D Y N A M I C SOIL-STRUCTURE INTERACTION: THEORY AND VERIFICATION by MUTHUCUMARASAMY YOGENDRAKUMAR B . S c . E n g . ( H o n s ) , University of Peradeniya, Sri L a n k a , M . A . S c , T h e University of British C o l u m b i a , A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L M E N T THE REQUIREMENTS FOR T H E DEGREE OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE (Department STUDIES of Civil Engineering) We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH January © Muthucumarasamy COLUMBIA 1988 Yogendrakumar. 1980 1983 1988 OF In presenting degree freely at this the available copying of department publication of in partial fulfilment University of British Columbia, for this or thesis reference thesis by this for his thesis and study. scholarly or her for of C\U1L purposes gain shall € K J G 1M£<C£-< ' M G DE-6f3/81) the requirements agree that agree may representatives. financial The University of British C o l u m b i a 1956 Main Mall Vancouver, C a n a d a V6T 1Y3 I I further permission. Department of be It not is be that the for Library an shall permission for granted by understood allowed advanced the that without make it extensive head of copying my my or written ABSTRACT A nonlinear effective stress method of analysis for determining the static and dynamic response of 2 - D embankments and soil-structure interaction systems is presented. method of analysis is incorporated in the computer program T A R A - 3 . The T h e constitutive model in T A R A - 3 is expressed as a sum of a shear stress model and a normal stress model. The behavior in shear is assumed to be nonlinear and hysteretic, exhibiting Masing behavior under unloading and reloading. T h e response of the soil to uniform all round pressure is assumed to nonlinearly elastic and dependent on the mean normal effective stresses. T h e porewater pressures required in the dynamic effective stress method of analysis are obtained by the M a r t i n - F i n n - S e e d porewater pressure generation model modified to include the effect of initial static shear. During dynamic analysis, the effective stress regime and consequently the soil properties are modified for the effect of seismically induced porewater pressures. A very attractive feature of T A R A - 3 is that all the parameters required for an anal- ysis may be obtained from conventional geotechnical engineering tests either in-situ or in laboratory. A novel feature of the program is that the dynamic analysis can be conducted starting from the static stress-strain condition which leads to accumulating permanent deformations in the direction of the smallest residual resistance to deformation. T h e program can also start the dynamic analysis from a zero stress-zero strain condition as is done conventionally in engineering practice. The program includes an energy transmitting base and lateral energy boundaries to simulate the radiation of energy which occurs in the transmitting field. T h e program predicts accelerations, porewater pressures, instantaneous dynamic deformations, permanent deformations due to the hysteretic stress-strain ii response, deformations due to gravity acting on the softening soil and deformations due to consolidation as the seismic porewater pressures dissipate. T h e capability of T A R A - 3 to model the response of soil structures and soil-structure interaction systems during earthquakes has been validated using data from simulated earthquake tests on a variety of centrifuged models conducted on the large geotechnical centrifuge at Cambridge University in the United K i n g d o m . T h e data base includes acceleration time histories, porewater pressure time histories and deformations at many locations within the models. T h e program was able to successfully simulate acceleration and porewater pressure time histories and residual deformations in the models. T h e validation program suggests that T A R A - 3 is an efficient and reliable program for the nonlinear effective stress analysis of many important problems in geotechnical engineering for which 2-D plane strain representation is adequate. TABLE OF CONTENTS Page Abstract iii Table of Contents iv List of Symbols ix List of Tables xii List of Figures xiii Dedication xxi Acknowledgements xxii Chapter 1 INTRODUCTION 1 1.1 Scope 7 1.2 Thesis Outline 8 Chapter 2 METHOD OF STATIC ANALYSIS IN TARA-3 10 2.1 Introduction 10 2.2 Finite Element Representation 11 2.3 Stress-Strain-Volume 12 Change Behavior 2.3.1 Shear Stress-Strain Relationship 13 2.3.1.1 C o m p u t a t i o n of Hyperbolic M o d e l Parameters 2.3.2 Volume Change Behavior 13 18 2.4 L o a d Shedding Technique 19 2.5 Shear Induced V o l u m e Change 22 2.6 Simulation of Construction Sequence 27 2.6.1 Introduction 27 2.6.2 M e t h o d of Analysis 28 Chapter 3 METHOD OF DYNAMIC ANALYSIS IN TARA-3 31 3.1 Introduction 31 3.2 Equations of M o t i o n 32 iv 3.3 Incremental Equations of M o t i o n 3.4 D y n a m i c Stress-Strain 33 Behavior 35 3.4.1 D y n a m i c Shear Stress-Shear Strain Behavior 35 3.4.1.1 C o m p u t a t i o n of Hyperbolic M o d e l Parameters 39 3.4.2 Volume Change Behavior 43 3.5 Formulation of Mass M a t r i x 43 3.6 Formulation of Stiffness M a t r i x 44 3.7 Formulation of D a m p i n g M a t r i x 46 3.8 C o m p u t a t i o n of Correction Force Vector 48 3.9 Residual Porewater Pressure M o d e l 49 3.9.1 M a r t i n - F i n n - S e e d M o d e l 50 3.9.2 Extension O f M - F - S M o d e l to 2-D Conditions 52 3.10 Evaluation of C u r r e n t Effective Stress System 52 3.10.1 Modification of Soil Properties 53 3.10.2 Estimation of M a x i m u m Residual Porewater Pressure 54 3.11 Interface Representation 56 3.11.1 Slip Element Formulation 56 3.11.2. Analysis Procedure 59 3.12 C o m p u t a t i o n of Deformation Pattern Chapter 4 60 INCORPORATION OF ENERGY TRANSMITTING BOUNDARY 62 4.1 Introduction 62 4.2 Review of Possible Transmitting Boundaries 64 4.3 Energy Transmitting Boundaries in T A R A - 3 66 4.4 Finite Element Formulation For Transmitting Base 66 4.5 Finite Element Formulation F o r Lateral Viscous Boundary 72 4.6 Effectiveness of Transmitting Base 76 4.7 Effectiveness of the Lateral Viscous Boundary 81 4.7.1 Linear Analysis 83 v 4.7.2 Nonlinear Analysis 88 4.7.3 Discussion 98 Chapter 5 S I M U L A T E D SEISMIC T E S T S ON C E N T R I F U G E 100 5.1 Introduction 100 5.2 Centrifuge Testing 101 5.3 Scaling Laws 102 5.4 Earthquake Simulation in Cambridge Geotechnical Centrifuge 103 5.5 M o d e l Construction 104 5.5.1 D r y M o d e l Construction 105 5.5.2 Saturated M o d e l Construction 106 5.5.2.1 M e t h o d 1 106 5.5.2.2 M e t h o d 2 107 5.6 Relative Density Estimation 107 5.7 Instrumentation 108 and Accuracy 5.7.1 Accelerometers 108 5.7.2 Porewater Pressure Transducers 109 5.7.3 Linearly Variable Displacement Transducers ( L V D T ' s ) 5.8 D a t a Acquisition and Digitisation Ill Ill 5.9 Centrifuge Flight 112 5.10 T y p i c a l Test D a t a 113 5.11 Centrifuge Tests Used in the Verification Study 117 Chapter 6 118 SOIL P R O P E R T I E S F O R TARA-3 A N A L Y S E S 6.1 Introduction 118 6.2 Shear and B u l k M o d u l i Parameters 119 6.3 Liquefaction Resistance C u r v e 120 6.4 Structural Properties 122 6.5 Slip Element Properties 123 Chapter 7 125 VERIFICATION BASED ON DRY M O D E L TESTS vi 7.1 Verification Study Based on Test Series L D O l 125 7.1.1 Centrifuge M o d e l in Test Series L D O l 125 7.1.2 M o d e l Response in Test L D O l 128 7.1.3 C o m p a r i s o n of Acceleration Responses of Test L D O l / E Q l 131 7.1.4 Comparison of Settlements in Test L D O l / E Q l 142 7.2 Verification S t u d y Based on Test Series L D 0 2 145 7.2.1 Centrifuge M o d e l in Test