p-y curve, z is the depth and k is the modules of subgrade reaction. The modulus of subgrade reaction is expressed as a function of soil density. Reese et al.(1974) values Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 36 of modulus of subgrade reaction were originally intended for wet soil conditions. Murchison and O'Neill (1984) extended it for dry soil conditions also. Murchison and ONeill(1984) proposed a single analytical function to derive the Reese et al. (1974) p-y curve. This equation was adopted by API(1993). The methods recommended by API(1993) for generating the p-y curves for sand are given in detail in Appendix 1. These curves are widely used in practice. The construction of p-y curves for clay is similar to that for sand. The most widely used procedure is based on the research by Matlock (1970). He suggested a parabolic curve for soft clay. API(1993) adopted this curve with some slight modifications. The methods recommended by API(1993) for generating the p-y curves for clay are given in detail in Appendix 1 3.1.2 Dynamic Behaviour 3.1.2.1 Linear Elastic Methods Single Pile under Pile head Loading : The lateral dynamic response of a single pile under pile head loading can also be studied using the Winkler model. The stiffness characteristics between the pile and the soil can be modeled as series of linear springs. The damping can be incorporated by placing a series of dashpots in parallel with the springs. These dashpots are to model the energy losses due to radiation of waves and due to hysteretic dissipation. Expressions for the spring coefficient and dashpot constants are available in Gazetas and Dobry (1984) and Rosset and Angelidas (1989). These expressions are frequency dependent and can directly be used in linear elastic methods to determine the pile response. Such analytical methods were developed by Makris and Gazetas (1992). Single Pile Under Base Excitation : The dynamic response of pile under earthquake loading is different from a pile under pile head loading. The pile head loading causes deformations of soil in Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 37 the vicinity of the pile called near field only. The earthquake loading causes deformations in both the near field soil and the far field soil. The far field soil, which is far away from the pile and is usually called the free field, is unaffected by the presence of the pile. The near field behaviour of the soil, which is due to the soil-pile interaction, can be modeled using the Winkler springs and dashpots. The free field behaviour is usually obtained using the site response model such as the one used in S H A K E (Schnabel et al., 1972). Flores Berrones and Whitman (1982) proposed a simplified procedure to analyze the dynamic response of an end bearing pile under harmonic excitation. This procedure is based on the Winkler model and the soil pile interaction in the near field was modeled by a series of linear springs. No dashpots were included in the model to represent the energy losses. An enhanced model of this type has been proposed by Makris and Gazetas (1992) who used dashpots in addition to springs. The prescribed models and procedures are linear in nature and are applicable to the response of a pile under harmonic loading only. Under seismic excitation, the pile exhibits highly nonlinear behaviour and the use of non linear p-y curves is essential in this type of Winkler spring model. 3.1.2.2 Nonlinear Methods The nonlinear seismic analysis of a single pile using the Winkler model requires nonlinear p-y curves that are appropriate for dynamic loading. The data on the dynamic loading of piles are limited and hence the p-y curves that are appropriate for dynamic loading are also limited. Some of the available data are from a very low frequency repeated cyclic loading. (Yan , 1990). The cyclic p-y curves derived from simulated wave loading of a pile represents an envelops of p-y curves after shaking. An assumption implicit in these methods is that the pile response is dominated by the effects of loading at the pile head and the lateral response is reasonably independent of frequency in the low frequency range. The most commonly used set of specification for constructing the p-y curves are from American Petroleum Institute (API, 1993) and the validity of the use of these curves for earthquake loading has not been checked. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 38 In the dynamic analysis using the Winkler model, tangent stiffness method can be incorporated to model the near field behaviour. In the tangent stiffness method, the nonlinear dynamic p-y curve is used as a back bone curve. The unloading and reloading curves are constructed either by simply reversing the back bone curve or using the Masing rule. In the tangent stiffness approach, the hysteric damping in the near field is automatically taken into account. The radiation damping can be simulated by placing a series of dampers in parallel with the springs along the length of the pile. The nonlinear analysis method developed by Matlock et al. (1979), which is implemented in the program S P A S M (Single Pile Analysis Under Support Motion) basically follows this procedure. The nonlinear analysis method developed by PAR(1994) and currently being used by CalTrans is also similar. Bridge piers and abutments are usually supported on pile foundations and they exhibit highly nonlinear behaviour during strong shaking mainly. The localized nonlinear behaviour of the foundation under strong shaking is one of the primary cause for the overall non linear behaviour of the piers and abutment. The nonlinear behaviour of the pile foundation is a complicated process due to the soil-pile-structure interaction and the nonlinear inelastic behaviour of the soil. The use of p-y curves and Winkler spring model has been very popular in the highway design engineering profession. For a long time, their use has been limited to the static analysis only. Recently a method has been adopted by CalTrans (California Transportation Agency) for the nonlinear analysis of pile foundation using the p-y curves. This method takes into account the superstructure interaction in addition to the soil-pile interaction in an approximate way. However, the effectiveness of this method has not been extensively verified so far. The use of Winkler model and the nonlinear p-y curves is the main method used to investigate the seismic behaviour of the pile foundations. Thus, a need arises to evaluate the effectiveness of this method. There are several factors which are needed to be studied including the appropriateness of available methods for constructing the p-y curves for dynamic analysis, the sensitivity of the form of p-y curve, various assumptions made in the analytical methods regarding the radiation and hysteric damping and the effect of free field motions on the pile response. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 3.2 SINGLE PILE UNDER SEISMIC EXCITATION 3.2.1 Finite Element Formulation for Winkler Model 39 A single pile is embedded in soil and subjected to base excitation. Fig. 3.4 shows the original and deformed shape of the pile at any time t. The governing differential equation of motion of the pile in the direction of shaking can be expressed as \ d4v E I — + pA ax4 { dt2 dt2 ) ( dv_dvt? dt dt , + k h (v-v f f ) = 0 (3.2) Where E- Youngs modulus of pile A - Cross section area of pile I - Second moment of area of pile p- Mass Density of pile k h- Soil reaction coefficient c- Equivalent dashpot coefficient v-Relative displacement of pile with respect to the base excitation vg-Base excitation vff-Relative free field displacement with respect to the base excitation v'-Absolute pile displacement v'=v+vg v' f f- Absolute free field displacement v' f f =v f f+vg Rearranging the terms in Eq. 3.2 gives . d2v dv pA +c— dt2 at l E I ^ + L v l ax4 h PA-3 \ dt2 ' at + k h v f f (3.3) A cubic displacement field is assumed in the direction of shaking. The displacement vector v can Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 40 Pile — • El, A, p j 3ZD 1 Near Field V Free Ftield Base Excitation Fig. 3.4 Winkler Soil Model for the Dynamic Analysis of a Single Pile Under Base Excitation Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 4 1 be expressed as v = S V J N J j = 1,4 (3.4) in which Nj are the shape functions and v, are the nodal displacements. Applying the Galerkin weighted residual procedure to Eq. 3.3, the element mass, damping and stiffness matrices and load vector can be obtained. 3.2.1.1 Element Stiffness Matrix The flexural stiffness matrix of the pile is expressed as K f = E l j N/Ny ' d x o 12 6L -12 6L EI 6L 4 L2 6L 2 L 2 L 3 -12 6L 12 -6L 6L 2 L 2 -6L 4L (3.5) The soil stiffness matrix is expressed as K EIJNjNjdx o " 156 22L 54 -13L k h L 22L 4 L 2 13L - 3 L 2 420 54 13L 156 -22L -13L - 3 L 2 -22L 4 L 2 (3.6) Complete element stiffness matrix can be written as Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 42 K p = K f + K EI L 3 12 6L -12 6L 6L 4 L 2 6L 2 L 2 -12 6L 12 -6L 6L 2 L 2 -6L 4L k h L 420 156 22L 54 -13L 22L 4 L 2 13L - 3 L 2 54 13L 156 -22L -13L - 3 L 2 -22L 4 L 2 (3.7) 3.2.1.2 Element Mass Matrix The consistent element mass matrix is expressed as L M p = pA JNjNjdx o (3.8) The lumped mass matrix is convenient compared to the consistent mass matrix as the off diagonal terms in the consistent mass matrix leads to additional storage, computational time and cost. Therefore, the lumped mass matrix in Eq. 3.9 is used. 1 0 0 0 0 0 0 0 0 0 0 0 in which m is the mass of the pile per unit length. M p = mL 3.2.1.3 Element Damping Matrix The damping in pile and the soil is represented by equivalent viscous type damping and the element damping matrix is expressed as Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 43 C p = c jNjNjdx o 156 22L 54 -13L cL 22L 4 L2 13L - 3 L 2 420 54 13L 156 -22L -13L - 3 L 2 -22L 4 L 2 (3.10) The determination of the equivalent viscous dashpot coefficient is explained in Section 3.2.8.3. 3.2.1.4 Element Load Vector The load vector associated with the base excitation is expressed as 1 = pAVgJNjdx P A V S 1 L/6 1 -L/6 (3.11) The load vector associated with the free field displacements is expressed as P d = k h v f f | N i d x k h v f f 1 L/6 1 -L/6 (3.12) The load vector associated with free field velocity is expressed as dv f f L, P v = c — fN.dx at J 1 c d y f f 2 at 1 L/6 1 -L/6 (3.13) Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves Hence, the complete load vector can be expressed as 44 p =pi+pd +pv = 5 v f f ) \ pAv g + k h v f f + c — jNjdx pAv + k h v f f + c dt 1 L / 6 1 - L / 6 1 (3-14) 3.2.2 Equation of Motion The global dynamic equilibrium equation is written in matrix form as [M]({v} + {v g }) + [C]({v}-{v f f}) + [K]({v}-{v f f}) = 0 (3.15) in which [M],[C] and [K] are global mass, damping and stiffness matrices and v,v v are relative nodal acceleration, velocity and displacement with respect to the base. v f f and v f f are free field displacement and velocity and is v gthe base acceleration. The global mass, damping and stiffness matrices and the global load vector are obtained by assembling the element matrices and load vectors following the conventional procedures. 3.2.3 Solution Scheme The dynamic equilibrium equations are usually solved using either the modal superposition method or the direct step-by-step integration method. The modal superposition method can be used for linear systems but not for nonlinear. The direct step-by-step integration method can be applied to both linear and nonlinear systems. Thus, the step-by-step integration method is used to solve the dynamic equations of equilibrium. The step-by-step integration method allows one to obtain the solution at time t+At by Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 45 knowing the properties and solution at time t. The global dynamic equilibrium equations at time t and t+At are given by Eqs. 3.16 and 3.17 respectively. [M]({v t} +{(v g) t}) + [C]({v t}-{(v f f) t}) + [K]({v t}+{(v f f) t}) = 0 (3.16) [M]({v t + A J + {(vp t + A J) + [CW = 0 (3.17) Assuming that the behaviour of the pile-soil system is incrementally elastic, the global dynamic equilibrium equations can be written in the incremental form as [M]({Av}+{(Avp}) + [C]({Av}-{(Avff)}) + [K]({Av} + {(Avff)}) = 0 (3.18) in which [C] and [K] are the average damping and average stiffness matrices over the small time interval At. [M] is the mass matrix that does not change with time. Av, Av, Av, A v g , Av f f , A v f f are the increments respectively over the time increment At. Usually, two distinct approaches are followed in soil-structure interaction analysis in solving the nonlinear dynamic equilibrium equations. One is the secant stiffness approach and the other is the tangent stiffness approach. In secant stiffness approach, [K] is assumed as the secant stiffness [K s ] at the beginning of the time interval At. In tangent stiffness approach, [K] is taken as the tangent stiffness [K T ] at time t. In both approaches, the average damping matrix is taken as the damping matrix at the beginning of the time interval At. In these secant and tangent stiffness approaches the actual nonlinear behaviour is approximated by a series of linear steps. Therefore, appropriate corrections are made at the end of the time step to maintain the global dynamic equilibrium. In the secant stiffness approach, the incremental equation is expressed as [M]({Av} + {(Avg)}) + [C t]({Av}-{(Av f f)}) + [(K s) t]({Av} +{(Av f f)}) = 0 ( 3 1 9 ) Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves In the tangent stiffness approach, the incremental equation is written as 46 [M]({Av} +{(Av g)}) + [C t]({Av}-{(Av f f)}) + [(KT) t]({Av} + {(Av f f)}) = 0 ( 3 2 0 ) Each of these equations represents a set of simultaneous equations that can be solved in the time domain using the numerical procedure developed by Wilson et al. (1973) or Newmark (1959). 3.2.4 Near Field Behaviour In linear elastic analysis, the pile-soil behaviour is modeled using a constant horizontal coefficient of subgrade reaction. This coefficient k h in Eq. 3.6 is kept as constant throughout the time domain analysis. In nonlinear analysis, k h is obtained from the nonlinear soil reaction p-y curves. 3.2.5 Free Field Behaviour It is apparent from Eq. 3.18 that, in order to solve the dynamic equilibrium equations, the free field displacement and the free field velocity are required. The free field response of the soil is obtained through a separate site response analysis and the method of this analysis is described in Section 3.3. 3.2.6 Equivalent Linear Approach Using Secant Stiffness The global secant stiffness matrix [K s ]is assembled from the elements stiffness matrices. The secant stiffness soil reaction coefficient, k h of each individual element is required in calculating the element stiffness matrix. This coefficient should be representative over the entire length of the beam element. In the current approach, the p-y curves are specified at the nodes of the element and Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 47 the secant soil reaction coefficient at the nodes are calculated based on the relative nodal deflection with respect to free field. The average value of coefficients at the nodes is assumed as the required secant soil reaction coefficient of the element. 3.2.7 Incrementally Linear Elastic Approach Using Tangent Stiffness The seismic loading imposes irregular loading pulses which consists of loading, unloading and reloading. As a result the p-y relationship in the near field can also be expected to show different behaviour during these phases. Adequate modeling of these different phases is essential in order to obtain the true dynamic response of the pile under seismic loading. However, the data on the p-y behaviour under random loading is scarce. In view of this, it has been decided to use the concepts which were proposed for the shear stress- shear strain behaviour during random loading and showed to be very effective in capturing the actual behaviour of the soil. In the current approach, the p-y behaviour during unloading and reloading is assumed to follow the extended Masing(1926) criterion proposed by Finn et al.(1977). This procedure is based on the method proposed by Masing (1926) for regular cyclic loading but extended for random loading. A somewhat similar approach is to be found in the program SPASM (Matlock et al., 1979). In the incrementally linear elastic approach, p-y curves during loading, unloading and reloading phases are constructed at each node and the tangent stiffnesses are calculated at element nodes based on these curves. The required tangent stiffness of the element is assumed as the average of the stiffnesses at the element nodes as in the secant approach. Construction of p-y curves under Random Loading: The initial loading phase is modeled using the original p-y curve such as the one shown in Fig. 3.5. This is also used as the skeleton curve. The subsequent unloading and reloading is modeled using the extended Masing Criterion. According to the Masing criterion, the unloading and reloading branches of a hysteresis loop are the same skeleton curve with origin transferred to the reversal point and the scaling for soil reaction and Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 48 deflection increased by a factor of two. The equation of virgin loading or skeleton curve is expressed as P = f(y) (3.21) The equation of the unloading from the point (Pr,yr) at which the loading reverses is expressed as in which f is the same function which describes the skeleton p-y relationship. The shape of the initial loading and unloading curve is shown in Fig. 3.6. The Masing criterion described above is suitable for constructing the p-y curves under regular loading only. It is not good enough for random loading. Finn et al. (1976) proposed rules for the extension of Masing concept for irregular loading. They suggested that the unloading and reloading curves should follow the prescribed skeleton loading curve when the magnitude of the previous maximum deflection is exceeded. In Fig. 3.6, the unloading curve begin the extension of the initial loading in the negative direction, ie. BC. In the case of general loading history, they assumed that when the current loading curve intersects the previous loading curve, the p-y curve follows the previous loading curve. Two examples are provided in Fig. 3.7 to illustrate these rules. 1. If loading path B C is continued, the loading path is assumed to be B C A M where A M is the extension of OA. 2. If unloading along path CPB is continued, then the unloading path will be A B P ' . 3.2.8 Formulation of Global Mass, Stiffness and Damping Matrices 3.2.8.1 Global Mass Matrix The global mass matrix is formulated from element mass matrix described in Eq. 3.9. Though the consistent mass matrix is more accurate, the lumped mass matrix is more convenient as it requires less storage and leads to less computational time compare to the consistent mass matrix without any appreciable loss of accuracy. The global lumped mass matrix is diagonal and it does not vary with time. (P-P r)/2 = f[(y-y r)/2] (3.22) Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 400 49 300 \-0.00 0.02 0.04 0.06 0.08 0.10 y (m) Fig. 3.5 A typical p-y Curve Suggested by API(1993) for Sand P A Pu Skeleton^--curve^^ (P„y,) /Unloading / Reloading/ 1/ Y Fig. 3.6 p-y Curve for Loading and Unloading Using Masing Rule A' Fig. 3.7(a) First Unloading Fig. 3.7(b) General Reloading Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 3.2.8.2 Global Stiffness Matrix 50 The global stiffness matrix is assembled from the element stiffness matrix described in Eq. 3.7 following conventional procedures of finite element analysis. The element stiffness matrix, K e consists of two parts, the pile flexural stiffness matrix, K f and the soil stiffness matrix, K s representing the pile-soil resistance, ie. The complete element stiffness matrix is written as K e = K f + K s In the current approach, the behaviour of the pile is assumed linear elastic and hence the flexural stiffness matrix is kept constant throughout the time domain analysis. The stiffness matrix representing the pile-soil interaction in the near field is varied depending upon the pile deflection level. In equivalent linear analysis, the method described in Section 3.2.6 is used to obtain the secant stiffness coefficient and, in the incrementally linear elastic approach, the method explained in Section 3.2.7 is used to calculate the tangent stiffness coefficient. 3.2.8.3 Damping Matrix In evaluating the dynamic response of pile, the energy losses due to the hysteretic behaviour of the soil and the radiation of energy by waves transmission should be taken into account. The damping due to the hysteretic behaviour of the soil is called the material damping. This damping is automatically taken into account in incrementally elastic analysis in which the p-y curve is followed during all phases of loading, unloading and reloading using the tangent stiffness approach. In equivalent linear analysis using the secant stiffness, damping must be included explicitly and is normally represented by equivalent viscous damping. The dashpot coefficient representing the hysteretic damping of the material is expressed as C m = 2Pskh-secan/W (3.23) Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 51 in which Ps is the damping ratio, k h . s e c a n t is the secant soil reaction coefficient and co is the circular frequency. This approximate expression for the equivalent dashpot coefficient was proposed by Gazetas(1984). The damping ratio adjacent to the pile varies depending upon the level of shear strain. A typical variation of shear strain versus damping ratio is shown in Fig. 3.8. Kagawa and Kraft (1980) proposed an approximate relationship for the average shear strain in terms of the pile deflection. This relationship is an extension of the Matlock(1970) relationship between pile deflection and normal strain and is expressed as (l+v)y Y = (3.24) in which y is the shear stain, v is the Poisson's ratio,D is the diameter of the pile and y is the pile deflection. In the current approach, the shear strain versus damping ratio is specified for each layer of soil and the dashpot coefficient representing the hysteretic damping of the soil is calculated using theEq. 3.23. Gazetas (1984) proposed that the dashpot coefficient representing the radiation energy losses be expressed as c r = 6 a o - 0 2 5 p s V s d (3.25) in which p s is the mass density of the soil, V s is the shear wave velocity and d is the diameter of the pile. a0(=cooWs) is the non dimensional frequency and co is the circular frequency. This expression was adopted for the current procedure. The radiation damping varies with the shear wave velocity which in turn depends on the level of shear strain in the near field. Thus, the strain dependency of the shear wave velocity should be taken into account. Since the shear wave velocity can be expressed in terms of the shear modulus Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 52 as G = p s V s 2 the strain dependence of the shear wave velocity can be readily the data like that in Fig. 3.8. The global damping matrix can be assembled form the element damping matrices following the conventional procedures. 3.2.9 Computation of Correction Force Vector In both secant stiffness approach and the tangent stiffness approach, the nonlinear behaviour of the soil is approximated by a series of linear steps. Therefore, at the end of each time step At, the computed pile deflection and the soil reaction at each node may not be compatible with the assumed p-y relationship. To restore compatibility, correction forces are applied at the end of each time step. The correction forces are calculated assuming that the calculated deflections are true deflections and global equilibrium is imposed. The correction force vector is expressed as {AF c o r r } = - [ M ] { A v } - [ C t ] { A v } - [ ( K s ) t ] A v + {AF) ( 3 2 6 ) This correction force vector is added to the right hand side of Eq. 3.19 in the next step of integration. 3.2.10 Free Field Response As shown in Eq. 3.20, the free field soil response is an important input parameter in the determination of single pile response under the earthquake loading. By definition, the free field soil behaviour is unaffected by the presence of the pile. It is usually evaluated using one or two dimensional wave propagation theories, lumped parameter models or the finite element method. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 53 Generally, the soil layer surrounding the pile is layered and exhibits highly nonlinear behaviour under strong earthquake loading. In evaluating the response of the free field soil, the inhomogeneous nature of the soil and the strain dependent behaviour of the moduli and damping should be taken into account. The site response analysis program 'Shake' developed by Schanabel et al. (1972) is usually used to obtain the site response under earthquake loading. This program can handle the layered soil deposit and uses equivalent linear approach. Idriss et al. (1983) proposed a finite element methodology to evaluate the nonlinear response of the soil deposit. This methodology is essentially followed in the current method to determine the free field repose parameter and is described below. 3.3 MODELING OF HORIZONTAL LAYERED DEPOSIT The free field soil is modeled by a column of plane strain rectangular element as shown in Fig. 3.9. The vertical degrees of freedom of the rectangular element are restrained thus allowing the nodes to translate in the horizontal direction only. The governing dynamic equation of motion of the system is expressed as [M f f ]{v} f f + [C f f]{v} f f + [K f f]{v f f} = - [M f f ]{ I}v g (3.27) in which [M f f ] , [C f f] and [K f f] are global mass, damping and stiffness matrices and v , v and v are acceleration, velocity and displacement relative to the base. v g is the base excitation and {1} is a unit vector. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 54 . 2 1.2 S h e a r S t ra in (%) Fig. 3.8 A Typical Variation of Shear Modulus and Damping Ratio With Shear Strain L a y e r e d S o i l Q u a d r e l a t e r a l C 1 <**> KVI /*\ n t D e p o s i t t lern eni (Vertical D O F are Restrained) Base Input Motion Fig. 3.9 Finite Element Model for the Analysis of Free Field Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 55 The damping of the system is assumed to be of Rayleigh type and the global damping matrix is assembled from the individual element damping matrix, [C] e which is expressed as [C]e=a[M]e+p[K]e (3.28) where [M] e and [K]e are the element mass and stiffness matrices, a and P are constants that depend on the damping ratio, A and the modal circular frequency, co and they are expressed as a=Aw p=A/co (3.29) The advantage of this method is that different degrees of damping can be assigned to each individual element according to the shear strain level. A typical variation of damping ratio with shear strain is shown in Fig. 3.8. The dynamic equilibrium equations are solved using the direct step-by-step integration procedure proposed by Wilson et al (1973). The secant stiffness approach is followed. The global dynamic equilibrium is ensured at each time step by doing the appropriate correction for the unbalanced force at the end of each time step. 3.4 INTRODUCTION TO THE COMPUTER PROGRAMS PILE-PY AND FREEFLD The procedure for nonlinear dynamic analysis of a single pile using the Winkler model and the p-y curves which was described in the previous sections was implemented in a computer program called PILE-PY. Options are available in this program to choose either the secant stiffness or tangent stiffness approach. The procedure for the nonlinear dynamic analysis of the free field was implemented in a computer program called FREEFLD. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 56 3.5 ANALYSIS USING SECANT AND TANGENT STIFFNESS APPROACHES 3.5.1 Introduction Two methods have been described for solving the nonlinear dynamic equilibrium equations; secant stiffness method and tangent stiffness method. In order to determine how these two methods compared with each other, a numerical study was carried out under two different peak base accelerations of O.lg and 0.3g. Problem Description : The single pile carrying a concentrated mass at the pile head, shown in Fig. 3.10 was chosen for the study. The pile is of diameter, d=0.5 m and length, L=15 m. The embedded length of the pile is 1 lm and the free standing length is 3 m. The Young's modulus of pile, Ep=22 GPa and the mass density of the pile, pp=26 kNm"3. The concentrated mass at the pile head, M=25 Mg and the mass moment of inertia, Im=25 Mgm 2 . The pile is embedded in a sandy soil having a relative density, Dr of 30% and a friction angle, (b=31°. The unit weight of the soil is y=20 kNm"3 and the Poissons ratio v=0.4. The shear modulus profile is assumed to follow the empirical relationship suggested by Seed and Idriss (1972) is expressed as G m „ - 21.7 k P max g a A 0.5 m J , (3.30) in which P a is the atmospheric pressure, aj is the mean effective stress and k g is a coefficient related to the relative density of the soil. The mean effective stress, oj is expressed as , (1+2K) , oj = (3.31) in which a 1 is the vertical effective stress and K is the lateral stress coefficient which is assumed V o as 0.5. The coefficient, k is assumed as Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 57 k = g 15+0.61D (3.32) in which D r is the relative density of the soil. 3.5.2 Model of the Pile and Soil The finite element model of the pile and the soil is schematically shown in Fig. 3.10. The soil layer was divided into eleven layers. The free field soil was modeled using rectangular plane strain elements. The embedded length of the pile was modeled using eleven beam elements and the free standing portion of the pile was modeled with three beam elements. The pile nodes and the rectangular elements nodes are taken to be at the same horizontal level. 3.5.3 Near Field Behaviour In the modeling of near field soil pile interaction, the soil reaction curves p-y recommended by API (1993) for loose dry sand were used. 3.5.4 Input Base Acceleration The input base acceleration used in the study was the first fifteen seconds of 1971 San Fernando earthquake recorded at the Caltech Seismological Lab, Pasadena California. The record is shown in Fig. 3.11. The record was modified by scaling to peak accelerations of 0.1 g and 0.3g for the in analysis. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 58 Near Field Pile Free Field v, v, Fig. 3.10 Soil-Pile Model Used in the Analysis o 15 .2? CD O o < 1.0 0.5 h 0.0 -0.5 -1.0 i 1 i 1 i 1 11 In 11 v i r " WV^T^^WW* 1 1 1 1 1 1 1 1 1 0 2 4 6 8 10 12 14 Time (sec) Fig. 3.11 Unit Normalized Input Acceleration Time History Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 3.5.5 Response of Pile Under Earthquake Excitation 59 The pile was first subjected to the record with a peak acceleration of 0.1 g and the response was evaluated using both the secant and tangent stiffness methods. The computed acceleration and displacement time histories of the superstructure mass from these two methods are shown in Fig. 3.12. Except for the small differences in the peak values, the responses from the two methods agree very well. Fig. 3.13 shows the comparison of the displacements and accelerations at the ground level of the pile. In this case also, a very good agreement can be observed between the two methods. The p-y response at 1-D and 2-D depths where D is the diameter of the pile were traced during analysis with the tangent stiffness method and is shown in Fig. 3.14. The narrower width of the p-y loops at these two depths show that the nonlinearity is not very significant. A second analysis was carried out under a higher peak acceleration of 0.3 g. The p and y time histories and the p-y hysteresis loops at 1-D depth from the analysis using the tangent stiffness method are shown in Fig. 3.15a through 3.15c. It is clearly evident that the near field soil at 1-D depth exhibits highly nonlinear behaviour. The hysteresis loops also show that the energy losses due to the hysteretic behaviour of the soil becomes significant when the peak base acceleration is increased from 0.1 g to 0.3 g. Fig. 3.16a through c show the p,y time histories and the p-y loops at the depth 2-D. The behaviour is similar to that at depth 1-D. However, the p-y loops are not as wide as at depth 1-D indicating less dissipation of energy and less nonlinearity in the behaviour of soil at 2-D depth. They are also steeper than the loops at 1-D depth. This is due to the initial stiffness of the p-y curve suggested by API(1993) which is directly proportional to the depth. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 60 Q J E (D o o. w 0.02 0.01 h 0.00 -0.01 -0.02 Non-Linear Analysis Using Secant Stiffness Tangent Stiffness 6 8 Time (sec) 10 12 14 Fig. 3.12a Comparison of Displacement at the Superstructure Mass 0.20 0.10 h 0.00 -0.10 Non-Linear Analysis Using Secant Stiffness Tangent Stiffness -0.20 Amax=0.1 g i ' i ' ' 111 in ni 1111111111111111111111111 0 6 8 Time (sec) 10 12 14 Fig. 3.12b Comparison of Acceleration at the Superstructure Mass Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 61 CU E 0 o ro o. w 0.010 0.005 h 0.000 -0.005 h -0.010 Non-Linear Analysis Using Secant Stiffness Tangent Stiffness ' ' ' ' ' 1111 Amax=0.1 g i i 11111111111 111111111111111 6 8 Time (sec) 10 12 14 Fig. 3.13a Comparison of Displacement of Pile at the Surface TO CZ o ro L _ <o <D O O < 0.20 0.10 h 0.00 -0.10 -0.20 Non-Linear Analysis Using Secant Stiffness Tangent Stiffness Amax=0.1 g _L J L j L _L 6 8 Time (sec) 10 12 14 Fig.3.13b Comparison of Acceleration of Pile at the Surface Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 62 Fig. 3.14b P-Y Hysteresis Loop at 2-D Depth Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 63 150 100 50 0 -50 -100 -150 (a) I I 1 Amax=0.3 g | I i -0.02 -0.01 0.00 y (m) 6 8 10 Time (sec) CL 150 100 50 0 -50 -100 -150 0.01 0.02 (C) j I \ \ i L Amax=0.3 g i l i l 6 8 Time (sec) 10 12 14 Fig. 3.15 p and y Time Histories and p-y loops at 1-D depth for Amax=0.3 g Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 64 CL 150 100 50 0 -50 -100 -150 -(a) -0.02 -0.01 0.00 y (m) £ 0.00 6 8 10 Time (sec) 150 CL 100 -50 -0 -50 -100 -150 *4 o 6 8 10 Time (sec) 0.01 0.02 (c) Amax=0.3 g J i I 12 14 Fig. 3.16 p and y Time Histories and p-y loops at 2-D depth for Amax=0.3 g Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 65 The comparison of acceleration and the displacement at the superstructure mass from both the tangent stiffness method and the secant stiffness method are shown in Fig. 3.17. It can be seen from these figures that the two approaches agree very well. The very good agreement between the results from the two methods under two different peak accelerations suggests that both methods would give similar results regardless of the levels of nonlinearity experienced by the soil during strong shaking. However one crucial difference must be noted between the two methods. The tangent stiffness method can capture any permanent or residual displacements that may occur during the earthquake loading. The secant stiffness method which is based on the equivalent linear behaviour cannot capture such displacements. The displacement response of the superstructure mass using the tangent stiffness approach shown in Fig. 3.12 and 3.17 do not show any permanent displacement. Thus, the responses from the two approaches are not affected by any residual displacement. 3.6 ANALYSIS OF CENTRIFUGE TEST OF SINGLE PILE A centrifuge test on a single pile was carried out at the California Institute of Technology by Gohl (1991). Detailed data on this centrifuge test is given by Gohl (1991) and Finn and Gohl(1987). Fig. 3.18 shows the single pile-soil-structure system used in the test. A stainless steel hollow tube was used as the pile. Eight pairs of foil type strain gauges were molded on the side of the pile to measure the bending strains. A mass was screwed to the clamp attached to the head of the pile to simulate the influence of the superstructure inertia forces that would act on the prototype pile. The location of accelerometers and light emitting diode used by the displacement sensors are shown in Fig. 3.18. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 66 a cu E CD O ro a. CO b 0.10 0.05 0.00 -0.05 h -0.10 6 8 Time (sec) 10 12 14 Fig. 3.17a Comparison of Displacement at the Superstructure Mass o ro V— CU O O < 0.80 0.40 h 0.00 -0.40 h -0.80 Non-Linear Analysis Using Secant Stiffness Tangent Stiffness Amax=0.3 g j I i L J I I I I L 6 8 Time (sec) 10 12 14 Fig. 3.17b Comparison of Acceleration at the Superstructure Mass Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 67 L.E.D Pile Head Clamp and Mass Soil Surface 16.5: i_L 9 + ^ 2 N o . l Location of Strain Gauges No. 4 No. 6 Base of Centrifuge Bucket No. 8 No. 1 Accelerometer No. 3 No. 5 Axial Strain Gauge No. 7 SCALE: 0 20mm Pile Tip Fig. 3.18 Details of Instrumented Single Pile Used in the Centrifuge Test Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 68 The prototype parameters of the single pile test are shown in Fig. 3.19. The pile is of diameter d=0.5712 m and length L=12.89 m. The unit weight, y and flexural rigidity of the pile, EI are 74.7 kNm"3 and 172614 kNm 2 respectively. The mass, m s and the mass moment of the inertia of the structural mass, \ attached to the pile head are 53.2 kN sec m and 53.11 kNsec2 m. The center of gravity of the attached structural mass is 0.99 m from the base of the pile head clamp and the free standing portion of the pile is 0.99 m. Loose sand having a void ratio of eG = 0.78 and the mass density, p s of 1.5 Mgm"3 was used in the test. The relative density is determined to be 38%. Gohl (1991) has showed that the low strain shear modulus of the soil can be approximated by the following Hardin and Black (1968) equation (2 973 -e ) 2 G = 3230— — (o 'f5 (3 33) max i , v m ' \-J.->~>J 1 + e o in which a  1 is the mean effective stress and e is the initial void ratio of the soil. The effective m o mean stress is expressed as ( 1+2K ^ o a / (3.34) in which oj is the effective vertical stress and K o is the lateral earth pressure coefficient, is 0.4 for loose sand. 3.7 RESPONSE OF PILE UNDER MODERATELY STRONG EARTHQUAKE EXCITATION 3.7.1 Introduction Centrifuge test No. 12 was conducted by Gohl(1991) using the acceleration record in Fig. 3.20 as input motion. The peak base acceleration was 0.158 g. This test was analyzed using the computer program PILE-PY. The same test was also analyzed by Wu(1994) and Wu and Finn, (1997b) using PILE3D. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 69 0.99 m 12 m Structural Mass m = 53.2 kNsec/m I = 53.11 kNsec2 m Pile El =172614 kNm2 L =12.89 m D =0.5712 m 'A Base Motion Fig. 3.19 Prototype Model ofthe Single Pile Test J i i i I i i i i I i i i i i i i i 10 15 20 Time (sec) 25 30 Fig. 3.20 Base Input acceleration time history Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 3.7.2 Dynamic Analysis Using PILE3D 70 The finite element model of the system used by Wu(1994) is shown in Fig. 3.21. The pile was modeled by fifteen beam elements including five elements modeling the free standing portion of the pile above the soil surface. The superstructure mass is treated as a rigid body and the motion is represented by a concentrated mass at the center of gravity. A very stiff beam element with flexural rigidity 1000 times that of the pile was used to connect the mass at the pile head. The soil was modeled by 450 brick elements. A nonlinear time domain analysis was carried out to determine the response of the soil-pile system. The variation of shear modulus and damping ratio with shear strain used in the study is shown in Fig. 3.22 (Gohl, 1991). The maximum damping ratio of the soil was taken as 25% following Gohl(1991) and the maximum shear modulus was calculated by the Eq. 3.33 The computed and measured acceleration and displacement time history at the top of structure are shown in Fig. 3.23 and 3.24. There is a good agreement between the measured and computed accelerations. The computed displacements are somewhat smaller than the measured displacements. The measured and computed bending moment time histories of the pile at the ground level and 3m depth are shown in Fig. 3.25. and 3.26. Satisfactory agreement can be observed between the measured and computed during the periods of strong shaking. Fig. 3.27 shows the comparison of maximum bending moment profile along the pile length. The computed maximum bending moments agree well with the measured maximum moments across the pile length. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 7 1 < • Shaking Direction Fig. 3.21 Finite Element Mesh Used in the PILE3D Analysis Shear strain (%) Fig. 3.22 Variation of Shear Modulus and Damping Ratio with Shear Strain Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 72 c o i _ 0) CD O O < 0.4 0.2 h 0.0 -0.2 -0.4 0 Measured Computed-PILE3D 10 15 20 Time (sec) 25 30 Fig. 3.23 Comparison of Measured and Computed Acceleration Time Histories at the Top of Structure _Q io i I i I i I i I i I i I I 0 5 10 15 20 25 30 Time (sec) Fig. 3.24 Comparison of Measured and Computed Displacement Time Histories at the Top of Structure Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 73 -600 0 Measured Computed-PILE3D 10 15 20 Time (sec) 25 30 Fig. 3.25 Comparison of Measured and Computed Bending Moment Time Histories at the Soil Surface Fig. 3.26 Comparison of Measured and Computed Bending Moment Time Histories at 3 m Depth Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 74 200 Bending Moment (kN.m) F i g . 3.27 Comparion o f M a x i m u m Bending Moment Profiles Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 75 3.7.3 Dynamic Analysis of Centrifuge Test Using the Winkler Model and the p-y Curves 3.7.3.1 Model of the Soil-Pile-Structure System The model of the soil-pile-structure system is similar to the one shown in Fig. 3.10. The sand deposit was divided into 0.25 m thick layers. The free field soil was modeled by rectangular plane strain elements. The embedded length of the pile was modeled using 47 beam elements. The free standing portion of the pile was modeled using 6 beam elements. The superstructure mass and the mass moment of inertia was assigned at the center of gravity of the mass as concentrated parameters. A very stiff beam element was used to connect the pile head to the center of gravity of the mass. The nodes of the pile elements were taken in such a way that they are at the same horizontal level as the rectangular elements modeling the free field soil. 3.7.3.2 Dynamic Response of Free Field The free field response was evaluated using the computer program FREEFLD. Nonlinear analysis was performed in the time domain to account for the changes in the modulus and damping. The shear strain dependency of the modulus and damping was taken into account using the relationships shown in Fig. 3.22. which were used also in the PILE3D analysis. The comparison of computed response by PILE3D and measured free field response is shown in Fig. 3.28 Apart from the slight difference in the peak values, there is a good agreement between the two responses. The computed responses from FREEFLD and PILE3D are shown in Fig. 3.29 The agreement between the results from the two program is excellent. 3.7.3.3 Dynamic Response of Single Pile using PILE-PY The free field displacements and the velocities from the FREEFLD analysis were used as input parameters. The near field nonlinear behaviour is modeled using the p-y curves. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 76 0 . 4 0 . 2 h 2 0.0 CD - 0 . 2 - 0 . 4 Measured Computed - PILE3D 0 5 1 0 1 5 2 0 2 5 Time (sec) Fig . 3.28 Comparison of Measured and Computed Free F ie ld Acceleration Time Histories 3 0 F i g . 