Series L D 0 2 145 7.2.2 M o d e l Response in Test L D 0 2 145 7.2.3 C o m p a r i s o n of Acceleration Responses of Test L D 0 2 / E Q 4 154 7.2.4 Comparison of Settlements in Test L D 0 2 / E Q 4 167 7.3 Verification Study Based on Test Series RSS110 173 7.3.1 Centrifuge M o d e l in Test Series RSS110 173 7.3.2 M o d e l Response in Test RSS110 175 7.3.3 Comparison of Acceleration Responses of Test R S S 1 1 0 / E Q 1 179 7.3.4 Comparison of Settlement in Test R S S 1 1 0 / E Q 1 185 7.4 Verification Study Based on Test Series RSS90 189 7.4.1 Centrifuge M o d e l in Test Series RSS90 189 7.4.2 M o d e l Response in Test RSS90 192 7.4.3 C o m p a r i s o n of Acceleration Responses of Test R S S 9 0 / E Q 2 195 7.4.4 Comparison of Settlement in Test R S S 9 0 / E Q 2 208 Chapter 8 VERIFICATION BASED ON SATURATED MODEL TESTS 8.1 Verification Study Based on Test Series L D 0 4 211 211 8.1.1 Centrifuge M o d e l in Test Series L D 0 4 211 8.1.2 M o d e l Response in Test L D 0 4 213 8.1.3 Comparison of Acceleration Responses in Test L D 0 4 / E Q 2 217 8.1.4 Comparison of Porewater Pressures in Test L D 0 4 / E Q 2 222 8.1.5 C o m p a r i s o n of settlements in Test L D 0 4 / E Q 2 228 8.2 Verification Study Based on Test Series RSS111 vii 230 8.2.1 Centrifuge M o d e l in Test Series RSS111 230 8.2.2 M o d e l Response in Test RSS111 231 8.2.3 Comparison of Acceleration Responses in Test R S S 1 1 1 / E Q 1 238 8.2.4 C o m p a r i s o n of Porewater Pressure Response in Test R S S 1 1 1 / E Q 1 8.2.5 Stress-Strain Behavior 258 8.2.6 C o m p a r i s o n of Displacements in Test R S S 1 1 1 / E Q 1 262 Chapter 9 SUMMARY AND CONCLUSIONS . . . . 250 270 9.1 S u m m a r y 270 9.2 Conclusions 271 9.3 Recommendations For Further Study 272 References 274 Appendix I 282 A p p e n d i x II 286 W2J LIST OF SYMBOLS a Constant b Constant B Bulk M o d u l u s B Tangent Bulk M o d u l u s [B] Strain Displacement M a t r i x c' Cohesion Intercept t Volume Change Constant c 2 Volume Change Constant Volume Change Constant c Volume Change Constant [c] G l o b a l D a m p i n g Matrix d D e p t h to the Center of Element 4 D Relative Density e Void Ratio r E Rebound Modulus G Shear M o d u l u s r G Tangent Shear M o d u l u s Gmax Shear M o d u l u s at Small Strain K Shear M o d u l u s Parameter t 2 Shear M o d u l u s Parameter Rebound K G K b Constant Shear M o d u l u s Constant Bulk M o d u l u s N u m b e r Shear M o d u l u s Parameter for C l a y Unit Shear Stiffness U n i t N o r m a l Stiffness m Global Stiffness M a t r i x ix Global Tangent Stiffness M a t r i x Rebound Constant Mass Global Mass M a t r i x Rebound Constant Bulk M o d u l u s Exponent Shape Function Shape F u n c t i o n Number of Elements N u m b e r of Failed Elements Atmospheric Pressure Correction Force Vector External Force Vector Incremental External Force Vector Radius of M o h r Circle Undrained Shear Strength Time Time T i m e Increment Porewater Pressure Porewater Pressure Vector Incremental Porewater Pressure Vector Volume Shear Wave Velocity Compression Wave Velocity Cartesian Coordinate Cartesian Coordinate Cartesian Coordinate a D a m p i n g Coefficient (3 D a m p i n g Coefficient A Critical D a m p i n g Ratio Natural Frequency 7 Shear Strain A m p l i t u d e Izy Shear Strain e«d Volumetric Strain Ae,«j Incremental Volumetric Strain P Mass Density 4>' Internal Friction A n g l e V Dilation Angle T Shear Stress maz T Shear Strength Major Principal Effective Stress M i n o r Principal Effective Stress "I M e a n Normal Effective Stress xt LIST OF TABLES Table N o . Title Page 4.1 Properties Selected For the E x a m p l e P r o b l e m 81 4.2 Linear Analysis: Free F i e l d Peak Accelerations 83 4.3 Nonlinear Analysis: Free F i e l d Peak Accelerations 92 5.1 Scaling Relations 102 5.2 Properties of M o d e l Sand 105 5.3 Centrifuge Test S u m m a r y 117 6.1 Porewater Pressure M o d e l Constants 122 6.2 Structural Properties 123 6.3 Slip Element Properties 124 7.1 Comparison of Peak Accelerations in Test L D O l / E Q l 142 7.2 Comparison of Settlements in Test L D O l / E Q l 145 7.3 Comparison of Peak Accelerations in Test L D 0 2 / E Q 4 162 7.4 Comparison of Settlements in Test L D 0 2 / E Q 4 173 7.5 Comparison of Peak Accelerations in Test R S S 1 1 0 / E Q 1 179 7.6 Comparison of Settlements in Test R S S 1 1 0 / E Q 1 189 7.7 Comparison of Settlements i n Test R S S 9 0 / E Q 2 209 8.1 Comparison of Settlements in Test L D 0 4 / E Q 2 230 8.2 Comparison of Peak Residual Porewater Pressures in Test R S S 1 1 1 / E Q 1 253 8.3 Comparison of Displacements in Test R S S 1 1 1 / E Q 1 268 xii LIST OF FIGURES Figure N o . Title Page 2.1 Stress Strain C u r v e For L o a d i n g and Unloading 15 2.2 Stress State of an Element 15 2.3 M o h r Circle Construction 17 2.4 M o h r Circle Construction 17 2.5 Corrected and Uncorrected M o h r Circles 20 2.6 Characteristic 23 2.7 Idealised Drained Behaviour 2.8 Variation O f Dilation A n g l e with M e a n N o r m a l Stress Drained Behaviour of Dense and Loose Sands (Adapted F r o m Robertson 23 1982) 25 3.1(a) Initial Loading C u r v e 38 3.1(b) M a s i n g Stress Strain Curves for L o a d i n g and Unloading 38 3.2 Hysteretic Characteristics 38 3.3 Shear M o d u l i of Sands at Different Relative Densities 3.4 Shear M o d u l i for Saturated ( A d a p t e d F r o m Seed and Idriss 1970) 41 Clays (Adapted F r o m Seed and Idriss 1970) 42 3.5 Simple Shear and T r i a x i a l Stress Conditions 55 3.6 M o h r Circle Construction 55 3.7 Definition of Slip Element 57 4.1 Boundary Stresses on a Discrete M a s s on Horizontal B o t t o m B o u n d a r y 70 4.2 Boundary Stresses on a Discrete M a s s on Vertical Lateral Viscous B o u n d a r y 74 4.3 Soil Property Profile 77 4.4 Reversed Spike Input M o t i o n 77 4.5 Surface Acceleration Response W i t h Rigid Base 79 4.6 Surface Acceleration Responses 80 4.7 Soil-Structure Interaction P r o b l e m 4.8 Linear Analysis - Distribution of Accelerations W i t h Rigid and Elastic Bases 82 xiii W h e n Roller Boundaries 4.9 are at D = 4 B 91 are at D = 2 0 B 93 are at D = 1 0 B 94 are at D = 4 B 95 are at D = 2 0 B 96 Nonlinear Analysis - Distribution of Accelerations W h e n Viscous Boundaries 4.19 90 Nonlinear Analysis - Distribution of Accelerations W h e n Viscous Boundaries 4.18 are at D = 1 0 B Nonlinear Analysis - Distribution of Accelerations W h e n Roller Boundaries 4.17 89 Nonlinear Analysis - Distribution of Accelerations W h e n Roller Boundaries 4.16 are at D = 2 0 B Nonlinear Analysis - Distribution of Accelerations W h e n Roller Boundaries 4.15 87 Linear Analysis - Distribution of Accelerations W h e n Viscous Boundaries 4.14 are at D = 4 B Linear Analysis - Distribution of Accelerations W h e n Viscous Boundaries 4.13 86 Linear Analysis - Distribution of Accelerations W h e n Viscous Boundaries 4.12 are at D = 1 0 B Linear Analysis - Distribution of Accelerations W h e n Roller Boundaries 4.11 84 Linear Analysis - Distribution of Accelerations W h e n Roller Boundaries 4.10 are at D = 2 0 B are at D = 1 0 B 97 Nonlinear Analysis - Distribution of Accelerations W h e n Viscous Boundaries are at D = 4 B 99 5.1 Layout of the Accelerometer Leads 110 5.2 Instrumentation of a Centrifuged M o d e l 114 5.3 T y p i c a l Test D a t a on Seismic Response of the M o d e l 115 6.1 Liquefaction Resistance C u r v e of Leighton B u z z a r d S a n d 121 7.1 Schematic of a M o d e l E m b a n k m e n t 126 7.2 Instrumented M o d e l E m b a n k m e n t 7.3 M o d e l Response in Test L D O l / E Q l 129 7.4 Input M o t i o n for Test L D O l / E Q l 130 7.5 C o m p u t e d and Measured Accelerations at the Location in Test