3.29 Comparison of Measured and Computed Free F ie ld Acceleration Time Histories Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 77 The p-y curves at different depths along the pile were generated using the procedures recommended by API(1993) and given in Appendix 1. API(1993) suggests that, for the loose sand used in the test, the initial modulus of subgrade reaction, kh= 15000 kN/m 3. However, by constructing the p-y loops experimentally from the test, Gohl(1991) showed that this initial stiffness is too high and recommended a value of 2500 kN/m 3 which is only l/6th of the value suggested by API(1993).. Several researchers suggested that the initial modulus subgrade reaction, k h vary with depth, H with k h H=SE m a x where E m a x is the Young's modulus of the soil and 6 is a constant. In Appendix 2, the range of 6 was investigated and it was found that the typical range of 6 is 0.9 to 1.3. Thus, three analyses are carried out here with kh= 15000 kN/m 3 which is the value suggested by API(1993) , kh=2500 kN/m 3 , a value recommended by Gohl(1991) and k h H=E m a x with 8=1. Three analyses are carried out here with kh=15000 kN/m 3 which is the value suggested by API(1993), k,=2500 kN/m 3, a value recommended by Gohl(1991) and k h H=E m a x with 8=1. These analyses were reported in Wu etal. (1998). 3.7.3.4 Results of Analyses with kh=15000 kN/m 3 The computed and measured accelerations and displacements at the top of structure are compared in Fig. 3.30 and 3.31 . These two figures show that the peak response is overestimated and the agreement between the measured and computed response is not good. The computed peak acceleration was 0.247 m/s2 and the measured peak acceleration was 0.187 m/s2. The comparison between the computed and measured bending moment time histories response at the soil surface and at 3m depth are shown in Fig. 3.32 and 3.33 respectively. Again, poor agreement between the two responses can be observed. The computed maximum moments at the soil surface and at 3m depth are 40% and 53% higher than the measured moments. Fig. 3.27.. shows the comparison of maximum bending moment profile along the length of the pile. The maximum moments are severely overestimated. The difference between the calculated and measured peak moments is approximately 63%. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 78 3 3 cz o ' CD o C D O O < 0.4 0.2 h 0.0 -0.2 h -0.4 Measured Computed-PILE-PY kh=15000 kN/m 3 0 10 15 20 Time (sec) 25 30 Fig . 3.30 Comparison of Measured and Computed Acceleration Time Histories at the Top of Structure F i g . 3.31 Comparison o f Measured and Computed Displacement Time Histories at the Top of Structure Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 79 600 E z CD E o CD C T3 C CD 00 -600 0 10 15 Time (sec) 20 25 30 Fig . 3.32 Comparison of Measured and Computed Bending Moment Time Histories at the Soi l Surface F i g . 3.33 Comparison of Measured and Computed Bending Moment Time Histories at 3 m Depth Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 3.7.3.5 Results of Analyses with kh=2500 kN/m 3 80 A second analysis was carried out with kh=2500 kN/m 3 as suggested by Gohl(1991). Fig. 3.34 shows the measured and computed accelerations at the top of the structure. In this case there is a very good agreement between the two. The measured and computed displacements shown in Fig. 3.35 also agree very well. Fig. 3.36 and 3.37 show comparison of bending moments at the soil surface and at 3m depth. At these two depths, the agreement between the measured and computed bending moments are also good especially in the region of large moments. Fig. 3.27 shows the comparison of maximum moment along the pile length. The difference between the measured and computed maximum moment up to 4m depth is small. The peak moment occurs approximately at the same location of the pile and difference between the measured and computed peak moment is only 16%. Beyond 4m depth, the maximum moments are overestimated. Fig. 3.38 shows the p-y hysteresis loops at two depths;0.25 m and 0.75m. These loops show that the behaviour is highly nonlinear and inelastic. The hysteresis loops at shallower depth are flatter than those at deeper depth. This is due to the form of p-y curves recommended by API(1993). API(1993) recommends that p-y curve initial stiffnesses should be proportional to the depth. 3.7.3.6 Results of Analyses with khH=E, A third analysis was carried out with k h H=E m a x where H is the depth and E m a x is the Youngs modulus. Fig. 3.39 and 3.40 show the comparison of measured and computed acceleration and displacement time histories respectively. The accelerations are underestimated during the period of peak response. However, the agreement between the two are fair. Poor agreement between the computed and measured displacements can be observed in Fig. 3.40. The displacements are severely underestimated during the period of peak response. Fig. 3.41 and 3.42 show the comparison between the measured and computed moment time histories at ground level and at 3m depth. The agreement between the measured and computed Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 81 0.4 ~ 0.2 0.0 -0.2 -0.4 0 Measured Computed - PILE-PY 10 15 20 Time (sec) 25 30 Fig. 3.34 Comparison of Measured and Computed Acceleration Time Histories at the Top of Structure Fig. 3.35 Comparison of Measured and Computed Displacement Time Histories at the Top of Structure Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 82 c CD E o CD c "O c CD CO -600 0 Measured Computed-PILE-PY 10 15 20 Time (sec) 25 30 Fig. 3.36 Comparison of Measured and Computed Bending Moment Time Histories at the Soil Surface Fig. 3.37 Comparison of Measured and Computed Bending Moment Time Histories at 3m Depth Fig. 3.38b p-y Hysteresis Loop at 0.75m Depth Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 84 0.4 ~ 0.2 c o ro CD O O < 0.0 -0.2 -0.4 0 Measured Computed - PILE-PY 10 15 Time (sec) 20 25 30 Fig. 3.39 Comparison of Measured and Computed Acceleration Time Histories at the Top of Structure Fig. 3.40 Comparison of Measured and Computed Displacement Time Histories at the Top of Structure Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 85 c CO E o CD c TD C CD GO -600 0 Measured Computed-PILE-PY 10 15 Time (sec) 20 25 30 Fig . 3.41 Comparison of Measured and Computed Bending Moment Time Histories at the Soi l Surface F ig . 3.42 Comparison of Measured and Computed Bending Moment Time Histories at 3m Depth Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 86 moments are poor. The peak moments are also severely overestimated. Fig. 3.43 show the comparison of measured and computed maximum moments along the pile during the excitation. Also shown is the computed moment when kh= 15000 kN/m 3 is used. The peak moments are severely overestimated up to the depth of 4m. However, the maximum moments are smaller than those predicted when kh=15000 kN/m 3 is used. Beyond 4m depth, the comparison is good and much better than those predicted when kh= 15000 kN/m 3 is used. 3.8 RESPONSE OF PILE UNDER LOW HARMONIC EXCITATION Gohl(1991) also carried out a centrifuge test under a harmonic acceleration of 0.04g. An analysis was carried out to simulate this test using the computer program PILE-PY. In this simulation, the modulus of subgrade reaction, k h was taken as 15000 kN/m 3 which is the value recommended by API(1993). The free field response was first evaluated and used in the analysis of pile. The measured peak free field acceleration is approximately 0.040g which is the approximate average over the steady state portion. The computed peak acceleration is 0.042g. The measured and computed peak acceleration of the pile top agreed very well. The measured peak acceleration is 0.045g and the computed peak acceleration is 0.044g. Fig. 3.44 shows the comparison of measured and computed maximum bending moment profile along the pile length. It can be seen from this figure that the measured and computed maximum bending moments agree very well. The difference between the peak measured and calculated peak moments is only 3.5%. Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 87 -600 -400 -200 0 200 Bending Moment (kN.m) Fig.3.43 Comparion of Maximum Bending Moment Profiles Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 88 Fig. 3.44 Comparion of Maxiumum Bending Moment Profiles Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 89 3.9 ANALYSIS OF A SINGLE PILE EMBEDDED IN CLAYEY SOIL USING WINKLER MODEL AND P-Y CURVES Having investigated a single pile embedded in a sandy soil using the Winkler model and p-y curves, a study was under taken to analyze the seismic response of a single pile embedded in a clayey soil. In this study the single pile embedded in a soft clay was simulated using the PILE3D and PILE-P Y and the results were compared. The API(1993) recommended p-y curves for soft clay was used in the PILE-PY simulation. This study is expected to shed some light on the appropriateness of the API(1993) recommended p-y curves for soft clay in the seismic response analysis. 3.9.1 Problem Description A single pile embedded in a 10m thick uniform soft clay was used in the seismic response study. The unit weight, y and the underained shear strength, S u of the soft clay, was taken as 18kN/m3 and 50 kPa respectively. The Young's modulus of the pile, E p is 30500 MPa and the diameter, D and the length of the pile, L are 0.5m m and 9m respectively. The single pile carries a single degree of freedom system at the pile top. The natural frequency of the SDOF system, was taken as 4 Hz and the mass of the SDOF system, M s was taken as 50 Mg. The pile top is considered fixed against the rotation. Fig. 3.45 shows the single pile and the properties of the soil and pile. 3.9.2 PILE-PY Model of the Single Pile The single pile was modeled using 27 beam elements 0.3m long. The springs and dashpots were placed at each nodal point and they were modeled using the API(1993) recommended p-y curves.. The p-y curves for the soft clay was generated following the procedures recommended by API(1993). The API(1993) recommendations for generating the p-y curves for soft clay are Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 90 Superstructure Parameters Mass, Ms= 50 Mg Natural Frequency of SDOF system, f = 4Hz Soil Properties Undrained Shear Strength, S u = 50 kPa Unit weight, y s =18 kN/m 3 Shear Modulus, G m a x = 50 MPa Pile Properties Flexural Rigidity, El =1310 kNm 3 Unit weight, y p Diameter, D Length, L E p /E s=500 Pp/ps = 1 - 4 L/d >15 v =0.4 = 25 kN/m 3 = 0.5m = 9 m Homogeneous Clayey Soil 7 WW V Pile •yW\AA Nonlinear Springs and Dashpots Fig. 3.45 Model and the Properties of Soil and Pile Used in the Analysis Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 91 described in Appendix 1. The strain that occurs at one-half of the maximum stress on the laboratory undrained compression tests of undrained soil samples, ec was taken as 0.1%. The depth below soil surface to bottom of reduced resistances zone , X r is calculated to be 5.40m. The dimensionless empirical constant, J was taken as 0.375. The free field soil was modeled using 27 quadrilateral elements. The variation of shear modulus and damping ratio with shear strain recommended by Sun et al(1988) were used for the clay. 3.9.3 PILE3D Model of the Single Pile In the PILE3D model, the single pile foundation was modeled using 990 elements and 1320 nodes. The soil layer was modeled using 980 brick elements and the pile was modeled with 9 beam elements. The pile head was fixed against the rotation as in the PILE-PY model. The same superstructure used in the PILE-PY was used here also. The variation of shear modulus and damping ratio with shear strain used are those suggested by Sun et al(1988) for soft clay with plasticity index 20-40% shown in Fig. 3.46 were used in the analysis. 3.9.4 Earthquake Input The first 20 seconds of the 1971 San Fernando earthquake recorded at Griffith Park Observatory, Los Angels California was scaled to a peak acceleration of 0.5g and used as the input base motion. This record is shown in Fig. 3.47 3.9.5 Results of Analysis Fig. 3.48 shows the comparison of bending moments calculated using the PILE-PY and Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 92 1 . 2 0 . 0 ~ | 1—i i i 11 I I 1—i i i 11 i i | 1—i M i n i 1 — i i i 1111| 1 — i i i 1111 0 . 0 0 0 1 0 . 0 0 1 0 . 0 1 0 . 1 1 1 0 Shear Strain (%) Fig. 3.46 Variation of Shear Modulus and Damping Ratio with Shear Strain - 0 . 6 - 1 — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — I 0 5 1 0 1 5 2 0 Time (sec) Fig. 3.47 Input Acceleration Time History Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 93 PELE3D models. The agreement between the two is very poor. The moments predicted by the PILE-PY are greater than those by the PILE3D except between 3.5m and 4.5m. The maximum moment predicted by the PILE-PY model is 50% lower than that by the PILE3D model. However, the maximum moment in both models occurs near the pile head. Fig. 3.49 shows the comparison of shear force profiles. The agreement between shear force profiles is also very poor. The maximum shear force by the PILE-PY is 49% lower than that by the PILE3D model. However, in both models, the maximum response occurs near the pile head. The shear forces predicted by the PILE-PY is lower than those by the PILE3D up to 2.2m and beyond which it is greater except between 4.5m and 5.5m. 3.10 DISCUSSION OF RESULTS 3.10.1 Single Pile in Sandy Soil PILE-PY along with FREEFLD and PILE3D were used to simulate the centrifuge test on a single pile in sandy soil under strong and low level of shaking. The computed response of the free field using both programs PILE3D and FREEFLD agreed very well with the measured response. The single pile response was also successfully simulated by PILE3D. However, the response predicted by PILE-PY did not agree well with the measured pile response. The acceleration of the superstructure was over predicted by 32% and the maximum moment in the pile was over estimated by approximately 63%. The reasons for the disagreement can be attributed mainly to the use of API(1991) recommended p-y curves and to the problems associated with the Winkler model and the model parameters. It can also be noted that the API(1991) curves were estimated on the basis of slow cyclic loading and may not be suitable for higher frequency seismic loading. It was found that the response of the pile is sensitive to the initial stiffness of the p-y curve of the form recommended by API(1991). The initial stiffness, kh=15000 kN/m 3 recommended by API(1991) appeared to be too stiff. Gohl(1991) suggested a reduced stiffness of kh=2500 kN/m 3 by attempting to match the experimental p-y loops with the API p-y curve. This reduced stiffness lead Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 94 Fig. 3.49 Comparison of Shear Force Profiles Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 95 to improvement in the results. In this case the difference between the maximum moment was 16% only. However, the comparison of maximum bending moment profiles along the pile showed that the improvement in the results are not adequate. This suggests that the current shape of the API(1991) curve may not be appropriate to capture the nonlinear seismic response of a single pile. In Winkler model of a single pile, Winkler type springs and dashpots are used to model the near field soil behavior. These springs and dashpots are attached to the pile at discrete points along the piles and their behavior are assumed to be independent of one another. In other words, the behaviour of one springs is unaffected by the behaviour of other springs. This is not true since the soil is a continuous medium and its behaviour is also continuous. Also the nonlinear hysteretic behaviour of the soil is modeled using parameters that are not fundamental properties of the soil. In PILE3D which gave good predictions, the nonlinear hysteretic behaviour of the soil is modeled using fundamental properties of soil and the soil is also treated as continuum. The API(1993) recommendations for generating p-y curves were originally based on some field tests and they may not be appropriate for all type of sand over a wide range of densities and friction angles. Murchison and ONeill(1984) carried out an extensive study to evaluate the p-y relationships in cohesionless soils. They used four different shape of p-y curves including the one recommended by API(1993). Their study included 24 full scale field tests on cohesionless oils; 14 static and 10 cyclic. The site conditions varied from very loose clayey sand to very dense sands. Their study found out that the API(1993) recommended p-y curves failed to give any good predictions and the error in the predictions were high. However, the p-y curves recommended by API(1993) was the best among the four. Murchison and 0'Neill(1984) concluded that the recommendations made by API(1993) to generate the p-y curves for sands are not adequate even for static and slow cyclic loading. They said " ...it is likely that other factors, not included in the p-y curve formulation are operative and this observation suggests that further study into the fundamental mechanisms of lateral pile soil interaction in cohesionless oils is warranted". Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 3.10.2 Single Pile in Clayey Soil 96 A single pile in a soft clay was analyzed using PILE3D and PILE-PY. API(1993) recommended p-y curves for soft clay were used in the PILE-PY analysis. The PILE-PY response did not agree with the PILE3D response/ The reason for the difference between the two model response can again be attributed to the inappropriateness of the API(1993) recommended p-y curves and the Winkler model it self. Gazioglu and O'Neill (1984) carried out an extensive study to evaluate the p-y relationships in cohesive soils. Their data base included 30 full scale tests on clayey soils; 21 static and 9 cyclic. The soil conditions varies from very soft clay to very stiff clay. They evaluated three different methods including the one recommended in API(1993) for soft clay. Their analysis showed that the API(1993) procedure for generating p-y curve for soft clay failed to give any good predictions and the error in the predictions are high even for static and slow cyclic loading. They concluded " ...the confidence in predicting deflections and moments ....is unfortunately rather poor. Suggesting that further research into the fundamental mechanism of lateral pile-soil interaction is warranted". 3.11 CONCLUSIONS A method was presented to obtain the nonlinear dynamic response of a single pile using a Winkler model and p-y curves. Secant stiffness and tangent stiffness methods were incorporated to capture the non linear response. A comparison study using the secant stiffness and tangent stiffness approaches under two different peak base accelerations showed that both approaches would give the similar results regardless of the different levels nonlinearity experienced by the pile under strong shaking in the absence of any residual displacements of pile. The following can be concluded from the analyses of a centrifuge test of a single pile in sandy Chapter 3 : Modeling of a Single Pile Using Winkler Model and p-y Curves 97 soil. The computed response of the free field using both programs PILE3D and FREEFLD agreed very well with the measured response. The single pile response under strong shaking was also successfully simulated by PILE3D. However, the strong motion response predicted by PILE-PY did not agree well with the measured pile response. The reasons for the disagreement can be attributed mainly to the use of API(1993) recommended p-y curves and to the problems associated with the Winkler model and the model parameters. The PILE-PY response under low level shaking agree very well with the measured response. The results of the analyses under both strong and weak shaking suggest that the API(1993) p-y curves for sand may not be appropriate for seismic loading except for very low level of shaking. The seismic response appears to be dependent on the slope of the initial part of the p-y curve. However, the adjustment of slope alone did not yield good results suggesting that the shape of the curve may be inappropriate for the seismic analysis. A comparison study of a seismic response of a single pile in soft clay was carried out using PILE3D and PILE-PY. The response computed by PILE-PY did not agree with the response predicted by PILE3D. The reasons for the difference between the two model response can again be attributed to the problems associated with the procedures used to generate the p-y curves for soft clay and the Winkler model. As in the case of sand, the procedures recommended by API(1993) may not be appropriate for soft clay also under seismic loading. The API (1993) recommended p-y curves were developed for driven piles. The piles in the centrifuge tests can not be considered as driven piles. Thus, there is a somewhat different environment around the test piles and it may have some influence on the results described above. However, it should be noted that there is a very good agreement under elastic conditions and very poor agreement under strong shaking. Hence it does not appear that the fact that the field test piles were driven has not much effect on the conclusion above. CHAPTER 4 Modeling of Bridges for Dynamic Analysis 99 CHAPTER 4: MODELING OF BRIDGES FOR DYNAMIC ANALYSIS The dynamic behaviour of bridges subjected to earthquake shaking has received considerable attention since 1971 San Fernando earthquake. The result of these research efforts has been a substantial increase in understanding of the factors that control the performance of the bridge structures during an earthquake. Based on this knowledge, the seismic design and retrofitting procedures have been updated substantially in the recent years. However, earthquakes continue to cause severe damages to bridges showing a clear need for further improvement in analytical and design procedures. In the 1989 Loma Prieta and 1995 Kobe earthquakes, severe damage to bridges was caused by foundation failures. This clearly showed that the bridge foundations have an important role to play in the overall seismic performance of a bridge structure. Recognizing this important fact, many researchers insist on the importance of including the foundations in the structural model of the bridge. Many design codes including AASHTO-83, ATC-6 and CalTrans suggest the use of a set of single valued discrete springs to represent the effect of foundations in the bridge model. In design practice, the stiffnesses of these springs have been usually selected on the basis of simple empirical rules or simplified procedures. Better insight into the seismic behaviour of bridge structures is required to evaluate these methods and to address any deficiencies in the current procedures. Bridges are usually founded on piles. The seismic behaviour of the pile foundations is very complex because of the dynamic soil-pile-structure interaction and the non-linear inelastic behaviour of the soil. Some of the significant developments in the modeling of pile foundations for dynamic response analysis were discussed in detail in Chapter 2 and 3. Various aspects of the dynamic response of pile foundations were also studied in these chapters. This chapter is devoted in studying the current methods used for modeling the foundations for dynamic response analysis of bridges and identifying limitations. Chapter 4 : Modeling of Bridges for Dynamic Analysis 100 4.1 REVIEW OF METHODS USED FOR THE DYNAMIC ANALYSIS OF BRIDGES Seismic motions mostly accelerations have been recorded on several bridges during an earthquake. These records have been utilized in many strong motion studies. Ambient vibration tests and forced vibration studies have also been carried out to study the dynamic characteristics of bridges during low and moderate level shaking. These strong motion and ambient and forced vibration data have been used in system identification studies to back calculate the stiffness parameters of the foundations during low and high levels of shaking. These studies have contributed significantly to the understanding of the behaviour of the bridges under earthquake loading. Some significant past studies are briefly reviewed in the following sections. These studies are grouped into the following five categories. 1. Stick Model Studies 2. System Identification Studies 3. Single Bent Model Studies 4. Rigid Body Model Studies 5. Detailed Finite Element Model Studies 4.1.1 Stick Model Studies Many of the researchers, whose main concern is the performance of the foundations, have used a stick model of the superstructure in their studies. In almost all of these studies, a set of single valued discrete springs without any dampers was used to represent the bridge foundations. However, the methods chosen to calculate the values of these springs varied. Different foundation models were used to compute the stiffnesses of these springs. A simple analytical model was developed by Wilson (1988) to describe the stiffness of non-skew monolithic highway bridge abutments for the seismic analysis of bridges. The model allows Chapter 4 : Modeling of Bridges for Dynamic Analysis 101 an evaluation of six equivalent discrete spring stiffnesses for the three translational and three rotational degrees of freedoms of the abutment. This model includes the effect of abutment walls, pile foundations and soil. These springs were incorporated at the foundations nodes on the stick model of the bridge. The stick model with the boundary springs is shown in Fig. 4.1 . The translational and rotational springs represent the combined effect of the abutment wall and the pile foundations. The individual elastic translational stiffnesses of end wall, wing walls and pile foundations were calculated first. Then the equivalent combined translational and rotational stiffnesses of the whole abutment were calculated by considering the force and moment equilibrium of the abutment. In this study, the end wall, wing walls and the pile caps were assumed as rigid. The vertical stiffness of rigid rectangular footing was taken as k z - Vertical Stiffness of footing E - Young's modulus of soil - Poisson's ratio a - Longitudinal dimension of rectangular footing I - An influence factor Linear elastic theory was used in deriving this expression (Poulos and Davis, 1974). The same formula was used to calculate horizontal translational stiffness of the end wall and wing walls The rotational stiffness of the walls and footing was obtained by using the computed translational stiffness in the following manner. K = Ea ( l -v 2 ) I (4.1) (4.2) 12 Translational stiffness of the wall Rotational stiffness of the wall Fig. 4.1 Stick model with the boundary springs used by Wilson (1988) Chapter 4 : Modeling of Bridges for Dynamic Analysis 103 L - length of the wall This study revealed that the abutment stiffness may have a significant influence on the overall dynamic performance of the bridge structure. The parametric study of the abutment stiffness indicated that the abutment systems tend to have greater stiffness in the vertical direction than in the horizontal direction. In contrast, the bridge decks have a greater stiffness in the transverse direction than in the vertical direction. This creates a situation in which displacement of the bridge model in the transverse direction is dominated by the displacement of the abutments and comparatively little bending deflection will occur in the bridge deck. In the vertical direction, the bending of the bridge deck will be the major contributor to the system displacement and the displacements of the abutments will be small. Though this study shed some light on the behaviour of abutments it had several drawbacks. Linear elastic theories were used in developing the model of the abutment. Thus, it is only useful in studying the bridge behaviour under low levels of shaking. Even in the elastic model, the horizontal stiffness of a rigid wall was considered same as the vertical stiffness which is not correct. Furthermore, the coupling effects between the translation and rotation were disregarded. Therefore a better method is required to calculate the stiffnesses of the walls and piles even for an elastic analysis. The damping behaviour of the foundations was not considered in this study. Mc Callen and Romstad (1994) studied the dynamic response of the Painter Street Overpass located in Northern California. This overpass is a short continuous two span box girder bridge. The abutments and the two column bents of this bridge are severely skewed and supported on pile foundations. This overpass was instrumented and shaken by several earthquakes for which records are available. It is interesting to note that this bridge was not severely damaged by any of these earthquakes even though the measured acceleration during the 1992 main shock reached 0.5 g at the free field and 1.0 g at the bridge deck. Eigen analysis and time history response analyses were carried using a stick model of the bridge. In this stick model, the abutment and the foundation flexibility were included in the form of discrete single valued springs. Only translational springs were used and hinge conditions were Chapter 4 : Modeling of Bridges for Dynamic Analysis 104 assumed at the supports. The spring stiffnesses of the pile foundations and the abutment were calculated following the CalTrans Design Guide recommendations. The stick model representation of the entire bridge system is shown in Fig. 4.2 The transverse mode frequency of this stick model was closer to the frequency observed by Romstad and Maroney (1990) who analyzed the response of the bridge using the past six earthquake records and estimated the natural frequencies of the system. However, when attempts were made to predict the time history response of the bridge system during a recorded event, it was found that high levels of modal damping had to be utilized in this type of linear elastic stick model to accurately capture the seismic response of bridge. The required modal damping were as high as 30 % in some modes. This bridge model used by Mc Callen and Romstad (1994) for predicting the strong motion response of a bridge, has many deficiencies. Elastic stiffnesses were not used in this study recognizing that the stiffness reduces during strong shaking. The reduced stiffness values were chosen by simply following the CalTrans Design Guide (1989). It can be noted here that the guide specifications are for design purposes and may be highly conservative. No fundamental analysis of foundation that considers the complex soil-pile-structural interactions was carried out to select values for the foundation springs by Mc Callen and Romstad (1994). During an earthquake, the stiffness of the foundation may vary because of the non-linear behaviour of the foundation soil. However, no attempts were made in this study to change the stiffness values depending upon the amplitude of shaking. Instead, reduced constant values were used for the whole duration of shaking. It is also important to note here that rotational or coupling springs were not used in the model and hinge conditions were assumed. During strong shaking, the damping characteristics of the foundation system may play an important role because of the non-linear hysteretic behaviour of the soil. The hysteretic damping may be substantial during strong shaking because of the development of large strains. However, the bridge model used by Mc Callen and Romstad (1994) failed to include the damping of the foundation. This may have been one of the reasons why high modal damping was required to reproduce the seismic response of bridge. Chapter 4 : Modeling of Bridges for Dynamic Analysis 105 T ongihiHinfll Fmhanlrmwit Soil Springs K= 4650 k/in V Pile Spring K=600 k/in Transverse/Lcmgitndinal Pile Springs K - 800 k/in •««. Transverse Embankment Soil Springs K=4860 k/in Fig. 4.2 Details ofthe Painter Street Overpass and the stick model with foundation springs used by Mc Callen and Romstad(1994) Chapter 4 : Modeling of Bridges for Dynamic Analysis 106 Wilson and Tan(1990) used a stick model to analyze the seismic response of the Meloland Road Overcrossing. This overcrossing is located in the high seismic area of California. This is a non-skew, two span box girder overpass supported on a single column and two monolithic abutments. Both the abutments and the center columns are supported on timber piles. In the stick model of the bridge, only linear translational springs were used to represent the transverse and vertical flexibility of the abutments. At the center bent foundation, only linear rotational springs were used to represent the flexibility of the pile foundations. The stick model of the bridge is shown in Fig. 4.3. In contrast to McCallen and Romstad's (1994) approach, in which the spring stiffnesses were calculated following the CalTrans Design Guide (1989), a more rational approach was followed to calculate the abutment stiffnesses. This approach is described in detail in Chapter 1". The rotational stiffness of the center pier was calculated using the techniques proposed by the Poulos and Davis (1974). Wilson and Tan(1990) first carried out a time history response analyses of the abutment soil system of the Meloland Over Pass using the 1979 Imperial Valley Earthquake records and a plane strain finite element model of the abutment. The computed response was compared with the measured response . Since only linear elastic analysis was carried out, good agreement between the two responses were obtained when the shear modulus of the soil was reduced by a factor of three and the damping was increased to 25%. These adjustments in the soil properties were made to capture the nonlinear hysteretic behaviour of the soil by an equivalent linear methods. The reduced value of shear modulus was later used to calculate the transverse and vertical stiffness of abutment. An eigen value analysis of the whole bridge structure was then carried out to identify the fundamental modes and the modal frequencies. The calculated modal frequencies and the modes were in good agreement with system identification studies carried out by the same authors. Chapter 4 : Modeling of Bridges for Dynamic Analysis 107 208 (63.4 m). I04'(3l.7m)- I04'(3l.7m)-I7'(5.2m) Abul. I Bent 2 ELEVATION PLAN Abut. 3 TYPICAL SECTION 6 7 8 9 Fig. 4.3a Details of Meloland Overpass and the stick model with foundation springs used by Wilson and Tan(l 990) Fig. 4.3b Finite element model of the Meloland Overpass Abutment used by Wilson and Tan(1990) Chapter 4 : Modeling of Bridges for Dynamic Analysis 108 Wilson and Tan's (1990) study on the Meloland Overpass provided a new approach for modeling the abutment system for vibration analysis in the transverse and vertical direction. However, even in this study, only singled valued linear springs were used and linear elastic analysis was carried out. To simulate the non-linear hysteretic behaviour of the soil under strong shaking, the shear modulus and damping of the soil were multiplied by factors to reduce the stiffness and to increase the damping. The selection of appropriate value of these factors are very important to arrive at a meaningful results when these types of simple linear models are used. It can be noted here that, in Wilson and Tan's( 1990) study, the results from the system identification study provided guidance in the selection of these factors. In practice, a design engineer does not have this kind of information. Chen and Penzien (1977) presented a detailed model for studying the behaviour of the short skewed highway bridges interacting with their supporting soils during earthquakes. Their mathematical model consists of (1) three dimensional solid finite elements representing the backfills and abutment walls, (2) linear elastic prismatic beam elements representing the bridge deck and the pier columns, (3) non-linear friction elements representing the discontinuous behaviour of separation impact and slippage at the interfaces between back fills and abutment walls and (4) discrete translational and rotational linear springs representing the foundation flexibility at the base of the supporting elements. Non-linear time history analysis was carried out to study the seismic response of bridges. The model used in this study is shown in Fig. 4.4. The abutment-soil system was modeled using a bi-linear stress strain law following the simple Mohr Coulomb yield criterion. The abutment did not have any pile foundation. The bridge deck and the abutment system were analyzed as a fully coupled system. Very coarse finite elements were used to model the abutment in order to reduce the computational effort and the computer storage. The pier foundation was modeled by a set of linear elastic springs and damping was ignored. This study provided some valuable insight into the non-linear behaviour of the bridge structure. The study emphasized that the foundation flexibility and abutments have significant influence on the overall response of the bridge structure and should be included in the bridge model. Chapter 4 : Modeling of Bridges for Dynamic Analysis 109 Chapter 4 : Modeling of Bridges for Dynamic Analysis 4.1.2 System Identification Studies 110 System identification has been used to determine the foundation stiffness and damping parameters of bridges for which recorded motions are available. Such data is useful for future design and checking the validity of the computational models. In system identification, model parameters such as the stiffness and damping are identified by fitting calculated response to the measured data by adjusting these parameters. A suitable optimization scheme is used in the identification process. The input data to this system identification process are usually obtained from ambient vibration tests or strong motion records. Sometimes, forced vibration tests such as the quick release ram test are carried out to provide the necessary data. The physical model employed in this process is normally a linear stick model with discrete springs representing the foundation flexibility. Meloland Overcrossing and Painter Street Overcrossing have been the subject of many system identification studies. Both were instrumented and experienced several earthquakes, for which records are available. Ambient vibration studies were carried out in both bridges by many researchers. A forced vibration study was also carried out in the Meloland Overpass. Several researchers including Gates and Smith (1983), Gates (1993), Wilson and Boon (1990), Crouse and Price (1993), Vrontinos et al (1993), Werner et al.(1993), Werner et al. (1993), Werner et al. (1987) studied the modal characteristics of these bridges and the accuracy of some proposed bridge models using the system identification process. They used ambient vibration data, forced vibration data and strong motion data in their analyses along with the several optimization algorithms. Their studies showed that the system identification process is an excellent tool in determining the model parameters such as the foundation stiffnesses and flexural and torsional rigidity of the bridge superstructure members. A better understanding of the dynamic behaviour of a bridge during both low level shaking and strong shaking resulted from these studies. Some of the conclusions are described briefly below. Werner, Beck and Levine (1987) from their study of the Meloland Bridge concluded that the Chapter 4 : Modeling of Bridges for Dynamic Analysis 111 abutments and soil embankments had a significant influence on the global bridge superstructure response. Wilson and Tan (1990) performed a system identification study on the Meloland Bridge using the records from 1979 Imperial Valley Earthquake. They used system identification techniques on windowed segments of the response records, and concluded that the frequency of vibration of abutments system tend to decrease during the strong portion of the earthquake ground motion, and subsequently increase to near the initial value after the strong portion of the ground motion subsided. The frequency during the first 0-4 seconds window was 2.52Hz seconds which was close to its elastic frequency of 2.56Hz and, in the next window 4-8 seconds during the stronger shaking period, it dropped to 1.39 Hz. In the next window 8-12 seconds, it increased to2.08 Hz. Wilson and Tan(1990) attributed the frequency shift to the softening of the embankment soil and they estimated that the frequency shift corresponded to a 50% reduction in the soil shear modulus. The authors also stated that the superstructure behaved linearly and the system non-linearities were confined to the soil foundation region. Werner, Beck and Nisar(1990) performed system identification studies on the Meloland bridge using the data generated by low amplitude, quick-release hydraulic ram excitation. Comparison of mode shapes determined from the low amplitude response with the mode shapes obtained from the strong motion data, indicated a significant difference in the mode shapes of the two data. The frequency of the fundamental transverse mode obtained from the low amplitude response was approximately 35% higher than the frequency obtained from the strong motion data. The amplitude dependent behaviour, and apparent stiffness reduction of the bridge system, indicated significant non-linear behaviour occurred during earthquake shaking. Gates (1990) carried out a similar study on the Meloland Overcrossing using both ambient vibration data and strong motion data. He also concluded in his study that the bridge exhibits significantly different response under ambient loading and under strong earthquake loading. The first few modes and the modal frequencies identified using the ambient vibration data and the 1979 Imperial valley earthquake data are presented in Table 4.1. This table shows that for all modes that could be measured, the frequency decreased with the increase in force level indicating a decrease in Chapter 4 : Modeling of Bridges for Dynamic Analysis 112 the stiffness. The primary cause of this difference was attributed to the non-linear behaviour of the soil at the abutment and at the foundations. Table 4.1 Comparison Frequencies from ambient data and strong motion data Mode Frequency (Hz) Period Change (%) Ambient Data Earthquake Data First Vertical 3.52 2.78 -21.0 Second Vertical 4.88 4.39 -10.0 First Transverse 3.42 2.49 -27.2 Second Transverse 4.32 5.03 31.3 First Longitudinal 4.98 2.69 -46.0 Werner et al. (1994) investigated the implications of the findings from system identification studies of the Meloland Bridge for the analysis of short bridges. System identification techniques were applied to identify parameters for stick type finite element model of the bridge. One of the important conclusions of this study was that high modal damping were necessary to adequately replicate the response of the bridge if a simple linear model is used. The stiffness of the abutment identified by these authors using strong motion data were only about one-fourth the stiffness obtained from low amplitude vibration data. Thus, the reduction in stiffness during the strong motion due to the non-linear behaviour of the soil was proved again. 4.1.3 Single Bent Model Studies An alternative method of modeling the bridge is to consider each bent of the bridge as a separate system. This idealization is very convenient for long bridges with multiple supports when the supports exhibit varying stiffness and damping characteristics and experience different support motions during an earthquake. This type of single bent model is also widely used by researchers and practicing engineers because of the following two main reasons. The model is relatively simple which reduces the computational effort significantly and allows a more detailed model for the foundation to be incorporated into the analysis. This type of model has been used to study the effect Chapter 4 : Modeling of Bridges for Dynamic Analysis 113 of soil-pile-superstructure interactions and the non-linear behaviour of the soil on the bridge response. Some of these studies are reviewed here in some detail. Makris et al. (1994) used a single bent model to study the seismic response of the Painter Street Overpass. This overpass is a continuous two span, cast-in-place, prestressed post tensioned box girder bridge. The abutments and the two columns bent are severely skewed and founded on groups of concrete piles. The span containing the center bent of the model considered in this study was modeled as six degrees of freedom lumped parameter system. The model is shown in Fig. 4.5 Each column of the center bent is supported on a rectangular pile group consisting 20 (5x4) piles. This foundation system was represented in the model by a linear lumped parameter system. The foundation model used by Makris et al.(1994) is superior to other foundation models in many aspects. Normally in the model of the foundation, only discrete springs are used to represent the stiffness of the foundation. The damping of the foundation and the cross-coupling effect between translation and rotation are not taken into account. In the foundation model used by Makris et al.