Series L D O l of A C C 1583 in Test L D O l / E Q l 127 132 xiv 7.6 C o m p u t e d and Measured Accelerations at the Location of A C C 1258 in Test L D O l / E Q l 7.7 133 C o m p u t e d and Measured Accelerations at the Location of A C C 1938 in Test L D O l / E Q l 7.8 134 C o m p u t e d and Measured Accelerations at the Location of A C C 2033 in Test L D O l / E Q l 7.9 135 C o m p u t e d and Measured Accelerations at the Location of A C C 1487 in Test L D O l / E Q l 7.10 136 C o m p u t e d and M e a s u r e d Accelerations at the Location of A C C 1908 in Test L D O l / E Q l 7.11 137 C o m p u t e d and M e a s u r e d Accelerations at the Location of A C C 1928 in Test L D O l / E Q l 7.12 138 C o m p u t e d and Measured Accelerations at the Location of A C C 2036 in Test L D O l / E Q l 7.13 139 C o m p u t e d a n d Measured Accelerations at the Location of A C C 988 in Test L D O l / E Q l 7.14 140 C o m p u t e d a n d Measured Accelerations at the Location of A C C 1225 in Test L D O l / E Q l 7.15 141 C o m p u t e d Shear Stress-Strain Response Near the Location of A C C 1583 in Test L D O l / E Q l 7.16 143 C o m p u t e d Shear Stress-Strain Response Near the Location of A C C 1932 in Test L D O l / E Q l 144 7.17 Settlement Pattern in Test L D O l / E Q l 146 7.18 Schematic of a M o d e l E m b a n k m e n t W i t h Surface Structure 147 7.19 Instrumented M o d e l i n Test Series L D 0 2 149 7.20 M o d e l Response in Test L D 0 2 / E Q 4 150 7.21 Input M o t i o n for Test L D 0 2 / E Q 4 152 7.22 Fourier S p e c t r u m of A C C 1544 Record in Test L D 0 2 / E Q 4 153 7.23 C o m p u t e d a n d M e a s u r e d Accelerations at the Location of A C C 1486 in Test L D 0 2 / E Q 4 7.24 155 C o m p u t e d and M e a s u r e d Accelerations at the Location of A C C 1487 in Test L D 0 2 / E Q 4 7.25 156 C o m p u t e d a n d Measured Accelerations at the Location of A C C 2033 in Test L D 0 2 / E Q 4 157 xv 7.26 C o m p u t e d and Measured Accelerations at the Location of A C C 1928 in Test L D 0 2 / E Q 4 7.27 158 C o m p u t e d and Measured Accelerations at the Location of A C C 1908 in Test L D 0 2 / E Q 4 7.28 159 C o m p u t e d and Measured Accelerations at the Location of A C C 1258 in Test L D 0 2 / E Q 4 7.29 160 C o m p u t e d and Measured Accelerations at the Location of A C C 1225 in Test L D 0 2 / E Q 4 7.30 161 C o m p u t e d and Measured Accelerations at the Location of A C C 1583 in Test L D 0 2 / E Q 4 7.31 164 C o m p u t e d and Measured Accelerations at the Location of A C C 1932 in Test L D 0 2 / E Q 4 7.32 165 C o m p u t e d and Measured Accelerations at the Location of A C C 1938 in Test L D 0 2 / E Q 4 166 7.33 Fourier Spectrum of A C C 1932 Record in Test L D 0 2 / E Q 4 168 7.34 Fourier Spectrum of A C C 1938 Record in Test L D 0 2 / E Q 4 169 7.35 C o m p u t e d and Filtered Accelerations at the Location of A C C 1932 in Test L D 0 2 / E Q 4 7.36 170 C o m p u t e d and Filtered Accelerations at the Location of A C C 1938 in Test L D 0 2 / E Q 4 171 7.37 Settlement Pattern in Test L D 0 2 / E Q 4 172 7.38 Schematic of a M o d e l E m b a n k m e n t W i t h E m b e d d e d Structure 174 7.39 Instrumented M o d e l in Test Series RSS110 176 7.40 M o d e l Response in Test R S S 1 1 0 / E Q 1 177 7.41 Input M o t i o n for Test R S S 1 1 0 / E Q 1 178 7.42 C o m p u t e d and Measured Accelerations at the Location of A C C 3479 in Test R S S 1 1 0 / E Q 1 7.43 180 C o m p u t e d and Measured Accelerations at the Location of A C C 3466 in Test R S S 1 1 0 / E Q 1 7.44 181 C o m p u t e d and Measured Accelerations at the Location of A C C 3477 in Test R S S 1 1 0 / E Q 1 7.45 182 C o m p u t e d and Measured Accelerations at the Location of A C C 3478 in Test R S S 1 1 0 / E Q 1 7.46 183 C o m p u t e d and Measured Accelerations at the Location xvi of A C C 3457 in Test R S S 1 1 0 / E Q 1 7.47 184 C o m p u t e d and Measured Accelerations at the Location of A C C 1225 in Test R S S 1 1 0 / E Q 1 7.48 186 C o m p u t e d and Measured Accelerations at the Location of A C C 1938 in Test R S S 1 1 0 / E Q 1 7.49 187 C o m p u t e d and Measured Accelerations at the Location of A C C 1572 in Test R S S 1 1 0 / E Q 1 188 7.50 Settlement Pattern in Test R S S 1 1 0 / E Q 1 190 7.51 Schematic of a 3-D M o d e l E m b a n k m e n t W i t h E m b e d d e d Structure 191 7.52 Instrumented M o d e l in Test Series R S S 9 0 193 7.53 M o d e l Response in Test R S S 9 0 / E Q 2 194 7.54 Input Motion for Test R S S 9 0 / E Q 2 196 7.55 C o m p u t e d and Measured Accelerations at the Location of A C C 988 in Test R S S 9 0 / E Q 2 7.56 197 C o m p u t e d and Measured Accelerations at the Location of A C C 1225 in Test R S S 9 0 / E Q 2 7.57 198 C o m p u t e d and Measured Accelerations at the Location of A C C 1583 in Test R S S 9 0 / E Q 2 7.58 199 C o m p u t e d and Measured Accelerations at the Location of A C C 1487 in Test R S S 9 0 / E Q 2 7.59 200 C o m p u t e d and Measured Accelerations at the Location of A C C 1544 in Test R S S 9 0 / E Q 2 7.60 202 C o m p u t e d and Measured Accelerations at the Location of A C C 1932 in Test R S S 9 0 / E Q 2 7.61 203 C o m p u t e d and Measured Accelerations at the Location of A C C 1486 in Test RSS90/EQ2 7.62 204 C o m p u t e d and Measured Accelerations at the Location of A C C 728 in Test R S S 9 0 / E Q 2 7.63 205 C o m p u t e d and Measured Accelerations at the Location of A C C 2033 in Test R S S 9 0 / E Q 2 7.64 206 C o m p u t e d and Measured Accelerations at the Location of A C C 734 in Test R S S 9 0 / E Q 2 207 7.65 Settlement Pattern in Test R S S 9 0 / E Q 2 210 8.1 Instrumented M o d e l in Test Series L D 0 4 212 xvii 8.2 M o d e l Response in Test L D 0 4 / E Q 2 214 8.3 Input M o t i o n for Test L D 0 4 / E Q 2 216 8.4 Original and Corrected Accelerations at the Location of A C C 2033 in Test L D 0 4 / E Q 2 8.5 218 C o m p u t e d and Measured Accelerations at the Location of A C C 2033 in Test L D 0 4 / E Q 2 8.6 219 C o m p u t e d and Measured Accelerations at the Location of A C C 1258 in Test L D 0 4 / E Q 2 8.7 220 C o m p u t e d and Measured Accelerations at the Location of A C C 1928 in Test L D 0 4 / E Q 2 8.8 221 C o m p u t e d and Measured Accelerations at the Location of A C C 1908 in Test L D 0 4 / E Q 2 8.9 223 C o m p u t e d and Measured Accelerations at the Location of A C C 1544 in Test L D 0 4 / E Q 2 8.10 224 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2252 in Test L D 0 4 / E Q 2 8.11 225 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2335 in Test L D 0 4 / E Q 2 8.12 225 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2255 in Test L D 0 4 / E Q 2 8.13 227 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2331 in Test L D 0 4 / E Q 2 8.14 227 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2330 in Test L D 0 4 / E Q 2 8.15 229 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 68 in Test L D 0 4 / E Q 2 229 8.16 Instrumented 232 8.17 M o d e l Response in Test R S S 1 1 1 / E Q 1 233 8.18 M o d e l Response in Test R S S l l l / E Q l 234 8.19 Input M o t i o n for Test R S S l l l / E Q l 237 8.20 M o d e l in Test Series RSS111 C o m p u t e d and Measured Accelerations at the Location of A C C 3479 in Test R S S l l l / E Q l 8.21 239 C o m p u t e d and Measured Accelerations at the Location of A C C 3466 in Test R S S l l l / E Q l 240 xvni 8.22 C o m p u t e d and Measured Accelerations at the Location of A C C 3478 in Test R S S l l l / E Q l 8.23 241 C o m p u t e d and Measured Accelerations at the Location of A C C 1938 in Test R S S l l l / E Q l 8.24 243 C o m p u t e d and Measured Accelerations at the Location of A C C 1900 in Test R S S l l l / E Q l 8.25 Original and Corrected Accelerations 244 at the Location of A C C 1572 in Test R S S l l l / E Q l 8.2G 245 C o m p u t e d and Corrected Accelerations at the Location of A C C 1572 in Test R S S l l l / E Q l 8.27 246 C o m p u t e d and Measured Accelerations at the Location of A C C 3436 in Test R S S l l l / E Q l 8.28 247 Original and Corrected Accelerations at the Location of A C C 3457 in Test R S S l l l / E Q l 8.29 248 C o m p u t e d and Corrected Accelerations at the Location of A C C 3457 in Test R S S l l l / E Q l 8.30 249 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2338 in Test R S S l l l / E Q l 8.31 251 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2631 in Test R S S l l l / E Q l 8.32 251 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2848 in Test R S S l l l / E Q l 8.33 252 C o m p u t e d a n d Measured Porewater Pressures at the Location of P P T 2626 in Test R S S l l l / E Q l 8.34 252 C o m p u t e d a n d Measured Porewater Pressures at the Location of P P T 2851 in Test R S S l l l / E Q l 8.35 255 C o m p u t e d a n d Measured Porewater Pressures at the Location of P P T 2628 in Test R S S l l l / E Q l 8.36 255 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2846 in Test R S S l l l / E Q l 8.37 256 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2855 in Test R S S l l l / E Q l 8.38 256 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2842 in Test R S S l l l / E Q l 8.39 257 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2255 in Test R S S l l l / E Q l 259 xix 8.40 8.41 Contours of C o m p u t e d Peak Residual Porewater Pressures Shear Stress-Strain Response at the Location of P P T 2338 in Test R S S l l l / E Q l 8.42 261 Shear Stress-Strain Response at the Location of P P T 2842 in Test R S S l l l / E Q l 8.43 261 Shear Stress-Strain Response at the Location of P P T 2851 in Test R S S l l l / E Q l 8.44 263 Shear Stress-Strain Response at the Location of P P T 2848 in Test R S S l l l / E Q l 8.45 264 Shear Stress-Strain Response at the Location of P P T 2846 in Test R S S l l l / E Q l 8.46 264 Measured cyclic displacement and accelerations at the Locations of L V D T 4457 and A C C 1938 in Test R S S l l l / E Q l 8.47 260 266 Frequency Dependent Characteristics of L V D T s (Adapted F r o m L a m b e and W h i t m a n 1985) 267 8.48 C o m p u t e d Deformation Pattern in Test R S S l l l / E Q l 269 A2.1 Definition of Slip Element 290 xx Dedicated to A p p a h , A m r n a h , Raju, Pappy. U m a and all in my family. ACKNOWLEDGEMENTS I would like to thank Professor W . D . L i a m F i n n for his helpful guidance, constructive suggestions and encouragement which enhanced the quality of the research work. I am also indebted to all my collegues in the department of civil engineering for providing a pleasant, arid cordial atmosphere during the studies. I am grateful to M r . R a m L i n g a m and family for providing a home away from home and for their helps on many occasions. Special thanks are due to my wife, U m a , for her support and patience shown throughout my studies. T h e U . S . Nuclear Regulatory Commission, Washington, D . C . , through the U . S . A r m y Corps of Engineers, Waterways Experiment Station, Vicksburg, Miss., financed the series of simulated earthquake tests on centrifuged model structures at Cambridge University, U . K . , which were used to validate the T A R A - 3 program. 1 am grateful to L . L . Beratan of U.S. N R C and R . H . Ledbetter of the U S A E for their interests and constructive suggestions and to Professor A . N . Schofield, S.Steedman, R. Dean and F . H . Lee of Cambridge University for innovative high-quality centrifuge tests. T h e financial assistance provided by the department of civil engineering in the form of graduate research assistantship and graduate teaching assistantship is greatly xxri appreciated. CHAPTER 1 INTRODUCTION Since the occurrence of damaging earthquakes in Niigata and Alaska in 1964, interests and research have been directed first towards understanding the phenomenon of liquefaction then were slowly shifted towards developing methods to assess the safety of critical facilities which are located in soils susceptible to liquefaction. T y p i c a l examples of such facilities are nuclear power plants, liquefied natural gas ( L N G ) plants, dams, embankments and pipelines. M o r e recently, methods have been introduced to cater to the needs of the offshore industry. Earlier assessments of the safety of soil structures subjected to seismic loading were based primarily on factors of safety along an assumed potential failure surface. However, the trend shifted from assessment in terms of factors of safety to one in terms of deformations. The latter method of assessment is believed to be more suitable as it allows the functional aspects of the structure to be incorporated in performance criteria. In the past, several methods were proposed to compute earthquake induced deformations in two-dimensional earth structures. T h e two methods of analysis which have found wide application in current engineering practice are Newmark's method of analysis (Newmark, 1965) and Seed's semi-empirical method of analysis (Seed et al, 1973; Seed, 1979). Newmark's method of analysis is based on the concept that no movement takes place 1 Chapter 1 : 2 along a potential sliding surface until the acceleration of the sliding mass exceeds the yield acceleration (Newmark, 1965). yield acceleration, integration. Whenever the acceleration of the sliding mass exceeds the the progressive displacement is calculated using the process of double While the determination of yield acceleration of the sliding mass is straight- forward, difficulties may arise in determining the representative mass, since the accelerations find representative acceleration of the sliding vary throughout the sliding mass. O n e of the simplest ways to acceleration is to take the average of the accelerations over the sliding surface. Alternatively, procedures developed by Makdisi and Seed (1978) could be employed. Newmark's method of analysis does not give deformation and strain fields of the earth structures. Rather, it gives an index of probable behavior which can be compared with indices of other earth structures which have behaved satisfactorily or unsatisfactorily during earthquakes structures, ( F i n n , 1987). Therefore, for the assessment of the safety of new types of as often found in offshore oil exploration, where experience with the application of this method is lacking, one has to be extremely careful in interpreting the index from the point of view of safety. Furthermore, since yield acceleration is calculated using in-situ initial properties, this method of analysis is only appropriate for materials which do not suffer significant strength loss during earthquake shaking. A n o t h e r limitation in this method of analysis is that it is applicable only to cases where the movement occurs along well-defined narrow failure zones. Such a failure mechanism may not occur in many cases as the deformations are often broadly distributed within the soil structures. However, as shown by G o o d m a n and Seed (1966), this method gives satisfactory results in situations where a well-defined failure mechanism exists. Therefore, while this m e t h o d of analysis remains a useful approach, it is not generally a satisfactory method to compute permanent deformations induced by seismic loading. Chapter 1 : 3 O n the other h a n d , Seed's method of analysis is a semi-analytical method in which data from a dynamic response analysis and data from cyclic triaxial tests are used to estimate potential displacements in the soil structures. T h e basic steps involved in this method of analysis are summarized below. 