(1994) dashpots were used to represent the damping of the foundation and the cross-coupling effect were also considered. Furthermore, the group effect was modeled using dynamic interaction factors and the frequency dependence of stiffness and damping were taken into account although this was based on elastic response. Despite these significant improvements, the main problem in the model was that it was still a purely linear elastic model. To account for the non-linear behaviour of the foundation soil during the strong shaking, the small strain stiffnesses of the single pile, which was found using elastic theory, were reduced by some arbitrary factors before calculating the group stiffnesses. The reduced stiffnesses were two to four times smaller than the small strain stiffnesses. The horizontal and cross-coupling stiffnesses were reduced more than the vertical stiffness because the soil strains near the soil surface that primarily influence the horizontal motions were considered larger than the strains at larger depth on which the vertical stiffness depends. Chapter 4 : Modeling of Bridges for Dynamic Analysis 114 Fig. 4.5 Single bent model of the Painter Street Overpass used by Makris et al., 1994) Chapter 4 : Modeling of Bridges for Dynamic Analysis 115 This study concluded that the unrealistic modeling of the foundation drastically affects the prediction of the superstructure response. It also suggested that a non-linear analysis of the problem could be more realistic in situations where the bridge structure is subjected to strong shaking. 4.1.4 Rigid Body Model Studies The in plane motion of a short and medium span bridges during strong shaking have been studied using the rigid body models. In these rigid body models, the bridge deck is usually modeled as a rigid plate or rigid bar. The abutments and piers are usually represented by springs and dashpots in this type of model. Maragakis and Jennings (1987) developed a rigid body model to study the in plane motions of skew bridges. In this model, the bridge deck was modeled as rigid bar. The bridge piers were modeled by bilinear springs and viscous dashpots. The abutment was modeled using bilinear hysteretic springs. In estimating the stiffness of the abutment soil system, a global yield criterion based on the Rankine theory of active stress and passive resistance was used. The discontinuities in the bridge deck were also modeled appropriately using springs and dashpots. This model was shown to be capable of capturing the rigid body motions of the deck especially the rotations arising from the skewness of the deck and the impact between the deck and abutment. Such impact is possible in case of seated abutments. 4.1.5 Detailed Finite Element Model Studies - Coupled and Uncoupled Analysis The dynamic interactions that influence the earthquake response of bridge structures are not only the soil-pile interaction but also the foundation-superstructure interaction. In current engineering practice, the soil-pile-superstructure interactions are accounted for by equivalent linear spring coefficients. Usually these spring coefficients are determined by analyzing foundation Chapter 4 : Modeling of Bridges for Dynamic Analysis 116 separately without any superstructure. This type of uncoupled analysis is correct if the foundation exhibits linear elastic behaviour. The implications of this uncoupled approach in strong motion analysis is not very clear and still being researched. The ideal approach for strong motion analysis is a fully coupled nonlinear analysis in which the foundation and the superstructure are analyzed together. A nonlinear three dimensional finite element method may be used in such coupled analysis. A full 3D finite element analysis of the coupled structure-foundation system, though ideal, is not generally feasible. Very large computational effort is required that is impractical most of the time especially in the case of long bridges with multiple supports. It can be noted here that in such analyses, each individual foundation of the bridge need to be modeled in detail to capture the complex soil-pile and group interactions during the strong shaking. The time history analyses will also be very time consuming and costly because of the nonlinear nature and size of the problem. The limited computational and storage facilities available in the personal computers and the cost involved prevents the routine use of such large scale rigorous analyses. A fully coupled analysis of the Painter Street Overpass using a large scale 3D finite element model was carried out at Lawrence Livermore Laboratory by Mc Callen and Romstad (1994) using a 3D non-linear finite element program called NIKE3D. This detailed finite element model of the Painter Street Overpass (LL Model) is shown in Fig. 4.6 Shell elements were used in this model to represent the box girder deck, beam elements represented the center pier columns and 3D solid elements were used for the center bent beam and the back walls. The individual piles below the abutment pile caps and center pier footing were modeled with beam elements and the original grade and embankment fills were modeled with 3D solid elements. The non-linear hysteretic behaviour of the soil was modeled using a simple Romberg-Osgood elasto-plastic model. In addition to the hysteretic damping of the soil that is automatically accounted for by the nonlinear stress strain law, some mass and stiffness proportional damping was also used. The L L model yielded good approximations to both the frequency content and amplitude of the bridge response, when time history analyses were carried out using the measured records. Chapter 4 : Modeling of Bridges for Dynamic Analysis 117 Baseline Model Soil Properties 2 5 Embankment Soil G m a x = 11.000 psi Y-100 lb/ft3 777 Center Bent Soil 0 ^ = 20.000 psi y-110 lb/ft3 V \ 77^ Original Grade Soil G m a x = 22,000 psi Y-120 lb/ft3 Boundary Original Grade Elevation Fixed Boundary Fixed Boundary Fig. 4.6 Detail finite element model of Painter Street Overpass used by Mc Callen and Romstad(1994) (LL model) Chapter 4 : Modeling of Bridges for Dynamic Analysis 118 This study concluded that a bridge of this type exhibits highly non-linear behaviour even when the superstructure remains essentially elastic. The overall system natural frequencies are sensitive to both the stiffness and inertia of the soil embankments. The soil plays a more significant role in the response during strong shaking than it would for low levels of shaking. This agrees with the findings by other researchers cited earlier. Comparison studies were also made between this L L model and a stick model. First, a natural vibration characteristics of the superstructure assuming fixed bases for the supports were calculated by both models without including the foundations or abutments. The identified modes and the modal frequencies are listed in Table 4.2. Table 4.2 Comparison Modal Frequencies between Stick Model and L L Model with Fixed Bases Mode Frequency (Hz) Discrepancy in % Stick Model Detailed Model First Vertical 2.74 2.93 6.5 Second Vertical 4.14 4.56 9.2 First Transverse 4.21 4.77 11.7 First Torsional 5.21 5.13 1.6 Second Torsional 6.66 6.32 5.4 First Longitudinal 8.27 7.95 4.0 This table shows that the stick model on a fixed base did not miss any important modes and it could capture even the torsional modes of the bridge. The modal frequencies were also very close with a maximum discrepancy of 11.7 %. This shows that the stick model may be adequate for capturing the essential dynamic characteristics of a bridge structure. Comparative analyses were then conducted including the foundations and abutments . In stick model, the abutment and foundations were represented by linear translational springs and the stiffnesses of these springs were computed following the CalTrans Design Guide (1989). The Chapter 4 : Modeling of Bridges for Dynamic Analysis 119 identified modes and the modal frequencies are given in Table 4.3. Table 4.3 Comparison of Modal Frequencies between Stick Model and Detail Model Mode Frequency (Hz) Discrepancy in % Stick Model Detail Model First Vertical 2.69 3.06 12.1 First Transverse 3.30 4.29 23.1 Second Vertical 4.00 4.45 10.1 First Longitudinal 4.72 5.28 10.6 First Torsional 5.24 5.03 4.2 Second Torsional 6.61 5.53 19.5 In this case also, the stick model captured all the important modes. However, the discrepancies between the modal frequencies were as high as 23%. This high discrepancy can be attributed to the following short comings of the stick model used in this study. First, no theoretical analysis of the foundations was performed to guide the selection of the spring stiffnesses. The stiffnesses were selected by simply following the CalTrans Design Guide and may not have been appropriate for the study. No rotational springs or cross-coupling springs were used. Although, the damping of the foundations is an important factor during strong shaking it was not included in the foundation model. The time history analyses using strong shaking records revealed that high modal damping as high as 20% were required in stick model analyses to model the measured response adequately. 4.1.6 Code Based Models CalTrans, ATC-6, AASHTO-83 and many other design codes recognize the importance of including the abutment and foundation flexibility in the structural idealization of bridge structures. They suggest that these flexibilities can be included in the form of linear springs. Chapter 4 : Modeling of Bridges for Dynamic Analysis 120 AASHTO-83 and ATC-6 models for the multimode spectral analysis of a relatively simple bridge without any irregularities in the superstructure is described below. Superstructure: The bridge super structure should be modeled as a three dimensional space frame with nodes selected to realistically model the stiffness and inertia effect of the structure. Each node should have six degrees of freedom; three translation and three rotation. Nodes should be at such points as the quarter spans in addition to the nodes at the end of the span. The structural mass should be lumped with at least three translational inertia terms. The mass should take into account structural elements and other relevant loads including but not limited to pier caps, abutments, columns and footing. Other loads such as the live loads may also be included. Substructure: The intermediate columns and piers should also be modeled as space frame members with six degrees of freedom, three translational and three rotational, at each node. Generally, for short stiff columns having length less than one-third of either of the adjacent span lengths, intermediate nodes are not necessary. Long flexible columns should be modeled with intermediate nodes at the one-third points in addition to the joints at the end of the structure. The eccentricity of the columns with respect to the superstructure should be considered in the model. The foundation conditions at the base of the columns and at the abutments can be modeled using equivalent linear spring coefficients. Due to the unavailability of more established theoretical procedures to calculate the abutment and foundation stiffnesses and due to the many complexities, the design codes do not recommend specific guide lines for calculating the stiffnesses of the foundations and abutments. Chapter 4 : Modeling of Bridges for Dynamic Analysis 121 4.2 CASE STUDIES: DYNAMIC BEHAVIOUR OF THE PAINTER STREET OVERPASS UNDER AMBIENT VIBRATION AND EARTHQUAKE SHAKING Two case studies to further our understanding of the dynamic behaviour of the Painter Street Over Pass under ambient vibration and under strong shaking will be presented in detail because they provide a clear picture of the effects of shaking level on the foundation stiffnesses and on the global response of the bridge. 4.2.1 Details of the Painter Street Overpass The Painter Street Overpass, located near Rio Dell in Northern California, is a continuous two span, cast in place, prestressed post tensioned concrete, box girder bridge. It is typical of concrete bridges constructed in 1973 and spans a four lane highway. This bridge is supported on integral abutments at the two ends and on an interior reinforced concrete, two column bent. This bent divides the bridge into two unequal spans of 119 feet and 146 feet. The width of the bridge is 52 feet. Both abutments and the bent are skewed at an angle of 38.9 °. The east abutment which is monolithically cast with the road deck and is supported on 14 driven 45 ton nominal capacity concrete friction piles. The west abutment is a seat type abutment and is founded on 16 driven 45 ton nominal capacity concrete friction piles. At the west abutment, the deck rests on a neoprene bearing strip which is a part of a design thermal expansion joint of the road deck. The center bent consists of two reinforced concrete columns, each column is supported on a group of 20 driven concrete piles. A sketch of the bridge is shown in Fig. 4.7 Ambient vibration tests were carried out by researchers from the University of British Columbia(UBC)on the Painter Street Overpass in 1994 (Finn and Wu, 1994 ). The data from these tests and the results of an earlier test by Gates and Smith in 1982 were used to investigate the dynamic behaviour of the bridge under low level shaking. A linear elastic stick model was used for Chapter 4 : Modeling of Bridges for Dynamic Analysis 122 the superstructure of the bridge and the foundation and the abutments were modeled using the linear elastic springs. The Painter Street Overpass was instrumented under the California Strong Motion Instrumentation Program by the California Division of Mines and Geology in 1977 and is identified as CSMJP station No. 89324. The location of the instruments which are used to measure the accelerations of the structure and the free field in the North/South direction and in the East/West direction are shown in Fig. 4.7. Strong motion instruments triggered during nine earthquakes dating from 1980 to 1992 and ranging in magnitude from 4.4 M L to 6.9 M L . The bridge structure survived during all nine earthquakes without any noticeable damage to the structure. Data from two of the earthquakes listed in Table 4.4 were used in a second study to investigate the behaviour of the bridge, abutments and foundations under earthquake shaking. A rigid body model of the superstructure was used in this study. Table No. 4.4 List of strong motions recorded at Painter Street Overpass No. Earthquake Depth (km) Magnitude M L Distance (km) Max. FreeField Acc.(g) Max. Structural Acc. (g) 1 Cape Mendocino 21 Nov, 1986 (Second Event) 18 5.1 26 0.144 0.35 2 Cape Mendocino /Petrolia 25 Apr, 1992 15 6.9 6.4 0.543 1.089 Chapter 4 : Modeling of Bridges for Dynamic Analysis 123 N-S Elevation Fig. 4.7 Details of the Painter Street Overpass and the Locations of Strong Motion Instruments Chapter 4 : Modeling of Bridges for Dynamic Analysis 124 4.3 CASE STUDY 1: DYNAMIC BEHAVIOUR OF THE BRIDGE UNDER AMBIENT VIBRATION 4.3.1 Introduction Modal analysis was carried out on the Painter Street Overpass to identify the mode shapes and the modal frequencies. These mode shapes and the modal frequencies were compared with the ones obtained from the ambient vibration tests of Ventura et al. (1995) and Gates and Smith (1982). The main objective of this study is to understand the role of foundation in the linear elastic behaviour of the bridge and to investigate the adequacy of available methods to calculate the elastic stiffness of the foundations. .3.2 Mathematical Model of the Bridge The stick model of the bridge superstructure with springs representing the abutment and the foundations used in the analysis is shown in Fig. 4.8. The bridge deck, the center bent columns and the bent cap beam are modeled using 3D frame elements. The actual structural properties were used for the bridge deck and the center bent columns. However, the sectional properties of the bent cap beam was artificially increased to prevent any unrealistic large deflections in the cap beam. The behaviour of all the structural components are assumed linear elastic. The bridge abutments, abutment foundations and the pier foundations are replaced with a set of translational and rotational springs. The stiffnesses of these springs are assumed constant. In the stick model of the bridge shown in Fig. 4.8. 27 three dimensional frame elements were used to model the bridge deck, center bent columns and the cap beams. Translational and rotational springs were used in all three directions; longitudinal, transverse and vertical to account for the flexibility of the abutments and pier foundations. Chapter 4 : Modeling of Bridges for Dynamic Analysis 125 44.62m 36.39 m Details of Spring Model © Fig . 4.8 Stick M o d e l o f the Painter Street Overpass Used in the M o d a l Analysis Chapter 4 : Modeling of Bridges for Dynamic Analysis 4.3.3 Stiffnesses of Springs 126 Fig. 4.9 shows the sketch of the Painter Street Over Pass abutment. At the east abutment, the end wall is supported on a two rows of 7 piles and the west abutment end wall is supported in two rows of 8 piles. Both end walls are not perpendicular the bridge axis but inclined at a skew angle of degrees 38.9°. Both abutments have wing walls and each wing wall is supported on a single pile footing at the end. The stiffness of the spring at the abutment end represents the resultant stiffness of the abutment end wall, wing walls and the pile foundations. The center bent has two piers and each pier is supported on 20(5x4) group piles. The stiff nesses of the springs at the bottom of the piers represent the stiffnesses of the (5x4) pile group foundation. 4.3.4 Calculation of Resultant Abutment Stiffness The elastic stiffness of the abutment end wall and the end wall pile foundations were calculated in the vertical and two horizontal directions; one parallel to the end wall which is inclined 38.9° degrees to the transverse direction and the other perpendicular to it. Then, the stiffnesses in the longitudinal and transverse direction were obtained by coordinate transformation. After that, by appropriately assembling the stiffnesses of the individual components, the resultant stiffnesses of the abutment in the longitudinal and transverse direction were obtained. Chapter 4 : Modeling of Bridges for Dynamic Analysis 127 Fig. 4.9 Sketch of the Painter Street Overpass East Abutment X-Longitudinal Direction X'-Perpendicular to / ' Abutment Envyafl y 3 8 . 9 ° \ ^Y'-Parallelto Y-Transverse Abutment Endwall Direction Fig. 4.10 Coordinate System Used in the Analysis Chapter 4 : Modeling of Bridges for Dynamic Analysis 1 2 8 The resultant (3x3) stiffness matrix of the abutment, K X Y Z in the x,y and z coordinate system can be expressed as K . xyz 0 0 0 0 0 0 K y 0 0 0 0 0 0 K z 0 0 0 0 0 0 K R X 0 0 0 0 0 0 K ry 0 0 0 0 0 0 (4.3) in which K X , K Y and K Z are translational stiffnesses in the x,y and z directions respectively and K „ , and are the rotational stiffnesses in the x,y and z directions respectively. These stiffnesses can be expressed in terms of the stiffnesses of the individual stiffness components as follows. K = X K y = K Z = K R X = % K „ WWPF K X T W + K / R + 2 K X l £ EW + PF + 2\T WWF + WWPF y 1 V y ^ y - ^ y K P F + K E W F + 2 K + 2 K W W P F z y z z K E W + K P F + K E W F + K WWF + K W P F j^ - E W + K PF + K- WWF + K - WWPF ry ry l v r y ^ r y g EW + g P F + K WW + K W W P F rx rz rz rz (4.4) In which K X E W , K Y E W - Translational stiffnesses of end wall in the x and y directions respectively K X P F , K Y P F , K Z P F - Translational stiffnesses of end wall pile foundation in the x,y and z directions respectively K X W W P F , K Y W W P F , K Z W W P F - Translational stiffnesses of wing wall pile foundation in the x,y and z directions respectively K Z E W F - Translational stiffnesses of end wall footing in the z direction Chapter 4 : Modeling of Bridges for Dynamic Analysis 129 K y W F , Kj™* - Translational stiffnesses of wing wall footing in the y and z directions respectively K j / ^ K , ^ , K r y E W - Rotational stiffnesses of end wall about x,y and z directions respectively K r x P F , K n , P F , K , / F - Rotational stiffnesses of end wall pile foundation about x,y and z directions respectively J^WWPF^WWPF^WWPF-Rotational stiffnesses due to the wing wall pile foundation about x,y and z directions respectively K , . X E W F - Rotational stiffnesses of end wall footing about z direction K r x W W F , Kf™p - Rotational stiffnesses of wing wall footing about y and z directions respectively K r 2 w w - Rotational stiffnesses of wing wall about z direction 4.3.5 Transformation of Stiffness The stiffness matrix of the abutment end wall and the pile foundations, K x y z in the x,y and z coordinate system can be obtained from the stiffnesses matrix, K x . y . z of the x',y' and z coordinate system using the transformation Kx y z = [TTXyzHT] (4.5) in which [T], the transformation matrix, can be expressed as cosa since 0 0 0 0 -since cosa 0 0 0 0 0 0 1 0 0 0 0 0 0 cosa sina 0 0 0 0 -sina cosa 0 0 0 0 0 0 1 (4.6) Chapter 4 : Modeling of Bridges for Dynamic Analysis 130 in which a is the angle of rotation from xy coordinate system to x'y' coordinate. Fig. 4.10 shows both coordinate systems. 4.3.6 Resultant Column Foundation Stiffness The resultant column foundation stiffness is the sum of stiffnesses from the (4x5) pile group and the pile foundation footing. As the pile group axes are parallel to x' and y' axes, the resultant stiffnesses were determined in these two directions and they were transformed to the x and y directions which are parallel to longitudinal and transverse directions respectively. Determination Pile Foundation Stiffnesses: The translational and rotational stiffnesses in the two horizontal directions and the vertical stiffness of pile foundation stiffnesses were determined using the Quasi-3D finite element computer program PILE3D (Wu and Finn, 1997a). The rotational stiffness of the abutment about the vertical z axis was determined using the following formula (Wilson, 1988). K T V B 2 in which is the rotational stiffness about vertical axis and is the translational stiffness of the abutment in the direction perpendicular to the abutment end wall and B is the width of the abutment. A l l the piles in the group are the same. The unit weight of the pile is 25.5 kN/m 3 and the Young's modulus is 22000 MPa. The diameter, cross sectional area and the second moment of area of the pile are 0.36m, 0.102m2 and 0.008244m4 respectively. The soil properties underneath the columns foundations were obtained from Lawrence Livermoore Laboratory(1994). The shear Chapter 4 : Modeling of Bridges for Dynamic Analysis 131 modulus profile underneath the column is shown in Fig. 4.11 At both abutments, the soil properties reported by Mc Callen and Romstad (1994) were used as they were not available from Lawrence Livermoore Laboratory (1994). Determination of End Wall and Wing Wall Stiffnesses: The wall stiffnesses due to pure translation and rotation can be determined using the following expressions recommended by Lam and Martin (1986). K T = 0.425 E s B K R = 0.072 E s B H 2 ( 4 8 ) in which K T and K R are translational and rotational stiffness of the wall, E s is the Young's modulus of the soil and B and H are the width and height of the wall respectively. The end wall and the wing wall stiffnesses were calculated using the expressions given in Eq. 4.8. Determination of Footing Stiffness: The vertical stiffnesses of a rectangular footing can be expressed as follows using the formula proposed by K E L v = 77""^ 7 <4-9) ( l - v z ) I in which K v is the vertical stiffness of the footing , E s is the Young's modulus of the soil, v is the poisson's ratio, L is the length of the rectangular footing and I is an influence factor. The pile cap was assumed to be not embedded and the frictional force at the base of the pile cap was neglected. The calculated individual pile foundation, walls and footing stiffnesses are summarized in Table 4.5, 4.6 and 4.7 respectively. Chapter 4 : Modeling of Bridges for Dynamic Analysis 132 Shear Modulus (MPa) 400 12 Fig. 4.11 Measured Shear Modulus Profile Underneath the Center Pier Chapter 4 : Modeling of Bridges for Dynamic Analysis 133 Table 4.5 Pile Foundation Stiffness Stiffness East Abutment West Abutment Center Pier Wing Wall K x (MN/m) 2510 2730 935 179 K y (MN/m) 2355 2550 921 179 K , (MN/m) 6447 6950 530 550 K„ (MN/rad) 112210 121090 11517 (MN/rad) 171960 185670 10220 (MN/rad) 91570 100200 1601 Table 4.6 Wall Stiffness Stiffness East Abutment West Abutment End Wall K x (MN/m) 701 701 K y (MN/m) 1076 1076 K„ (MN/rad) 1331 1590 (MN/rad) 2044 2442 Wing Wall K y (MN/m) 691 691 (MN/rad) 603 655 Table 4.7 Footing Stiffness Component Stiffness (MN/m) End wall Footing 1670 Wing Wall Footing 191 Pier Footing 630 Chapter 4 : Modeling of Bridges for Dynamic Analysis 134 Table 4.8 summarizes the resultant stiffnesses of the abutments and the center pier foundations. Table 4.8 Resultant Stiffness of the Abutments and the Center Pier Foundation Stiffness East Abutment West Abutment Center Pier K x (MN/m) 3570 3790 930 K y (MN/m) 4350 4550 920 K , (MN/m) 9590 10100 5930 K„ (MN/rad) 242500 251800 97370 (MN/rad) 329000 343400 104430 (MN/rad) 253300 261900 1600 4.3.7 Modal Frequencies and Mode Shapes 4.3.7.1 Determination of Modal Frequencies and Mode Shapes An eigen analysis was carried out to find out the modal frequencies and the mode shapes. Fig. 4.12 through 4.17 show the first 6 modal frequencies and the nodes shapes. The mode shapes are plotted as an amplitude in the three directions; longitudinal, transverse and vertical The first vertical mode shape is shown in Fig. 4.12. The mode shown in Fig. 4.13 is primarily the second vertical mode. However, there is some movement in the transverse direction also. The mode shown in Fig. 4.14 is primarily the first transverse mode but has some vertical movement. It can clearly be seen from this figure that there is a significant lateral movement of the abutments due to the flexibility of the abutment and its foundation. Chapter 4 : Modeling of Bridges for Dynamic Analysis 1 3 5 1.0 CD •a f 0.5 E < | 0.0 CO E o ^ -0.5 •1.0 - • — Transverse • • • — Longitudinal - • — Vertical I- i- ~m Mode 1 : 3.09 Hz J i I i I i I i L 0 10 20 30 40 50 60 70 80 Distance from East Abutement (m) Fig. 4.12 Computed First Mode Shape and the Modal Frequncy — • — Transverse - — • — - Longitudinal — • — Vertical o 2 -0.5 -'c => i n _ Mode 2: 4.58 Hz 0 20 40 60 80 Distance from East Abutment (m) Fig. 4.13 Computed Second Mode Shape and the Modal Frequncy Chapter 4 : Modeling of Bridges for Dynamic Analysis 136 "O "5. E < TJ CD co E i o c -0.5 -1.0 Mode 3: 5.14 Hz - • — Transverse ••— Longitudinal - • — Vertical J L j L J i L J_ 0 10 20 30 40 50 60 70 Distance from East Abutement (m) 80 Fig. 4.14 Computed Third Mode Shape and the Modal Frequncy CD TJ _=J ~cx E < TJ CD CO 03 E c 1.0 0 » — Transverse i — Longitudinal • — Vertical Mode 4: 8.37 Hz I i L_ 20 40 60 Distance from East Abutment (m) 80 Fig. 4.15 Computed Fourth Mode Shape and the Modal Frequncy Chapter 4 : Modeling of Bridges for Dynamic Analysis 137 1.0 CD f 0.5 E < | 0.0 cc E o -0.5 ZD -1.0 - • — Transverse • • — Longitudinal - • — Vertical Mode 5 : 8.75 Hz 7^ _L 0 10 20 30 40 50 60 70 Distance from East Abutement (m) 80 Fig . 4.16 Computed Fifth Mode Shape and the M o d a l Frequncy 0 20 40 60 80 Distance from East Abutment (m) F i g . 4.17 Computed Sixth Mode Shape and the M o d a l Frequncy Chapter 4: Modeling of Bridges for Dynamic Analysis 138 The mode shown in Fig. 4.15 is also primarily a vertical mode but there is some movement in the longitudinal direction. The mode shown in Fig. 4.16 is primarily the first longitudinal mode but with some vertical movement. This mode also shows some movement at both abutments due to the flexibility of the abutments. The second transverse mode is shown in Fig. 4.17. Table 4.9 summarizes the modal frequencies and the mode shapes. Table 4.9 Computed modal frequencies and mode shapes Mode No Frequency (Hz) Mode Shape 1 3.09 First Vertical 2 4.58 Second Vertical/Torsional 3 5.14 First Transverse 4 8.37 Third vertical/Torsional 5 8.75 First Longitudinal/Torsional 6 9.70 Second Transverse 4.3.7.2 Comparison of Modal Frequencies and Mode Shapes Fig. 4.18 shows a comparison between the computed first vertical mode shape and that by Ventura et al. (1995) and Gates and Smith (1982). There is a very good agreement. The computed modal frequency, 3.09 Hz, is 9% and 14% lower than the frequencies reported by Ventura et al. (1995) and Gates Smith (1982) respectively. The second vertical mode shape comparison shown in Fig. 4.19 also closely agrees with the mode shape measured by Ventura et al. (1995). However, there is a slight difference in the first span of the bridge deck. Gates and Smith (1982) also show a similar trend. The computed frequency, 4.58 Hz is slightly lower than the frequency reported by Ventura et al. (1995) which is 4.92 Hz. Ventura et al. (1995) identified this mode as a torsional mode. Chapter 4 : Modeling of Bridges for Dynamic Analysis 139 The computed and measured first transverse modes are compared in Fig. 4.20. There is a reasonable agreement. The computed mode shape shows less movement at the east abutment than was measured by Ventura et al. (1995). The mode shape by Gates and Smith (1982) shows much greater movement at the abutments than the computed one. These larger movements suggest that the model used in the present study is too stiff at the abutments. The computed frequency, 5.14 Hz is 25% and 14% greater than the frequencies reported by Ventura et al. (1995) and Gates and Smith (1982) respectively. The discrepancy between the measured and computed frequency can be attributed to the uncertainties in the soil properties used in the calculation of abutment springs. Though the properties underneath the columns were available from Lawrence Livermore Laboratory (1994), they were not available at the abutments. The soil properties at the abutments were taken same as those assumed by Mc Callen and Romstad (1994). The second transverse modes are compared in Fig. 4.21. The measured mode shape of Ventura et al. (1995) is similar to the computed mode shape. However, the measured mode shape did not show any movement at the west abutment end. In the computed mode shape, the movements at both ends are nearly the same. The measured model frequency by Ventura et al. (1995) is only 12% lower than the computed frequency which is 9.70 Hz. Despite the uncertainties in the soil properties, the low vibration frequencies and mode shapes agrees reasonably well with the measured frequencies and mode shapes. Chapter 4 : Modeling of Bridges for Dynamic Analysis 140 CD ~o E < TD CD CO 16 E 0.0 o z -0.5 —•— - Computed (— Measured ZD --Ventura et al. (1995) -1.0 — • Gates and Smith (1982) 1 st Transverse Modd J i I i I i L _ 0 10 20 30 40 50 60 70 Distance from East Abutement (m) 80 F i g . 4.18 Comparison o f Measured and Computed First Transverse Mode Shapes F i g . 4.19 Comparison of Measured and Computed Second Transverse Mode Shapes Chapter 4 : Modeling of Bridges for Dynamic Analysis 4.3.8 Fixed Base Model vs Flexible Base Model 141 An eigen analysis was carried out using the stick model with fixed bases. In this analysis the flexibility of the foundation was ignored. Table 4.10 shows the comparison of model frequencies and the differences between the model frequencies of flexible and fixed base models. There is a small difference between the flexible and fixed base modal frequencies of modes 1,2 and 4 that are primarily vertical. It shows that the foundation flexibility does not influence the vertical vibration of the bridge very much. However, there is a significant difference between the flexible and fixed base modal frequencies of modes 3 and 6 which are primarily transverse and mode 5 which is primarily longitudinal. The greater differences show that the foundation flexibility has significant influence on the transverse and longitudinal vibration of the bridge. The reasons for different degrees of influence of the foundation flexibility on the different modes of vibrations are described below. Table 4.10 Comparison of Fixed Base Model and Flexible Base Model Frequencies Mode Number Frequency (Hz) Flexible Base Model Fixed Base Model Difference(%) 1 3.09 3.16 2.3 2 4.58 4.80 4.8 3 5.14 6.92 34.6 4 8.37 8.88 6.1 5 8.75 12.18 39.2 6 9.70 12.41 27.9 Chapter 4 : Modeling of Bridges for Dynamic Analysis 142 -0.5 -1.0 —•— -Computed Measured — •-Ventura et al. (1995) • Gates and Smith (1982) 1 st Vetical Mode 0 10 20 30 40 50 60 70 80 Distance from East Abutement (m) Fig. 4.20 Comparison of Measured and Computed First Vertical Mode Shapes cu TJ E < TJ Qi CO "co E o c => 1.0 -0.5 -1.0 — • — - Computed Measured Ventura et al. (1995) • Gates and Smith (1982) 0 I 2nd Vertical Mode I i L _ 20 40 60 Distance from East Abutment (m) 80 Fig. 4.21 Comparison of Measured and Computed Second Vertical Mode Shapes Chapter 4 : Modeling of Bridges for Dynamic Analysis 143 It was shown in Table 4.8 that the vertical stiffnesses of the abutments and the foundations are much higher than the stiffnesses in the two horizontal directions. But, the bridge deck is more rigid in the two horizontal directions than it is in the vertical directions. Thus, the greater flexibility of the bridge abutments compared to the bridge deck influences the vibration of the bridge in the transverse and longitudinal directions. The significant movements of the abutments observed in the transverse modes 3 and 6 and the longitudinal mode 5 confirm this fact. The bridge deck is much more flexible in the vertical directions than the abutment. This fact is confirmed by the negligible movement of the abutments observed in the vertical modes 1,2 and 4. Thus, vertical vibration of the bridge is not influenced much by the foundation flexibility. This comparison study of the flexible and fixed base models show that, unless the abutment and foundation flexibility are included in the bridge model, the response of the bridge structure in the longitudinal and transverse direction cannot be captured. Chapter 4 : Modeling of Bridges for Dynamic Analysis 144 4.4 CASE STUDY 2: DYNAMIC BEHAVIOUR OF THE BRIDGE UNDER EARTHQUAKE SHAKING 4.4.1 Rigid Body Model of the Painter Street Over Pass A rigid body model of the Painter Street Overpass originally proposed by Goel and Chopra (1994) was used with some modification to investigate the seismic behaviour of the Painter Street Overpass (Finn et al., 1998). The modified model is shown in Fig. 4.22 and described below in detail. Modeling of Abutment Soil System: The abutment consists of the abutment end wall, wing walls and the pile foundation In the model, the abutment soil-system is replaced by a spring-damper system. The spring represents the stiffness of the abutment and the damper accounts for the material and radiation damping. Since the east abutment is monolithically cast with the road deck the spring-damper systems are used in both directions; perpendicular to and along the east abutment. The west abutment is a seat type abutment which is retrained in the transverse direction and allowed to move in the longitudinal direction. The road deck rests on a neoprene strip. Two assumptions are made regarding this seat type abutment. One is that the frictional resistance produced during strong motion in the longitudinal direction on this strip is negligible. Second is that the gap between the deck and the abutment does not close even during the strong motion. It can be noted here that the maximum relative displacement occurred in the East/West direction during the 1992 Main shock was only about 1.05 inches. Based on these two assumptions, the abutment force in the longitudinal direction is assumed as zero. The abutment force in the transverse direction is not zero as the motion in this direction is restrained. This abutment force was resolved in two perpendicular directions; one along the abutment skew direction and the other perpendicular to the direction and is represented by a spring damper system as shown in Fig. 4.22. Chapter 4: Modeling of Bridges for Dynamic Analysis 145 Fig. 4.22 Rigid Body Model of the Painter Street Overpass Used in the Analysis Chapter 4 : Modeling of Bridges for Dynamic Analysis 146 In the model proposed by Goel and Chopra (1994), at the west end of the abutment, the abutment force in the direction perpendicular to the direction of skew was assumed as zero. The drawings of the bridge does not support this assumption. They show that the bridge deck is free to move in the longitudinal direction at the seated west abutment. Thus the abutment force should be zero in the longitudinal direction rather than in the direction perpendicular to the skew and the model was modified as described earlier. Modeling of the Center Bent: The center bent is parallel to the abutments. It consists of two reinforced concrete columns and each column is supported on 20 concrete friction piles. The approximate height of these columns is 24 feet and the minimum cross sectional dimension is 5 feet. Each column is represented in the model by two linear elastic springs; one normal to the bent and the other along the bent. The stiffness of the spring which represents the concrete column at the bent depends on two factors. One is the flexural rigidity of the column and the other is the boundary conditions at the end. During the strong motions, the column may crack in flexure which will reduce the flexural rigidity of column and alter the end conditions, ie. the ends will not remain fully fixed against rotation. As a result of these effects, the actual stiffness of the column may be much less than that of the uncracked stiffness. In recognition of this, in this study, the cracked stiffness of the column is taken as 25% of the uncracked stiffness (Collins and Mitchell, 1987). Check on the Rigid Body Model Assumption of the Deck: A rigid plate was used to model the bridge deck. A check was made using the strong records to validate the rigid body assumption. During the 1992 Main Shock, motions of the bridge in the North/South direction were recorded at the east and west ends of the deck and at the top of center bent. From the recorded motions at the two ends of the deck, the motions of the deck at the top of center bent were estimated by linear Chapter 4 : Modeling of Bridges for Dynamic Analysis 147 interpolation. This interpolated motions were compared with the measured motions at the top of center bent. This comparison is shown in Fig. 4.23. The very good agreement between the linearly interpolated motions and the measured motions suggests that the deck moves as a rigid body. 4.4.2 Analytical Procedure Equations of Equilibrium: The formulation of the problem follows that proposed by Goel and Chopra (1994). The three equations of dynamic equilibrium for the system of Fig. 4.22 are f i + f D + f s = 0 (4.10) in which fj is the vector of inertia forces, fD is the vector of damping forces and f s is the vector of spring forces; fD is formed from the damping forces at the abutments fD 1,fD 2>fD 3 and f D 4 . f s is formed from the spring forces at the abutments f s l,fS 2>fS3 and f S 4 and the spring forces at the columns f S 5 , f S 6 , f S 7 and f s g . The three components of the inertia force vector f^fj and f,e are computed from the mass properties determined from the structural plans, and recorded accelerations. The forces at the column springs f S 5 , f S 6 , f S 7 and f s g can be directly calculated by multiplying the stiffness with relative displacements. These relative displacements are computed by subtracting the free field displacements from the displacement of columns at the top. Thus, the column forces in the dynamic equilibrium equations are taken as known parameters. The only unknowns in the dynamic equilibrium equations are the abutment forces. Abutment forces and the displacements: The abutment force consists of both the spring force and the damping force. However, no attempt is made in this study to separately calculate these two forces. Instead, the resultant abutment forces are calculated by solving the three dynamic Chapter 4 : Modeling of Bridges for Dynamic Analysis 148 0 2 4 6 8 10 12 14 Time (sec) Fig. 4.23 Comparion of Measured and Linearly Interpolated Displacement (from the Measurements at the Ends of Deck) of Bent Top, Main Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 149 equilibrium equations at each time instant. The force in each column spring is determined from its computed structural stiffness and displacement. At each time instant, the displacement in the spring damper system modeling the abutment soil system or the column spring is the relative displacement between the structure and the ground. This relative displacement is obtained by subtracting the free field displacement from the motion of the top of the abutment. The motion of the top of the abutment can be obtained from the recorded motions of the rigid road deck. 4.4.3 Analysis of Response of the Bridge under the Main Shock 4.4.3.1 Force-Displacement Behaviour of Abutments The response of the bridge was analyzed under the Cape Mendocino/Petrolia, Main Shock of 1992 using the procedures outlined above. The time histories of force and displacement during the first fifteen seconds of the Main Shock are shown in Fig. 4.24 and 4.25. A l l the time histories show that there is a phase difference between the resultant abutment force and the relative displacement. The phase difference is due to the damping in the system. The maximum force normal to the abutment was 1770 kips and the corresponding displacement was 1.05in. The maximum force and the displacement along the abutment were higher at West abutment compare to the East. Along the west abutment, the maximum force and the displacement were 1275 kips and 1.96 in respectively and the same along the east abutment were 790 kips and 0.99 in. 4.4.3.2 Hysteresis Loops The hysteresis loops during the earthquake can be obtained by plotting the abutment force against displacement for many time intervals. The force-displacement behaviour of the east and west Chapter 4 : Modeling of Bridges for Dynamic Analysis 150 2000 1000 0 -1000 -2000 Force Deformation Spring 1: Normal to East Abutment I i I i I i L_ 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 0 8 Time (sec) 10 12 14 1500 750 0 -750 h -1500 Force Deformation Spring 2: Along East Abutment I i I i L 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 6 8 Time (sec) 10 12 14 o. Q E Fig. 4.24 Force and Displacement Time Histories of East Abutment Fig. 4.25 Force and Displacement Time Histories of West Abutment Chapter 4 : Modeling of Bridges for Dynamic Analysis 151 abutments for the first fifteen seconds of shaking during Main Shock, 1992 is shown in Fig. 4.26 and 4.27. The force-displacement plots clearly indicate non-linear inelastic behaviour of the abutment soil system during the earthquake It is evident from these figures that the stiffnesses of the abutments depend on displacement and are highly variable in time. 4.4.3.3 Displacement Dependency of Stiffness The force-displacement behaviour of the abutment was plotted for each cycle of displacement by isolating the displacement cycles at zero displacement. Often these plots do not show a complete hysteresis loops because of the variable phase difference between the force and the displacement. However, the more or less complete hysteresis loops are utilized in this analysis to determine the stiffnesses. The stiffness of the abutment continuously changes over the duration of a loop. Thus, an average value of stiffness is calculated by drawing a straight line across the loop as shown in Fig. 4.28 and taking the slope of this line as the average stiffness. This average stiffness value is assigned to one half of the total displacement which is the sum of displacements in the positive and negative directions. The variation of this average stiffness with the displacement is shown in Fig. 4.29 and 4.30. This figure clearly shows the displacement dependency of stiffness. In all three figures, it can be seen that a substantial drop in stiffness occurs from its high initial elastic stiffness value to a much lower value for a small increase in displacement. However, after this drop, the rate of reduction in stiffness with displacement becomes very small. In the case of stiffness normal to the east abutment, the major drop occurred over a displacement of about 0.2 in. The major drop in stiffnesses along the abutments occurred over about 0.15 in of displacement. Chapter 4 : Modeling of Bridges for Dynamic Analysis 152 2000 1000 "Jo" Q. o u_ -1000 -2000 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 Displacement (in) Spring 1: Normal to East Abutment, Main Shock, 1992 1500 _1500 i i i i I i I i L i i i i -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 Displacement (in) Spring 2: Along East Abutment, Main Shock, 1992 Fig. 4.26 The Force Displacement Loops for 0-15 sec during Main Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 153 1000 500 h V) O o h -500 -1000 -2.0 -1.0 0.0 1.0 2.0 Displacement (in) Spring 3: Normal to West Abutment, Main Shock, 1992 1500 to a . co o -2.5 -1.3 0.0 1.3 2.5 Displacement (in) Spring 4: Along West Abutment, Main Shock, 1992 Fig. 4.27 The Force Displacement Loops for 0-15 sec during Main Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 154 1500 1000 "w 500 CO u o 1 £ -500 -1000 -1500 - 4.92 i •5.50 sec - ! -^ ' / r O l / \ / -y 1 i 1 r i i -i -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Displacement (in) Spring 1: Normal to East Abutment, Main Shock, 1992 Spring 2: Along East Abutment, Main Shock, 1992 Fig. 4.28 Determination of Stiffness from Force Displacement Loop Chapter 4 : Modeling of Bridges for Dynamic Analysis 155 0.0 0.2 0.4 0.6 0.8 1.0 Displacement (in) Spring 1: Normal to East Abutment, Main Shock, 1992 Fig. 4.29 The Stiffness Displacement Relationship for East Abutment during Main Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 156 co Q. 60000 40000 co to cu «| 20000 CO • • •—*• • j I LJJ i I i I i I i_ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Displacement (in) Spring 3: Normal to West Abutment, Main Shock, 1992 co Q. 1*: co co CD c it 60000 40000 20000 0 0.0 0.2 0.4 0.6 0.8 Displacement (in) 1.0 1.2 Spring 4: Along West Abutment, Main Shock, 1992 Fig . 