1) Determine the pre-earthquake condition that exists in the soil structures by performing a static finite element analysis. 2) Select design earthquake motions appropriate for the site where the soil structure is situated. 3) Perform a d y n a m i c response analysis to determine the time histories of dynamic shear stresses throughout the soil structure resulting from the design motions. 4) A p p l y the computed time history of stresses to representative samples and observe the effect in terms of strains and porewater pressures. Plot contours of strains and porewater pressure data. These allow interpolation of the strain and porewater pressure d a t a for other elements so that strain and porewater pressure data are developed for all the elements. 5) Determine the m i n i m u m factor of safety against total failure by limit equilibrium methods with the assigned strengths of elements consistent with the pressure d a t a observed in the laboratory tests. porewater Chapter 1 : 4 6) Assess the overall deformations from the observed strains in the laboratory sam- ples, if the soil structure is found to be safe against a total failure. In current practice, the dynamic response analysis in step 3 is conducted using equivalent linear elastic analyses. using an iterative elastic In these analyses, the nonlinear behavior of soil is accounted approach so that the soil properties (i.e., shear modulus a n d damping) are compatible with the computed strains. However, as pointed out by Desai a n d Christian (1979), the iterative equivalent linear elastic m e t h o d , like any other iterative approaches, suffers f r o m the fact that the solutions obtained are not unique a n d are dependent on the assumed properties for the first iteration. T h e other limitation of equivalent linear methods is that these may overestimate the seismic response of soil structures comprising nonlinear hysteretic materials due to the phenomenon known as pseudo-resonance (Finn et al, 1978). T h i s occurs if the fundamental period of the input motion coincides with the fundamental period of soil structures as defined by the final set of compatible properties in the iterative method. Moreover, analyses are conducted in terms of total stresses so that the progressive effect of seismically induced porewater pressures are not reflected in stresses a n d accelerations. Detailed studies conducted by F i n n et al (1978) on one dimensional problems indicate that total stress methods overestimate porewater the seismic response when the seismically induced pressures exceed about 30% of the effective overburden pressures. Therefore, there is reason to believe that at least similar overestimation may occur between the total stress and effective stress methods for two-dimensional problems. There are several techniques available to compute the deformation field from the strain data obtained in step 4. T h e modulus reduction technique proposed b y Lee (1974) a n d the strain harmonising technique proposed by Serf! et al (1976) are c o m m o n ones. In the Chapter 1 : 5 strain harmonising technique, the strain potentials obtained through laboratory testing are converted to shear stresses. T h e corresponding nodal forces are applied as loads in a static analysis to compute compatible deformations. be the seismic deformations. T h e resulting deformations are assumed to A s pointed out by Siddharthan (1984), this approach gives rise to a set of inconsistent assumptions. First, the computed strains in the last iterations of the equivalent linear elastic analysis are ignored as being not correct but the stresses are assumed to be correct. T h i s violates the one to one relationship of stresses and strains for a given loading. Secondly, although the final strains computed in the last iteration are assumed not to be correct the strains in the previous iteration procedure are used in the process of obtaining strain compatible soil properties as if they were correct. Since the deformation field is obtained through a pseudo-static analysis, the time vari- ation of the deformation field cannot be obtained. Also, the Seed approach does not take into account porewater of the deformation that results from dissipation of the seismically induced pressures. D y n a m i c effective stress models are available to compute seismic deformations directly in two-dimensional problems. M a n y are two-dimensional elastic-plastic Biot's equations (Biot, 1941) for coupled fluid-soil systems. incorporated in commercially available programs. is D Y N A F L O W (Prevost, 1981). models based on However, few of these have been T h e most general program of this type While the elastic-plastic models offer the most complete description of the soil response, they are difficult to use and the soil properties required in some of t h e m are difficult to measure. T h e y also make very heavy demands o n computing time. Furthermore, there has been no extensive validation of these methods of analyses. While Newmark a n d Seed methods of analyses are suitable to earth structures such as embankments a n d dams, they are not appropriate for analysing soil-structure interaction effects. D y n a m i c soil-structure interaction during earthquakes is a very complex phenomenon Chapter 1 : 6 because of the nonlinear response of soil to strong shaking. T h e interaction becomes even more complex if the soil is saturated and large seismically induced porewater pressures are generated which alter the strength and stiffness of the soil. T h e most commonly used program, in current engineering practice, for the analysis of soil-structure interaction systems is F L U S H (Lysmer et al, 1975). It is an equivalent linear finite element analysis in the frequency d o m a i n and as such it cannot model certain important phenomena in soil-structure interaction such as relative displacements at the soil-structure interface, uplift during rocking, transient and permanent deformations, the progressive effects of increasing porewater pressures and the hysteretic behavior. T o model these phenomena and to obtain reliable estimate of seismic response, nonlinear dynamic effective stress analysis in the time d o m a i n is necessary. Therefore, it is indeed necessary to develop an efficient, practical and reliable m e t h o d of analysis to compute seismic response of soil structures and soil-structure systems. interaction T h i s need has been already recognised by the N a t i o n a l Research C o u n c i l of the U n i t e d States. T h e state-of-the-art for analysing permanent deformations was assessed in a report on eathquake engineering research by the N a t i o n a l Research C o u n c i l of the U n i t e d States ( N R C 1982) as follows: " M a n y problems in soil mechanics, such as safety studies of earth dams, require that the possible permanent deformations that could be produced by earthquake shaking of prescribed intensity and duration be evaluated. Where failure develops along well-defined failure planes, relatively simple elasto-plastic However, if permanent models may suffice to calculate displacements. deformations are distributed throughout the soil, the problem is much more complex and practical, reliable methods of analysis are not available." Consequently, N R C recommended that active research should be directed toward de- Chapter 1 : veloping practical and reliable methods to compute seismic deformations 7 ( N R C 1982 and 1985). 1.1 Scope A s a first step towards achieving the N R C goal, Siddharthan and F i n n developed a d y n a m i c nonlinear effective stress method of analysis a n d incorporated it into the computer program T A R A - 2 (Siddharthan a n d F i n n , 1982). A very limited verification of this method of analysis has been reported (Siddharthan, 1984). T h i s thesis undertakes to enhance T A R A - 2 a n d to provide an extensive verification of the method of analysis. T h e enhanced version of the method of analysis has been incorporated in T A R A - 3 (Finn et al, 1986). One of the major problems i n validating d y n a m i c response analysis is the lack of data f r o m suitably instrumented structures i n the field. Some limited validations have been reported for the limited but practical case of the level ground conditions (Finn et al, 1982; Iai et al, 1985). M o s t of the methods are often validated using data from element tests such as cyclic triaxial or simple shear tests. A l t h o u g h this type of validation is an important first step, it is inadequate because in these tests either the stress or strain is prescribed and both are considered homogeneous. Therefore, the tests do not provide the rigorous test of either the constitutive relations or the robustness of the computational procedure that would be made possible by data from an instrumented structure in the field with inhomogeneous stress a n d strain fields. H a v i n g this i n m i n d , the U n i t e d States Nuclear Regulatory Commission ( U S N R C ) , through the U n i t e d States A r m y Corps of Engineers ( U S A E ) , sponsored a series of centrifuge model tests to provide data for the verification of the method of analysis incorporated in TARA-3. T h e tests were conducted on the large geotechnical centrifuge at Cambridge University i n the U n i t e d K i n g d o m by D e a n and Lee (1984) and Steedman (1985 and 1986). Chapter 1 : T h e centrifuge 8 models were of a variety of structures with foundations of both dry and saturated sands. T h e comprehensive data base generated the centrifuged models included acceleration through the simulated earthquake tests on time history at selected locations within the sand foundation and on the structure, porewater pressure time history at selected locations within the saturated sand foundation and deformations along the surface of structure and sand foundation. 1.2 Thesis Outline Chapter 2 deals exclusively with the method of static analysis. T h e formulations, basic assumptions and the stress strain model are discussed. Approximate ways of handling some of the limitations are also presented. Chapter 3 discusses extensively the important aspects of the dynamic nonlinear effective stress method of analysis. T h e finite element formulation, the numerical treatment and the porewater pressure generation model are presented in detail. Chapter 4 is entirely devoted to the introduction and implementation of energy transmitting boundaries into the method of analysis. T h e effectiveness of different boundaries are discussed and examples of the performance of the more useful types are presented. T h e principles of centrifuge testing and its applicability for validation of numerical analysis are briefly discussed in Chapter 5. geotechnical centrifuge and associated tion, d a t a acquisition, instrumentation In particular, aspects related procedures are briefly mentioned. to Cambridge Model construc- and related accuracy and model tests selected for the T A R A - 3 verification study are also discussed. T h e selection of soil parameters and other relevant data required for the analyses are Chapter 1 : 9 summarised in Chapter 6. T h e verification of the predictive capability of T A R A - 3 using data from the model tests on dry and saturated sand foundations is presented in Chapter 7 and Chapter 8 respectively. T h e summary and the conclusions drawn f r o m this research are given in Chapter 9. CHAPTER 2 M E T H O D OF S T A T I C A N A L Y S I S IN TARA-3 2.1 Introduction For a complete analysis of the response of a soil-structure system subject to earthquake loading, it may often be necessary to first conduct a static analysis to determine the stressstrain state of the system prior to the earthquake. T h e knowledge of the in-situ stress-strain state is essential since soil properties such as stiffness and strength which govern the response of the system to earthquake loading depend on these in-situ stress-strain states. In general, in order to determine in a realistic manner the behavior of the soil structure system to any load, it is necessary to make simplifying assumptions, particularly, regarding the modelling of soil behavior, structural behavior and the site. T h e significant assumption regarding the geometric modelling of the soil structure system is that the three dimensional nature of the system can be adequately represented by a transverse a state of plane strain exists. engineering structures cross section in which T h i s assumption is often useful since many such as earth embankments and dams approximate geotechnical conditions of plane strain. The method of static analysis incorporated in T A R A - 3 takes into account the nonlinear stress dependent behavior of the soil to loads. 10 Furthermore, the soil behavior depends on Chapter 2 : the loading path. Therefore, a m e t h o d of analysis that simulates the construction where an additional layer of elements is added at each step is incorporated. 11 sequence In this way, it may be possible to follow the actual sequence of construction loading in a simplified manner. Provision is also included to analyse an earth structure using only one layer, the so-called gravity switch on analysis. C o m p a r i s o n of this analysis with that based on the construction sequence can be found in Serff et al (1976), Desai and Christian (1979) a n d Naylor a n d Pande (1981). T h i s chapter deals with aspects related to modelling of soil behavior, the simplified assumptions a n d the basic framework for conducting static analysis. 2.2 Finite Element Representation The region of interest is approximated that are connected through n o d a l points. node isoparametric are also permissible. by an assembly of a finite number of elements T h e type of element used i n T A R A - 3 is the 4 quadrilateral element with 8 degrees of freedom. Triangular elements T h e unknowns are the horizontal and vertical displacements at each node of the element. T h e interpolation function that describes the variation of the unknown displacement within the element i n terms of nodal displacements is such that it produces a linear variation in strain within the element. Such an element is found to predict strains and stresses accurately in typical problems. Also, this type of element is useful as it can model the geometry of soil structures quite accurately. T h e incremental matrix equation, including the effect of porewater pressures, governing the static response of the system (see A p p e n d i x I) is: [K }{A} = {AP} t where, - [K*]{AU} (2.1) Chapter 2 : 12 [Kt] = the global tangent stiffness matrix, {A} = the incremental nodal displacement {AP} vector, = the incremental nodal force vector, [K*] = the matrix associated with porewater pressures. {At/} vector. = the incremental porewater pressure The stiffness matrix [Kt] in equation (2.1) depends on the tangent m o d u l i . T h e stress- strain and the volume change behavior assumed in the analysis to obtain tangent moduli are described in the following section. 