4.30 The Stiffness Displacement Relationship for West Abutment during M a i n Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 4.4.3.4 Time Variation of Stiffness 157 In order to investigate variation of stiffness with time, the stiffness values were plotted against time for the first fifteen seconds of shaking. Since the computed stiffness value is an average value of stiffness over the time duration of the loop, the stiffness is plotted at the time corresponding to the middle of the loop duration. These stiffness time history plots are shown in Fig. 4.31 and 4.32 together with the corresponding displacements. It is clearly evident from these figures that the stiffness of the abutment significantly changes during the same earthquake. At the initial phase of shaking the abutments tend to be stiff since the displacement is very low. When the shaking becomes intense, a substantial drop in stiffness occurs and it is associated with increased displacement. Towards the end of the earthquake, when the shaking is reduced and the displacements are small, the stiffness recovers. 4.4.4 Analysis of Response Using Power Spectral Density and Transfer Functions 4.4.4.1 Introduction When a structure is subjected to an excitation, it will respond strongest near the resonant frequencies. This can be seen in the plots of the power spectral density (PSD) computed from the acceleration records where the resonances appear as peaks. For lightly damped structures, with essential modes of vibration, these peaks will correspond to the damped natural frequencies of the structure. In the present study, the power spectral density of structural and free field accelerations in both longitudinal and transverse directions were computed to identify the resonant frequencies. To investigate the relationship between the relative motion of the structure and the input acceleration, the transfer function between these two was also computed. Chapter 4 : Modeling of Bridges for Dynamic Analysis 158 80000 £T 60000 w w 40000 tn <D C it w 20000 • S t i f f n e s s • D e f o r m a t i o n .CP B i rnirji i Di I i i i i I i i 0.0 5.0 10.0 T i m e ( s e c ) 1.0 0.8 _ c 0.6 | E <o 0.4 ro Q. tn 0.2 3 0.0 15.0 Spring 1: Normal to East Abutment, Main Shock, 1992 tn o. tn tn a> c it •4—* to 80000 60000 40000 H 20000 0 0.0 _5L • S t i f f n e s s a D e f o r m a t i o n •• • # • J I I I J I L 1.0 0.8 0.6 0.4 h 0.2 0.0 5.0 10.0 T i m e ( s e c ) 15.0 CD E o JS CL tn Q Spring 2: Along East Abutment, Main Shock, 1992 Fig. 4.31 The Stiffness Displacement Relationship for East Abutment during Main Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 159 80000 £T 60000 "to Q. ^ 40000 H <0 co c st co 20000 0.0 • Stiffness • Deformation • • • • n j L j i L 1.0 0.8 0.6 S E CO 0.4 ro CL (A 0.2 S 0.0 5.0 10.0 Time (sec) 15.0 Spring 3: Normal to West Abutment, Main Shock, 1992 to g. JX. in V) CO c i t CO 80000 60000 40000 20000 0 0.0 J3 i_ • Stiffness • Deformation D I I 1.0 0.8 h 0.6 0.4 0.2 0.0 5.0 10.0 Time (sec) 15.0 c CO E CO o _ro C L (fl Spring 4: Along West Abutment, Main Shock, 1992 F i g . 4.32 The Stiffness Displacement Relationship for West Abutment during M a i n Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 4.4.4.2 Response in the Longi tudinal Direction 160 Fig. 4.33 shows the unit normalized PSD of the structural accelerations and the free field acceleration. The fundamental frequency of the bridge in the longitudinal direction is 3.61 Hz (Gates and Smith, 1982). No significant peak in the PSD graph can be seen at this frequency even though there is some input at this frequency. However, clear amplification could be observed at much lower frequency range. For example, between 2-2.5 Hz. Fig. 4.34 shows the amplitude of transfer function between the relative displacement and the free field acceleration. This figure shows that, for frequencies greater than 2.8 Hz, the transfer function is nearly zero. Even at the fundamental frequency, the transfer function fails to show any significant value. However, at lower frequencies than 2.8 Hz, significant peaks could be observed. 4.4.4.3 Response in the Transverse Direction Fig. 4.35 shows the unit normalized PSD of structural average acceleration and the free field acceleration in the transverse direction. According to the U B C ambient vibration test data, the fundamental frequency in the transverse direction is 4.08 Hz. The PSD graph does not show any peak at this frequency. Fig. 4.36 shows the amplitude of transfer function between the relative displacement and the input acceleration in the transverse direction. It can be seen from this figure that there is no amplification of motion near the fundamental frequency. For frequencies greater than 2 Hz, the amplitude of transfer function is nearly zero though there is some transfer at low frequency range. The same analysis described above was repeated for the much lower shaking intensity associated with the Nov. 21, 1986, Cape Mendocino, After Shock (ML=5.5). The unit normalized PSD and the amplitude of Transfer function plots for motions in the longitudinal as well as transverse directions are shown in Fig. 4.37 through 4.40. No significant peaks occur near the Chapter 4 : Modeling of Bridges for Dynamic Analysis 161 (/) C CU Q "ro i o CO CL CO I CD o CL T3 0) .52 ro E East End of Deck(Ch11) Free Field (CM 2) Frequency (Hz) Fig . 4.33 Comparison of Uni t Normalised Power Spectral Density o f Deck and Free F ie ld Mot ion in the Longitudinal Direction, M a i n Shock, 1992 F i g . 4.34 Variation of the Amplitude of Transfer Function with Frequncy M a i n Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 162 & 1.2 co c Oi a ro i— o CD Q . CO i— CD o D_ T3 0) CO "ro E 1.0 0.8 0.6 0.4 0.2 0.0 Transverse Direction Average Motion of Deck (Ch4,7&9) Free Field (Ch14) Frequency (Hz) F i g . 4.35 Comparison of Unit Normalised Power Spectral Density o f Deck and Free Fie ld Mot ion in the Transverse Direction, M a i n Shock, 1992 o c CO c ro o CD Q . E < 0.10 0.05 0.00 Transverse Direction- Average Motion of Deck Frequency (Hz) Fig . 4.36 Variation of the Amplitude o f Transfer Function with Frequncy M a i n Shock, 1992 Chapter 4 : Modeling of Bridges for Dynamic Analysis 163 CO c cu Q " r o -5 cu C L CO 1— cu o CL T3 CU V) "ro E 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Longitudinal Direction East End of Deck (Ch11) Free Field (CM 2) Frequency (Hz) F i g . 4.37 Comparison of Uni t Normalised Power Spectral Density o f Deck and Free F ie ld Mot ion in the Longitudinal Direction, Nov21,1986 After Shock c o o c 3 c ro o cu T3 C L E < 0.10 0.05 0.00 Frequency (Hz) F i g . 4.38 Variation of the Amplitude of Transfer Function with Frequncy Nov21,1986 AfterShock Chapter 4 : Modeling of Bridges for Dynamic Analysis 164 & 1.2 CO c <u Q ra "G O) CL CO 1 CD o D. X J (D (fl "ro E 1.0 0.8 0.6 0.4 \-0.2 h 0.0 Average Motion of Deck ( Ch4 , 7 & 9 ) Free Field (Ch14) Transverse Direction Frequency (Hz) Fig. 4.39 Comparison of Unit Normalised Power Spectral Density of Deck and Free Field Motion in the Transverse Direction, Nov21,1986 After Shock Fig. 4.40 Variation of the Amplitude of Transfer Function with Frequncy Nov21,1986 AfterShock Chapter 4: Modeling of Bridges for Dynamic Analysis 165 fimdamental frequencies which were determined during the ambient vibration test. But, they clearly show peaks at much lower frequencies. Clearly the nonlinear behaviour of the soil even during the aftershock motions was sufficient to move the period of peak response way from the ambient vibration frequencies. The results suggest that the ambient vibration data are not appropriate for getting the periods of the peak response to strong shaking. The primary value is providing data for calibrating the initial state of the bridge before nonlinear behaviour sets in. 4.5.3 Limitations of the Current Modeling of Bridge Foundations Dynamic response studies of the Painter Street Overpass using the ambient vibration data and strong motion data showed that the modal periods estimated using the strong motion data were substantially greater than those from the ambient vibration data. Studies on the Meloland Overpass also showed a similar trend. The increase in periods were attributed to reduction in the foundation stiffness during strong shaking. The rigid body model study on the Painter Street Overpass using the strong motion data confirmed the substantial reduction in the foundation stiffness during strong shaking. In many system identification studies in which optimized stiffness values of foundation springs were used to capture the measured strong motion response showed that the optimized stiffnesses were substantially less than the elastic small strain stiffnesses. In these studies also, the reason for the reduction in stiffness was attributed to the non-linear behaviour of the foundation soils. Recognizing the fact the foundation stiffnesses are reduced during strong shaking, in the current modeling of bridge foundations, the small strain stiffnesses are reduced by a factor and then used. Many analytical studies such as Makris et al. (1994) and Wilson and Tan (1990) that used the reduced stiffnesses were successful in predicting the strong motion response. However, there remain Chapter 4 : Modeling of Bridges for Dynamic Analysis 166 several unanswered questions. How to find out the appropriate reduction factors ? What factors influence these reduction factors and how ? Can these factors be generalized or is it problem specific ? Is it appropriate to use a single factor for the whole duration of shaking ? These questions address some of the limitations in the current methodologies. Most current modeling of bridge foundations failed to incorporate the foundation damping in their analytical model, ie. Only springs are used and dashpots are omitted while modeling the foundations. It was shown by many researchers that foundation soils exhibit hysteretic behaviour during strong shaking and it is important to simulate their energy dissipation characteristics in the model. This can be done by including viscous dampers in the foundation model in addition to the springs. Some models recognize the importance of foundation damping and include it in the foundation model. However, even in these models, the strain dependent characteristics of the damping was not modeled. Instead, constant values are used. During strong shaking, the behaviour of the soil is usually non-linear and inelastic. The shear modulus of the soil changes continuously during strong shaking depending upon the level of shear strain. Fig. 4.41 shows a typical variation of shear modulus with shear strain. The shear modulus decreases with increasing shear strain. As the stiffness of the foundation mainly depends upon the shear modulus of the soil a continuous change in the stiffness also can be expected during shaking. The damping ratio of the soil also depends on the level of shear strain. A typical variation of damping ratio with the shear strain is shown in Fig. 4.42. It can be seen from this figure that the damping shows behaviour compare to the modulus. Therefore a continuous variation in damping also can be expected during strong shaking. Current modeling procedures failed to model such continuous variation of stiffness and damping and this is a major short coming in the current procedures. Since a single valued springs and dampers are normally used to model the foundations, often linear elastic analysis procedures are followed to determine the bridge response. A non-linear analysis procedure is what is required. This is another limitation in the current procedures. Chapter 4 : Modeling of Bridges for Dynamic Analysis 167 0.0001 0.001 0.01 0.1 1 Shear Strain (%) Fig. 4.41 The Variation of Shear Modulus and Damping Ratio with Shear Strain Chapter 4 : Modeling of Bridges for Dynamic Analysis 4.6 CONCLUSIONS 168 Two case studies were undertaken on the dynamic response of the Painter Street Overpass under ambient vibration and strong earthquake shaking. A stick model of the superstructure was used for the ambient vibration study and a rigid body model for the strong shaking analysis. In the ambient vibration analysis, the foundations were modeled using the elastic procedures and found to be adequate to capture the linear elastic behaviour of the bridge. Fixed base and flexible base model analyses showed that foundation flexibility influences the vibration of the bridge in the transverse direction even under ambient vibration. However, The vertical vibration is not affected much by the foundation flexibility at low level of shaking. The rigid body model study which utilized strong motion records from the bridge showed that the bridge abutments have an important role to play in the seismic behaviour of the bridge. The abutments exhibited highly nonlinear hysteretic behaviour under strong shaking and the stiffness dropped drastically during strong shaking. Analysis of the records showed that the fundamental frequencies under string shaking were much smaller than those from ambient vibration analysis. The reason for the reduction is attributed to the highly nonlinear behaviour exhibited by the bridge foundations. From the review and the case studies, the limitations in the current modeling of bridge foundations were identified. The main limitation was found to be the inappropriate modeling of the foundations due to the unavailability of a suitable method to capture the variation of stiffness and damping during strong shaking. 169 CHAPTER 5 Modeling of Pile Foundation as a Lumped Parameter System 170 CHAPTER 5 : MODELING OF PILE FOUNDATION AS A LUMPED PARAMETER SYSTEM FOR NONLINEAR ANALYSIS 5.1 I N T R O D U C T I O N If the nonlinear behaviour of the pile foundation can be modeled using a lumped parameter system it can easily be incorporated into a stick model without much computational effort. A method is presented herein to model the pile foundation as a lumped parameter system for nonlinear analysis. The lumped parameter system consisting of a set of springs and dashpots. The springs and dashpots represent the stiffness and damping of the pile foundation respectively. In order for the lumped parameter model to replicate the nonlinear behaviour of the pile foundation under earthquake shaking, the springs stiffness and dashpot parameters should also be nonlinear. A method is described to obtain the nonlinear stiffness and damping using PILE3D and PILIMP programs. Fig. 5.1a shows a pile group foundation and the superstructure and Fig. 5.1b shows the corresponding lumped parameter model. In the lumped parameter model, the pile foundation is represented by a set of springs and dashpots and it is subjected to a set of motions called the foundation motions at the base of the springs and dashpots. The modeling of the pile foundation as a lumped parameter system for the nonlinear analysis requires the determination of the stiffness and damping of the springs and dashpots and the foundation motions. The stiffness, damping and the foundation motion should reflect the nonlinear behaviour of the pile foundation. Methods are described below to determine these parameters using PILE3D and PILIMP. 5.2 INPUT MOTIONS The pile cap motions are used as input motions to the lumped parameter model of the pile Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 171 Superstructure &xxxxxxxxxxxx Pile Group Fig. 5.1a Pile-Soil-Superstructure System o Lateral Cross K xe K X , C X —3]-Superstructure Coupling Rotational K e - C e Springs and Dampers e Fig. 5.1b Lumped Parameter Model of Soil-Pile-Superstructure System Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 111 foundation. These motions are determined by analyzing the pile foundation without the superstructure. These motions may include both the translation and rotation components and reflect both the effects of kinematic and inertial interaction. These aspects of the foundation motion are examined below. Effect of kinematic interaction: During excitation, a flexible pile may simply follow the seismic motion of the ground. However, a more rigid pile can resist and hence modify the soil deformation. As a result the incident waves are scattered and the seismic excitation to which the structure base is subjected differs from the free field motion due to kinematic interaction. Fan et al. (1993) and Wu (1994) showed that the effect of kinematic interaction is important only when the E p / E s ratio is greater than 10000, where Ep and Ej. are the Young's modulus of the pile and soil respectively,. The ratio, E p / E s is a measure of the flexibility of the pile relative to the soil. In practice most laterally loaded piles are indeed flexible (Raul and Whitman, 1982, Gazetas and Dobry, 1984, Kuhlmeyer, 1979, Velez et al. ,1983, Randolph, 1981, Krishnan, 1982). Thus, the effect of kinematic interaction on the foundation motion can be neglected for most piles. Also elastic finite element analysis has shown that the effect of kinematic interaction becomes unimportant when the excitation frequency is less than 1.5 times the foundation frequency of the free field soil. (Wass and Hartmann, 1984, Gazetas, 1984). Thus even for rigid piles the effect of kinematic interaction can be relatively minor for low excitation frequencies. Effect of Inertial Interaction: Soil properties remain constant during linear elastic analysis and therefore the additional strains caused by the inertial interaction will not have any effect on the properties of the foundation soils. Thus the foundation motion determined in the absence of superstructure inertia is strictly valid for elastic foundation soil only. However, in almost all practical analyses of bridge structures including the nonlinear analyses, the free field motion is taken as the input motion to the bridge superstructure. During strong shaking the additional strains caused by the inertial interaction cause degradation in moduli and increase in damping. In the lumped parameter model analysis, the effects of the inertial interaction on the foundation motions are neglected and the extent of the error caused by this depends on the relative degradation potential of Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System the free field shaking and the inertial interaction. 173 5.3 NONLINEAR PILE CAP STIFFNESS AND DAMPING The first step in trying to incorporate nonlinear effects is to track the changes that occur in stiffness and damping during earthquake shaking. Such a procedure is outlined below for determining the nonlinear pile cap stiffness and damping as a function of time. Step 1: The seismic response analysis of the pile foundation is carried out in the time domain using PILE3D. During this analysis, the strain compatible shear moduli and the damping ratio in each soil element are traced as a function of time. Step 2: The pile head impedances are then computed using PILIMP through successive elastic analysis at each preselected time at which the shear moduli and damping ratio for each soil element are available from step one. This two-step procedure gives the time varying stiffness and damping of pile foundations during strong shaking. 5.4 SEISMIC SUPERSTRUCTURE RESPONSE The nonlinear response of the superstructure is obtained by solving the dynamic equilibrium equations of the lumped parameter system in the time domain. The foundation motion described in Section 5.2.1 is used as the input motion to the lumped parameter system. The time histories of stiffnesses and damping of the pile foundation are used directly in the time marching solution procedure to determine the superstructure response Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 174 5.5 NUMERICAL STUDIES OF NONLINEAR RESPONSE 5.5.1 Introduction Numerical studies are performed to demonstrate that the nonlinear stiffness and damping time histories determined using PELE3D and PILIMP can reproduce the nonlinear behaviour of the pile foundation and the free field motions are good approximation to the pile head motions. The effect of inertial interaction of superstructure on the stiffness and damping of the pile foundations and its effect on the lumped parameter model response are investigated. An approximate procedure that uses the effective shear moduli and damping of the free field in the computation of the stiffness and damping of the pile foundations is also investigated. A simplified problem of a pile foundation carrying a concentrated mass at the pile head is considered in the numerical studies. The pile foundation system is modeled as a two-degrees of freedom system with translation and rotation in the plane of shaking. 5.5.2 Lumped Parameter Model of a Single pile 5.5.2.1 Problem Description The single pile carrying a concentrated mass at the pile head and the equivalent two-degrees of freedom model are schematically shown in Fig. 5.2. The pile is of diameter D and length L and considered to be a linear elastic beam with constant Young's modulus E p and mass density p p . The foundation soil is homogeneous with Poisson's ratio, v constant material density p s and constant material damping ps. Unless otherwise noted the following typical values are chosen for the parameters. Ep/Es=1000, pp/ps=1.4, L/D >15,v=0.4, p\=0.05. The concentrated mass and the mass moment of inertia were taken as 50 Mg and 100 Mg m 2 respectively. The input horizontal base acceleration record used in this analysis is shown in Fig. 5.3. The peak acceleration amplitude of 0.54 g. The variation of shear moduli and damping with the shear strain is shown in Fig. 5.4. The damping ratio has an upper limit of 25%. Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 175 Free Field Mus illltllhuWiillnllhi/ | 5 I I 1 | | I H 1 I ! ! H 1 Fig. 5.2a Single Pile with a Lumped Mass in a Homogeneous Media Fig. 5.2b Equivalent 2-DOF Lumped Parameter Model Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 176 0.6 o 2 0.0 _CD CD 8 -o.2 < -0.4 -0.6 A = 0.5 q max a l 1 I 1 l < 6 8 Time (sec) 10 12 14 Fig . 5.3 Input Horizontal Acceleration Used in the Quasi-3D Finite Element Analysis Shear Strain (%) Fig . 5.4 The Variation of Shear Modulus and Damping Ratio with Shear Strain Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 111 E m o 34.0 m Axis of Symmetry PLAN E in o End Elevation Fig. 5.5 Finite Element Mesh Used in the Single Pile Analysis Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 178 The response of mass and the free field were determined using PIL3D in a fully coupled analysis. The finite element mesh used in the analysis is shown in Fig. 5.5. Because of symmetry, only one half of the problem is considered. The mesh consists of 1080 nodes and 770 brick elements representing the soil. The soil continuum is divided into 11 horizontal layers and the layer thickness is gradually reduced in the direction from bottom to the top. The pile is modeled using 8 beam elements. 5.5.2.2 Pile Head Impedances During strong shaking, the time histories of stiffness and damping were obtained using the procedure described in Section 5.2.2 Fig. 5.6a show the variation of stiffness with time. The stiffness decreases drastically as the level of shaking is increased. It reached the minimum value when the acceleration at the pile head attained its maximum value at about 5 seconds. Then the stiffness rebounds as the level of shaking is reduced. The reduction in the lateral stiffness is much greater than that in the rotational or cross coupling stiffnesses. The lateral stiffness reduced to a minimum value of 64 MN/m from its initial value of 373 MN/m; a reduction to 17% from the initial value. The rotational stiffness reduced to 205 MNm/rad from its initial value of 340 MN.m/rad; a reduction to 60%. The initial and minimum value of the cross coupling stiffness are 234 and 85 MN/rad respectively the corresponding percentage of reduction to 36. Fig. 5.6b shows the variation of damping with time. As the level of shaking increases, the damping also increases due to the increase in the hysteretic damping of the soil. When the shaking level reduces the damping also reduces. Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 179 F ig . 5.6a Time histories o f Stiffnesses During Strong Shaking — © — Lateral (MN/m) — - Q — Cross-Coupling(Mn.rad) —A— Rotational(MN.m/rad) 0 2 4 6 8 10 12 Time (sec) Fig . 5.6b Time Histories o f Damping During Strong Shaking Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 180 5.5.2.3 Equivalent Lumped Parameter Model Response The equivalent 2-DOF lumped parameter model of a single pile carrying a concentrated mass is shown in the Fig. 5.2b. The time histories of the pile head impedances were used to represent the properties of the springs and dashpots. The input motion to the lumped parameter model was taken as the free field motion that was obtained during the PILE3D analysis. The solution procedure adopted in determining the response of the 2DOF system is described below. For the 2DOF system lumped parameter model shown in Fig. 5.2b, the two degree of freedoms are translation x and rotation about y axis. The system is subjected to translational acceleration in the x direction. The equation of motion of the 2DOF system is expressed in the matrix form as m 0 , 0 I, x 1 c c ^ L x C x 6 V c x 8 c e ; 6 k ( ^ x xo Ke \ x mx f I 0 ) in which m -Mass I -Mass moment of inertia x, x, x - Displacement, velocity and acceleration in the x direction 0,0,6 - Rotation, rotational velocity and rotational acceleration about y axis x f - Input acceleration in the x direction k x , k x 6 , k e - Translational, cross-coupling and rotational stiffnesses c x , c x 9 , c e - Translational, cross-coupling and rotational dampings The equations of motions are solved in the time domain using the Wilson-theta method (Wilson et al., 1973). While solving the equations by step-by-step integration procedure, the Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 181 unbalanced force is calculated at the end of each step and added in the next step to maintain the dynamic equilibrium. The solution procedure adopted in evaluating the response of the 2-DOF lumped parameter model was implemented in a computer program LUMPILE. The acceleration and displacement of the mass by the PHJE3D and L U M P I L E analyses are shown in Fig. 5.7a and 5.7b respectively. Very good agreement can be observed between the response of these two models over the whole duration except near the peaks. The lumped parameter model underestimated peak displacement by about 11%. The slight difference in the peaks can be attributed to the assumptions made regarding the input foundation motion. The good agreement between the two model solutions shows that the lumped parameter system is capable of replicating the nonlinear response of the pile foundations under strong shaking provided that the input motion to the lumped parameter model can be approximated as the free field motion. This allows us to represent the pile foundations with an equivalent lumped parameter system in the superstructure model of the structure such as bridges and buildings. 5.5.3 Approximate Determination of Nonlinear Pile Head Impedance The effects of earthquake shaking on the soil properties of free field can be obtained using the 1-D ground motion analysis programs such as "SHAKE" (Schnabel et al., 1972) and DESRA (Lee et al. ,1975). The equivalent linear method proposed by Seed and Idriss (1967) is implemented in the ' S H A K E ' computer program. This program gives a vertical distribution of shear moduli and damping ratios compatible with the effective shear strain levels (65% maximum strain in each layer) developed during the earthquake. Sometimes the pile head impedances are based on these effective shear moduli and damping ratios. Any elastic method that can handle a layered deposit can be used to determine the impedances. It can be noted here that this type of 1-D inelastic ground motion analysis considers neither the soil-pile interaction effect nor the superstructure inertial interaction effect. Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 182 0.6 -0.6 PILE3D Analysis Model with variable stiffness and damping (Shear moduli and damping from PILE3D) J i I i L J L J I L 6 8 10 12 Time (sec) 14 Fig. 5.7a Comparison of Acceleration Response of a Single pile Under Strong Shaking c CD E CD O 03 Q. to Q 0.03 0.02 -0.01 -0.00 -0.01 -0.02 h -0.03 0 PILE3D Analysis Model with variable stiffness and damping (Shear moduli and damping from PILE3D) _L 6 8 Time (sec) 10 12 14 Fig. 5.7b Comparison of Displacement Response of a Single Pile Under Strong Shaking Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 183 To study the use of "SHAKE" based pile head impedance, first, a 1-D ground motion analysis was performed using 'SHAKE' to obtain the strain compatible effective shear moduli and damping. The computed effective shear moduli and damping were used to calculate the pile head impedances using the PILIMP program. The computed stiffness and damping are schematically illustrated in Fig. 5.8a and 5.8b. These figures also show the variation of stiffness and damping as a function of time that were obtained during the nonlinear Quasi-3D finite element analysis. As expected the stiffnesses are lower than the initial low strain stiffnesses and the damping is higher than the initial low strain damping. However, it is evident form these figures that the constant impedances obtained using the "SHAKE" are not good average values for the whole duration of shaking. These average constant pile head impedances were utilized in the 2-DOF lumped parameter model to evaluate the earthquake response. The relative acceleration and the relative displacement response of the lumped parameter model are compared with those from the PILE3D analysis in Fig. 5.9a and 5.9b. Poor agreement is observed between these two responses. There are two reasons for the poor response. The 2DOF impedances are based on constant effective moduli and damping and the effects of inertial interactions were neglected. 5.5.4 The Effect of Superstructure Interaction The effect of superstructure inertial interaction was determined by repeating the PILE3D analysis without the concentrated mass. The effect of superstructure inertial interaction on the pile head impedance is illustrated in Fig. 5.10a and 5.10b where the stiffness and damping with and without the inertial interactions are shown. The stiffnesses much less with inertial interaction. Higher damping was observed during high levels of shaking with inertial interaction. Otherwise, the damping was less. Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 184 400 0 2 4 6 8 10 12 14 Time (sec) F i g . 5.8a Comparison of Stiffness Time Histories Computed Directly from PILE3D and by Us ing the M o d u l i and Damping from S H A K E Analysis 100.0 Rotational Lateral* Cross-Coupling — © — Lateral (MN/m) —E3— Cross-Coupling(Mn.rad) —A— Rotational(MN.m/rad) J i I i I i L 6 8 10 12 Time (sec) 14 Fig . 5.8b Comparison of Damping Time Histories Computed Directly from PILE3D and by Using the M o d u l i and Damping from S H A K E Analysis Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 185 PILE3D Analysis Model with variable stiffness and damping (Shear moduli and damping from SHAKE) 0 2 4 6 8 10 12 14 Time (sec) Fig . 5.9a Comparison o f Acceleration Response of a Single Pi le Under Strong Shaking c CD E CD O ro CL to Q 0.02 0.01 h 0.00 n -0.01 h -0.02 - "_n_n_A.AAAAfl ni^  ft r> A dA ' — PILE3D Analysis Model with variable stiffness and damping (Shear moduli and damping from SHAKE) . - •••'vvvTVVVVVV'vVvWvj n 1 i 6 8 Time (sec) 10 12 14 Fig . 59b Comparison of Displacement Response of a Single Pi le Under Strong Shaking Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 186 500 E 400 \-co 300 h fn 200 | -B w 100 h -Q— Without Mass With Mass j i L J i L _L A. 0 2 4 6 8 10 12 14 Time (sec) Fig . 5.10a The effect of Inertial Interaction on the Lateral Stiffness F i g . 5.10b The Effect o f Inertial Interaction on the Cross-coupling and Rotational Stiffnesses Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 187 The dynamic impedance time histories that were obtained without including the effect of superstructure mass were used in the lumped parameter model to evaluate the relative acceleration and relative displacement response. These responses were compared with the response of the lumped parameter model obtained in the previous section in which the response was evaluated using a constant set of impedance from "SHAKE" properties. The comparisons between these two responses are shown in Fig. 5.1 la and 5.1 lb. A fairly good agreement was observed between them. This suggests the impedances from shake properties are good approximations for analysis if inertial interaction is not significant. 5.5.5 The Effect of Cross Translation-Rotational Impedance Commercial computer programs that are widely used by practicing engineers to evaluate the dynamic response of bridge structures do not incorporate the cross-coupling stiffness and damping terms in the foundation impedances. In order to demonstrate the effect of the cross coupling terms, elastic analyses were carried out using PILE3D and the 2-DOF lumped parameter model under harmonic excitation with frequency of 2 Hz and amplitude 0.5 g. When the cross-coupling terms were present in the lumped parameter model, the difference between the two model responses was negligible. When the cross-coupling terms were omitted in the lumped parameter model, its steady state relative displacement amplitude was about 60% less than the same from the PILE3D response. However, no significant error was observed in the phase. 5.6 CONCLUSIONS A method was presented to model the pile foundation as a lumped parameter system consisting a set of springs and dashpots and to carry out a nonlinear soil-pile-structure interaction Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 188 0.5 2 0.0 CD -0.5 1 Model with variable stiffness and damping (Shear moduli and damping from SHAKE) Model with variable stiffness and damping (Shear moduli and damping from PILE3D analvsis WITHOUT TH E MASS} k i , i If 1 1 • 1 0 2 4 6 8 10 12 Time (sec) Fig . 5.11a Comparison of Acceleration Response o f a Single Pi le Under Strong Shaking 0.010 0.005 h 0.000 r i -0.005 -0.010 Model with variable stiffness and damping (Shear moduli and damping from SHAKE) Model with variable stiffness and damping (Shear moduli and damping from PILE3D analysis WITHOUT THE MASS) 6 8 Time (sec) 10 12 14 F i g . 5.1 l b Comparison of Displacement Response o f a Single Pi le Under Strong Shaking Chapter 5 : Modeling of Pile Foundation as a Lumped Parameter System 189 analysis. Analyses with and without the inertial interaction showed that the inertial interaction is important and should be taken into account as it alters the stiffness of the pile foundations during strong shaking. An approximate procedure that uses a constant reduced stiffness and increased damping was evaluated. In this procedure the stiffness and damping were computed using a equivalent set of shear moduli and damping ratio for the soil obtained from a " S H A K E " analysis. This approximate procedure gave acceptable results only when the inertial interaction from the superstructure is not important. CHAPTER 6 Seismic Response Analysis of Bridges 191 CHAPTER 6 : SEISMIC RESPONSE ANALYSIS OF BRIDGES A computational model is presented here for the nonlinear analysis of the bridge structure. The model is developed by incorporating the lumped parameter model of the pile foundation described in Chapter 5 into the 3D stick model of the superstructure. The behaviour of the superstructure is treated as elastic and the behaviour of the foundations is treated as nonlinear and inelastic. The dynamic equilibrium equations are formulated using finite element method and they are solved in the time domain to obtain the nonlinear response of bridge structure. The model is applied to the three span continuous box girder bridge used as an example in the seismic design guide published by A A S H T O (1983). 6.1 GENERAL COMPUTATIONAL MODEL The bridge superstructure, piers or columns and the abutment wall are modeled as space frame members. The foundations are modeled using equivalent springs and dashpots. 6.1.1 Space Frame Member Two noded 3D beam elements are used as space frame members. This beam element has six degrees of freedom at each node; three translational along the local axes x,y and z and three rotations about the same axes. Consequently, the beam element has twelve degrees of freedom and the resulting element matrices will be of dimension 12x12. Fig. 6.1 shows a typical beam element with its twelve nodal coordinates. Chapter 6 '. Seismic Response Analysis of Bridges 192 Node i 6.1.1.1 Element Stiffness Matrix The deformations considered in the development of the element stiffness matrix are those caused by torsion about x axis, bending about two principal axes of the cross section y and z, axial force in the direction x and transverse shear in the directions y and z. Thus, the 12x12 stiffness matrix for the 3D beam element will be as shown in Eq. 6.1 Chapter 6 : Seismic Response Analysis of Bridges 193 EA 1 12EI2 l 3(l+* y) 12EIy l 3 ( l + * z ) 0 GJ 1 -6EI -EA 1 -6EIZ l2(l+<f>) -12EIz l 3(l+* y) i2o+<i>z) 0 0 -12EIy 13(1+<]>Z) 0 -6El y 12(1+(|>Z) 0 0 0 -GJ (4+4>z)EIy l(i+4>z) 0 0 0 6EIy l2(l+<t>z) 0 (4+<t>y)EIz 1(1 +<l>y) EA 1 6EI -12EI z— 0 -l2(l+<|>y) l3(l+(|>y) 0 0 0 o 1(1-<1>Z) 0 0 0 0 12EIy l3(l+<t>z) 0 6EI„ GJ 1 0 6EI l 2 ( l + cb) Q -6EIZ 1(1+^) l2(l+<j>y) (4±(j)z)EIy i 2 d +4>z) " i(i+<t>z) 0 0 o (4±c|>y)EIz 1(1+cb) (6.1) In this element stiffness matrix (j) and (j)z are shear deformation parameters given by 4>v *7 12EI z G A l 2 sy 12EI 24(1+v)-sy [A \ 2 (6.2) y _ G A 1 2 = 24(1+v)- f j where 1 - Element length A - Cross-sectional area A s y - Equivalent shear area in y direction A s z - Equivalent shear area in z direction Iy - Second moment of area about y axis I^  - Second moment of area about z axis Chapter 6 '. Seismic Response Analysis of Bridges 194 J - Torsional constant E - Modulus of elasticity G - Modulus of elasticity in shear u - Poisson's ratio Y y - Radius of gyration about y axis y z - Radius of gyration about z axis 6.1.1.2 Element Mass Matrix The lumped mass matrix of a 3D beam element is simply a diagonal matrix in which coefficients corresponding to translatory and rotational displacements are equal to one half of the total inertia of the element and while the coefficient corresponding to flexural deformations are equal to zero. This diagonal mass matrix leads to a significant saving in the computer storage and computational time compared to the consistent mass matrix. The degree of accuracy obtained through the use of lumped mass matrix is considered to be good enough for all practical purposes. The lumped mass matrix is expressed as [ M J = ml/2 <1 1 1 1 I-/m 1 1 1 I-/m> (6.3) 6.1.2 Lumped parameter model of the Pile Foundation Nonlinear PILE3D and PILIMP analyses of the pile foundation are carried out to obtain time histories of pile head stiffness and damping. From these time histories, the stiffness and damping matrix of the lumped parameter system representing the foundation can be formed as follows. Chapter 6 : Seismic Response Analysis of Bridges 195 [KpilJ6x6 0 k y 0 0 k. k 0 0 k x-ry ry 0 k x_„ 0 0 k r x 0 0 0 0 0 0 [^pilJ6x6 0 c y 0 0 c, c 0 0 c x-ry ry 0 c 0 0 c x-rx rx 0 0 0 0 0 0| where x,y - Two orthogonal horizontal directions z - Vertical direction k x >cx Translational stiffness and damping in the x direction ky >Cy Translational stiffness and damping in the y direction k z ,cz Vertical stiffness and damping in the z direction k r a >Crx Rocking stiffness and damping about x axis kfy >cry Rocking stiffness and damping about y axis k C -x-ry '^ x-ry Cross horizontal-rocking stiffness and damping k c -y-rx >wy-rx Cross horizontal-rocking stiffness and damping A l l stiffnesses and dampings are time dependent. The torsional stiffness of the pile foundations about the vertical axis is not considered in the present study. 6.1.3 Equations of Motion The equation of motion at time t for the bridge subjected to rigid base excitation can be expressed as [M]{v} +[C] t{v}+[K] t{v} = [M]{I}v b (6.5) v b in which is the rigid base excitation and {1} is a column vector of 1. v, v and v are the relative acceleration, velocity and displacement respectively. [M] is the global mass matrix and [C] t and Chapter 6 '. Seismic Response Analysis of Bridges 196 [K] t are the time dependent global damping matrix and the global stiffness matrix respectively. 6.1.4 Formulation of Global Mass, Stiffness and Damping Matrices 6.1.4.1 Global Mass Matrix The global mass matrix [M] is assembled from the individual lumped mass matrices of the space frame elements described in Section 6.1.1.2. The inertia effects of the bridge deck, columns or piers and abutment walls are represented by the time invariant lumped mass matrix. The global mass matrix is diagonal which will lead to significant savings in the computer storage and computational time. The diagonal elements of the global mass matrix can be stored in vector form. 6.1.4.2 Global Stiffness Matrix The global stiffness matrix [K] t is assembled from the individual space frame element stiffness matrices described in Section 6.1.1.1 and the foundation element stiffness matrix described in Section 6.1.2. The space frame element stiffness matrix is constant and it represents the stiffness of the bridge deck, pier or columns and the abutment wall. The nonlinear stiffness of the pile foundation is represented by the time dependent foundation stiffness matrix. The global stiffness matrix is banded and symmetric and only the lower half band is stored in vector form during computations. 6.1.4.3 Global Damping Matrix The global damping matrix is formed from the element damping matrices of the space frame and the pile foundation. The structural damping of the space frame is assumed to be of the Raleigh type. It is expressed as a linear combination of mass matrix [m] and stiffness matrix [k]. [c] = a[m]+p[k] (6.6) Chapter 6 : Seismic Response Analysis of Bridges 197 where a and P are scalar quantities. These scalar quantities can be determined by considering any two modes of the system and the corresponding frequencies as follows. (AjCOj-lcD.) a = 2co.G). J (c^-coj2) (A.co.-A.(o.) ( 6 J ) p = 2co.co. J ( o ^ - W j 2 ) where A ; and Aj are the damping ratios in the r* and j t h mode and co (and co jare the corresponding circular modal frequencies. The modal frequencies can be determined through an eigen analysis of the bridge structure. In the case of pile foundations, hysteretic and radiation damping are important, especially in strong motion analysis and they should be taken into account in a realistic manner. The pile foundations are represented in the mathematical model of the bridge as lumped parameter system and the damping matrix of this lumped parameter system is given in Eq. 6.4. This element damping matrix can directly be used in forming the global damping matrix of the bridge structure system. 