2.3 Stress-Strain-Volume Change Behavior The stress strain relationship of structural elements is assumed to be linearly elastic. T h i s assumption follows f r o m the fact that the structural elements remain elastic for the range of stresses encountered during the loading. However, to model the nonlinear behavior of soils, an incrementally elastic approach has been adopted. T h e soil is assumed to be isotropic and elastic during the load increment and therefore the stress-strain can be described in terms of any pair of elastic constants. tangent bulk modulus, Bt, have been selected. for soils because special test procedures other. Tangent shear m o d u l u s , G , and These moduli are particularly t appropriate are available to evaluate one independent of the T h e selection of these m o d u l i also facilitates stresses and strains. relationship the imposition of good controls on For example, at failure, the shear modulus could be reduced to a small value (almost to zero) and the bulk modulus could be maintained at a higher value (Serff et al 1976; Wedge 1977; Vaziri-Zanjani 1986). A p a r t from this, the selection of Gt and B t has another distinct advantage strain problems in dynamic analysis as described in C h a p t e r 3. for plane Chapter 2 : 13 2.3.1 Shear Stress-Strain Relationship The shear stress-strain relationship of many soils under drained and undrained condi- tions is found to resemble a hyperbola. M a n y researchers have used hyperbolic stress strain relationships (Kulhawy et al 1969; D u n c a n and C h a n g 1970; Serff et al 1976). Part of the reason for its popularity is that it is a simple model and its parameters can be obtained using conventional laboratory testing. In T A R A - 3 , the relationship between shear stress, r, and shear strain, 7, in terms of the hyperbolic model parameters, G and r , is given by m a z max r = (1 + (2.2) 9™ where, Gmax = m a x i m u m shear modulus as 7 —> 0, max — appropriate T Fig. ultimate shear strength. 2.1 shows the shear stress-strain curves applicable during the loading, unloading and reloading phases. 2.3.1.1 Estimation of Hyperbolic Model Parameters The hyperbolic parameters in equation (2.2) depend on many factors so that computa- tion should at least reflect the influence of the most important factors. For sandy soils and silts, the m a x i m u m shear modulus, Gmax, depends primarily on the mean normal effective stress, a' , m relative density, D , of the following expressions r and previous stress history. T h i s is estimated using either depending on the option invoked: G mta = K G P a (OCR)" C-fY' CL 2 (2-3) Chapter 2 : 14 in which KG = shear modulus constant for a given soil, O C R = overconsolidation ratio, k = a constant dependent on the plasticity of the soil, P a = atmospheric pressure, or, Gmax = 1000 Ki (p\ 'J ' 1 2 (OCR) k (inpsf) (2.4) in which, K2 = a constant which depends on the type of soil a n d relative density. Equation and equation (2.3) is similar to the equation proposed by H a r d i n a n d Drnevich (1972) (2.4) is similar to the expression proposed by Seed a n d Idriss (1970) for computation of Gmax f ° the dynamic analysis. r For clayey soils, Gmax is computed using the expression: (2.5) u in which, Kday — 5 U a constant for a given clay, = undrained shear strength of the clay. For sandy soils, the value of Tmax depends on the current governing failure and the path by which failure is brought stress state, the criterion about in the soil mass. It is usually assumed that the failure is governed by the M o h r - C o u l o m b criterion which is defined by the parameters, effective cohesion, c', and angle of internal friction, <f>'. In practice, it is widely assumed that failure in a soil element with current stress state, as shown i n F i g . 2.2, is brought about by increasing the major principal effective stress, a' , 1 Chapter CO U t/3 QJ Unloading C/3 (1 + Reloading Shear Strain 2 1 Stress Strain Curve For Loading and U n l o a d i n g Fig. 2.2 Stress State of an Element 2 Chapter while holding the minor principal effective stress, tional triaxial testing conditions. 16 T h i s follows from conven- U n d e r this assumption, the value of T max under the current stress state shown i n F i g . smaller M o h r circle represents constant. 2 : for an element 2.2, can be c o m p u t e d using F i g . 2.3. T h e the initial stress state of the element a n d the larger circle represents the failure state. T h e radius, R, of the larger M o h r circle which touches the failure envelope can be computed as: c' cos <p' + R = — ( i a' - w 3 sin <f>' ) — ( 2 - 6 ) Therefore, in this case, Tmax = R However, if it is assumed that (2.7) is the value of shear stress at failure on the failure plane, then Tmax = R COS <f> (2.8) In field conditions, the soil mass may not follow a path similar to the triaxial conditions as assumed i n the above derivations. It is sometimes assumed that the soil mass fails in a manner i n which the mean normal stress remains constant ( H a r d i n and Drnevich 1972). U n d e r this condition, for a plane strain problem, the centre of the M o h r circle remains fixed. Therefore, the circle that represents the failure c a n be drawn by simply enlarging the initial M o h r circle until it touches the failure envelope (see F i g . 2.4). In this case, the radius, R, of the M o h r circle representing failure c a n be obtained as, R = c cos<t>' + C* + ay ) sin<£' (2.9) Li therefore, Tmax=R (2.10) Fig. 2.4 M o h r Circle Construction Chapter 2 : A s described earlier, if it is assumed that T 18 is the shear stress at failure on the failure max plane, then Tmax = R COS (j)' (2-H) T h e two failure options are included in T A R A - 3 a n d one should invoke the option appropriate to the problem that is being analysed. 2.3.2 Volume Change Behavior T h e tangent bulk modulus, Bt, is assumed to be a function of mean normal effective stress only. T h e value of Bt at any stress level is given by, B t = K b P a & ) (2.12) n *a in w h i c h , K{, = bulk modulus number, n = bulk modulus exponent, P a = atmospheric pressure. T h e parameters K^ and n in equation triaxial test d a t a following procedures (2.12) can be determined using conventional proposed by D u n c a n et al (1978, 1980). T h e y can also be obtained from isotropic consolidation tests as described by Byrne (1981). T y p i c a l values of Kf, vary between 300 and 1000 depending on the relative density of the soil a n d soil type. Tables of K\, and n applicable to normal sands are presented by Byrne (1981) a n d B y r n e a n d C h e u n g (1984). Chapter 2 : 19 2.4 Load Shedding Technique T h e stresses computed by incremental elastic analysis at any stage of loading or unloading must be checked continuously to ensure that they do not violate the failure criterion. A technique known as load shedding (Desai and C h r i s t i a n 1979; Byrne and Janzen 1984) is employed to redistribute excess stresses in an element to other elements in a sub-failure state whenever the failure criterion is violated. T h i s technique has been already applied sucessfully in the past for analysis of underground openings (Desai and Christian and of tunnels and shafts (Byrne and Janzen 1984). 1979) T h e deformations computed by the load shedding technique has been found to be in good agreement with closed form solutions (Byrne and Janzen 1984). T h e first step involved in this technique is to determine the correcting stresses in each of the elements that have stress states violating the failure criterion. In T A R A - 3 , the correcting stresses are computed assuming a constant mean normal stress condition which is similar to the approach suggested by Byrne a n d Janzen (1984). F i g . 