6.1.5 Modes and Modal Frequencies The free vibrational modes and the corresponding modal frequencies can be determined by solving the standard eigen value problem, [[K] t-co 2[M]]v = 0 (6.8) where [K] t and [M] are global stiffness and mass matrices, v is the mode shape vector and a> is the corresponding modal frequencies. The global stiffness matrix [K] t is a time dependent parameter due to the nonlinear behaviour of the pile foundations and as a result the free vibration frequencies and mode shapes also vary with time. By carrying out eigen analysis at discrete time interval, the time dependent modal frequencies and the mode shapes can be determined. Chapter 6 '. Seismic Response Analysis of Bridges 198 6.1.6 Dynamic Solution Approach The coupled equations of motion of the bridge structure system subjected to rigid base excitation can be expressed in matrix form as follows. [M]{v} + [C] t {v} + [K] t {v} = -[M]{I}v b (6.9) In Eq. 6.9, the stiffness and damping matrices are not constant but vary with time. Thus, it is solved using direct step-by-step integration. By considering the equations of motion at time t and t+At, the equation of motion in the incremental form can be expressed as [M]{Av}+[C]TAv}+[K] t{Av} = -[M]{I}Av b ( 6 . i 0 ) Wilson-0 method is used to solve the above equation to determine the incremental response. During the time marching solution, dynamic equilibrium should be ensured at the end of each step by checking the equation of motion described in Eq. 6.9. In the present approach, the unbalanced force AF U is calculated at the end of each step. This unbalanced force is expressed as A F u = - [M]{Av}- [C] t {Av}- [K] t {Av}- [M]{I}Av b ( 6 . l l ) This unbalanced force is then added to the right-hand side of Eq. 6.11 in the next step of integration to satisfy the dynamic equilibrium. The finite element method described so far to obtain the nonlinear response of a bridge structure was implemented in a computer program called BRIDGE-NL. 6.2 NUMERICAL STUDIES ON THE AASHTO CODE BRIDGE 6.2.1 Introduction A three span continuous box girder bridge structure having six piers and two abutments was chosen for the numerical studies. This bridge is used as an example in the guide to the seismic design of bridges published by the American Association of State and Transportation Highway Officials( AASHTO, 1983). Chapter 6 : Seismic Response Analysis of Bridges 199 Details of Superstructure and Piers :The bridge structure and its dimensions and are shown in Fig. 6.2 . The sectional and physical properties of the superstructure and the piers are given in Table 6.1. Table 6.1 Properties of bridge superstructure and the pier Property Bridge Deck Bridge Pier Length, L or Height, H (m) 114.6 7.62 Cross-sectional area, A (m2) 11.43 1.21 Second moment of area about x axis, Ix (m4) 1.01 0.2244 Second moment of area about y axis, L, (m4) 565.65 0.1122 Second moment of area about z axis, \ (m4) 4.55 0.1122 Youngs Modulus, E c (kPa) 2.07e+07 2.07e+07 Mass per unit length, m (Mg/m) 30.37 3.20 Details of Pier Foundations : Each pier is supported on a group of sixteen (4x4) concrete piles. The piles are of diameter, d=0.36 m and length, L=7.2 m. The spacing of the pile is s=0.90 m. The Young's modulus and mass density of pile are E=22,000 MPa and p=2.6 M g m"3 respectively. The soil beneath the foundation is assumed to be a nonlinear hysteric continuum with unit weight, y=18 kN m~3 and Poisson's ratio, v=0.35. The low strain shear modulus of the soil varies as the square root of the depth with values of zero at the surface and 213 MPa at 10 m depth. The initial and maximum potential damping ratios are assumed to be 2% and 25% respectively. The variation of shear moduli and damping ratio with shear strain are those recommended by Seed and Idriss (1970) for sand. Chapter 6 '. Seismic Response Analysis of Bridges 200 Fig. 6.2 Details of AASHTOQ983) Bridge and the Pile Foundations Chapter 6 '. Seismic Response Analysis of Bridges 201 Input Acceleration Record : The input acceleration record used in the study was the first 20 seconds of N-S component of the free field acceleration recorded at CSMIP Station No. 89320 at Rio Dell, California during the April, 25, 1992 Cape Mendocino Earthquake (Fig. 6.3a). The power spectral density of this acceleration record in Fig. 6.3b shows that the predominant frequency of the record is approximately 2.2 Hz. 6.2.2 Finite Element Model of the Bridge Structure Structural Model: The idealized three dimensional space frame model of the bridge is shown in Fig. 6.4. The bridge deck was modeled using 13 beam elements and each pier was modeled by 3 beam elements. The cap beam that connects the tops of adjacent piers was modeled using a single beam element. The sectional and physical properties of the deck and the piers are those provided in the AASHTO guide. The sectional properties of the cap beam were selected so that the rigidity of the bridge deck is properly modeled. Pier Foundation Model : The pier foundation is modeled using a set of nonlinear springs and dashpots derived using P1LE3D and PILIMP analyses. Abutment End Connections : At the abutments the deck is free to translate in the longitudinal direction but restrained in the transverse and vertical directions and rotation of the deck end is allowed about all three axes. 6.2.3 Nonlinear Stiffness and Damping of Pile Foundation Using PILE3D and PILIMP The time histories of stiffnesses and dampings of the pile foundation was determined using the procedure described in Chapter 5. As all the pier foundations are identical, only a single pile foundation needs to be analyzed. Only one half of the system was included in the analysis because Chapter 6 '. Seismic Response Analysis of Bridges 202 c n c o 03 _CD CD O O < 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 , 1 , 1 i 1 1 , 1 , 1 , 1 , 8 10 12 Time (sec) 14 16 18 20 Fig . 6.3a Input Acceleration Time History TJ OT CD c . « CD ro Q E 75 ^ CD ~ Q . c : CO CD O C L 0.5 0.0 --I t 'A A V 4 6 Frequency (Hz) 10 Fig . 6.3b Unit Normalised Powerspectral Density of Input Acceleration Chapter 6 '. Seismic Response Analysis of Bridges 203 B e n t ! Bent 2 Fig. 6.4 Stick Model of the Bridge with the Foundation Springs and Dashpots Chapter 6 : Seismic Response Analysis of Bridges 204 of symmetry and the direction of shaking was assumed to be along one of the principal axes of the pile foundations. The principal axes of the pile foundations and that of the bridge structure were similar. The soil layer overlying a hard stratum at 10m was divided into 10 layers with varying thicknesses. The thickness decreased towards the surface because soil-pile interaction effects are stronger at shallower depth. A total of 900 brick elements was used to model the soil around the piles and 64 beam elements were used to model the piles. The acceleration record shown in Fig 6.3. was taken as the free field surface acceleration. The input acceleration at the rigid stratum (at 10m depth) was calculated by deconvolution using the computer program, 'SHAKE' (Schnabel and Seed, 1972). The variation of lateral, cross-coupling and rocking stiffnesses with time is shown in Fig. 6.5. 6.2.4 Stiffness and Damping Based on "SHAKE" Moduli And Damping Ratio The effect of earthquake loading in the free field soil was obtained using 'SHAKE' (Schnabel and Seed, 1972). This equivalent linear analysis program gives a vertical distribution of shear moduli and damping ratios that are compatible with an effective strain of 65% of the maximum shear strain developed in each layer. The pile head impedances based on these moduli and damping ratio will reflect the effects of free field conditions on the pile head impedances. 1-D ground motion analyses were carried out under the peak free field accelerations of 0.5g as in the PILE3D analysis. The soil layers in the PILE3D model was also used. Fig 6.5a shows the shear moduli profile and Fig. 6.5b shows the damping ratio profiles. Fig. 6.5 a also shows the initial shear moduli profile. Chapter 6 : Seismic Response Analysis of Bridges 205 to CO CD cz CO 1200 1000 800 600 400 200 Using Moduli and Damping From S H A K E Analysis Lateral (MN/m) Cross-Coupling (MNm/rad) From Direct PILE3D Analysis Lateral (MN/m) Cross-Coupling (MNm/rad) 0 2 4 6 8 10 12 14 16 18 20 Time (sec) Fig. 6.5a Lateral and Cross-Coupling Stiffness Time Histories for Amax=0.5g CO co CD c CO "co c o CD -4—' O 8000 6000 F 4000 h 2000 From Direct PILE3D Analysis Using Moduli and Damping from S H A K E Analysis 8 10 12 Time (sec) 14 16 18 20 Fig. 6.5b Rotational Stiffness Time Histories for Amax=0.5g Chapter 6 '. Seismic Response Analysis of Bridges 206 The constant elastic stiffness based on the 'SHAKE' moduli is shown in Fig. 6.6. It can be seen that the lateral stiffness based on the 'SHAKE' moduli are greater than the stiffnesses from PILE3D during the period of strong shaking but are otherwise smaller The same trend was observed in the cross-coupling and rocking stiffnesses also. The stiffnesses from the PILE3D analyses and those based on the 'SHAKE' moduli are summarized in Table 6.2. The minimum stiffness value from the P1LE3D analysis and the stiffnesses using the 'SHAKE' moduli are given as percentage of initial low strain stiffness. Under a peak acceleration of 0.5 g, the minimum PILE3D lateral stiffness was 43% of the initial stiffness and the lateral stiffness based on 'SHAKE' moduli was 68%. The reduction in cross-coupling and rocking stiffness were also significant. Table 6.2 Summary of Stiffnesses Method Translational Cross-coupling Rotational Vertical Stiffness Stiffness Stiffness Stiffness (MN/m) (%)' (MNm/rad) (%y (MN/rad) (%)' (MN/m) (%)' Method 1 868 410 6181 4202 Method2 373 43 237 58 3337 54 1890 45 Method3 590 68 299 73 4388 71 2563 61 1 As a percentage of low strain stiffness Method 1: Stiffness based on the initial low strain shear moduli Method 2: Minimum stiffness based on the shear moduli from the PILE3D analysis Method 3: Stiffness based on the shear moduli from the S H A K E analysis 6.2.5 Seismic Response of Bridge Under Transverse Earthquake Loading The bridge structure was analyzed under transverse earthquake loading with different foundation conditions to study the influence of foundations using the computer program BRIDGE-NL. Chapter 6 : Seismic Response Analysis of Bridges 207 Fig. 6.6a Shear Modulus Profiles Fig. 6.6b Damping Ratio Profie from S H A K E Chapter 6 '. Seismic Response Analysis of Bridges 208 The idealized model of the bridge structure is shown in Fig. 6.7. The free field acceleration was used as the input acceleration and the peak acceleration was set to 0.5 g. The different foundation conditions used in the analyses are listed below. 1. Rigid foundation and fixed base condition is assumed. 2. Flexible foundation with elastic stiffness and damping 3. Flexible foundation with time dependent stiffness and damping 4. Flexible foundation with stiffness and damping based on the 'SHAKE' moduli and damping ratio. The fundamental transverse mode frequency of the bridge structure with fixed base was found to be 3.18 Hz. This frequency agreed with the frequency quoted in the AASHTO-83 guide . When the foundations were modeled using the low strain stiffnesses in Table 6.2, the flexible base model frequency was found to be 3.08Hz , very close to the fixed base frequency. This indicates an acceptable structural model. The stiffness time histories shown in Fig. 6.6, were used to obtain the time history of first transverse mode frequency shown in Fig. 6.8. The fixed base and the linear elastic stiffness model frequencies are also shown for comparison. When variable stiffnesses are used, the frequency gradually reduced from its elastic value of 3.08Hz at the beginning to the value of 3.01Hz during strong shaking. Then, with the reduction in the level of shaking, the frequency gradually increased to its elastic value. When the elastic stiffnesses based on the 'SHAKE' moduli were used, the fundamental frequency of the bridge was 3.05Hz. This frequency is also shown in Fig. 6.8 for comparison. Fig. 6.8 shows that the maximum change in frequency was only 3% when the foundation was included in the bridge model. The reason for this is that the columns are very flexible compared to the foundations (The ratio of fixed end pier stiffness to the elastic lateral stiffness of the foundation is 7% only) and, for this bridge model, the columns control the fundamental frequency of the bridge. When the column stiffness was increased to 50% of the elastic lateral stiffness of the foundation major changes in the fundamental frequencies occurred for different foundation conditions. In this Chapter 6 '. Seismic Response Analysis of Bridges 209 X Bridge Deck > '1 Abutment 1 Bent 1 Bent 2 Abutment 2 Vertical -y Direction z- Transverse Direction Fig. 6.7 Bridge Model Used in the Transverse Vibration Chapter 6 '. Seismic Response Analysis of Bridges 2 1 0 Transverse Vibration Kcolumn = Y °/0 Kfoundation Amax = 0 5 9 3.6 3.4 3.2 3.0 Constant stiffness and damping based on the initial shear moduli and damping ratios Constant stiffness and damping based on the shear moduli and damping ratios from 'SHAKE' analysis Variable stiffness and damping based on the shear moduli and damping ratios from 'PILE3D' analysis Fixed Base Frequency = 3.18 Hz 2.8 J i i i_ J i i i_ _i i i i_ J I I L_ 10 Time (sec) 15 20 Fig. 6.8 The Effect of Support Conditions on the First Mode Frequency Chapter 6 '. Seismic Response Analysis of Bridges 211 case, the fixed base model fundamental frequency was 5.82Hz. At low strain initial stiffness the fundamental frequency reduced to 4.42Hz, a 24% reduction from the fixed base model frequency. For the strain dependent stiffnesses, the frequency reduced to a minimum value of 3.97Hz during strong shaking, a 32 % reduction from fixed base frequency. When the foundation stiffnesses were based on the shear moduli from the 'SHAKE' analysis the frequency reduced to 4.18Hz, a 28 % change from the fixed base model frequency. Fig. 6.9 shows the variation in fundamental modal frequency with time for different foundation conditions. The frequency results under different foundation conditions and pier stiffnesses are summarized in Table 6.3. This table shows the importance of including the foundations in the structural model and the effect of different foundation conditions. The effects of the factors such as the relative stiffness between the pier and foundation and the level shaking are also shown. Table 6.3 Summary of First Transverse Mode Frequencies under Different Support Conditions Case First Mode Frequency Fixed Supports (Hz) Flexible Supports Method 1 Method 2 Method3 (Hz) (%)' (Hz) (%y (Hz) (%y Case 1 Kc/K F =7% 3.183 3.080 3 3.013 5 3.046 4 Case 2 K C /K F =50% 5.822 4.425 24 3.972 32 4.183 28 1 As percentage of fixed base frequency Method 1: Foundation stiffness and damping are based on the low strain shear moduli and damping. Method 2: Minimum foundation stiffnesses and damping are based on shear moduli and damping from PILE3D analysis Method 3: Foundation stiffnesses and damping are based on shear moduli and damping from S H A K E analysis Chapter 6 '. Seismic Response Analysis of Bridges 212 N IE >. O c CD cr CD CD TD O CD > to c CO to 5.5 5.0 h 4.5 4.0 3.5 Transverse Vibration Kcolumn Kfoundation Amax = 50 % = 0.5g Constant stiffness and damping based on the initial shear moduli and damping ratios Constant stiffness and damping based on the shear moduli and damping ratios from 'SHAKE' analysis Variable stiffness and damping based on the shear moduli and damping ratios from 'PILE3D' analysis Fixed Base Frequency = 5.82 Hz 10 Time (sec) 15 20 Fig. 6.9 The Effect of Support Conditions on the First Mode Frequency Chapter 6 '. Seismic Response Analysis of Bridges 213 The time history response analysis of the bridge structure was also performed under different foundation conditions. The response of the bridge deck at the bent 2 (Node no. 5 in Fig 6.4) was evaluated with and without including the foundation flexibility. When the foundation flexibility was included in the model, the strain dependent stiffness and damping were used in the analysis. The effect of including the foundation flexibility is shown in Fig 6.10. There is a dramatic change in the deck displacement during the strong shaking when the foundation flexibility is included in the model. The peak displacement increased from 7mm to 17mm. An elastic analysis was carried out using the low strain elastic stiffness and damping instead of using the variable stiffness and damping. The displacement response of the deck at the bent 2 (Node No. 5 of Fig. 6.4) is shown in Fig. 6.11. The response for variable stiffness and damping from Fig. 6.10 are also shown for comparison. The elastic response underestimates the peak displacement. The stiffness and damping based on the 'SHAKE" moduli and damping ratio account for the effect of earthquake in the free field. An elastic analysis was carried out using this stiffness and damping and the deck response at the bent 2 was compared with the response shown in Fig. 6.10. The comparison is shown in Fig 6.12. Only a slight difference between the two responses can be observed from this figure that shows that the stiffness and damping based on the 'SHAKE' moduli and damping ratio are good average of the variable stiffness and damping evaluated from PILE3D. It should be noted here that, in the PILE3D analysis of the pile foundations, the effect of the superstructure interaction was not taken into account. The effect of superstructure interaction on the stiffness of the pile foundation and the frequency response of the bridge are studied later in a subsequent section. Chapter 6 '. Seismic Response Analysis of Bridges 214 co E c g "co 0) CD O O < Time (sec) Fig . 6.10a Comparison o f Acceleration Response of the Bridge Deck c CD E cu o 03 C L CO Q 0.02 0.00 -0.02 Fixed Supports Spring and Damper Supports with variable Stiffness and Damping j I I L J I I L _L J L Amax=0.5g J I I L 10 Time (sec) 15 20 Fig . 6.10b Comparison of Displacement Response ofthe Bridge Deck Chapter 6 : Seismic Response Analysis of Bridges 2 1 5 1 2 tn E c o ro i_ 0 0 o o < Variable Stiffness and Damping Based on Shear Moduli and Damping Ratio from PILE3D Analysis Constant Stiffness and Damping Based on Initial Shear Moduli Damping Ratio Fig. 6.1 l a Comparison of Acceleration Response of the Bridge Deck 0 . 0 2 c E 0 o _ro CL W b 0 . 0 0 -0.02 1 0 Time (sec) 15 2 0 Fig. 6.1 lb Comparison of Displacement Response of the Bridge Deck Chapter 6 '. Seismic Response Analysis of Bridges 216 CM CO c o ro i _ 0 0 o o < 12 8 7 4 1 Variable Stiffness and Damping Based on Shear Moduli and Damping Ratio from PILE3D Analysis Constant Stiffness and Damping Based on Shear Moduli Damping Ratio from SHAKE Analysis Fig. 6.12a Comparison of Acceleration Response of the Bridge Deck Fig. 6.12b Comparison of Displacement Response of the Bridge Deck Chapter 6 '. Seismic Response Analysis of Bridges 217 6.2.6 Seismic Response of Bridge Under Longitudinal Earthquake Loading The idealization of the bridge to study the effect of earthquake loading in the longitudinal direction is shown in Fig. 6.13. The bridge deck ends are supported on rollers at the abutments and therefore abutments do not contribute to the longitudinal stiffness of the bridge. This will simulate a situation where the abutments are of seated type and the expansion gaps do not close under earthquake loading. The frequency response of the bridge was analyzed under the foundation conditions described earlier. The same free field acceleration was used with peak acceleration set to 0.5 g. As the pier foundations are of square type (4x4 pile group), the same stiffness and damping results described in Section 6.2.3 may be used. The fundamental longitudinal mode frequency of the fixed base model was 4.37Hz, when the ratio of pier to foundation stiffness was 50%. The initial low strain fundamental frequency is 3.01Hz, a 31% reduction from the fixed base model frequency. When the stiffnesses were based on the 'SHAKE' moduli, the frequency further reduced to 2.77Hz, a 37% change from the fixed base model frequency, and 8% change from the elastic foundation model frequency. When the variable stiffnesses from PILE3D shown in Fig. 6.5 were used, the frequency reached a minimum value of 2.55Hz, a 42% change from the fixed base frequency and a 15% change from elastic foundation model frequency. Fig. 6.14 summarizes all the frequency results under different foundation conditions. Fig. 6.15 shows the displacement and acceleration response of bridge deck at bent 2 (Node No.5 in Fig. 4) for the fixed base model and the flexible base model with variable stiffness and damping. The two responses are significantly different and the response is highly underestimated when rigid base model is used. The frequency results under different foundation condition and two different stiffnesses are summarized in Table 6.4. When the columns are relatively soft compare to foundation the Chapter 6 '. Seismic Response Analysis of Bridges 218 Bridge Deck Vertical -y Direction x- Longitudinal Direction Pier Fig. 6.13 Bridge Model Used in the Longitudinal Vibration Chapter 6 '. Seismic Response Analysis of Bridges 219 4.0 Longitudinal Vibration Kcolumn _ gQ Kfoundation Amax = 0 5 9 3.5 3.0 2.5 2.0 Constant stiffness and damping based on the initial shear moduli and damping ratios Constant stiffness and damping based on the shear moduli and damping ratios from 'SHAKE' analysis Variable stiffness and damping based on the shear moduli and damping ratios from 'PILE3D' analysis Fixed Base Frequency = 4.37 Hz 10 Time (sec) 15 20 Fig. 6.14 The Effect of Support Conditions on the First Mode Frequency Chapter 6 : Seismic Response Analysis of Bridges 220 CM CO E, c o "•4—» CD _CU CU O O < 15 5 h -15 Fixed Supports Spring and Damper Supports with variable Stiffness and Damping Amax=0.5g J L _L J I I L 10 Time (sec) 15 20 Fig. 6.15a Comparison of Acceleration Response of the Bridge Deck c CD E cu o TO C L CO b 0.05 0.00 -0.05 Fixed Supports Spring and Damper Supports with variable Stiffness and Damping Amax=0.5g J L 10 Time (sec) 15 20 Fig. 6.15b Comparison of Displacement Response of the Bridge Deck Chapter 6 : Seismic Response Analysis of Bridges 221 maximum change in frequency reached 14%. When the piers are relatively stiff compare to the foundation, the maximum change in frequency increased to 42%. This shows that the flexibility of the foundation plays a greater role in the response of the bridge when the piers are stiff compared to the foundations. Table 6.4 Summary of First Longitudinal Mode Frequencies under Different Support Conditions Case First Mode Frequency Fixed Supports Flexible Supports Method 1 Method 2 Method3 (Hz) (%)' (Hz) (%)' (Hz) (%)' Case 1 K C / K F = 7 % A m a x=0.5g 1.659 1.525 8 1.434 14 Case 2 Kc/K F =50% Am a x=0.5g 4.365 3.013 31 2.550 42 2.769 37 1 As percentage of fixed base frequency 6.2.7 Seismic Soil-Pile-Superstructure Interaction Analysis of Bridge 6.2.7.1 Mathematical Model for the Superstructure In analyzing the effect of earthquake loading on the foundation stiffness and damping, the superstructure interaction should be taken into account in addition to the soil-pile interaction. The analyses described so far considered soil-pile interaction only. As such, in those analyses, the foundation was analyzed separately without any superstructure. The ideal way of capturing the effects of superstructure interaction is to consider the bridge structure and the foundation as a fully coupled system in the finite element analysis. However, such fully coupled analysis is not practically feasible because it requires enormous amount of computational storage, effort and time. An approximate way of including the effect of superstructure interaction which is used by CalTrans Chapter 6 '. Seismic Response Analysis of Bridges 222 for design (Abghari and Chai, 1995) has been adopted. The approximate model is shown in Fig. 6.16. In this model, the superstructure is represented by a single degree of freedom (SDOF) system. The mass of the SDOF system is assumed to be the static portion of the superstructure mass carried by the foundation. The stiffness of the SDOF system is selected so that the SDOF system has the first mode period of the fixed base bridge structure . 6.2.7.2 Superstructure Model Parameters The approximate approach will be demonstrated by the analysis of the center pier at bent 2. The fundamental transverse mode frequency of the fixed base model was found earlier as 5.82Hz. The corresponding mode shape is shown in Fig. 6.17. The static portion of the mass carried by the center pier at bent 2 is 370 Mg. The superstructure can be represented by a SDOF system having a mass of 370 Mg at the same height as the pier top and frequency 5.82 Hz. The corresponding stiffness of the SDOF system is 280 MN/m. 6.2.7.3 Seismic Response of the Bridge When the Effect of Superstructure Interaction on the Foundation is included A coupled soil-pile-structure interaction analysis can be carried out using PILE3D by incorporating the SDOF model into the finite element model of the pile foundation. The time histories of stiffnesses with and without the superstructure are shown in Fig. 6.18. The reduction in lateral stiffness was higher throughout the shaking when the inertial interaction was included. A similar effect was noted in the rotational and cross-coupling stiffnesses. When inertial interaction was included, the lateral stiffness reached a minimum of 188MN/m which is 78% lower than the initial value. This minimum was 20% lower than the minimum that was attained when the inertial interaction was not included. Chapter 6 '. Seismic Response Analysis of Bridges 223 E q u i v a l e n t S D O F s y s t e m M superstructure "^"superstructure te Fig. 6.16 Model used for the Pile-Soil-Superstructure Interaction Analysis using PILE3D Abutmentl Abutment4 Bent2 Bent3 Modal Frequency = 5.82Hz Fig. 6.17 First Transverse Mode Shape of the Bridge Chapter 6 : Seismic Response Analysis of Bridges 224 Transverse Vibration Kcolumn _ Kfoundation Amax = 0-5 g 1000 — 800 h ^ to CO CD c i t CO CD ro 600 k= 400 200 Shear Moduli and Damping Ratios from SHAKE Analysis Direct Non-Linear Analysis by PILE3D Without Superstructure With Superstructure - 1 — i — i — i — r ~ 0 5 T 1 1 1 1 1 1 1 1 1 1 1 r~ 10 15 20 Time (sec) Fig. 6.18 The effect of Superstructure Interaction on the Lateral Stiffness of the Pile Foundation Chapter 6 '. Seismic Response Analysis of Bridges 225 An eigen analysis of the complete bridge structure was carried out with the newly found stiffness of the foundations. The variation in first mode transverse frequency with time is shown in Fig. 6.19. This figure also shows the frequency variation for the case in which the inertial interaction was not considered. The frequency reached a minimum of 3.62Hz when the inertial interaction was included and 3.97 Hz when the interaction was ignored. Fig. 6.20 shows the effect of superstructure interaction on the time histories of acceleration and displacement. When the superstructure interaction effect is included it leads to greater acceleration and displacement. The increase in peak displacement is approximately 72%, a major increase. 6.3 PARAMETRIC STUDY 6.3.1 Introduction In this section a parametric study is conducted using a simplified bridge model to quantify the role foundation stiffness and other key parameters on the seismic response of a simple two span bridge. The results of the study are useful in selecting approximate methods for analysis and in understanding the global response of different types of bridges in a general way. The two span bridge is shown in Fig. 6.21. It has a single column bent at the center and two seated type abutments at each end. The center bent is supported by a pile group foundations. This bridge is similar to the one studied by Lam and Martin (1986) and Imbsen and Penzien (1984). 6.3.2 Mathematical Model of the Bridge A simplified mathematical model of the .bridge is shown in Fig. 6.22. The bridge deck is represented by a concentrated mass, M s corresponding to the static load carried by the pier and by a mass moment of inertia, I s at the top of the pier. The flexibility of the pile foundations is represented by a set of linear springs with lateral stiffness, K F L , cross-coupling stiffness, and the rocking stiffness, K F R Chapter 6 : Seismic Response Analysis of Bridges 226 5.5 5.0 Transverse Vibration Kcolumn _ gQ cyQ Kfoundation Amax = 0 5 9 Without Superstructure Effect With Superstructure Effect Fixed Base Frequency = 5.82 Hz 4.5 h 4.0 3.5 J I L J I I I I I I I l_ J I I l_ 10 Time (sec) 15 20 F i g . 6.19 The Effect o f Superstructure Interaction on the First Mode Frequency Chapter 6 '. Seismic Response Analysis of Bridges 221 Time (sec) Fig. 6.20a The Effect of Superstructure Interaction on the Acceleration Response of the Bridge Deck Fig. 6.20b The Effect of Superstructure Interaction on the Displacement Response of the Bridge Deck Chapter 6 : Seismic Response Analysis of Bridges 228 Bridge Deck Seat Type Abutment Bridge Pier Pile Foundation Fig . 6.21 A Typical two Span Bridge Mass of Superstructure Rotational Deck- K s r Stiffness Bridge Pier Rotational Foundation- «F Stiffness F K S A -Lateral Abutment Stiffness K F -Lateral Foundation Stiffness |) K F -Cross-Coupling \ / L R Foundation Stiffness Fig . 6.22 Mathematical M o d e l o f the Bridge Used in the Parametric Study Chapter 6 '. Seismic Response Analysis of Bridges 229 As the ends of the bridge deck are supported on seat type abutments there will be horizontal forces on the bridge deck whenever the gap between the deck and the abutment closes. This mechanism of force development is modeled by a linear spring with stiffness, K S A . If the bridge deck is rigid the condition at the top of the bridge pier can be considered as fixed. If it is flexible the rotational resistance of the bridge deck at the top of the pier can be represented by a rotational spring with stiffness, K S D . Stiffness matrix of the pier, K p : The (4x4) stiffness matrix of the pier can be expressed in terms of the Young's Modulus, E, second moment of area, I and the height, L as, 12 6L -12 6L EI 6L 4 L2 -6L 2 L 2 L 3 -12 -6L 12 -6L 6L 2 L 2 -6L 4 L 2 (7.12) Mass matrix of the pier, M p : The (4x4) consistent mass matrix of the pier can be expressed in terms of the height, L and the mass per unit length, m as M p 156 22L 54 13L mL 22L 4 L2 13L 3 L 2 120 54 13L 156 22L 13L 3 L 2 22L 4 L 2 (7.13) Pile foundation stiffness matrix, K F : The stiffness of the pile foundation is represented by equivalent compliance springs and the corresponding 2x2 stiffness matrix of the pile foundation can be expressed as Chapter 6 '. Seismic Response Analysis of Bridges 230 K V F v F K L LR LR K R (7.14) Stiffnesses of the abutment and bridge deck: The lateral stiffness provided by the abutment to the deck is K S L and rotational stiffness of the deck is K S R Global stiffness matrix of the bridge, K G : By assembling the individual stiffness matrices, the global stiffness matrix can be obtained and it is expressed as K G _ 12EI /L 3 +K S , 6EI/L 2 -12EI/L 3 6EI/L 2 6E1/L 2 4EI/L+K S , 12EI/L 3 -6EI /L 2 6EI/L 2 2EI/L -6EI/L 2 12EI7L 3 +K F L -6EI/L 2 +K F L R 2EI/L -6EI7L 2 +K F L R 4EI/L+K F L R (7.15) Global mass matrix of the bridge, M G : By assembling the individual mass matrices, the global mass matrix can be obtained and it is expressed as M 39mL/30+M s l l m L 2 / 6 0 27mL/60 13mL2/120 l l m L 2 / 6 0 mL 2 /30 2 +I s 13mL2/120 mL 3 /40 27mL/60 13mL2/120 39mL/30 l l m L 2 / 6 0 13mL2/120 mL 3 /40 l l m L 2 / 6 0 mL 3 /30 (7.16) The free vibration frequencies of the bridge model, co can be obtained by solving the following equation. | [ K G ] - [ M G ] * c o 2 | = 0 Chapter 6 : Seismic Response Analysis of Bridges 231 6.3.3 Effect of Foundation Flexibility 6.3.3.1 Introduction When the foundation flexibility is included in the bridge model it alters the fundamental period of the bridge structure. The ratio, Tp/TF which is the ratio of the period of fixed base structure over the period of the flexible base structure is the basic parameter used to assess the effect of foundation flexibility on the bridge response. The ratio, Tp/TF depends on the stiffness of the pier, K p , height of the pier, H p , mass of the superstructure , M s , the lateral foundation stiffness, K L F and the rotational foundation stiffness, K F R . The ratio also depends on the lateral stiffness of the deck due to abutment resistance , K S A , and the rotational stiffness of the deck, K S D . By nondimensionalizing these parameters, the dependency on their individual magnitudes can be removed and their effect on the period ratio, Tp/TF can be assessed. The column 1 in Table 6.5 lists the parameters and the column 2 shows how they are nondimensionalized for the parametric study. Table 6.5 Nondimensional parameters chosen for the parametric study Parameter Nondimensional Parameter Foundation flexibility Tp/TF Lateral stiffness of the foundation K P / K F L Rotational stiffness of the foundation K P * H 2 / K F R Mass of the superstructure M7pd 3 Height of the pier H p /d Lateral abutment stiffness K S A / K F L Rotational deck stiffness K V K F L In Table 6.5, d is the diameter of the foundation pile. 6.3.3.2 Foundation Stiffness Parameters The stiffness of the pile group foundation was determined using the computer program PILIMP. Chapter 6 '. Seismic Response Analysis of Bridges 232 A (2x2) pile group and a (3x3) pile group were chosen for the study. Each of this pile group was founded on three different soil profiles; one with parabolic G m a x and two with uniform G m a x where G m a x is the shear modulus of the soil. Table 6.6 summarizes the six cases for which the foundation stiffnesses were calculated using the program PILIMP. In Table 6.6. E p and E s are the Youngs modulus of the pile and soil respectively, S is the spacing between piles, D is the diameter of the pile, L is the length of the pile and p p and p s are the densities of the pile and soil respectively. Table 6.7 shows the foundation calculated stiffnesses for the six different cases. Table 6.6 Foundation Parameters chosen for the study Case No of Piles Gmax Profile Stiffness Spacing Depth Density Ratio,E p/E s Ratio, S/D ratio, L /D Ratio, pp/ps 1 (2x2) Uniform 100 5 15 0.7 2 (2x2) Uniform 1000 5 15 0.7 3 (2x2) Parabolic 312 5 15 0.7 4 (3x3) Uniform 100 4 15 0.7 5 (3x3) Uniform 1000 4 15 0.7 6 (3x3) Parabolic 312 4 15 0.7 Table 6.7 The foundation stiffnesses calculated for the six different cases. Case Lateral Stiffness Cross-Coupling Rotational Stiffness (MN/m) Stiffness (MNm/rad) (MN/rad) 1 704 364 2440 2 507 118 1244 3 273 99 996 4 1075 578 7375 5 858 177 4169 6 507 178 3412 Chapter 6 '. Seismic Response Analysis of Bridges 233 6.3.3.3 Superstructure and Pier Parameters The mass of the superstructure, the height of the pier and the diameter of the pier were varied. Table 6.8 summarizes the range of parameters used in the parametric study. Table 6.8 Superstructure and pier parameters Parameter Values Mass of superstructure, M (Mg) 500, 750, 1000 Young's modulus of pier, E (MPa) 25000 Density of pier, p(Mg/m3) 2.5 Height of pier, H (m) 5, 10, 15, 20 Diameter of pier, D (m) 1.25, 1.50, 1.75 6.3.4 Effect of Different Parameters on Bridge Response 6.3.4.1 Effect of Nondimensional Stiffness Parameter, K P / K F L The period ratio, T p / T F is plotted against the stiffness ratio, K P / K F L in Fig. 6.23a. Clearly the foundation flexibility can have a major effect on the period of the structure. When the foundation stiffness is of the same order as the pier stiffness the period ratio, T p / T F is approximately 0.6. The results shown in Fig. 6.23 are replotted on a semi logarithmic scale in Fig. 6.24. This figure indicates that the decrease in the period ratio, T p / T F is linearly related to the logarithmic of the stiffness ratio, K P / K F L for the range of parameters considered in this analysis. 6.3.4.2 Effect of Nondimensional Stiffness Parameter, K P * H 2 / K F R Fig. 6.25 shows the effect of nondimensional stiffness parameter K P * H 2 / K F R , on the period ratio Tp/T F . The trend is similar to the one induced by the first parameter, K P / K F L . However, the drop in the period ratio, T p / T F is not as dramatic as it was observed for K P / K F L . Chapter 6 '. Seismic Response Analysis of Bridges 234 co Od TJ g cu 0-— I > 1 ' 1 ' 1 ' — 0.0 1.0 2.0 3.0 4.0 5.0 Nondimensional Stiffness Parameter, K S P / K F L Fig. 6.23 Effect of Nondimensional Parameter, K S p / K F L o n the Period Ratio 0.3 —j i i i i i > 111 i i i i 11111 1—i—i i 11111 1—i—i i 11111 0.001 0.01 0.1 1 10 Nondimensional Stiffness Parameter, K S P / K F L Fig. 6.24 Effect of Nondimensional Parameter, K S p / K F L o n the Period Ratio Chapter 6 '. Seismic Response Analysis of Bridges 235 6.3.4.3 Effect of Nondimensional Parameter, H p/d The height of the pier also an important parameter which affects the period of the superstructure itself. Higher the pier, the more flexible is the superstructure compare to the foundation. Fig. 6.26 shows that the nondimensional parameter, H p /d is small the period ratio is also small and when it increases the period ratio also increases. 6.3.4.4 Effect of Nondimensional Stiffness Parameter, MVpd3 This nondimensional parameter reflects the effect of superstructure mass, M s . Fig. 6.27 shows the effect of this parameter on the period ratio. As expected, the change in this parameter does not change the period ratio. 6.3.4.5 Effect of Nondimensional Stiffness Parameter, K S A / K F L Fig. 6.28 shows the effect of the nondimensional stiffness parameter K S A / K F L on the period ratio. This parameter reflects the effect of abutment lateral stiffness. When there is no lateral resistance from the abutments, the period ratio is less than one. When the lateral abutment stiffness is introduced it increases the period ratio dramatically and brings it close to one. 6.3.4.6 Effect of Nondimensional Stiffness Parameter K S , / K F L The nondimensional parameter, K ^ K ^ reflects the effect of the rotational stiffness of the bridge deck induced by the flexibility of the bridge deck. Fig. 6.29 shows the effect of this parameter. The increase in the nondimensional parameter, K S R / K F L causes only a slight increase in the period ratio. Chapter 6 '. Seismic Response Analysis of Bridges 236 i ' 1 1 1 1 r 0.0 1.0 2.0 3.0 4.0 5.0 Nondimensional Stiffness Parameter, K s p * H 2 / K F R Fig. 6.25 Effect of Nondimensional Stiffness Parameter, K s p * H 2 / K F R on the Period Ratio 1.0 20 40 60 80 100 Nondimensional Parameter, HP/d Fig. 6.26 Effect of Nondimensional Parameter, H p / d on the Period Ratio Chapter 6 '. Seismic Response Analysis of Bridges 237 co IX. TJ o ' i CD D_ 1.0 0.9 H 0.8 0.7 0.6 0.5 0.4 0.3 5000 A A A A A A A A A A A A A A A A A A A A * * * * * * * * * * * * * * * * * * * * • • • • • • • • • • • • • • • • • • • • H=10m,d=1.5m -#— Case 1 (2x2) Ep/Es= 1000 Case 2(2x2) Ep/Es=100 — Case 3(2x2) Parabolic Es - ± — Case 4(3x3) Ep/Es= 1000 nfr— Case 5(3x3) Ep/Es= 100 1 " 1 1 1 10000 15000 20000 Nondimensional Parameter, M s /pd 3 25000 Fig . 6.27 Effect o f Nondimensional Parameter, M s / p d 3 on the Period Ratio 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Nondimensional Stiffness Parameter, K S A / K F L Fig . 6.28 Effect o f Nondimensional Stiffness Parameter, K S A / K F on the Period Ratio Chapter 6 '. Seismic Response Analysis of Bridges 238 1.0 0.9 0.8 0.7 H 0.6 0.5 0 Case 1(2x2) Ep/Es=1000 Case 2(2x2) Ep/Es=100 Case 3(2x2) Parabolic Es Case 4(3x3) Ep/Es=1000 Case 5(3x3) Ep/Es=100 Ms=750 Mg,H=10m,d=1.5m ~\ 1 1 r 5 10 15 20 Nondimensional Stiffness Parameter, K S D / K F L Fig. 6.29 Effect of Nondimensional Stiffness Parameter, K s D / K F L o n the Period Ratio Chapter 6 '. Seismic Response Analysis of Bridges 239 6.4 CONCLUSIONS A method was developed for the three dimensional nonlinear analysis of bridges. A 3D space frame stick model was used to model the superstructure and the foundations were modeled using nonlinear springs and dashpots. Analysis on the AASHTO(1983) bridge showed that the fundamental frequency of the bridge is reduced during strong shaking due to the degradation of the foundation stiffness. The reduction in the stiffness of the foundation stiffnesses is mainly caused by the nonlinear behaviour of the soil. It was also found that the inertial interaction alters the stiffness of the foundation during strong shaking and as a result the fundamental frequency of the bridge is changed. Significant differences were observed between the fixed base and flexible base model responses when the pier of the foundation is relatively stiff compared to the foundation. When the pier is flexible the difference between the response of the two models were minimal. A parametric study was conducted to investigate the effect of the foundation and superstructure stiffness parameters on the dynamic response of a bridge showed the following. The foundation flexibility elongates the period of the structure. The lateral foundation stiffness has significant effect on the period of the structure when its magnitude is not extremely high and comparable to the stiffness of the pier. The decrease in foundation stiffness causes a dramatic increase in the period of the structure. The rotational stiffness of the foundation also has the same effect as the lateral stiffness. However, its effect is not as significant as the one induced by the lateral foundation stiffness. The superstructure mass does not have any effect on the period ratio as the analysis is linear elastic. The height of the pier has significant effect on the period of structure . When the pier height increases it makes the superstructure flexible and reduces the effect of the foundation stiffness. The lateral abutment stiffness, if present, has important effect on the period of the structure. The increase in the lateral abutment stiffness drastically reduces the effect of foundation flexibility and brings the period of the structure closes to the fixed base period. The rotational stiffness of the deck has minimal effect on the period of the structure. CHAPTER 7 Stiffness and Damping of Bridge Abutments 241 C H A P T E R 7: S T I F F N E S S A N D D A M P I N G O F B R I D G E A B U T M E N T S In the design of bridges, sometimes simplified boundary conditions are assumed at the abutment ends. When the vibration of the bridge in the longitudinal direction is considered roller supports are assumed at the abutment ends and when the vibration in the transverse direction is considered pinned end conditions are assumed. These simplified boundary conditions may be appropriate in cases where the actual condition is close to the simplified boundary condition. An example of this situation is a seat type abutment with a gap between the abutment and bridge deck that does not close during shaking. However, the simplified boundary conditions may not be appropriate if the flexibility of the abutments play an important role in the dynamic behaviour of the bridge. The monolithic abutment of a short stiff bridge is an example of this situation. In these cases the effects of the abutments on the response can be modeled by incorporating a set of springs and dashpots in the structural model of the bridge to represent the stiffness and damping parameters. Therefore methods are required to determine these stiffness and damping parameters of bridge abutments. ATC-18 report (Rojahn et al., 1997) states that the state of knowledge and the ability to accurately model abutments was significantly behind that of columns and foundations. It also states that for many bridges, abutment performance would have significant impact on the overall response of a bridge at different levels of shaking. ATC-18 recommends that the future projects should be focused on developing abutment models for various levels of shaking, directions of shaking (i.e. transverse and longitudinal) and different design strategies. Detailed analyses of various types of abutments covering all the aspects are beyond the scope of this research. However, methods are presented herein model the abutments for shaking in the transverse, vertical and longitudinal directions. Abutments may exhibit nonlinear behaviour under strong shaking. A method is also presented to analyze the abutments under strong shaking and to determine the stiffness, damping and fundamental frequency time histories using a simplified model of the abutments. Chapter 7 : Stiffness and Damping of Bridge Abutments 242 7.1 TRANSVERSE AND VERTICAL STIFFNESS AND DAMPING OF ABUTMENTS Fig. 7.1 shows a typical two span bridge with abutments and approach embankment. An abutment usually consists of an end wall and two wing walls. The wing walls are normally long and flexible and they are buried in the abutment-embankment soil. Thus, for all practical purposes, when the transverse and vertical vibration is considered, the abutment embankment soil system can be modeled as a trapezoidal soil wedge as shown in Fig. 7.2. This trapezoidal wedge model was originally proposed by Wilson and Tan (1990) Wilson and Tan (1990) developed analytical expressions for the static stiffnesses of the trapezoidal wedge assuming linear elastic behaviour. The proposed expressions for transverse stiffness , k t and the vertical stiffness, kv per unit length of the abutment are , _ 2sG In l + 2 s * * 2sE w (8.1) In w, in which E and G are Youngs modulus and shear modulus of the soil, w is the top width, H is the height and s is the side slope. Wilson and Tan (1990) showed that the stiffnesses from these two expressions agree with the stiffnesses from a plane strain finite element analyses. The difference between the two solutions was less than 20% and the finite element solutions were lower than the solutions from the proposed analytical expressions. Wilson and Tan (1990) used system identification analysis to study the frequency response of the Meloland Over Pass Abutment during strong shaking. The analysis used the accelerations recorded on the bridge during the 1979 Imperial Valley earthquake. System identification showed that the first transverse frequency of the abutment did not remain constant but decreased as the level of shaking increased. During the first 4 seconds(0-4) of shaking the frequency was 2.52 Hz and it Chapter 7 : Stiffness and Damping of Bridge Abutments 243 Bridge Deck •Embankment Soil -Bridge Abutment Fig. 7.1 A Typical Two Span Bridge and the Abutment-Embankment Soil System I Fig. 7.2 A Typical Abutment-Soil System Chapter 7 : Stiffness and Damping of Bridge Abutments 244 reduced to 1.39 Hz during the next 4 seconds(4-8) of stronger shaking. It again increased to 2.08Hz during the next 4 seconds (8-12) as the level of shaking diminished. The computed frequency of the abutment under elastic conditions was 2.56 Hz, very close to the computed elastic frequency. Thus, there was a 56% reduction in the frequency during the strong shaking periods (4-8 sees). Wilson and Tan (1990) also carried out a linear elastic time history response analysis of the abutment using the plane strain finite element model shown in Fig. 7.3. They found that it was necessary to reduce the modulus by a factor of three and increased the damping ratio of the soil to 25% to get a good agreement between the measured and computed response. The factors are a measure of the effects of soil nonlinearity on response. Clearly methods are required to capture the nonlinear behaviour of the abutment and to investigate the time dependent characteristics of the stiffness and damping during strong shaking. These methods are developed in this chapter 7.2 LONGITUDINAL STIFFNESS AND DAMPING OF ABUTMENTS Maragakis (1986) presented an approach to determine the elastic static longitudinal and rotational stiffness of the abutment by assuming the abutment to be rigid wall and so neglecting the deformations due to bending and shear. The effect of backfill soil was represented by a set of Winkler springs. The model used by Maragakis(1986) is shown in Fig. 7.4. Maragakis and Siddharthan(1988) extended this approach to determine inelastic static longitudinal and rotational stiffness of the abutment. They used global yielding criterion based on Rankine theory of active thrust and passive resistance was used to model the inelastic behaviour of the soil. Lam and Martin(1986) presented the following simplified expressions for the longitudinal and rotational stiffness of a rigid wall abutments. Longitudinal K L = 0.425 E s B Rotational K p = 0.072E B H 2 K S (8.2) Chapter 7: Stiffness and Damping of Bridge Abutments 245 Fig. 7.3 Plane Strain Finite Elelement Model of MRO Abutement Abutement Wall - V W V H Soil Springs ^ A A A / ^ t -WW\A—1 t - W V W - ^ V V V V H 1—vWW" - W W V ^ t f—WWW-rr - A V V V H Fig. 7.4 Model Used by Maragakis(1986) Chapter 7 : Stiffness and Damping of Bridge Abutments 246 in which H is the height of the wall, E s is the Youngs modulus of soil and B is the width of the abutment wall. The translational spring with K L stiffness and the rotational spring with K R stiffness are placed at 0.37H height from the bottom of the wall. The expressions for the stiffnesses were derived from the pressure diagrams shown in Fig. 7.5 corresponding to pure translation and pure rotation of the wall. The pressure diagrams used by Lam and Martin(1986) are those recommended by Matheson et al.(1980). Finn (1963) presented solutions for the stresses behind a translating and rotating wall. The translational and rotational stiffness based on his solutions closely agree with the expressions presented by Matheson et al. (1980). The methods described so far to determine the abutment stiffnesses are suitable for static analysis only and the methods described above do not consider the nonlinear behaviour of the soil. A new method is presented in this chapter that can be used to determine the nonlinear stiffness and damping of the abutment. The abutment is modeled as a flexible vertical wall and the stiffness and the damping are determined at the top of wall. Typical models of the abutment wall-backfill systems are shown in Fig. 7.6. In Fig. 7.6a, the wall and the retained soil is underlain by a rigid base. In this case, the condition at the base of the wall can be considered either fixed or elastically restrained. In Fig. 7.6b, the retained soil is underlain by a soil deposit and then by a rigid base. The condition at the bottom of the wall in this case can be considered either fixed or free. 7.3 DYNAMIC IMPEDANCES OF VERTICAL WALL: ELASTIC RESPONSE 7.3.1 Modeling of Abutment Wall-soil System The soil is modeled using plane strain rectangular elements and the wall using beam elements as shown in Fig. 7.7 Chapter 7 : Stiffness and Damping of Bridge Abutments 247 o 1.0 DIMENSIONLESS STRESS, E$ 6 (A) W A L L T R A N S L A T E D Fig. 7.5 Pressure Diagrams for Pure Translation and Rotation ofthe Wall (Lam and Martin, 1986) Chapter 7 : Stiffness and Damping of Bridge Abutments 248 Wall Soil Rigid Base 3 Wall Fig. 7.6 Type of Problems Considered in the Vertical Wall Analysis H=4m 1*0.5 ni L=20 m Ew/Es=? *,=? Gs=18 Mpa vs=100 m/s YvAs =1-4 H/t =8 Uniform Soil Rigid Base -A -A •A •A Fig. 7.7 Finite Element Mesh Used in the Vertical Wall Analysis Chapter 7 : Stiffness and Damping of Bridge Abutments 249 Rectangular Soil Element: Conventional four node rectangular elements are used. However, in the derivation of the element stiffness matrix, the complex modulus is used to include the hysteretic damping of the soil. The complex modulus E* is expressed as E* = E (l+2ft) (7.3) in which E is Youngs modulus of the soil, X is the damping ratio and i is equal to 7-1. For the element mass matrix , a diagonal lumped mass matrix is used in which one fourth of the element mass to each node. Beam element: A two node beam element with 2-DOF, translation and rotation, at each node are used to model the abutment wall. The nodes of the beam element in the abutment wall-soil system is shared by the adjacent soil elements. Thus, the stiffness at the beam nodes is composed of the stiffness of the wall and the stiffness of the soil. The stiffness matrix of the wall element is expressed as 12 6L -12 6L EI 6L 4 L2 -6L 2 L 2 L 3 -12 -6L 12 -6L 6L 2 L 2 -6L 4 L 2 (7.4) in which EI is the flexural rigidity of a unit length of the wall and L is the height of the wall. The consistent mass matrix of the wall element is expressed as 156 22L 54 -13L mL 22L 4 L2 13L - 3 L 2 420 54 -13L 156 -22L -13L - 3 L 2 -22L 4 L 2 (7.5) in which m is the mass of the wall per unit height. The radiation damping is assumed to be proportional to the velocity and the equivalent dashpot coefficient is expressed as Chapter 7 : Stiffness and Damping of Bridge Abutments 250 c r = y^Gp/a-v) (7.6) in which G is the shear modulus of the soil, p is the mass density of the soil and is u the poissons ratio of the soil. The damping matrix of the wall element is expressed as 156 22L 54 -13L c r L 22L 4 L 2 13L - 3 L 2 420 54 -13L 156 -22L -13L - 3 L 2 -22L 4 L 2 (8.7) in which c r is the radiation dashpot coefficient. The global mass, stiffness and damping matrices are obtained by assembling the element mass, stiffness and damping matrices. The global dynamic equilibrium equation of the abutment wall-soil system can be expressed as [M]{v} + [C]{v} + [K*]{v} = {P} (7.8) in which v,v,vand P are the acceleration, velocity, displacement and the external force respectively. [M], [C] and [K *] are global mass, damping and complex stiffness matrices. 7.3.2 Determination of Impedance of the Abutment Wall The method for determining the lateral, cross-coupling and rotational impedances of the abutment wall are same as those described for a single pile in Chapter 2, Section 2.3.3. Chapter 7 : Stiffness and Damping of Bridge Abutments 7.3.3 Numerical Studies 7.3.3.1 Introduction 251 An elastic analysis was carried out to determine the impedance of a vertical wall on a rigid base and fixed at the bottom. The vertical wall is of thickness, t, and the Young's modulus and Poisson's ratio are E w and v w respectively. The unit weight of the wall is yw- The uniform soil deposit is of height, H and it is underlaid by a rigid base. The unit weight of the soil is ys and the Young's modulus and the Poisson's ratio are E s and v s respectively. The critical damping ratio of the soil deposit is X. 7.3.3.2 Finite Element Mesh The finite element mesh is shown in Fig. 7.7. A total of 96 quadrilateral elements were used to model the soil and 8 beam elements were used for the wall. The soil layer was divided into an equal thickness of 0.5 m. Finer mesh was used near the wall. The bottom boundary is fixed and the right lateral boundary which is placed 20m away from the wall is fixed in the horizontal direction only. 7.3.3.3 Wall and Soil Parameters The dimensions and the wall and soil parameters used in the analysis are shown in Fig 7.7 except the ratio EJES and X. Three different analyses were carried out with the following values of BJES and X. Table 7.1 Parameters, E^E,. and X used in the analysis Case E w / E s X(%) 1 1000 5 2 10000 5 3 1000 15 Chapter 7 : Stiffness and Damping of Bridge Abutments 7.3.3.4 Impedances of the Vertical Wall 252 The impedances of the vertical wall are presented in nondimensional form. The lateral, cross-coupling and rotational stiffnesses and damping are normalized with E st,E st 2,E st 3 respectively. The non dimensional frequency is expressed as (cot/vs) in which co is the circular frequency and v s shear wave velocity of the soil. The maximum non dimensional frequency considered in the calculation is approximately 0.3 which corresponds to 10Hz. Fig. 7.8 shows the variation of the nondimensional stiffnesses with the nondimensional frequency. The stiffnesses reduce slightly with an increase in the nondimensional frequency within the range of frequency considered. When the ratio, EJES is increased by 10 fold, the stiffness at the wall head also increases. The increase in damping from 5% to 15% does not alter the dynamic stiffness much. Fig. 7.9 shows the variation of damping with the nondimensional frequency. When the critical damping ratio is 5%, the damping is almost a constant up to a nondimensional frequency of 0.2. After that it shows a bell shape. The effect of a 10 fold increase in the E w / E s ratio does not cause much difference in the nondimensional damping. When the critical damping ratio is increased from 5% to 15%, the nondimensional damping also increases. It remains almost a constant up to 0.2 and after that it increases with a slight oscillation. 7.4 DYNAMIC IMPEDANCES OF TRAPEZOIDAL ABUTMENTS 7.4.1 Finite Element Model Plane strain finite elements are used to model the abutment wedge soil system. Fig. 7.10 shows a typical finite element model. Rectangular or triangular or both rectangular and triangular elements can be used to model the abutment. Chapter 7 : Stiffness and Damping of Bridge Abutments 253 tn to c !t CO "ro i_ ro 50 w 40 2 30 -20 -10 0.00 Ew/Es=1000, X=5% Ew/Es=1000, \=15% Ew/Es=10000, Ji=5% T T 0.05 0.10 0.15 0.20 0.25 0.30 Non-Dimensional Frequency ((o*t/Vs) 0.35 tn U J tn tn to rz CO Q. O O • tn tn o O 200 150 K S 100 h 50 Ew/Es=1000, )i=5% Ew/Es=1000, X=15% Ew/Es=10000, )i=5% - 1 1 1 1 1 1 1 1 1 1 1 1— 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Non-Dimensional Frequency, <o*t/Vs) 0.35 « tn in <D I CO fo c o ro -.—< o 1000 800 600 h 400 200 h Ew/Es=1000, X=5% Ew/Es=1000, X=15% Ew/Es=10000, X=S% -I 1 1 1 1 1 1 1 1 1 1 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Non-Dimensional Frequency, (co*t/Vs) 0.35 Fig. 7.8 Variation of Nondimensional Lateral, Cross-Coupling and Rotational Stiffness with Dimensionless Frequency Chapter 7 : Stiffness and Damping of Bridge Abutments 254 in UJ, "x x O CL E CO Q "co cu -.—' CO CM in x O di CL E CO D D) "5. O O i in in o V— O 2 h 12 8 h 4 h Ep/Es=1000, \=5% Ep/Es=1000, X=15% Ep/Es=10000, \=5% ~ i — 1 — i — 1 — r ~ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Non-Dimensional Frequency (ca*t/Vs) 0.35 Ep/Es=1000, X=5% Ep/Es=1000, X=15% Ep/Es=10000, X=S% 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Non-Dimensional Frequency, co*t/Vs) 0.35 6 ro_ * in w i _ i _ o c5 c CL £ CO Q CO c o » ro o or 4 h ^ 2 0.00 Ep/Es=1000, \=5% Ep/Es=1000, \=15% Ep/Es=10000, X=5% T T T 0.05 0.10 0.15 0.20 0.25 0.30 Non-Dimensional Frequency, (co*t/Vs) 0.35 Fig. 7.9 Variation of Nondimensional Lateral, Cross-Coupling and Rotational Damping with Dimensionless Frequency Chapter 7 : Stiffness and Damping of Bridge Abutments 255 28 35 42 49 56 63 70 *1 27 34 41 48 55 62 69 \ 76 / 20 26 33 40 47 54 61 68 75 \ 81 / 14 19 25 32 39 46 53 60 67 74 80 \ 85 / 9 13 18 24 31 38 45 52 59 66 73 79 84 \ 88 5 8 12 17 23 30 37 44 51 58 65 72 78 83 87 \ 90 2 4 7 11 16 22 29 36 43 50 57 64 71 77 82 86 89 \ 91 Horizontal Scale i > 1 1 1 m Scale 1H:2V 0 5 1 0 Fig. 7.10 Finite Element Mesh for the Analysis of a Trapezoidal Abutment Chapter 7 : Stiffness and Damping of Bridge Abutments 256 The element stiffness and mass matrices of the rectangular element are same as those described for the backfill soil in the vertical wall case. The complex element stiffness matrix and the lumped mass matrix of the triangular element are derived similarly. By assembling the element complex stiffness and mass matrices of the finite element system appropriately, the global complex stiffness matrix , [K *], and the global mass matrix, [M] can be obtained. Hence, the global dynamic equilibrium equation of the trapezoidal abutment wedge system can be expressed as in which v and v are the acceleration and displacement respectively and [P] is the external load vector. 7.4.2 Determination of Impedance The transverse impedance of the abutment wedge is defined as the complex amplitude of the harmonic transverse force that has to be applied at the top of abutment in order to generate a harmonic motion with unit amplitude in the transverse direction under the condition of zero vertical displacement. Under harmonic loading P(t) =P Qe l t o t , the displacement vector is of the form v =v oe 1 C 0 t. By substituting for v,v and P , Eq. 8.8 is rewritten as [M]{v} + [K*]{v} = [P] (8.9) [[K*]-co 2 [M]]{v 0 } = {Po} [ K J {v0} = {P0} (8.10) in which [K d y ] = [ [K*]-co 2 [M]j . According to the definition of transverse impedance, it can be found by applying a unit Chapter 7 : Stiffness and Damping of Bridge Abutments 257 transverse displacement to the top of abutment wedge while keeping the vertical displacement of the top of the abutment at zero. Eq. 8.10 is re written as {Pi h}' [ K d y ] ' ( V ) • = < {P/} {v/}_ {Pj'}. i = l,n j=N-n (8.11) in which, n - number nodes in at the top of abutment N - total number of nodes {Vj*1} - unit vector of size n representing the horizontal displacement of the top of abutment {v ; v} - null vector of size n representing the vertical displacement of the top of abutment {v/} - vector of size (N-n) representing the displacements of all the nodes except the ones at the top {P.h} - vector of size n representing the horizontal forces at the top of abutment {Pj11} - vector of size n representing the vertical forces at the top of abutment {Pjh} - vector of size (N-n) representing forces of all the nodes except the ones at the ones at the top The transverse impedance of abutment K T is expressed as K T = S ? h i = l (8.12) Similarly, the vertical impedance of the trapezoidal abutment can be determined by applying a unit vertical displacement to the top of abutment while keeping the transverse displacement of the top of abutment zero. Chapter 7 : Stiffness and Damping of Bridge Abutments 7.4.3 Elastic Impedances of Meloland Overpass Abutments 7.4.3.1 Introduction 258 The Meloland Overpass located near E l Centra in Southern California is a two span reinforced concrete box girder bridge. The bridge is supported on two monolithic abutments at the ends and on a single columns at the Center. The span length of the bridge is 31.7 m and the central column height is 5.2m. The elevation view of the bridge superstructure, abutments and the bridge approaches are similar to Fig. 7.1. The cross section of the bridge abutments is of trapezoidal shape with a top width, w=14.6m , height, H=7.3 m and side slope, s=l. The wings walls of the bridge are flexible and 6m long. They are buried into the embankment soil. The soil-structure interaction between the wingwall and the soil are ignored considering the high flexibility of the wingwall and the entire trapezoidal abutment is assumed to be made up of soil only (Wilson and Tan, 1990). 7.4.3.2 Finite Element Model The finite element mesh is as shown previously in Fig. 7.10. The abutment is Modeled using 66 quadrilateral elements and 12 triangular elements. The bottom nodes are assumed to be fixed in both horizontal and in the vertical directions. A l l the other nodes except the ones at the top of the abutment to which the bridge deck is connected are allowed to move in both horizontal and vertical directions. The top nodes are constrained to have same deformations. 7.4.3.3 Soil Properties Results of test borings at the Meloland Road Overpass site indicated that the soil is mostly clay with traces of silt and sand. The standard penetration resistance was approximately 14blows/ft. Chapter 7 : Stiffness and Damping of Bridge Abutments 259 Based on the site investigation, Wilson and Tan(1990) suggested the following set of uniform properties for the bridge abutment soil. Soil unit weight, y=15.7 kN/m 3 , Youngs Modulus, E=19MPa and the Poisson's ratio v=0.3. These properties were used in the present analysis also. The critical damping of the soil was assumed as 5%. 7.4.3.4 Elastic Impedances Lateral Stiffness and Damping Fig. 7.11 shows the variation of lateral stiffness. The lateral stiffness reduces from its static stiffness value to zero near its first transverse fundamental frequency. After that it shows oscillatory response with zeros near the fundamental frequencies. Vertical Stiffness and Damping Fig. 7.12 shows the variation of vertical stiffness. The vertical stiffness also shows similar response initially. It reduces from its static stiffness value to zero near its first vertical fundamental frequency. Comparison of Static Stiffnesses The static transverse and vertical stiffness of the Meloland Overpass abutment were calculated using the expressions given in Eq. 8.1 by Wilson and Tan(1990) and reported as 160 M N / m and 418 MN/m respectively. In the calculation of stiffnesses, the effective length of the abutment was taken as 6m which is the length of the wingwall in the longitudinal direction. From the computed dynamic impedances, the static stiffnesses which is the stiffness at zero frequency can be obtained. The computed static transverse stiffness is 135 MN/m and the computed vertical stiffness is 384 MN/m. The difference between the computed transverse stiffness and that from the simplified expression of Wilson and Tan(1990) is 16%. The difference between the vertical stiffnesses is 8%. Chapter 7 : Stiffness and Damping of Bridge Abutments 260 E x X CO cu c i t CO "co 4 6 Frequency (Hz) • Modal Frequencies! 8 10 Fig. 7.11 Variation of Lateral Dynamic Stiffness with Frequency 4000 p 3000 E-2000 V 1000 ~r 0 T -1000 | -2000 -3000 -4000 0 Modal Frequencies! _L _L 4 6 Frequency, (Hz) 8 10 Fig. 7.12 Variation of Vertical Dynamic Stiffness with Frequency Chapter 7 : Stiffness and Damping of Bridge Abutments 261 7.5 NONLINEAR STIFFNESS AND DAMPING AND SEISMIC RESPONSE OF MELOLAND OVERPASS ABUTMENT 7.5.1 Introduction A method has already been presented to determine the frequency, stiffness and damping of the bridge abutment under elastic conditions. However, the bridge abutment may not remain elastic under strong shaking. To investigate the behaviour of the bridge abutment under strong shaking, an equivalent linear finite element analysis of the abutment of the Meloland1 Overpass was carried in the time domain. Th results of this analysis were then used to study the frequency, stiffness and damping response of the abutment under strong shaking. 7.5.2 Method of Analysis 7.5.2.1 Time history response the abutment was Modeled using quadrilateral and triangular finite elements. The global dynamic equilibrium equation was solved in time domain using the Wilson-theta method. During the analysis, the shear modulus and damping ratio in each finite element was updated depending on the level of shear strain in that element. At time t+At, the shear modulus and damping ratio were calculated based on the shear stain at the previous time step t using the curves of shear modulus and damping ratio versus shear strain. (Seed and Idriss, 1970). 7.5.2.2 Frequency and Stiffness and Damping Response During the time domain analysis, eigen value analyses were carried out at regular time intervals to determine the time histories of the first mode transverse and vertical frequencies of the abutment resulting in time histories of frequencies. The time histories of shear modulus and damping Chapter 7 : Stiffness and Damping of Bridge Abutments 262 ratio in each element was traced during the time domain analysis. These time histories of properties were then used to calculate the time histories of stiffness and damping of the abutment, using the method described in Chapter 5. 7.5.2.3 Finite Element Mode l , Mode l Parameters and the Earthquake Input Fin i te Element M e s h : Fig. 7.10 shows the finite element mesh used in the analysis. The mesh contains 66 quadrilateral elements and 12 triangular elements. A l l the nodes except the ones at the top and bottom were allowed to move in both vertical and horizontal directions. Top nodes were constrained to have equal deformations and the bottom nodes were restrained in both directions. Earthquake Input: The 1971 San Fernando earthquake recorded at Caltech Seismological Laboratory was scaled to a peak acceleration of 0.2g and was used as the input motion. Fig. 7.13 shows the input acceleration record. The input acceleration was applied at the base of the abutment. Properties of the soil : The elastic properties of the soil described in Section 7.4.3.3 were used here also. For the shear modulus degradation, the curve recommended by Sun et al.(1988) for the stiff clay with Plasticity Index of 20-40 was used. For variation of the damping ratio, the average curve for clay recommended by Sun et al.(1988) was used. 7.5.3 Results of Analysis 7.5.3.1 Acceleration Response Fig. 7.14 shows the acceleration response at the top of the abutment. The base input acceleration 0.2g is slightly amplified to 0.22 g at the top. The frequency content of the abutment acceleration shows a longer period response than the base acceleration input due to the effects of nonlinearity. Chapter 7 : Stiffness and Damping of Bridge Abutments 263 0.3 c g co _S cu o o < 0.1 0.0 -0.1 -0.2 -0.3 rV^^ A I^lll/lLi Hlfl/l I III Amax=0.2g fTVW^yiniiyuifl N nyi 11 IU i 1 i 1 i 1 Ul ll IU« if^ » 'If * T'^vW** 1 ' 1 ' 1 ' 1 ' 1 6 8 10 Time (sec) 12 14 Fig. 7.13 Input Base Acceleration Time History 0.30 - | 0.20 -0.10 -cz o 2 0.00 -<D CO -O O < -0.10 --0.20 --0.30 -Amax=0.22g ">—r 2 -i | i | i | i | i |— 6 8 10 12 14 Time (sec) Fig. 7.14 Acceleration Time History at the Top of Abutment Chapter 7 : Stiffness and Damping of Bridge Abutments 7.5.3.2 Stiffness and Damping Response 264 Fig. 7.15 shows the variation in transverse stiffness of the abutment with time. The initial stiffness drops gradually and reaches the lowest value of 38 MN/m during strong shaking which is only a 28% of the initial stiffness. The transverse stiffness of the abutment then increases as the level of shaking diminishes. The variation of vertical stiffness with time shown in Fig. 7.16 shows similar behaviour as the transverse stiffness. The vertical stiffness drops to 28% of the initial stiffness during strong shaking. Fig. 7.17 and 7.18 show the variation of transverse and vertical damping with time respectively. The damping is very high during the strong shaking period due to the high strains and the strain dependency of damping. 7.5.3.3 Frequency Response Fig. 7.19 shows the variation of the first transverse mode frequency with time. The initial frequency of the abutment is 2.59 Hz . This frequency corresponds to the linear elastic properties of the soil. This frequency agrees very well with that reported by Wilson and Tan(1990) which is 2.56 Hz. The variation of transverse stiffness with frequency showed that the stiffness initially diminishes with increase in frequency reaching zero near the fundamental frequency of the abutment. After that the stiffness shows oscillatory response reaching zeros at the natural frequencies. The vertical stiffness also shows similar trend. The first transverse mode frequency of the abutment gradually reduces with time with increasing levels of shaking and reaches the lowest value of 1.38 Hz at about 7 seconds, a 47% reduction from its elastic value.. The frequency then increases with diminishing acceleration. The reduction in the frequency during the strong periods of shaking can be attributed to the 72% reduction in the stiffness of the abutment caused by the nonlinear behaviour of the soil. The system identification study by Wilson and Tan(1990) showed a similar trend. Chapter 7 : Stiffness and Damping of Bridge Abutments 265 3.0 x 0 2 4 6 8 10 12 14 Time (sec) Fig. 7.15 Variation of First Transverse Mode Modal Frequncy with Time Fig. 7.16 Variation of First Vertical Mode Modal Frequncy with Time Chapter 7: Stiffness and Damping of Bridge Abutments 266 Fig. 7.18 Variation of Vertical Stiffness with Time Chapter 7 : Stiffness and Damping of Bridge Abutments 267 The variation of first vertical mode frequency is shown in Fig. 7.20. The vertical mode frequency reduces from its initial vale of 4.09 Hz to 2.16 Hz during strong shaking, a 48% reduction from the initial value caused by 72% drop in the vertical stiffness. 7.6 CONCLUSIONS A method was presented to determine the stiffness and damping of the of the abutments assuming either linear elastic behaviour or nonlinear response. The abutment soil system is modeled as trapezoidal soil wedge using plane strain soil elements and the analysis is carried out in the frequency domain. Computed static stiffnesses of the abutment of the Meloland Overpass using the new method agreed well with the analytical expressions proposed by Wilson and Tan(1990). If the response is nonlinear, a non linear analysis of the abutment is first carried out in the time domain to determine the variation shear modulus and damping ratios as a functions of time. These functions are then used to obtain time histories of stiffness and damping. The method was tested by analyzing the response of the Meloland Overpass to the 1971 San Fernando Earthquake. The computed transverse and vertical stiffnesses dropped 72% during strong shaking which resulted in a 47% drop in the transverse frequency and 53% drop in the vertical frequency. The trend in the frequency time history agrees closely with the one derived from system identification analysis of recorded response of the bridge by Wilson and Tan(1990). It should be noted here that the effect of inertial interaction due to the bridge deck on the abutment was not taken into account in the analysis. Additional strains may occur due to the inertial interaction and as a result the stiffnesses of the abutments may further be reduced as observed in the analysis of pile foundations in Chapter 6. Further parametric studies are desirable to investigate this aspect. A method was also presented to determine the longitudinal stiffness and damping of the abutment. In this case the abutment soil system is modeled as a vertical wall and backfill. Plane strain finite elements were used to model the back fill soil and the wall was modeled using beam Chapter 7 : Stiffness and Damping of Bridge Abutments 268 elements. A typical wall with a rigid base at the bottom of the wall was analyzed under two different values of Ew/Es ratio. The lateral cross coupling and rotational stiffnesses of the wall remain approximately constant up to a nondimensional frequency of 0.3 which corresponds to 10Hz exciting frequency. The increase from Ew/Es from 1000 to 10000 resulted in 8 fold increase in the lateral stiffness. Chapter 7 : Stiffness and Damping of Bridge Abutments 269 25 20 H 15 $ 1 0 5 H Time (sec) Fig. 7.19 Variation of Transverse Damping with Time Fig. 7.20 Variation of Vertical Damping with Time 270 CHAPTER 8 Evaluation of Simplified Pile-Superstructure Model 271 CHAPTER 8 : SIMPLIFIED PILE-SUPERSTRUCTURE MODEL [SPSM] CalTrans (Abghari and Chai, 1995) has adopted a very simplified model of a bridge -foundation system that facilitates taking nonlinear soil behavior and inertial interaction between foundation soils and superstructure into account. The model is shown in Fig. 8.1. There are two major simplifications. The foundation pile group is represented by a single pile that supports a concentrated mass corresponding to its proportion of the total static force carried by the group. The mass is supported in a SDOF system with a period equal to the first mode period of the bridge assuming fixed supports in the mode of interest. The function of the SDOF system is to model approximately the inertial contribution of the superstructure to the response of the pile foundation. The interaction between the soil and the pile is modeled using Winkler springs and dashpots with properties equivalent to p-y curves. In Chapter 3, it was shown that seismic response analyses based on p-y curves may be unreliable when not calibrated for site and loading conditions. However the representative pile concept is a very attractive computational feature. Therefore the concept is evaluated below by treating the foundation soil as a nonlinear continuum in both representative pile and full pile group analyses using PILE3D. In this way, the representative pile concept can be tested against full group analysis under identical foundation conditions. 8.1 EVALUATION OF SPSM The following strategy is followed in evaluating the effectiveness of the SPSM model based on CalTrans concepts: 1. Analyze the SPSM model of the bridge and foundation using PILE3D. 2. Analyze the system comprised of the pile group carrying a concentrated mass corresponding to the entire static load carried by the foundation. 3. Evaluate SPSM by comparing the results in steps 1 and 2. Chapter 8: Simplified Pile-Superstructure Model 272 SDOF System Model of the Superstructure Soil Strata Pile Near Field Soil Free Soil Rield 7WWV Nonlinear Springs and Dashpots Fig. 8.1 Simplified Pile Superstructure Model (SPSM) Chapter 8: Simplified Pile-Superstructure Model 273 In both analyses, following CalTrans procedure, no rotation of the pile cap was allowed at first. Later this requirement was relaxed to show the effects of pile cap rotation. The foundation soil (Fig. 8.2) is a layer of uniform soft clay,10m thick. The unit weight y = 18kN/m3 and the undrained shear strength, S u = 50kPa. The variations in shear modulus and damping ratios recommended by Sun et al. (1988) for soft clay with plasticity index between 20% and 40% were used in the equivalent linear analyses. 8.2.2 Pile Foundations In the SPSM the representative pile is a circular reinforced concrete pile with a diameter D = 0.5m and length L = 9m. The Young's Modulus of the pile is E p = 30500MPa. To provide data for evaluating the effectiveness of the SPSM model fully coupled analyses were conducted on two different pile groups, (2x2 and 4x4), with pile spacing at 2D. The piles were identical in properties to the representative pile. 8.2.3 Computational Models The soil layer was modeled using 980 brick elements. Each pile was modeled using 9 beam elements. A concentrated mass, M = 50Mg, was selected to account for superstructure inertia in SPSM; correspondingly 200Mg and 800Mg masses were selected for the 2x2 and 4x4 pile groups. A l l masses were supported in such a way that the frequency of the SDOF was 4Hz. 8.2 MODEL PROPERTIES 8.2.1 Soil Chapter 8: Simplified Pile-Superstructure Model 274 Superstructure Parameters Mass, M s= 50 Mg Natural Frequency of SDOF system, f = 4Hz Soil Properties Undrained Shear Strength, S u = 50 kPa Unit weight, y s =18 kN/m 3 Shear Modulus, G m a x = 50 MPa Pile Properties Flexural Rigidity, El =1310 kNm 3 Unit weight, y p = 25 kN/m 3 Diameter, D Length, L 0.5 m 9 m E p/E s=500 Pp/ps =1.4 L/d >15 v =0.4 Homogeneous Clayey Soil Pile •AAAAA Nonlinear Springs and Dashpots Fig. 8.2 Simplified Pie-Superstructure Model and the Properties of Soil and the Pile Used in the Analysis Chapter 8: Simplified Pile-Superstructure Model 8.2.4 Input Motion 275 The first 20 seconds of the 1971 San Fernando earthquake recorded at Griffith Park in Los Angeles, scaled to a peak acceleration of 0.5g was used as base input motion. 8.3 RESULTS OF SPSM AND FULL GROUP ANALYSES Fig. 8.3a shows the comparison of bending moment profiles for the two different pile groups and the single pile . In the case of (4x4) pile group the bending moment shown is the bending moment of the pile at the corner. The bending moment profiles from the (2x2) and (4x4) analyses do not deviate much from the moment profile in the single pile analysis. The difference in the peak moment is less than 8% only. Fig. 8.3b shows the comparison of shear force profiles. The shear force profiles also show the similar behavior as the bending moment. Fig. 8.4a and 8.4b show the maximum deflection and maximum rotations profiles. The deflection and rotation profiles from the different group piles do not show much deviation from those of the single pile. It is evident from the Figs. 8.3 and 8.4 that the group effect, if there is any, was not a significant factor in the response of the different pile groups. In order to determine the group effect, the time histories of stiffnesses of the single pile and pile groups were determined using the computer program PILE3D. The procedure to determine these stiffness time histories were described in Chapter 5. The stiffness changes with the time due to the nonlinear behavior of the soil during shaking. The computed initial elastic lateral stiffnesses and minimum stiffnesses are shown in Table 8.1. It is evident from this table that the loss in stiffnesses during the periods of strong shaking are substantial. The reduction in the single pile stiffness is 81% and the same in the group stiffnesses is 76%. Chapter 8: Simplified Pile-Superstructure Model 276 Fig. 8.3a Comparison of Maximum Bending Moment Profiles Fig. 8.3b Comparison of Maximum Shear Force Profiles Chapter 8: Simplified Pile-Superstructure Model 277 Maximum Deflection (m) 0.00 0.05 0.10 0.15 Q. CD O r> 1 1 1 1 / /'' pp * //' / / / / / / / m * if i ilJ « wf Single Pile /// V if B 2x2 Group — 4x4 Group 0 2 4 6 8 10 Fig. 8.4a Comparison of Maximum Deflection Profiles Q. Qi Q Maximum Rotation (rad) 0.00 0.01 0.02 0.03 0.04 0.05 0 6 H 8 10 i 1 i 1 i ' r V Single Pile 9 • 2x2 Group 4x4 Group Fig. 8.4b Comparison of Maximum Rotation Profiles Chapter 8: Simplified Pile-Superstructure Model 278 Table 8.1 Lateral stiffnesses of single pile and pile groups Type of Pile Foundation Initial Elastic Lateral Stiffness (MN/m) Minimum Lateral Stiffness (MN/m) Reduction in Stiffness (%) Single Pile 183 35 81 (2x2) Group Pile 444 106 76 (4x4) Group Pile 921 228 76 Table 8.2 Group effect corresponding to initial elastic and minimum lateral stiffnesses Type of Pile Foundation Group Effect, K g r o u p / (n*K s i n g l e ) Initial Elastic Stiffnesses Minimum Stiffnesses (2x2) Group Pile, K 2 x 2 0.61 -0.76 (4x4) Group Pile, K 4 x 4 0.31 -0.41 n is the total number of piles in t ie pile group. Table 8.2 shows the group factor, calculated as a ratio of group stiffness over the single pile stiffness multiplied by the number of piles in the pile group. It was 0.61 for (2x2) pile group and 0.31 for (4x4) pile group under elastic conditions. The group effect corresponding to the minimum stiffnesses for the (2x2) pile group is 0.76 and for the (4x4) pile group is 0.41. The same spacing was used in both pile groups and the group effect increased with the number of piles in the pile group as expected. Also, there is a reduction in the group effect during the periods of strong shaking at which the pile foundations reached their minimum stiffnesses. This is in keeping with the general perception that the range of pile to pile interaction is reduced as the soil behavior becomes non linear. It should be noted here that the level of nonlinearity experienced by each element in the different pile foundations may have been different during shaking and the assessment of the group effect using the minimum stiffnesses is an approximate estimate only. Despite the group interaction effects, the bending moment and shear force responses were not significantly affected. The reason for this behavior is explained below. The fundamental frequencies of the three pile foundation systems were determined corresponding to the initial stiffnesses and the minimum stiffnesses that occur during the time Chapter 8: Simplified Pile-Superstructure Model 279 strongest shaking. The frequencies are shown in Table 8.3. Although the group effebts on foundation stiffnesses of the pile groups were significant, the difference in the global system frequencies of the pile groups were not significantly different from the frequency of the single pile system. This is due to the fact that the global system frequencies result from the combined serial stiffnesses of the superstructure and pile foundation rather than the stiffness of the pile foundation alone. In this study the superstructure stiffness as reflected in the predominant system frequency. The similarity in frequencies of the different foundation models is responsible for the similarity in outputs. CalTrans adjust the results of the SPSM for group effects using a group reduction factor. Since group effects need to be incorporated in the dynamic analysis, their use after the event is not appropriate. Table 8.3 First mode frequency of the superstructure-pile foundation system Type of Structure First Mode Frequency (Hz) Initial Minimum Single Pile-Superstructure 3.66 1.89 (2x2) Group Pile -Superstructure 3.45 1.82 (4x4) Group Pile -Superstructure 3.07 1.61 Results suggest that the SPSM model concept for the analysis of pile foundations is only valid if the system frequency of the representative pile model and the frequency of the full foundation model are approximately the same. This can only occur when the stiffness of the support structures of the superstructure dominates the system frequency. For this to happen, the support structures must be much more flexible than the pile foundation in the degree of freedom under consideration. Parametric studies using many earthquakes with different frequency contents would be desirable to explore how full foundation system frequencies compare with representative pile Chapter 8: Simplified Pile-Superstructure Model 280 system frequencies for typical bridges and foundation soils. The study would allow the reliability of the representative pile concept to be evaluated. As pointed out earlier, in practice, the soil is represented by p-y curves. These curves have been shown to be unreliable in predicting dynamic response (Chapter 3 Section 3.10) and also unreliable in analyses of static and slow cyclic loading (Murchison and O'Neill, 1984, Gazioglu and O'Neill, 1984 ). The conclusion about dynamic analysis is based on analyses of a limited number of centrifuge tests. In view of the widespread use of p-y curves for dynamic analyses, it would seem necessary to investigate their reliability by more centrifuge tests and field studies. The problems associated with system frequencies and p-y curves can be avoided if PILE3D or any other program is used which can treat the soil as non-linear continuum and model the whole pile group conveniently enough for engineering practice. 8.4 EFFECTS OF PILE CAP ROTATION In line with the CalTrans procedure, the evaluation analyses were conducted with the assumption that the pile cap was fully restrained against rotation. The analyses were repeated without preventing rotation of the pile cap. In these latter analyses, the only restraint on the pile cap is provided by the moments on the pile cap caused by the axial forces in the piles. Profiles of maximum bending moments and maximum shear forces are shown in Figs. 8.5a and 8.5b. In this case there is very poor agreement between results of analyses based on the representative single pile concept and the full group analyses, even though the foundation soils are treated identically for all three systems. Profiles of deflections and rotations are shown in Figs. 8.6a and 8.6b. It would seem that the differences in computed rotations may be primarily responsible for the differences in moments and shears from the representative pile and full pile foundation group analyses. It can be noted here that the group reduction factor usually do not take into account the condition of the pile cap. Therefore the representative pile concept is probably not appropriate for Chapter 8: Simplified Pile-Superstructure Model 281 Q. 0 Q Maximum Bending Moment (kNm) 0 1000 2000 3000 — Single Pile — 2x2 Group --• 4x4 Group 10 Fig. 8.5a Comparison of Maximum Bending Moment Profiles (Pile cap is not fixed against Rotation) Fig. 8.5b Comparison of Maximum Shear Force Profiles (Pile cap is not fixed against Rotation) Chapter 8: Simplified Pile-Superstructure Model 282 sz 0 Q Maximum Deflection (m) 0.00 0.04 0.08 0.12 - Single Pile ---«— 2x2 Group —•-— 4x4 Group Fig. 8.6a Comparison of Maximum Deflection Profiles (Pile cap is not fixed against Rotation) sz -4—» Q. Qi Q Maximum Rotation (rad) 0.00 -0.02 0.04 0.06 0.08 Single Pile • f§ 2x2 Group — 4x4 Group Fig. 8.6b Comparison of Maximum Rotation Profiles (Pile cap is not fixed against Rotation) Chapter 8: Simplified Pile-Superstructure Model 283 small pile groups that cannot effectively restrain the pile cap. However for large pile groups, the restraint of the pile cap by the axial forces in the pile may be sufficient to approximate affixed condition. Parametric studies to establish a limiting pile group size are necessary. 8.5 CONCLUSIONS Parametric studies have shown that the CalTrans concept of using a single representative pile model of a pile foundation group is valid provided that the system frequency of the representative pile system and that of the full foundation model are sufficiently close, the pile cap does not rotate and the foundation soil is treated as a continuum. In practice the interaction between pile and soil has been modeled using p-y curves. Analyses in Chapter 3 showed that p-y curves may be very unreliable representations of soil behavior for dynamic analyses. However the PILE3D analysis is quite fast and stable, so there is no need to rely on p-y curves in their present form. Also PILE3D can model the pile group as a group and the problem associated with system frequency does not arise. The parametric studies were repeated without maintaining a fixed pile cap. In this case the single pile concept did not give good results. Even the largest group analyzed, the 4x4 group did not provide sufficient restraint of the pile cap by the axial forces in the piles. Very large pile groups probably would provide sufficient restraint. A limiting group size could be determined by parametric studies. 284 CHAPTER 9 Coupled Analysis of Bridge Superstructure and Pile Foundation 285 CHAPTER 9 : COUPLED ANALYSIS OF BRIDGE SUPERSTRUCTURE AND PILE FOUNDATION 9.1 INTRODUCTION A fully coupled analysis of superstructure and foundations is the ideal way to obtain the seismic response of a bridge structure. However, it is not always feasible in engineering practice due to the complexity of the problem and due to the limitations in the computer resources currently available such as memory and speed. Thus, in practice, several assumptions are made to reduce the complexity of the problem and to approximate the coupled behaviour of the superstructure and foundations. A simplified model of the foundation is used while analyzing the superstructure with a detailed model and vice versa while analyzing the foundation. In many superstructure models of the bridge, the foundation is modeled as a lumped parameter model consisting of a set of springs and dashpots. These springs and dashpots are expected to duplicate the behavior of the pile foundation. However, the accuracy of this model mainly depends on the way spring stiffness and dashpot dampings are determined. Various issues related to the determination of the stiffness and damping were dealt with in detail in Chapter 5 and 6. Sometimes uncoupled analyses of foundations and superstructure are also carried out in practice. An analysis of a bridge superstructure with fixed base conditions is basically an uncoupled analysis. In this analysis, the flexibility of the foundations is not taken into account, assuming that its effect on the superstructure is not significant. The foundation is also sometimes analyzed without including a model for the superstructure assuming that the effect of superstructure interaction on the foundation is not significant. Some common models of the superstructure and pile foundation are also given in Table 9.1. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 286 Table 9.1 Details of the simplified models of a bridge structure Model No Superstructure Pile Foundation Input Motion Model Type Model Type 1 Space Frame or Plane Frame Linear Elastic Rigid Base (Foundation is not modeled) Free Field 2 Space Frame or Plane Frame Linear Elastic Springs and Dashpots Linear Elastic Free Field 3 Space Frame or Plane Frame Linear Elastic Springs and Dashpots Equivalent Linear elastic Free Field 4 Space Frame or Plane Frame Linear Elastic Springs and Dashpots Nonlinear Free Field Models 1 through 4 and the methods to obtain the spring stiffness and the damping have been described in detail in Chapter 4, 5 and 6. In model 1, the effect of the foundation is completely ignored. In models 2 and 3, the soil-pile and pile-pile interactions are taken into account. The nonlinear hysteretic behavior of the soil is not modeled in model 2. In model 3, this behaviour is recognized by reducing the linear elastic stiffness and also by increasing the damping. The models 2 and 3 are elastic and, hence, the superstructure interaction is not important and ignored. In model 4, in which the behaviour of the foundation is nonlinear, the superstructure interaction is taken into account in addition to the soil-pile and pile-pile interactions. For interaction analysis, the superstructure is represented in this model by an equivalent single degree of freedom system(SDOF). The mass in the SDOF system is taken as the contributory mass of the superstructure carried by the pile foundation concerned and the period of the SDOF is taken as the first fundamental period of the superstructure. The stiffness of the SDOF is calculated based on this mass and the period. Model 4 also recognizes the time dependent effect of the stiffnesses and damping due to the nonlinear hysteretic behavior of the soil during an earthquake. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 287 In the models listed in Table 9.1, the behavior of the superstructure and foundation are not fully coupled. A fully coupled finite element model of the superstructure and foundation would require handling of very large size matrices in the solution process. This process is limited by the available computer storage and the speed. This limitation is overcome in a new method proposed below in which the finite element models of the superstructure and foundation are handled separately during the solution process, thus reducing the computer storage required, and the coupling between the superstructure and foundation is ensured through an iterative process. 9.2 FULLY COUPLED SOLUTION APPROACH In the new approach, the superstructure and the pile foundations are treated as two subsystems. The interactions between the two systems are transmitted through the motions and the dynamic forces and the moments at the joints between the superstructure and pile foundations. Finite element methods are used to form the dynamic equilibrium equations of the two systems that are described in detail later. These equations are assembled for the two systems separately and solved separately one at a time. As the two system of equations are solved separately the equilibrium and compatibility at the joints between the two systems are enforced through an iterative procedure. The treatment of superstructure and foundations as two subsystems greatly reduces the storage requirements in the finite element solution process. This subsystem approach is similar to the one described by Cai et al.(1995,1998). However, in the present method, a different model is proposed for the foundation subsystem. Also, in the proposed method, the equilibrium and compatibility at the joints between the superstructure and the foundations is strictly enforced unlike in the method proposed by Cai et al.(1995,1998). The new approach is schematically illustrated in Fig. 9.1. The steps involved in the solution process are described in the flow chart shown in Fig. 9.2 and described in detail below. In the new approach, the dynamic equilibrium equations are solved in the time domain. First the time history of earthquake is divided into n steps with equal step of At. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation * 288 Superstructure Pile Foundation Time=dt Input : Fs(dt)=0, Ms(dt)=0 Output: xls(dt), x^(dt) 3 * > j i Input :xis(dt),xrs(dt) W Output: Fs(dt), Ms(dt) Fig. 9.1b Time=2dt Input: Xg(dt) Fig. 9.1a Input : Fs(dt), Ms(dt) Output: xjs(2dt), xrs(2dt) Input: Xg(2dt) Fig. 9.1c 47f, Input : xjs(2dt), xrs(2dt) "Output: Fs(2dt), Ms(2dt) Fig. 9.Id Fig. 9.1 Schematic Diagram for the Soil-Pile-Structure Interaction Analysis (After Cai etal., 1995) Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 289 Calculate the base motion I Solve the FOUNDATION SUBSYSTEM Input 1) Base motion and 2) Support shear forces and moments ifnot timestep Siteration =1 output: Pile head translations and rotations Solve the STRUCTURAL SUBSYSTEM Input : Pile head translations and rotations Output: Support shear forces and momnts <~ If Iterations >^ Y e s JNo 'Check for Conevergence of support forces and moments |Yes Move to the next time step Fig. 9.2 A n Iterative Procedure for the Coupled Dynamic Analysis of Superstructure and Pile Foundation Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 290 Step 1: at time At, the pile head dynamic forces are set to zero and the foundation subsystems are analyzed separately under the base motion at time At, x g b(At). Resulting pile head translations, x L p (At) and rotations, x R p (At) at this time step are calculated (Fig. 9.1a). Step 2: at time At, the superstructure subsystem is analyzed under the multi-support excitations, x L p (At) and x R p (At) and the resulting support shear forces, F s(At) and moments, M s(At) are calculated (Fig. 9.1b). Step 3: at time 2At, the foundation subsystems are analyzed under the pile head forces which are same as the support forces, F s(At) and M s(At), and the base motion at time 2At, xg(2At) (Fig. 9.1c). Step 4: at time 2At, the superstructure subsystem is analyzed under the multi-support excitations x p and the resulting support shear forces F s(2At) and moments Mm(2At) are calculated (Fig. 9. Id). Step 5: The force residual ratio, r|F and the moment residual ratio, r|M which are defined below, is calculated. If these ratios are less than the allowable tolerance, the calculations are continued to the next time step. Otherwise, the steps 3 to 5 are repeated until the solution converges. £ ( F s ( 2 A t ) - F s ( A t ) ) *100 < Toler *100 < Toler £ ( M s ( 2 A t ) - M s ( A t ) ) p ( M s ( 2 A t ) in which Toler is the allowable tolerance in percentage. The new coupled solution approach is implemented in the computer program SOISTR. 9.3 MODELING OF FOUNDATION SUBSYSTEM The foundations subsystem consists of the pile group supporting the superstructure. Each pile group is treated separately during the solution process and it is assumed that there is no interaction between the different pile foundations. The latter assumption is true when the pile > Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 291 foundations are sufficiently far apart. Each pile group is modeled using the nonlinear quasi-3D finite element model implemented in the program PILE3D. The interaction of the superstructure with the pile foundations is represented by dynamic shear forces and moments at the pile heads. PDLE3D, which was originally written to handle base motion only, was modified to handle the forces and moments in addition to the base motion. In the foundation subsystem analysis, the translations and rotations of the pile head are computed and are used as input in the superstructure subsystem analysis, The PILE3D can be extended to handle forces and moments by including them in the global equilibrium equation as follows. [M]{x} + [C]{x}+[K]{x} = [M]{I}{x g } + {F} [M] - global mass matrix [C] - global damping matrix [K] - global stiffness matrix (9.2) {x} - displacement relative to the base {xg} - base excitation {F} -external force and moment In this global equilibrium equation, the external force and moment vector, {F} will have nonzero components at the pile head degrees of freedoms only. The time domain solution of the global dynamic equilibrium equation is obtained using the Wilson-theta method. 9.4 MODELING OF SUPERSTRUCTURE SUBSYSTEM In the superstructure subsystem, 3D beam elements are used to model the structural members and their behavior is assumed linear elastic. The 3D beam element has two nodes with 6 degrees of freedom (3 translations and 3 rotations) at each node. The structural damping was assumed to be Rayleigh damping. This type of space frame model was described in Chapter 7 to obtain the dynamic response of the superstructure under a rigid base excitation. This method is extended here to handle multi-support excitations following the method proposed by Imbsen and Penzien (1986). Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 292 The multi-support excitations consists of both translations and rotations and they can vary spatially from one foundation to the other. 9.5 DYNAMIC ANALYSIS PROCEDURE FOR A STRUCTURE WITH MULTI-SUPPORT EXCITATIONS The equation of motion of a structure with N degrees of freedom can be expressed as [M]{u t} +[C]{u}+[K]{u}={F} (9.3) in which [M] is the global mass matrix, [C] is the global damping matrix , [K] is the global stiffness matrix, {F} is the external force vector, ii is the absolute or total acceleration vector, ii is the relative velocity vector and {u} is the relative displacement vector. If the external excitations do not come from the supports, no distinction between the absolute and relative displacements are necessary. For earthquake type support excitations, the total or absolute displacement can be expressed in terms of the relative displacement u and the pseudo-static displacement u p s as {u1} = {u}+{u p s} (9.4) in which the pseudo static displacement u p s is the static response caused by the support movement u g and it can be expressed as {u p s} = [B]{ug} (9.5) in which [B] is a constant influence coefficient matrix. For rigid support excitations, in which all the support points are rigidly connected to the ground, u g consists of three translational components only and can be expressed as {ug }={ugx ,VLW U ^ } t. The corresponding influence coefficient matrix [B] is an (Nx3) matrix and can be expressed as [B]=[{bx} ,{by},{b z}] in which vectors ({b;}, i=x,y or z) have N components, each of which is equal to unity for the i * component and equal to zero for other components. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 293 For multi-support excitations, the ground excitation vector {ug} has N f components where N f is the total number of foundation degrees of freedom. The corresponding influence coefficient matrix can be determined by solving the following static equilibrium equations. [[K] [K f] = 0 (9.6) in which [K] is a (NxN) stiffness matrix corresponding the N active degrees of freedom. [Kf ]is a ( N x N f ) stiffness matrix coupling the N active degrees of freedom with the N f support degrees of freedom. Then the (NxN f) size influence coefficient matrix [B] can be expressed as [B] = -[[K]-'[K f] (9.7) By substituting Eq. 9.7 in Eq. 9.5 and then in Eq. 9.4, the total or absolute displacement can be written as {u1} = {u} + [B]{u g } {u1} = { u J + f - t K ] " 1 ^ ] ] ^ } (9.8) By substituting Eq. 9.8 in Eq. 9.3, the global dynamic equation of motion can be expressed as [M]{u}+[C]{u}+[K]{u} = {F}-[M][B]{u g} where [B] = -[K]" '[K f ] (9.9) This equation is solved in the time domain using the Wilson-theta method to obtain the relative dynamic response. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 9.6 DETERMINATION OF SUPPORT REACTIONS 294 The support reactions that consists of both shear force and moments need to be determined as they are used as input in foundations subsystem analysis. The support reaction under multi-support excitations can be expressed as {F} = [K/ tuVlKffH^} (9.10) in which {F} is an (N f xl) support reaction vector and [Kjf] is a (N f x N f) size stiffness matrix corresponding to the support degrees of freedoms. By substituting for u l, from Eq. 9.8 , Eq. 9.10 can be re written as {F} = [K f ] T j{u} - [ K ] - 1 [K f] {ug}} + [K f f]{u g} (9.11) The support reaction vector {F} can be found after solving the equation of motion(Eq. 9.11) for the relative dynamic response vector {u}. For rigid support excitations, the expression for the force vector {F} reduces to the following equation. {F} = [K f]T{u} (9.12) The program that can handle multi-support excitation is named SPFR-MS. 9.7 NUMERICAL STUDIES 9.7.1 Introduction Numerical studies were carried out to demonstrate the capability of the proposed method and Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 295 to identify the problems associated with the usual uncoupled analyses of superstructure and the pile foundations. 9.7.2 Problem Description A typical example of a single column bridge bent supported on (5x5) group pile foundations is chosen for the numerical study (Fig. 9.3). 9.7.2.1 Superstructure The superstructure consists of a single column and a concentrated mass at the top of column. The concentrated mass represents the mass contribution of the bridge deck. The fixed base frequency of the superstructure is taken as 3.75 Hz. The concentrated mass and the height of the column are taken as 375 Mg and 7m respectively. These are some typical values for the bridges in California (Abghari and Chai, 1995). 9.7.2.2 Soil Data The layered soil profile consists of a 4m sand layer underlain by 10m stiff clay. Below 14m, the soil is considered to be extremely stiff. The water table is located at 1 m below the surface. The relative density of the top sand layer, Dr is approximately 60% and the corresponding k m a x is taken as 50. The friction angle of the sand, (J) is taken as 30 degrees. The shear strength of the stiff clay, S u is assumed as 72 kPa. The unit weight, y of both the sand and the clay are taken as 18 kNm"3. The initial shear modulus of the sand and the stiff clay are calculated using the following formulas. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 296 # Concentrated Mass= 376 Mg 7m Sand Dr=60% k2=50,<t>=30° 12m Stiff Clay Su=72 kPa 2m Superstructure O O O O O O O O O O O O © ooooo ooooo o-T< iameter, D=0.3m JJpacing, s=0.9m (5x5) Pile Group Pile Foundation V//////////////////////////////////////// Fig. 9.3 Details of the the Superstructure and the Foundation Used in the Analysis Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 297 / , \ 0.5 G, max where G, k. max max initial shear modulus a constant depends on the relative density 1000 S u for clay (9.13) max a. S m u atmosphereic pressure mean effective stress shear strength The modulus and damping variation with shear strain used for the sand and the stiff clay are shown in Fig. 9.4a (Sun et al., 1988). The maximum damping of the sand and the clay are taken as 25% of critical damping. Fig. 9.4b shows the initial shear modulus profile. 9.7.2.3 Pile Foundation Data The pile group foundation consists of (5x5) pile group and a pile cap. The diameter, D and the length of a pile, L are 0.3m and 12m respectively. The spacing between the piles s, is 0.9 m. The flexural rigidity of the pile, EI is assumed as 11000 kNm 2 . The dimension of the pile cap is (4.6mx4.6mxl.lm) and the weight of the pile cap is taken as 75 Mg. 9.7.2.4 Earthquake Input The Cape Mendocino/Petrolia earthquake of April 25, 1992 recorded at the Painter Street Overpass, Rio Dell was used as the input acceleration. The recorded peak acceleration of 0.54g was scaled to 0.2g and applied at the 14m depth. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 298 1.2 Shear Strain (%) Fig. 9.4a Variation of Shear Modulus and Damping Ratios with Shear Strain Gmax (MPa) 0 20 40 60 80 100 120 10 H 12 -14 -Fig. 9.4b Maximum Shear Modulus Profile Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 9.7.3 Modeling of Superstructure Subsystem 299 The 7m column was modeled using seven beam elements and the concentrated mass was assigned at the top of the column. The beam stiffness was calculated to match the fixed base frequency of 3.75 Hz while keeping the length at 7m. The condition at the base of the column or at the top of the pile head was taken as fixed against rotation in line with the practice of CalTrans. 9.7.4 Modeling the Pile Foundation Subsystem The finite element mesh used in the pile foundation analysis consists of 1356 elements and 1575 nodes. The 14m thick sand and stiff clay layers were divided into 14 layers with equal thickness of lm. The soil was modeled using 1176 brick elements and each pile was modeled using 12 beam elements. The pile cap mass was assigned at the top of pile head. 9.7.5 Soil-Pile-Superstructure Interaction Analyses 9.7.5.1 Introduction The following four different types of soil-pile-structure interaction analysis were carried out to bring out all the aspects of coupled soil-pile-superstructure interactions. 1. Fully coupled analysis of entire structure using the PILE3D program 2. Iterative coupled analysis of superstructure and pile foundation as two subsystem using SOISTR program 3. Uncoupled analysis of superstructure and pile foundation using SPFR-MS and PIJLE3D programs 4. Analysis of pile foundation only using the program PILE3D under the base input and without any dynamic shear force at the pile head. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 300 9.7.5.2 Fully Coupled Analysis of Complete Structure Using PILE3D Program The PILE3D program is capable of analyzing the entire structure (ie. superstructure and pile group foundation)as a single system. In order to compare the accuracy of other types of analyses, in which the superstructure and the foundation are dealt as two subsystems, a PILE3D analysis of entire structure was carried out first. This analysis is a fully coupled analysis and it does not invoke any iterative or successive coupling procedures. 9.7.5.3 Iterative Coupled Analysis Using SOISTR Program An iterative coupled analysis was carried out to obtain the approximate coupled seismic response of the entire system using the procedure described previously. The condition at the pile head or at the base of the column was considered fixed against rotation in line with the CalTrans practice. Thus, at each step, the lateral displacement at the pile head was calculated due to base acceleration and dynamic shear force at the pile head. These pile head displacements were applied to the superstructure at each time step and the resulting dynamic shear force at the base of the structure was calculated. The tolerance on the values of successive dynamic shear forces was set as 1% and a maximum of 100 iterations were allowed at each step. The calculations were continued for the entire time history of 20 seconds. 9.7.5.4 Uncoupled Analysis of Superstructure and the Pile Foundation Using SPFR-MS and PILE3D Program In this uncoupled analysis, first the superstructure was analyzed with its base fixed using the program SPFR-MS. The free field acceleration was used as the input acceleration to the superstructure and the resulting dynamic shear force at the base of the column was calculated. Then, the pile group foundation was analyzed under both the base acceleration and the dynamic shear force using the program PILE3D. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 301 9.7.5.5 Analysis of Pile Group Foundation Only Using the Program PILE3D In this analysis, the pile group foundation was analyzed under the base input only. No dynamic shear force was applied at the pile head and the effect of the superstructure was simply ignored. The four different analyses are summarized in the following Table 9.2. Table 9.2 Summary of the four types of analyses No Superstructure Pile Foundation Type of Coupling Input Output Input Output 1 Base acceleration input; No subsystem analysis Fully coupled 2 Pile head displacements Dynamic force at the base Base acceleration and Pile head dynamic force Pile head displacement Iterative Coupling 3 Free Field acceleration Dynamic force at the base Base acceleration and Pile head dynamic force Uncoupled 4 Not modeled Base acceleration only Uncoupled 9.7.6 Results of Analysis 9.7.6.1 Iterative coupled approach vs Fully coupled approach Fig. 9.5 shows the comparison of acceleration and displacement time histories of the pile head between analyses No. 1 and 2. There is a very good agreement between the fully coupled analysis and the iterative coupling analysis. Fig. 9.6 shows the comparison of the time histories calculated at the top of superstructure. Here again there is a very good agreement between the two analyses. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 302 3 c o » _a> cu o o < 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0.0 Analysis 1: Analysis of Entire Structure Using PILE3D Analysis 2: Iterative Fully Coupled Analysis of Superstructure and Pile Foundation as Two Subsystems 5.0 10.0 Time (sec) Analysis 1 Analysis 2 15.0 20.0 F i g . 9.5a Comparison of Pile Cap Acceleration Time Histories c CD E CD O _co Q . CO Q 0.15 0.10 0.05 H 0.00 -0.05 -0.10 -0.15 0 10 Time (sec) Analysis 1 Analysis 2 15 20 F i g . 9.5b Comparison o f Pile Cap Displacement Time Histories Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 303 3 c g i _ _CD CD O O < 2 1 0 -1 Analysis 1: Analysis of Entire Structure Using PILE3D Analysis 2: Iterative Fully Coupled Analysis of Superstructure] and Pile foundation as Two Subsystems 0 ~ T ~ 5 10 Time (sec) Analysis 1 Analysis 2 15 20 Fig. 9.6a Comparison of Superstructure Acceleration Time Histories c CD E CD O JS CL CO 0.15 0.10 H 0.05 0.00 -0.05 -0.10 -0.15 0 5 10 Time (sec) Analysis 1 Analysis 2 15 20 Fig. 9.6b Comparison of Superstructure Displacement Time Histories Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 304 The small discrepancy between the two analyses is due to the tolerance level allowed in the iterative coupling analysis. A tolerance of 1% in the dynamic force was used in the analysis. Fig. 9.7 shows the comparisons of maximum bending moment and shear force profiles along the center pile in the pile group between the fully coupled analysis and the iterative coupled analysis. The maximum moment and the shear force from the iterative coupled analysis agrees very well with those from the fully coupled analysis and the difference between the two is less than 3%. The very good agreement shown in these figures demonstrate the capability of the iterative coupling approach. 9.7.6.2 Uncoupled Approach vs Fully Coupled and Iterative Coupled Approach In the uncoupled approach, the superstructure and pile foundation were analyzed separately. In the analysis of superstructure, fixed base conditions were assumed and the free field acceleration shown in Fig. 9.8a was used as the input acceleration. The resulting dynamic shear force at the base of the super structure in this uncoupled analysis of superstructure is shown in Fig. 9.8b. This force was compared with the dynamic force that was calculated at the base of the superstructure during the iterative coupled analysis. This comparison is also shown in Fig. 9.8b. There is a poor agreement between the two force time histories. However, the difference between the peak forces is only 12%. The peak dynamic force in the iterative coupled approach was 2843 kN and it occurred at 5.0 sec. The peak dynamic force in the uncoupled approach was 2493 kN and it occurred at 5.34 sec. The comparison between the power spectra of the two forces is shown in Fig. 9.8c. This figure shows that there is a significant difference between the frequency content of the two forces. The dynamic force in the iterative coupled approach shows a low frequency response compared to the dynamic force in the uncoupled approach. Fig. 9.9 shows the comparison between acceleration and displacement time histories of the top of superstructure between the fully coupled and uncoupled analyses. A very poor matching between the two records can be observed from Fig. 9.9. The peak displacement is severely underestimated in the uncoupled analysis and it was 52% less than that from the fully coupled analysis. It can be noted here that in the uncoupled analysis of superstructure, fixed based conditions were assumed and Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 305 Analysis 1: Analysis of Entire Structure Using PILE3D Analysis 2: Iterative Fully Coupled Analysis of Superstructure and Pile foundation as Two Subsystems Q. CD Q Bending Moment (kNm) 0 50 100 150 J i L 10 -4 12 Analysis 1 Analysis 2 Fig. 9 . 7 a Comparion of Maximum Bending Moment Profiles CL CD Q Shear Force (kN) 40 80 Analysis 1 Analysis 2 Fig. 9 . 7 b Comparison of Maximum Shear Force Profiles Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 1.0 306 3 c g co i _ _CD CD O O < 0.5 0.0 -0.5 -1.0 A A A A J B A A J L A I I I 1 - • v v / N ^ w v y W y w I 1111— 0 10 Time (sec) 15 20 Fig . 9.8a Free Fie ld Acceleration Time History Used as an Input in the Superstructure Analysis CD p CO CD SZ 00 -4000 0 Analysis 2: Iterative coupled Analysis of Superstructure and Pile Foundation as Two Subsystems Analysis 3: Fully Uncoupled Analysis of Superstructure and Pile Foundation as Two Subsystems T 5 10 Time (sec) Analysis 2 Analysis 3 15 20 Fig . 9.8b Comparison of Dynamic Shear Force Time Histories Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 307 Analysis 2: Iterative coupled Analysis of Superstructure and Pile foundation as Two Subsystems Analysis 3: Fully Uncoupled Analysis of Superstructure and Pile Foundation as Two Subsystems Analysis 2 Analysis 3 0 2 4 6 8 10 Frequency (Hz) Fig. 9.8c Comparison of Unit Normalised Power Spectral Densities of the Dynamic Shear Forces Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 308 3 c o "-t—> CD o o < c CD E CD O J5 Q . CO Q Analysis 1: Analysis of Entire Structure using PILE3D Analysis 3: Fully Uncoupled Analysis of Superstructure and Pile foundation as two subsystems 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 Analysis 1 Analysis 3 0 5 10 15 20 Time (sec) Fig. 9.9a Comparison of Superstructure Acceleration Time Histories 0.15 0.10 0.05 0.00 -0.05 -0.10 H -0.15 Analysis 1 Analysis 3 0 10 Time (sec) 15 20 Fig. 9.9b Comparison of Superstructure Displacement Time Histories Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 309 the flexibility of the foundation was ignored. The large discrepancy between the two responses show that the flexibility of the foundations is important and the uncoupled analysis of superstructure under fixed base conditions will not give acceptable results. Uncoupled analysis of the pile foundation was carried under the base motion and the dynamic shear force which was calculated in the uncoupled analysis of superstructure was applied at the pile head. Thus, though the analysis is of pile foundation is uncoupled, the effect of superstructure interaction was approximated by applying the fixed base dynamic shear force at the pile cap. Fig. 9.10 shows the comparison of acceleration and displacement time histories of the pile cap between the fully coupled and uncoupled analysis of foundation. There is a very poor agreement between the acceleration time histories from the fully coupled and uncoupled analyses. The displacement time histories also show disagreement especially after 6 seconds. However, the peak displacement of the pile cap from the uncoupled analysis agrees well with that from the fully coupled analysis. Fig. 9.11 shows the comparisons of maximum bending moment and shear force profiles along the center pile in the pile group between the fully coupled analysis and the uncoupled analysis of pile foundations. The maximum bending moment and the shear forces are only slightly over estimated in the uncoupled analysis. The maximum moment and maximum shear force in the pile calculated from the uncoupled analysis are 11% and 8% greater than those from the fully coupled analysis. It was shown earlier, though the dynamic force time histories from the fully coupled and uncoupled analyses were different, the peak dynamic force was nearly the same. This may have been the reason why the maximum displacement at the pile cap, the maximum bending moment and shear force in the pile from the uncoupled analyses agree well with those from the uncoupled analysis.. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 310 c g _CD CD O O < Analysis 1: Analysis of Entire Structure Using PILE3D Analysis 3: Fully Uncoupled Analysis of Superstructure and Pile foundation as Two Subsystems 0 Analysis 1 Analysis 3 y v vv y v v y y 10 Time (sec) 15 20 Fig. 9.10a Comparison of Pile Cap Acceleration Time Histories 0.15 - i 0.10 -0.05 -CD E CD 0.00 -O CC _ Q. to -0.05 -b --0.10 --0.15 -0 Analysis 1 Analysis 3 1 ii V I 5 10 Time (sec) 15 20 Fig. 9.10b Comparison of Pile Cap Displacement Time Histories Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 311 Analysis 1: Analysis of Entire Structure Using PILE3D Analysis 3: Fully Uncoupled Analysis of Superstructure and Pile Foundation as Two Subsystems Q. CU Q Bending Moment (kNm) 0 50 100 150 0 A 1 1 1 L Analysis 1 Analysis 3 Fig. 9.11a Comparion of Maximum Bending Moment Profiles Shear Force (kN) 0 20 40 60 80 0 n i i 1 1—f-f—| 1— Q. CU Q 10 12 Analysis 1 Analysis 3 Fig. 9.1 lb Comparison of Maximum Shear Force Profiles Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 312 9.7.6.3 Pile Foundation Alone vs Fully Coupled Analysis of Superstructure and Pile Foundation An analysis of pile group foundation without the superstructure was carried out. In this analysis no dynamic force was applied at the pile head and the effect of the superstructure was completely ignored. The computed pile head acceleration and the displacement time histories were compared with the time histories of the fully coupled analysis. These comparisons are shown in Fig. 9.12. It is clearly evident from this figure that the displacement and the accelerations are significantly underestimated when the effect of the superstructure is ignored. The peak displacement in the pile foundation analysis is 44% less than that from the fully coupled analysis. Fig. 9.13 shows that comparison of maximum bending moment and shear force profiles along the center pile in the pile group. The moment and shear force are severely underestimated in the uncoupled analysis of pile when the superstructure interaction is completely ignored. The discrepancy between the moment is 89% and the same between the shear force is 86%. This analysis clearly shows that the effect of superstructure interaction must be taken into account. 9.7 CONCLUSIONS An iterative finite element method was developed for the coupled analysis of bridge superstructure and foundation. As part of the new method, first a method was developed to determine the dynamic response of the bridge superstructure using a space frame model under multi support excitations and to determine the support reactions. Then, the Quasi-3D finite element method in PILE3D was extended to allow the analysis of pile foundations under the base excitation and dynamic pile cap forces. These two methods were then linked through the iterative procedure to determine the coupled response of the bridge. Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 313 3 c g *-4—» 2 J D CD o o < -1.5 Analysis 1: Analysis of Entire Structure Using PILE3D Analysis 4: Analysis of Pile Foundation Only Using PILE3D 10 Time (sec) Fig. 9.12a Comparison of Pile Cap Acceleration Time Histories 0.15 c CD £ CD O ro CL CO Q 10 Time (sec) 20 Fig. 9.12b Comparison of Pile Cap Displacement Time Histories Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 314 3 c o 2 _CD CD O O < Analysis 1: Analysis of Entire Structure Using PILE3D Analysis 4: Analysis of Pile Foundation Only Using PILE3D Analysis 1 Analysis 4 10 Time (sec) Fig . 9.12a Comparison of Pile Cap Acceleration Time Histories c CD E CD O JS Q . CO Q 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0 10 Time (sec) 15 20 Fig . 9.12b Comparison of Pile Cap Displacement Time Histories Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 315 The new iterative coupled method was verified by carrying out a nonlinear seismic response analysis on a a (5x5) pile group foundation carrying a SDOF system as superstructure and comparing the response with the fully coupled response. The fully coupled response was obtained using PILE3D as the whole pile foundation-SDOF system can be modeled in PILE3D. Comparisons of acceleration and displacement responses of the superstructure and the pile cap and maximum bending moment and shear force profiles along the center pile showed that the new method is capable of capturing the fully coupled response of the pile foundation and superstructure with prescribed accuracy. Because of the ease with which uncoupled analyses can be conducted, the difference between the coupled and uncoupled analysis were explored. Two types of uncoupled analyses were conducted. First the simplest type of analysis was conducted. The superstructure was analyzed as a fixed base structure and foundation effects were ignored. The comparisons of the acceleration and displacement time histories of the superstructure between the uncoupled and coupled analyzed showed that the uncoupled analysis fails to give acceptable results. The peak displacement in the uncoupled analysis was severely underestimated by about 52%. This shows that the flexibility of the foundation is important and an uncoupled analysis using a fixed base may not give the correct response. Next uncoupled analysis of the pile foundations was carried out using base motion. In this uncoupled analysis, the superstructure interaction was taken into account by applying also the time dependent dynamic force at the pile cap. The dynamic force is from the uncoupled analysis of the superstructure on a rigid base. The dynamic shear force at the base of superstructure determined from the uncoupled analysis of superstructure did not agree with that from the fully coupled analysis. A power spectral analysis showed that the dynamic shear force from the uncoupled analysis shows relatively higher frequency response compared to the fully coupled response. However, the peak dynamic force from the uncoupled analysis agreed well with that from the fully coupled analysis. This resulted in the similar peak response of the pile foundation in the uncoupled analysis for the case analyzed. The peak displacement of the pile cap and the maximum bending moment and shear force profiles along the pile from the uncoupled analysis agreed well with those from the fully coupled analysis. However, the time histories were very different except in the region of peak Chapter 9 : Coupled Analysis of Bridge Superstructure and Pile Foundation 316 response. This shows that the uncoupled analysis of pile foundation may give correct peak response but not the time history response if the superstructure interaction is taken into account in the uncoupled analysis. An uncoupled analysis of pile foundation was carried out under the base motion. In this analysis, the effect of superstructure was completely ignored and no dynamic force was applied at the pile cap. This uncoupled analysis of foundation severely underestimated the peak response of the pile cap. The difference between the peak displacement response was 44% and the difference between the peak moments and shear forces in the piles were also very high. This shows that the effect of superstructure interaction in the foundation is important and should be taken into account. CHAPTER 10 Effective Stress Analysis of Pile Foundations 318 CHAPTER 10 : EFFECTIVE STRESS ANALYSIS OF PILE FOUNDATIONS 10.1 INTRODUCTION Liquefaction has been one of the key causes of damage and failure in pile foundations in major earthquakes. Liquefaction in granular soils occurs due to the development of porewater pressures under undrained conditions during earthquake shaking. An increase in porewater pressure reduces the effective stresses in soils. Deformations in the soils are controlled by the effective stresses rather than by total stresses. Therefore effective stress analysis is required to obtain the seismic response of pile foundations in potentially liquefiable sites. The basis for effective stress analysis is a reliable model for the generation of porewater pressures under the cyclic loading. Many models have been developed for the generation of porewater pressures in the cohesionless soils under cycling loading.(Martin et al., 1977, Finn and Bhatia, 1984, Seed et al., 1976, Ishibashi et al., 1977, Ishihara et al. 1975, Ishihara et al., 1977, Kagawa and Kraft, 1981, Bazant and Krizek, 1976, Mroz et al., 1978, Zienkiewicz et al., 1978). The effective stress response analysis of pile foundations is a difficult task due to the complexity of the problem. Some attempts have been made by Kagawa et al.(1997) to analyze a single pile under effective stress conditions using a lumped parameter model and the p-y curves. However, an effective stress method for the analysis of pile groups that can effectively model the soil-pile-structural interactions in addition to the generation of porewater pressure due to the cyclic loading had not been developed prior to this study. A method for the effective stress analysis of pile foundation was developed in this study by incorporating a porewater pressure model into the PILE3D (Wu and Finn, 1997a and 1997b). Chapter 10 : Effective Stress Analysis of Pile Foundations 319 10.2 M-F-S POREWATER PRESSURE MODEL 10.2.1 Introduction The M-F-S model was developed in 1975 by Martin, Finn and Seed for the generation of porewater pressure under cyclic loading. Since its development this model has been extensively verified and improved at the University of British Columbia through various research studies. (Finn et al., 1977, Finn et al., 1986, Bhatia, 1982, Byrne, 1991). 10.2.2 Porewater Pressure under Undrained Conditions Consider a sample of saturated sand under a vertical effective stress, o v ' . Let the increment in volumetric compaction strain due to grain slip caused by a cycle of shear strain, y, during a drained cyclic simple shear test be evd. Let the increment in porewater pressure caused by a cycle of shear stain, y, during an undrained cyclic simple shear test starting with the same effective stress system be Au. Then, assuming that water to be incompressible, for fully saturated sand, the porewater pressure increment Au, is expressed as in which E r is the one dimensional rebound modules of sand at a vertical stress o v ' . E r is expressed Au = E r A e vd (10.1) as /. l -m / Nn-m (10.2) mK r (o V 0 ) in which a v o ' is the initial effective stress and K,., m and n are rebound constants. It is assumed in the M-F-S model that the increments in porewater pressure, Au, that develop in a saturated sand under cyclic shear strains are related to the volumetric strains increments, e v d that Chapter 10 : Effective Stress Analysis of Pile Foundations 320 occur in the same sand under drained conditions with the same shear strain history. This assumption has been verified by an extensive laboratory program involving drained and undrained tests on normally and overconsolidated sands. (Finn, 1981, Bhatia, 1982). M-F-S model was first implemented in the 1-D effective stress site response analysis programs DESRA (Finn et al. 1977). It was later extended for 2-D conditions and implemented in the 2-D nonlinear finite element program TARA3 (Finn et al., 1986 ). The application of these two program in several studies showed that M-F-S model can satisfactorily predict the rate of porewater pressure generation under simulated earthquake loading. 10.2.3 Original Four Parameter Model for the Volumetric Stains under Drained Conditions The original M-F-S model is based on the simple shear data and would simulate the earthquake loading under level ground conditions. In this model, under simple shear conditions, the volumetric strain increment, Ae v d per cycle of shear strain, is expressed as a function of the total accumulated volumetric strain, ev d, from the previous cycles and the amplitude of shear strain cycle, Y-in which C l 5 C 2 , C 3 and C 4 are volume change constants. These volume change constants depend on the type of sand and the relative density. A simplified version of the Eq. 10.4 was proposed by Byrne (1992) that contains only two volume change constants instead of four. (10.3) ( Y + C 4 e v d ) Ae = O.SyCjexp (10.4) half cycle Y Chapter 10 : Effective Stress Analysis of Pile Foundations 321 in which e v d m f c y c l e is the incremental volumetric strain per half cycle of shear strain, y, and e v is the accumulated volumetric strain from the previous cycles. Q and C 2 are volume change constants that depend only on the relative density. 10.2.4 Parameters of the M-F-S Model The rebound constants K,, m and n can be obtained from the rebound tests in a consolidation ring. The volume change constants C, and C 2 can be obtained from the drained test data. However, in practice, the parameters, K , , C\ and C 2 are usually obtained by calibrating the model against the field liquefaction resistance curve which is established from Standard Penetration Data or Cone Penetration Data (Finn, 1993). 10.3 EFFECTIVE STRESS ANALYSIS OF PILE FOUNDATIONS By suitably incorporating the M-F-S porewater pressure model into the dynamic analysis procedure of PILE3D, the total stress analysis program PILE3D was extended to carry out effective stress analysis. Under vertically propagating shear waves due to unidirectional shaking, the soil mainly undergo shear deformations in the horizontal plane in the direction of shaking. These shear strains y x y deformations are considered in the generation of porewater pressure using Eqs. 10.1, 10.2 and 10.4. Details of how porewater pressures are calculated using this model during dynamic analysis are to be found in Finn et al.(1977). The modified version of PILE3D with M-F-S porewater pressure generation model is called PILE3D-E. Chapter 10 : Effective Stress Analysis of Pile Foundations 10.4 MODIFICATIONS OF SOIL PROPERTIES 322 During seismic response analysis the shear modulus and shear strength are modified to be consistent with the current effective stresses using Eqs. 10.5 and 10.6 G a max _ V G a maxo vo 0.5 (10.5) X a max V X a maxo vo (10.6) Here G m a x 0 ,tm a x 0 and o'^ are the initial values of modulus, shear strength and vertical stress and G m a x , x m a x and o' v are the current values. 10.5 DYNAMIC ANALYSIS OF CENTRIFUGE TEST OF SINGLE PILES AND PILE GROUPS 10.5.1 Centrifuge Tests Dynamic centrifuge tests of pile supported structures in liquefiable sand were performed on the large centrifuge at University of California at Davis. The models consisted of three structures supported by single piles, one structure supported by a 2x2 pile group and another structure supported by 3x3 pile group. The arrangement of structures and instrumentation is shown in Fig. 10.1. Model tests were performed at a nominal centrifugal acceleration of 30g. Full details of the centrifuge tests can be found in Wilson et al. (1997). The soil profile consists of two level layers of Nevada sand, each approximately 10m thick. Nevada sand is a uniformly graded fine sand with a coefficient of uniformity of 1.5 and mean grain size of 0.15 mm. Sand was air pluviated to relative densities of 75-80% in the lower layer and 55% Chapter 10 : Effective Stress Analysis of Pile Foundations 323 GP1 GP2 n Porepressure 3 - Displacement SP1 > »• Strain Gauge > Accelerometer v v v Fig. 10.1 Layout of Single pile and (2x2) and (3x3) Group Pile Models for the Centrifuge Test Chapter 10 : Effective Stress Analysis of Pile Foundations 324 in the upper layer. Prior to saturation, any entrapped air was carefully removed. The container was then filled with hydroxy-propyl methyl-cellulose water under vacuum. The viscosity of the pore fluid is about ten times greater than that of pure water. Saturation was confirmed by measuring the compressive wave velocity from the top to the bottom of the soil profile. Four structural systems were used. Two systems were supported by single pile and two systems were supported by 2x2 and 3x3 pile groups. The test arrangement is shown in Fig. 10.1 The single pile system(SPl), (2x2) pile group(GPl) and (3x3) pile group (GP2) are studied here in detail. The model dimensions and the arrangement of bending strain gauges in these three systems systems (SP1, GPI and GP2) are shown in Figs. 10.2 ,10.3 and 10.4. Table 10.1 Prototype parameters of single and group pile systems Model Parameter Single Pile SP1 2x2 Group Pile, GPI 3x3 Group Pile, GP2 Pile Outer Diameter (m) 0.667 0.667 0.667 Inner Diameter (m) 0.523 0.523 0.523 Length (m) 16.76 16.8 16.8 Area (m2) 0.135 0.135 0.135 Bending Stiffness (MNm 2) 417 417 417 Pile Spacing (m) N A 2.667 2.667 Pile Cap Side Length (m) N A 4.647 7.314 Thickness (m) N A 2.286 2.286 Embedded N A Yes Yes Superstructure Mass (Mg) 49 233 468 The superstructures above the pile cap in the (2x2) group GPI and (3x3) group GP2 were not simple. The superstructures in both system consist of a steel tube, two stiff flanges, an aluminum spacer and a cylindrical mass. The sketch of the superstructure is shown in Fig. 10. 5a. The steel tube was braised into the stiff flanges at both ends of the tube. The cylindrical mass was connected Chapter 10 : Effective Stress Analysis of Pile Foundations 325 4 bSG4 a SG:Strain Gauge PSG1 SG2 SG3 Mass=1.82 kg All dimensions are in cm model scale Fig. 10.2 Instrumented Pile for Single Pile Test Mass=8.63 kg Column Mass= 6.46 kg Fixed Base Period=60 Hz , 15.49 A,B: Strain Gauge 1 1 1 i 1 1B2 1 3B3 i JB4 1 • B5 V V CSJ A ( D T 3.30 8.89 All dimensions are in cm model scale Fig. 10.3 Instrumented Test Piles and Details of Superstructure for (2x2) Group Pile Test Chapter 10 : Effective Stress Analysis of Pile Foundations 326 Strain Gauge Mass=17.34 kg Column Mass= 7.14 kg Fixed Base Period=60 Hz 24.38 1 — i f o o o o o o <i> o o 3.3 8.89 All dimensions are in cm model scale Fig. 10.4 Instrumented Test Piles and Details of Superstructure for (3x3) Group Pile Test Chapter 10 : Effective Stress Analysis of Pile Foundations 327 to the top end of the tube and the aluminum spacer was placed in between the top cylindrical mass and the top flange. The masses of each component and their locations were available. However, their stiffness properties were not available. Wilson et al. (1997) reported that the fixed base frequency of the whole superstructure in both GP1 and GP2 systems when it is fixed at the base of the columns was 2 Hz. In modeling the superstructure for analysis, the steel tube was modeled using beam elements. The cylindrical mass, stiff flanges and aluminum spacers were not explicitly modeled. Instead their masses were assigned at their center of gravity. The sections of steel tube within the stiff flange, aluminum spacers and the cylindrical masses were treated as rigid. The stiffness of the steel tube between the flanges was treated as flexible. However, the stiffness of the beam element that models this flexible part of the steel tube was not known. It was derived using the information that the fixed base frequency of the superstructure is 2Hz. A separate finite element eigen value analysis of the fixed base superstructure was carried out for this purpose. The stiffness of the beam element was back calculated from an elastic eigen value analysis that gives the first fixed base frequency of the superstructure as 2Hz. It can be noted here that the superstructure was not explicitly modeled due to lack of information. The simplified model of the superstructure shown Fig. 10.5b was used in the verification study. A total of sixteen shaking events, two small step wave events and fourteen earthquake events were performed at a centrifuge acceleration of 30 g. For earthquake events, the ground motions taken from strong motion records at Santa Cruz in the 1989 Loma Prieta Earthquake and Port Island in the 1995 Great Hanshin Awaji (Kobe) Earthquake were used. Table 10.2 shows the input motion parameters of the two events selected for a detailed study in this thesis. Table 10.2 Input motion parameters of the selected earthquake events Event Earthquake Record Amplification Factor Peak Acceleration (g) I Santa Cruz 12.0 0.49 K Santa Cruz 4.0 0.11 Chapter 10 : Effective Stress Analysis of Pile Foundations 328 •Hi Mass Aluminium Spacer Flange Steel tube Flange Concentrated Mass of Superstructure Block Concentrated Mass of Aluminium Spacer Rigid Beam Elements Flexible Beam Element Rigid Beam Element Fig. 10.5a Sketch of Superstructure Fig. 10.5b Model of Superstructure Fig. 10.5 Sketch of the Superstructure Used in the (2x2) and (3x3) Pile Group Tests and the Model of Superstructure Used in the Analysis Chapter 10 : Effective Stress Analysis of Pile Foundations 10.5.2 Effective Stress Dynamic Analysis of Single Pile 10.5.2.1 Introduction 329 An effective stress dynamic analysis was carried out using PHJE3D-E with M-F-S pore water pressure model to predict the response of the single pile system SP1 under the simulated earthquake events. 10.5.2.2 Finite Element Model of the Single Pile-superstructure System The finite element mesh used in the analysis is shown in Fig. 10.6 The finite element model consists of 1649 nodes and 1200 soil elements. The upper sand layer which is 9.1 m thick was divided into 11 layers and the lower sand layer which is 11.4 m thick was divided into 9 layers. The single pile was modeled with 28 beam elements, 17 elements were within the soil strata and 11 elements were used to model the free standing length of the pile above the ground. The superstructure was treated as a rigid body and its motion is represented by a concentrated mass at the center of gravity. A rigid beam element was used to connect the superstructure to the pile head. 10.5.2.3 Soil and Pile Properties The small strain shear moduli, G m a x were estimated using formula proposed by Seed and Idriss (1970). G = 21.7 k P max " - m a x ^ a m V P a , (10.7) in which k ^ is a constant which depends on the relative density of the soil, o'm is the initial mean effective stress and P a is the atmospheric pressure. The constant k m a x was estimated using the formula suggested by Byrne (1990). The calculated G m a x profile is shown in Fig. 10.7a. The equivalent linear stress -strain model used in PILE3D-E accounts for the change in shear moduli and Chapter 10 : Effective Stress Analysis of Pile Foundations 330 Plan View Elevation Direction of Shaking i i i i i i i i i i i 0 5 10 Geo Scale Fig. 10.6 Finite Element Mesh Used in the Single Pile Analysis Chapter 10 : Effective Stress Analysis of Pile Foundations 331 damping ratios due to dynamic shear strains. For the shear strain dependancy of the shear moduli and damping of the soil, the curves proposed by Seed and Idriss (1970) for sand and shown in Fig. 10.7b were used. The friction angles of the loose and dense sand were taken as 35 and 40 degrees respectively. The pile was treated as a linear elastic element. The elastic properties of the pile listed in Table 10.1. 10.5.2.5 Porewater Pressure Mode l Parameters The volume change constantsC, and C 2 and the rebound constant, k 2 were estimated based on the relative density of the soil and typical values were used for exponent constants m and n. Table 10.3 M-F-S Porewater pressure model parameters Layer Relative Density, Dr c, C 2 m n k2 Upper Layer 55% 0.34 1.18 0.43 0.62 0.035 Lower Layer 75-80% 0.13 3.01 0.43 0.62 0.015 10.5.2.6 Ear thquake Input Mot ion Analysis were carried out for the two earthquake events listed in Table 10.3. The measured acceleration at the west side of the base was used as the input acceleration. Wilson et al.(1995) showed that the measured acceleration at the east and west base of the container agreed very well. Fig. 10.8 show the unit normalized input acceleration of 1989 Loma Prieta Earthquake recorded at Santa Cruz. Chapter 10 : Effective Stress Analysis of Pile Foundations 332 G m a x (MPa) 0 20 40 60 80 100 120 140 Fig. 10.7a Gmax Profile 1.2 i 0.0001 0.001 0.01 0.1 1 10 Shear Strain (%) Fig. 10.7b Variation of Shear Modulus and Damping Ratios with Shear Strain Chapter 10 : Effective Stress Analysis of Pile Foundations 333 1.0 Centrifuge Test: CSP3; Sanata Cruz EQ Amax for Test l=0.49g and for Test K=0.11g _1_0 I 1 ' 1 " 1 ' 1 I 0 5 10 15 20 Time (sec) F i g . 10.8 Input Acceleration Time Histories Used in the Centrifuge Test o f Single and (2x2) and (3x3) Group Piles Chapter 10 : Effective Stress Analysis of Pile Foundations 334 10.5.3 Results of Single Pile Analysis 10.5.3.1 Test I Test I was carried out with a peak base acceleration of 0.49 g. Porewater Pressure Response: The computed and measured porewater pressures at four different depths are shown in Fig. 10.9a-d. There is a very good agreement between the measured and computed responses at depths 1.14 m and 6.78 m. At depths 4.56 m and 12.06 m the porewater pressure responses are slightly underestimated by about 5-10%. It is evident from the responses at depths 1.14 m and 4.56 m that the porewater pressure reached more than 90% of the initial effective overburden pressure suggesting a possible liquefaction of surficial layers. The maximum porewater pressure at the depth of 6.78m in the loose sand was about 65%. At 12.06 m, the measured and computed responses show that the maximum porewater pressure response was only about 20-25%. Bending Moment Response: Fig. lO.lOa-c shows the comparisons of measured and computed bending moment time histories at three different depths. Generally there is a very good agreement between the measured and computed time histories. Fig. 10.11 shows the measured and computed maximum bending moment profiles with depth. The maximum bending moment is somewhat overestimated between l m and 6m depth. Acceleration Response: Fig. 10.12a shows the acceleration response in the free field. The measured response shows very high frequency components with some of the peaks exceeding 0.4g. Wilson et al. (1995) reported that occurrence of such high frequency spikes are a phenomenon in centrifuge tests only in the event of liquefaction and are not observed in the field. As the measured field acceleration records do not usually show frequencies more than 10Hz the measured free field acceleration record was low pass filtered at 10 Hz. The comparison between the filtered free field acceleration record and the computed acceleration record is shown in Fig. 10.12b Good agreement can be observed between the measured and the computed accelerations. Chapter 10 : Effective Stress Analysis of Pile Foundations 335 (0 or 3 to c/> d> i Q _ d> i o 0_ 0 5 10 15 20 Time (sec) Fig. 10.9a Comparison of Porewater Pressure Time Histories at D=1.14 m (0 3 in (/) a> i — Q -(1) i O Q. 10 Time (sec) 15 20 Fig. 10.9b Comparison of Porewater Pressure Time Histories at D=4.56 m .2 100 3 in in a> I Q_ o a. 15 20 0 5 10 Time (sec) Fig. 10.9c Comparison of Porewater Pressure Time Histories at D=6.78 m .2 100 (0 3 in Ui 0) L_ CL E o o_ 50 h 10 Time (sec) 15 20 Fig. 10.9d Comparison of Porewater Pressure Time Histories at D=12.06 m Chapter 10 : Effective Stress Analysis of Pile Foundations 336 Test: CSP3-I; Amax=0.49g; Santa Cruz EQ Measured Computed 1.5 E z ~ 0.0 cu E o -1.5 I 10 Time (sec) 15 20 Fig. 10.10a Comparison of Moment Time Histories at D=0.76 m 1.5 ~ 0.0 CD E o -1.5 '/I ' \ 'AVv fa. :\ An A A - / i , i ;/ H/ lU W w ^ ^ ^ ^ V a w/ 4/ V/[ y V " " I 1 10 Time (sec) 15 20 Fig. 10.10b Comparison of Moment Time Histories at D=1.52 m 1.5 - 0.0 CD E o -1.5 ih " i'1 Ik i\ rA\ A A A '"v-./-^ - / _ i , i V V V 1 * 1 10 Time (sec) 15 20 Fig. 10.10c Comparison of Moment Time Histories at D=2.29 m Chapter 10 : Effective Stress Analysis of Pile Foundations 337 at c o ra \ 0) CD o o < Test: CSP3-I; Amax=0.49g; Santa Cruz EQ Bending Moment (MNm) - 2 - 1 0 1 Q. Q i r Surface -5 0 5 10 15 20 Fig . 10.11 Comparison of M a x i m u m Bending Moment Profiles Measured Computed Test: CSP3-I; Amax=0.49g; Santa Cruz EQ Measured Computed 1.00 0.00 -1.00 Meaured acceleration is unfiltered 15 0 5 10 Time (sec) F i g . 10.12a Comparison o f Free Field Acceleration Time Histories 20 c o ro 0 O O < 0.60 0.30 0.00 -0.30 -0.60 Measured acceler ation is lowpass filtered with 10 Hz frequency i j i . /ifii411L^ iL\iilAjiiiiA.lfi • ^ i i , 1 Ti'ii jj if v|f 7 y^ *v |f i i i 0 5 10 15 20 Time (sec) Fig . 10.12b Comparison of Free Fie ld Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 338 Fig. 10.13 shows the acceleration responses of the superstructure. There is a very good agreement between the measured and computed accelerations. The measured and computed pile head acceleration time histories are shown in Fig. 10.14. Though the computed response was somewhat soft, the agreement between the measured and the computed was reasonably good. 10.5.3.2 TestK The peak base acceleration in this test was O.lg Porewater Pressure Response:. Fig. 10.15a-c show the computed and measured porewater pressures at three different depths. Due to the low level of input base acceleration, the porewater pressure less than 20% of the initial effective overburden pressure in the loose layer. In the dense layer the response was negligible. The agreement between the measured and computed are reasonably good. Bending Moment Response: Fig 10.16a-c shows the comparison of measured and computed bending moment time histories at three different depths. There is a very good agreement between the measured and computed time histories except between 10 and 15 seconds. Fig. 10.17 shows the comparison of measured and computed maximum bending moment profile with the depth. The maximum bending moment is only slightly underestimated. Acceleration Response: Fig. 10.18 a,b and c show the accelerations in the free field, at the pile head and at the superstructure respectively. There is an excellent agreement between the measured and computed accelerations in the free field and at the pile head. The computed superstructure response also agree well with the measured response. Chapter 10 : Effective Stress Analysis of Pile Foundations 339 3 c o ro 0) <D O O < Test: CSP3-I; Amax=0.49g; Santa Cruz EQ 0.40 0.00 -0.40 Measured Computed 0 5 10 15 20 Time (sec) Fig. 10.13 Comparison of Superstructure Acceleration Time Histories c o ro k_ a> CD O O < 0.40 0.00 -0.40 10 Time (sec) 15 20 Fig. 10.14 Comparison of Pile Head Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 340 ro or CD </> </> co cL co \ o Test: CSP3-K; Amax=0.11g; Santa Cruz EQ r 100 h 50 10 Time (sec) Measured Computed 15 20 Fig. 10.15a Comparison of Porewater Pressure Time Histories at D=4.56 m CO or S> w co CL CO o CL 100 50 h 10 Time (sec) 15 20 Fig. 10.15b Comparison of Pporewater Pressure Time Histories at D=6.78 m r 100 CO 9? 3 V) V) CO CL CO 1 o 0_ 50 10 Time (sec) 15 20 Fig. 10.15c Comparison of Porewater Pressure Time Histories at D=20.07 m Chapter 10 : Effective Stress Analysis of Pile Foundations 3 4 1 Test: CSP3-K; Amax=0.11g; Santa Cruz EQ 0 . 5 ^ 0 . 0 co E o - 0 . 5 1 0 Time (sec) 1 5 Measured Computed ift A I * V / I 71 l l \ \ 11 >l'\ A : "Hi i V* WWW VI 1 , 1 1 2 0 Fig. 10.16a Comparison of Moment Time Histories at D=0.76 m 0 . 5 T= 0 . 0 CO E o - 0 . 5 '.A i \ 1 P i i J \A A r> r. A 'A 1 \vlAlr\i\ 1 If* *l tU V'V %y 11 ii 1/1 <? i;/ ni i • i 11 J !l If* 3V V 1 1 1 / \ / vs// i' V V v ' v « ff V/ 'V 1 0 Time (sec) 1 5 2 0 Fig. 10.16b Comparison of Moment Time Histories at D=l.52 m 0 . 5 ~ 0 . 0 CO E o - 0 . 5 >\ ! 'Jrl • \r\ a > l\ < /'\ 1 / \r J i Ji A t\ in A A a .7\ A 1 •' AA.'A fl / \ A A~/R ' / \ //vi i i i i '•1/ !• ll it <» » 1 i 1 0 Time (sec) 1 5 2 0 Fig. 10.16c Comparison of Moment Time Histories at D=2.29 m Chapter 10 : Effective Stress Analysis of Pile Foundations 342 Test: CSP3-K; Amax=0.11g; Santa Cruz EQ Bending Moment(MNm) -0.4 -0.2 0.0 0.2 0 i 1 1 1 1 1 1 1 Q. CD Q 10 12 - Measured w • Computed Fig. 10.17 Comparison of Maximum Bending Moment Profiles 3 c o •+—• CO fc_ _CD CD O O < Test: CSP3-K; Amax=0.11g; Santa Cruz EQ 0.20 0.00 -0.20 10 Time (sec) 15 Measured Computed 20 Fig. 10.18a Comparison of Free Field Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 343 c g ro i _QJ CD O O < Test: CSP3-K; Amax=0.11g; Santa Cruz EQ 0.20 0.00 -0.20 Measured Computed H I i i i i 0 5 10 15 20 Time (sec) Fig . 10.18b Comparison of Superstructure Acceleration Time Histories 3 c o 2 CD o o < 0.15 0.00 -0.15 i i i , i 10 Time (sec) 15 20 Fig . 10.18c Comparison o f Pile Head Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 10.5.4 Effective Stress Dynamic Analysis of (2x2) Group Pile 10.5.4.2 Finite Element Model of the (2x2) Group Pile-superstructure System 344 The finite element mesh used in the analysis is shown in Fig. 10.19 The finite element model consists of 1493 nodes and 1127 elements. The upper sand layer was divided into 9 layers and the lower sand layer was divided into 9 layers. Each pile was modeled with 15 beam elements. The pile cap was modeled with 16 brick elements and treated as rigid body. The superstructure was treated as a rigid body and its motion is represented by a concentrated mass at the center of gravity. The column carrying the superstructure mass was modeled using linear elastic beam elements. As the stiffness of this column element was not reported, it was calculated based on the fixed base period of the superstructure reported by Wilson et al. (1997) as 2 Hz. The same soil and pile properties and porewater pressure model parameters that were used in the single pile analyses were used in the pile group analyses also. The acceleration time history shown in Fig. 10.8 was used as the input motions. 10.5.5 Results of (2x2) Group Pile Analysis 10.5.5.1 Test I Acceleration Response: Fig. 10.20 shows the computed and measured acceleration responses at the pile cap. There is a good agreement between the two acceleration time histories. Fig. 10.21 shows the acceleration responses at the superstructure. The matching is not as good as the matching of time histories at the pile cap. The discrepancy between the measured and the computed are partly due to the simplifications made in the model of the superstructure which was described in detail earlier. Bending Moment Response: Fig. 10.22a-b show the measured and computed moment time histories at two depths 2.55m and 4.08 m. They compare very well up to eleven seconds. After that there is clearly a shift between the two time histories. This is shift is due to the presence of a residual Chapter 10 : Effective Stress Analysis of Pile Foundations 345 Plan View Axis of symmetry Elevation View ~* • 1 1 1 1 1 1 1 1 1 1 1 Direction of Shaking 0 5 10 Geo Scale Fig. 10.19 Finite Element Mesh of the (2x2) Pile Group Foundation Chapter 10 : Effective Stress Analysis of Pile Foundations 346 3 e o ro CO o o < Test: CSP3-I; Amax=0.49g; Santa Cruz EQ 0.40 0.00 -0.40 10 Time (sec) Measured Computed 15 20 Fig. 10.20 Comparison of Pile Cap Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 341 Test: CSP3-I; Amax=0.49g; Santa Cruz EQ 1.0 - 0.0 co E o -1.0 10 Time (sec) Measured Computed o „ r>r*.~j\f\ | A ft. AM A . AM A ll II I(\MI ll M A I ~yrvwy IMvW l^wHI I 1 i i 15 20 Fig . 10.22a Comparison of Moment Time Histories at D=2.55 m 1.0 - 0.0 CO E o -1.0 10 Time (sec) 15 20 Fig . 10.22b Comparison of Moment Time Histories at D=4.08 m Chapter 10 : Effective Stress Analysis of Pile Foundations 348 moment in the measured time history. Such residual moment cannot be predicted by the analysis based on equivalent linear approach. Fig. 10.23 a-b show the comparisons when the residual moment is removed from the measured moment time history. Excellent matching can be observed between the two time histories during the entire period of shaking. Fig. 10.24 shows the measured and computed bending moment profiles. The profiles match very well. 10.5.5.2 Approximate Effective Stress Analysis of Test I An approximate effective response of the pile foundation can be obtained using a total stress method if the total stress analysis is started from the weakened or liquefied soil conditions. This approximate method is described in the following steps below. • The effective stress condition at the end of an earthquake in the free field soil is established using some available 1-D (DESRA, Finn et al., 1977) or 2-D method (TARA3, Finn et al., 1986). Thus, in each soil layer in the free field, the soil properties will be consistent with the level of pore pressure developed at the end of the earthquake. Alternatively, the liquefaction potential of each layer of soil in the free field can be assessed based upon their standard penetration resistance(SPT) or cone penetration resistance(CPT) and the layers that have the potential to liquefy under the earthquake shaking can be assigned the liquefied soil properties. • Dynamic analysis is performed using the total stress analysis program PILE3D using the weakened soil properties. In order to assess the effectiveness of the approximate effective stress method, an analysis on Test I was carried out using the approximate method. In this analysis, first, an effective stress analysis of free field soil was carried out to determine the maximum porewater pressures in each layer at the end of shaking. Secondly, the shear modulus and the shear strength of each layer was reduced in consistent with the level of porewater pressure at the end of the earthquake shaking. Chapter 10 : Effective Stress Analysis of Pile Foundations 349 E Z Test: CSP3-I; Amax=0.49g; Santa Cruz EQ Measured Computed 1.0 •E 0.0 CD E o -1.0 Rc-ssidual Mo tally ment Removed from Measured Time History i i i W i 1 15 5 10 Time (sec) Fig. 10.23a Comparison of Moment Time Histories at D=2.55 m 20 1.0 •£ 0.0 CO E o -1.0 15 0 5 10 Time (sec) Fig. 10.23b Comparison of Moment Time Histories at D=4.08 m 20 Test: CSP3-I; Amax=0.49g; Santa Cruz EQ Bending Moment (MNm) -0.5 0.0 0.5 1.0 1.5 C L CO Q 10 12 Measured Computed Fig. 10.24 Comparison of Maximum Bending Moment Profiles Chapter 10 : Effective Stress Analysis of Pile Foundations 350 For the layers which liquefied during the analysis, the stiffness and strength was reduced to the 5% of the original values. In the third step, the modified free field soil properties were assigned as the initial properties of all the elements at the same depth and a total stress analysis of the pile foundation was carried out using the PILE3D program. Acceleration Response: Fig. 10.25 also shows the computed free field acceleration time histories from both the effective stress and the approximate effective stress analyses. There is a significant difference between the two time histories up to 11 seconds and after that they agree very well. It can be noted here that the surficial layers during the effective stress analysis liquefied approximately at 11 seconds. After 11 seconds, the rate of development of porewater pressure in the nonliquefied sub surficial layers were negligible. In contrast, in the approximate effective stress analysis, liquefied properties were used for the surficial layers up to 3.5m depth from the beginning of shaking. Because of this, the approximate effective stress analysis showed a much softer low frequency response up to 12 seconds. After that, in both analyses, the soil conditions were approximately the same and they showed a similar response. Bending Moment Response: Fig. 10.26 shows the computed bending moment time histories from both the effective stress analysis and the approximate effective stress analysis. Up to approximately 10 seconds, the peaks in the computed bending moments from the approximate analysis were consistently higher than the peaks in the from the effective stress analysis. After 10 seconds, both time histories agreed quite well. Fig. 10.27 shows the computed bending moment profiles from both the effective stress analysis and approximate effective stress analysis. It can be seen from this figure that, in the approximate analysis, the moments are underestimated up to the depth of 5.5m and beyond 5.5m they are overestimated. However, the difference between the maximum moments which occurred approximately at the same location is 8% only. Chapter 10 : Effective Stress Analysis of Pile Foundations 351 DJ C o 2 0) CD O O < 0.50 0.00 -0.50 Test: CSP3-I; Amax=0.49g; Santa Cruz EQ 1 1 . 1 Computed- Effective Stress Analysis Computed- Approximate Effective Stress Analysis l 15 20 0 5 10 Time (sec) Fig . 10.25 Comparison of Free Fie ld Acceleration Time Histories 1.00 - 0.00 cu E o -1.00 Computed- Effective Stress Analysis Computed- Approximate Effective Stress Analysis * \/ i» 'A* it tf * 1 1 1 * i 15 20 Fig . 0 5 10 Time (sec) 10.26 Comparison of Bending Moment Time Histories at D=2.29m Test:CSP3-l; Amax=0.49g; Santa Cruz EQ Bending Moment (MNm) -0.5 0.0 0.5 1.0 1.5 4 h C L Q 12 • Computed- Effective Stress Analysis • Computed- Approximate Effective| Stress Analysis F i g . 10.27 Comparison of M a x i m u m Bending Moment Profiles Chapter 10 : Effective Stress Analysis of Pile Foundations 10.5.5.3 Total Stress Analysis of Test I 352 It is of interest to compare the result from total stress analysis with those from effective stress analysis. Bending Moment Response: Fig. 10.28 shows the measured and computed maximum bending moment profiles with the depth. This figure also shows the bending moment profile from the effective stress analysis. It is clearly evident from this figure that the bending moments are significantly under estimated in the total stress analysis. The difference between the maximum bending moment from total and effective stress analyses is 19%. Fig. 10.29 shows the measured and computed bending moment time histories at 2.55 m depth. This figure shows that, at this depth, the bending moment is consistently under estimated in the total stress analysis. Acceleration Response: Fig. 10.30 shows the acceleration response at the pile cap. The accelerations are consistently overestimated in the total stress analysis during the shaking. There is a difference of 46% between the total stress and effective stress peak accelerations. 10.5.5.4 Linear Elastic Analysis of Test I In order to study the effects of nonlinear behaviour of the soil and the effect of porewater pressures a linear elastic total stress analysis of Test I was carried using the PILE3D. Bending Moment Response: Fig. 10.31 shows computed maximum bending moment profiles from the linear elastic analysis and the nonlinear effective stress. The measured maximum moments are also shown in Fig. 10.31 Fig. 10.32 shows the time histories of moment from the linear elastic and effective stress analyses at 2.55 m depth . It can be seen from these figures that the moment is severely underestimated in the linear elastic analysis. There is a 65% difference between the maximum moments. Chapter 10 : Effective Stress Analysis of Pile Foundations 353 Test:CSP3-l; Amax=0.49g; Santa Cruz EQ C L CO Q Bending Moment (MNm) -0.5 0.0 0.5 1.0 1.5 12 Measured -Q Computed-Effective Stress • m-- Computed-Total Stress Fig . 10.28 Comparion of M a x i m u m Bending Moment Profiles E Z e CO E o 3 c o ro i _ 0) CO o o < Test: CSP3-I; Amax=0.49g; Santa Cruz EQ Effective Stress Total Stress 0.4 0.0 -0.4 1 Ii A l '1 'fl AA i\ iH f \ . JJ*_JC3IA A fS 1 r i 1 i 1 n i i 10 Time (sec) 15 20 Fig. 10.29 Comparison of Moment Time Histories at D=2.55 m 0.4 0.0 -0.4 1 i . > I 1 j . Is i J . j A A i i i i 1 1 u 1 I 10 Time (sec) 15 20 Fig. 10.30 Comparison of Pile Cap Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 354 Acceleration Response: Fig. 10.33 shows the comparison of acceleration time histories at the pile cap. The peak acceleration is severely overestimated in the linear elastic analysis. The peak acceleration from the linear elastic analysis is 93% higher than the measured peak acceleration. 10.5.5.5 Tes tK In this test Santa Cruz earthquake record scaled to a peak acceleration of 0.1 g was used as the base input. Fig. 10.34a shows the computed and measured acceleration responses at the pile cap. The computed acceleration agrees very well with the measured acceleration. Fig. 10.34b shows the superstructure acceleration time history. Given the limitations of the superstructure model used in the analysis and described in the previous section, the agreement between the measured and the computed acceleration time history is acceptable. Fig. 10.35 shows the measured and computed maximum bending moment profiles. The calculated moment agrees fairly well with the measured maximum moment profile. The comparison of bending moment time histories at two depths are shown in Fig. 10.36a and 10.36b. Acceptable agreement can be observed between the measured and computed time histories. 10.6 EFFECTIVE STRESS DYNAMIC ANALYSIS OF (3X3) GROUP PILE 10.6.1 Finite Element Model of the (3x3) Group Pile-superstructure System The finite element mesh used in the analysis consists of 2262 nodes and 1819 soil elements. The upper sand layer was divided into 9 layers and the lower sand layer was divided into 9 layers. Each pile was modeled with 15 beam elements. The pile cap was modeled with 36 brick elements and treated as rigid body. The superstructure was treated as a rigid body and its motion is represented by a concentrated mass at the center of gravity. The column carrying the superstructure mass was modeled using the beam elements and is treated as linear elastic material. As the sniffiness Chapter 10 : Effective Stress Analysis of Pile Foundations 355 Test:CSP3-l; Amax=0.49g; Santa Cruz EQ Bending Moment (MNm) -0.5 0.0 0.5 1.0 1.5 0 CL 0) Q 12 1 I r Measured 0. Computed-Effective Stress - - B - - Computed-Linear Elastic Fig. 10.31 Comparion of Maximum Bending Moment Profiles Test: CSP3-I; Amax=0.49g; Santa Cruz EQ Effective Stress Linear Elastic E 0.4 z 0.0 d) E o 2 -0.4 Ui * " i l l 1 i p i ' n; W WW ^ ^^v W W W 1 i 10 Time (sec) 15 20 Fig. 10.32 Comparison of Moment Time Histories at D=2.55 m 0.6 3 0.3 e o 0.0 a> a> o o -0 .3 < -0.6 10 Time (sec) Fig. 10.33 Comparison of Pile Cap Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 356 3 c o ro \ CD 0) O O < Test: CSP3-K; Amax=0.11g; Santa Cruz E 0.20 0.00 -0.20 10 Time (sec) 15 Measured Computed 20 Fig . 10.34a Comparison of Pile Cap Acceleration Time Histories 3 c o "•4—» ro 0) CD O O < 0.30 0.00 -0.30 / l i / t i i f l j i N ' y ft fcj A r 1 ft'A-A'vA A A1 VI irl« • V/ B 5/ u VI Vt! 'V V/ « J w >l/ 1/ V " 9 u • 1 v V *•» 1 1 i 10 Time (sec) 15 20 Fig . 10.34b Comparison of Superstructure Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 357 Test:CSP3-K; Amax=0.11g; Santa Cruz EQ Bending Moment (MNm) -0.6 -0.4 -0.2 0.0 0.2 0 r ~ ~ i i — i — n — i i i i 4 h 12 -• Measured Computed E o Fig . 10.35 Comparison of Maximum Bending Moment Profiles Test: CSP3-K; Amax=0.11g; Santa Cruz EQ Measured Computed 0.2 0.0 -0.2 15 20 5 10 Time (sec) Fig . 10.36a Comparison of Moment Time Histories at D=2.55 m 0.2 ~ 0.0 E o -0.2 ' V U w 10 Time (sec) 15 20 Fig. 10.36b Comparison of Moment Time Histories at D = 4.08 m Chapter 10 : Effective Stress Analysis of Pile Foundations 358 of this column element was not reported, it was calculated based on the fixed base period of the superstructure reported by Wilson et al. (1997). 10.6.2 Results of (3x3) Group Pile Analysis 10.6.2.1 Test I Fig. 10.37a shows the acceleration time histories at the pile cap. Very good agreement can be observed between the measured and computed acceleration time histories. Fig. 10.37b shows the acceleration time histories at the superstructure. At least part of the difference between the measured and computed acceleration time histories can be attributed to the simplifications made in the modeling of the superstructure. Fig. 10.38a shows the bending moment time histories at 2.29m depth. Computed and measured time history compares well up to eleven seconds. After that the measured time history clearly shows a residual moment. As the model used in the analysis was an equivalent linear model it did capture any residual moment. This resulted in a shift between the measured and computed moments. When the residual moment is removed from the measured time history, the computed moment agrees well with the measured moment (Fig. 10.38b). 10.6.2.2 TestK Fig. 10.39a and 10.39b show acceleration time histories at the pile cap and superstructure respectively. There is a very good agreement between the measured and calculated acceleration time history at the pile cap. Fig. 10.39c shows the bending moment time histories at 2.29m depth. Good agreement can be observed between the measured and calculated moment time history. The peak moments were identical. At least part of the difference between the measured and computed can be attributed to the simplifications made in the modeling of the superstructure. Chapter 10 : Effective Stress Analysis of Pile Foundations 359 o 5 JH co o o < Test: CSP3-I; Amax=0.49g; Santa Cruz EQ 0.40 0.00 -0.40 10 Time (sec) Measured Computed 15 20 Fig. 10.37a Comparison of Pile Cap Acceleration Time Histories c o _co CO o o < 1.00 0.00 -1.00 ii I - L A A A A M I J1 iAl //iv; A • f . •'• ft f l / Vffl rWA A/tt ft A AA #\ / U A M A JV AI • ^ff\fylv\j 1 I ju j l i / ' l l/'Ai / l l / u i i ' v ilr 'V/<w V j M f V ^ i •1 V * 1 1 1 1 1 1 10 Time (sec) 15 20 Fig. 10.37b Comparison of Superstructure Acceleration Time Histories Chapter 10 : Effective Stress Analysis of Pile Foundations 360 Test: CSP3-I; Amax=0.49g; Santa Cruz EQ 1.00 •g 0.00 CD E o -1.00 Measured Computed i \' P N l l b f l E I J »A1 ft Y_V—"i/™ 1 /'NV. * A tj 1 / 'VM - S I V i vl!'Af\ M . T ^ i l t v * V r , J /-i i IK/ • !!" • ii '. 1 '.VIS ft Ml 1* '\i vr-vy-v,'-v?1/ rr"*7j \irJ •V v * '.V • i 0 5 10 15 20 Time (sec) Fig. 10.38a Comparison of Bending Moment Time Histories at D=2.29m 1.00 •£ 0.00 CD E o -1.00 h A J A g - * iVi< ft « A'I RIY IV 11 •II i • mi f\*,+V ittinA ft It V 11 .flit 1 P\\ HMl fit 11 ?! Residual Measured Moment Removed < • It i iii HI tt A * 1 In M.n I.V En • A. A i 1 • w f v ^ M W T, 1/ K fVAtllfl X11 It HI V / l' la W ' H.' 1 1 , 1 # y ii ti u ft V. •* • • U L' • If V » l l " B « 1 , 1 , 10 Time (sec) 15 20 Fig. 10.38b Comparison of Bending Moment Time Histories at D=2.29 m Chapter 10 : Effective Stress Analysis of Pile Foundations 361 D) C o ro cu o o < Test: CSP3-K; Amax=0.11g; Santa Cruz E Measured Computed 0.20 -0.20 10 Time (sec) 15 Fig. 10.39a Comparison of Pile Cap Acceleration Time Histories 0.40 0.00 -0.40 10 Time (sec) 15 20 20 Fig. 10.39b Comparison of Superstructure Acceleration Time Histories 0.20 E z •g 0.00 Qi E o -0.20 15 20 0 5 10 Time (sec) Fig. 10.39c Comparison of Bending Moment Time Histories at D=2.29m Chapter 10 : Effective Stress Analysis of Pile Foundations 10.7 CONCLUSIONS 362 A new method was developed for the effective stress analysis of pile foundations. This method is an extension of the total stress analysis, PILE3D, Wu and Finn(1997). The M-F-S (Martin, Finn and Seed, 1975) porewater pressure model was used for generation of porewater pressure.. A series of centrifuge tests were carried out at the University of California at Davis on a single pile and (2x2) and (3x3) group piles in a liquefiable soils under low level and strong shaking. The test data with peak accelerations ranging from O.lg to 0.55 g were selected for the verification study and the results corresponding to a peak acceleration of 0.1 lg and 0.49 g are presented in the thesis. The new method was able to capture the time histories of porewater pressure response at different depths with acceptable accuracy under both low level and strong shaking The difference between the measured and computed maximum porewater pressures were less than 20%. In the simulation of single pile test, the new method was able to capture the single pile response under low level and strong shaking. The computed and measured bending moment time histories at different depths agreed very well. The time histories of accelerations at the pile head and at the superstructure were captured with acceptable accuracy. The new method was also used to simulate (2x2) and (3x3) group piles. It was shown that the new method can capture the group responses as well. An approximate effective stress analysis was carried out on (2x2) pile group under strong shaking. This approximate method showed that reliable time histories of accelerations and moments cannot be obtained by this method as, in this method, the total stress analysis starts from the weakened or liquefied conditions of the soil. The time histories were quite different until the soil layers that had the potential to liquefy did liquefy. However, the approximate method gave the maximum accelerations and moments that were close to the valued obtained from the effective stress analysis. Chapter 10 : Effective Stress Analysis of Pile Foundations 363 Total stress nonlinear analysis of (2x2) pile group under strong shaking was carried out under strong shaking to assess the need for the effective stress analysis. The total stress analysis in which the softening of the soil due to the development of porewater pressure was not taken into account over predicted the pile cap accelerations and under predicted the bending moments. The difference between the peak pile cap acceleration was about 46% and the difference between the maximum moment was about 19%. The peaks in the time histories of moments showed that the moments are consistently underestimated during shaking in the total stress analysis. The significant difference between the accelerations and moments showed that a total stress analysis in which the reduction in effective stresses in the soil and the resulting softening of soil were not taken into account cannot reliably predict the response of the pile under string shaking. A linear elastic analysis was also carried out using PILE3D and the results were compared with the measured effective stress response. Significant differences observed between the measured and computed accelerations and bending moments. The differences were much higher than that were obtained in the total stress analysis. The linear elastic analysis do not take into account of the nonlinear hysteretic behaviour of the soil and effect due to the development of porewater pressure in soils. The amount of difference showed that the soil exhibited highly nonlinear hysteretic behaviour during strong shaking which was not taken into account in the linear elastic analysis. Thus the linear elastic analysis is obviously not suitable if the soil shows highly nonlinear hysteretic behavior and leads to the development of high porewater pressures. CHAPTER 11 Summary and Conclusions 365 CHAPTER 11: SUMMARY AND CONCLUSIONS This thesis dealt with the one of the important and complex problem in soil-structure interaction, the seismic response of bridge foundations which are normally supported on pile foundations and their effects on the overall response of bridges. A substantial portion of thesis was devoted in developing a better understanding of the seismic behaviour of the pile foundations and their effects on the overall response of the bridge structure. Current methods for modeling the pile foundation for bridge analysis were evaluated and improvements were suggested. New methods were developed to effectively capture the seismic behaviour of the pile foundations In order to develop a better understanding of the dynamic behaviour of bridge foundations, two case studies were carried out on the two span Painter Street Overpass located in Rio Dell, California using the low amplitude ambient vibration data and strong motion records. Elastic analyses of the foundations was found to be adequate to evaluate behaviour exhibited by the bridge under ambient vibration. A comparison study using a fixed base model and flexible base model analyses showed that the foundation flexibility influences the vibration of the bridge in the transverse direction even under ambient vibration. However, the vertical vibration is not affected much by the foundation flexibility at low level of shaking. Analysis of bridge using strong motion records from the bridge itself showed that the bridge abutments exhibited highly nonlinear behaviour and the stiffness dropped drastically. The fundamental frequencies under strong shaking were much smaller than those from ambient vibration analysis as a consequence of the reduced stiffness under strong shaking. An inability to model this nonlinear behaviour reliably in a way that can be incorporated into existing software for analyzing the superstructures is a major problem for bridge designers. As a solution to this problem, a lumped parameter model of the foundations using springs and dashpots was developed with nonlinear spring stiffnesses and dashpot parameters corresponding to the pile cap impedances. The lumped parameter model was shown to be modeling the effects of pile foundations on the superstructure. Chapter 11: Summary and Conclusions 366 Studies with and without the inertial interaction effects of the superstructure using the lumped parameter model showed the inertial interaction significantly affects the stiffness of the pile foundations during strong shaking and therefore should be taken into account. Following a procedure used sometimes in practice, the stiffness and damping were computed using the effective shear moduli and damping ratios for the soil obtained from a " S H A K E " analysis of the freefield. This approximate procedure gave acceptable results only when the inertial interaction from the superstructure is ignored. A new bridge model was proposed to determine the nonlinear seismic response of bridges. In the new model, the lumped parameter model of the pile foundations was incorporated into a three dimensional space frame model of the superstructure. The method was implemented in a computer program called BRIDGE-NL. The dynamic behaviour of the bridge in the transverse and the longitudinal directions were analysed under earthquake loading using BRTDGE-NL. The time histories of the fundamental frequencies were traced during the analysis. They showed that the fundamental frequencies reduce during strong shaking due to the reduction in the foundation stiffnesses. The reduction in the foundation stiffnesses is mainly caused by the nonlinear behaviour of the foundation soil. Studies on the effect of the inertial interaction showed that it alters the stiffness of the foundation significantly during strong shaking and as a result the fundamental frequency of the bridge is also altered. The role of foundation flexibility on the dynamic behaviour of the bridge was investigated by comparing the fixed base model response of the bridge with the flexible base model response. Significant differences were observed between the two response when the pier of the foundation is relatively stiff compared to the foundation. When the pier is flexible the difference between the response of the two models were minimal. A parametric study was carried on a two span single pier bridge to investigate the effect of various foundation and superstructure stiffness parameters on the dynamic response of a bridge. The following were learnt from the parametric study. Chapter 11: Summary and Conclusions 3