2.5 shows the offensive stress state in terms of the M o h r circle for an element. T h e stress state {a} of an element which violates the M o h r - C o u l o m b failure criterion is given by, (2.13) T h e assumption of constant mean normal stress condition for a plane strain problem implies that the centre of the M o h r circle remains fixed. Therefore, the centre of the corrected M o h r circle should be coincident with the centre of the uncorrected M o h r circle as shown in F i g . 2.5. T h e corrected M o h r circle should also touch the failure surface defined by c' and <f>'. Chapter 2 : 20 Chapter T h e overstresses 2 : 21 { A c t } is given by, Aa z {Aa} = I Aa (2.14) } y Ar xy Using geometric principles, it can be shown that, A . X = ( ^ - ^ ) A - 2 (2.15) Rune ^y^C-^ )-^ 1 2 (2-16) Rune R ±Tz» = T * - Z (2-17) 3 L Rune in which, Ry = radius of the corrected M o h r circle (yield circle) Rune = radius of the uncorrected M o h r circle. Ry and Rune can be computed as, R y = c' cos4>' + (* a + ) sm4>' + Tl Gy (2.18) and Rune=\jC-^^Y T h e second step is to redistribute these overstresses are capable of accepting additional loads. (2.19) to adjacent stable elements that T h i s is achieved following procedures proposed by Byrne a n d Janzen (1984). In this procedure, the overstresses are converted to equivalent nodal forces, {Af }, acting on the corresponding nodes of the elements using the expression cor (see A p p e n d i x I), {Near} = jJj ( ^> A & (2.20) Chapter T h e global nodal force vector, {AF }, cor 2 : 22 is calculated taking the contribution from all the failed elements as shown below: (2.21) where, Nf is the total number of failed elements. e T h e stresses, strains and deformations resulting from the nodal force application is added to the existing values. 2.5 Shear Induced Volume Change T h e volume change behavior described in section 2.3.2 is only due to change in the mean normal effective stress. T h a t is, only the increment in volumetric strain, Ae , vm re- sulting from a change in the mean normal effective stress, A(r' , is included. B u t in soils m volumetric strains can also occur due to changes in shear stresses. Experimental evidence for such behavior has been reported in detail in several studies using different test equipment. Examples are studies by Lee (1965) based on the drained triaxial tests a n d V a i d et al (1981) based on the drained simple shear tests. F i g . 2.6 shows the characteristic drained behavior of initially loose a n d dense samples in a simple shear device. T h e samples exhibit volume reduction for small strains followed by volume expansion with an approximate constant rate for a considerable range of strain. Finally, at very large strains, they both exhibit a constant volume condition. In order to fit this behavior into an analytical formulation, the behavior is idealised as shown i n F i g . 2.7. In this, it is assumed that there is no shear induced volume change until a shear strain level given by y . 0 After the exceedence of y , the dilation is assumed to be governed by the 0 Chapter 2 Characteristic Drained Behaviour of Dense and Loose Sands Shear Strain Fig. 2.7 Idealised Drained Behaviour Chapter 2 : 24 constant rate (Hansen 1958). T h a t is, A7 = - sin v (2.22) in which, A e ^ = increment of the shear induced volume change, A 7 = increment of shear strain, v — dilation angle defining the dilation rate (Hansen 1958). T h e final phase where the constant volume condition is reached is not modelled. T h i s may not be an important concern since the strains at which this condition occurs are usually very large. T h e dilation angle is dependent on the density and increases with increasing relative density. A l s o , it is dependent o n the level of mean normal effective stress. It is observed from the study carried out by Robertson (1982) that the variation of dilation angle, v, with mean normal effective stress for a number of different sands at a given relative density lies on a narrow b a n d when plotted in a semi-logarithm plot as shown in F i g . 2.8. Note that the data in F i g . 2.8 is for a relative density, D = 80% only. F o r analytical purposes, the r variation of dilation angle, i>, versus the logarithm of mean normal effective stress can be assumed to be linear for a given relative density. T h i s , along with the idealisation shown in F i g . 2.7, forms the framework for inclusion of shear induced volume change in T A R A - 3 . T h e r e are several methods one could adopt to include shear induced volume changes. T h e most straightforward m e t h o d would be to introduce appropriate terms in the elasticity matrix [D] that would reflect the coupling between shear stress a n d the volume change. T h i s approach will result i n an unsymmetrical stiffness matrix and hence additional computational effort. T h e m e t h o d adopted in T A R A - 3 is to treat the problem in the same tf> ' |degrees)_ 327 35 Chattahoochee Sand Mol Sand Monlerey Sond 37 33 35 Glacial Sand SATAF Leighton Buzzard Sand 3 2 T Vesie ond dough I96B DeBeer 1965 Villel and Mitchell 1981 Hirshfield ond Poulos 1963 Baldi et al. 1981 Colo 1967 -•+0 0.5 MEAN Fig. 2.8 10 I NORMAL 50 STRESS, <T m 100 kg/cm' Variation O f Dilation Angle with M e a n N o r m a l Stress (Adapted From Robertson 1982) 500 1000 Chapter way as temperature 2 : 26 variations are handled in structural mechanics (Zienkiewicz et al 1967; B y r n e 1981). In this m e t h o d , since the elasticity matrix \D] is unchanged, the stiffness m a trix remains symmetrical. T h e basic steps involved in the approach are summarised below: Step 1 T h e incremental stresses and strains in all elements resulting for the load increment are calculated, ignoring the effect of shear induced volume change. Step 2 T h e dilation angle is c o m p u t e d based on the new mean normal effective stress. T h e variation of dilation angle with mean normal effective stress supplied as the input is used for this purpose. W i t h the calculated dilation angle, Aef, is calculated f r o m equation (2.22). Step 3 A e ^ is split into Aef a n d Ae d to form the dilational strain vector as, {Ae } = 1/3 Ae d 0 d v \ (2.23) where a and 8 are constants which may be varied to cover the likely range of strain response. Step 4 T h e incremental n o d a l forces corresponding to {Ae^} are computed using the expression, ( A p p e n d i x I), { A / } = Iff [B\* [D\ {Aej} dV (2.24) Step 5 T h e global nodal force vector in step 4 is added to the incremental load in step 1 to give the new applied load. F o r this new load, the strain and stress increments, A e and Ac, are Chapter 2 : 27 calculated. For the stress increment, the following equation is used. {Aa} = [D] {{Ae} - {Ae }} d (2.25) 0 Step 6 Step 2 to 5 are carried out until the convergence occurs in stress a n d strain increments under the applied incremental loads or until a specified number of iterations. 2.6 Simulation of Construction Sequence 2.6.1 Introduction M a n y geotechnical engineering structures are constructed sequentially. ples are earth construction embankments and dams. For a realistic T y p i c a l exam- solution to these problems, the sequences should be simulated as carefully as possible. In the cases involv- ing large volumes of earthworks, it is often impractical to simulate the actual construction sequences partly because of the complexity involved and partly because of the computer storage and cost requirements. Therefore, in practice, limited number of construction steps. the problems are analysed using a For the cases involving materials that exhibit non- linear stress strain behavior, the computed stresses are relatively insensitive to the number of layers employed, but the computed displacements are quite sensitive to the number of layers ( K u l h a w y et al 1969; Desai and Christian 1979). T y p i c a l l y 10 to 15 layers have been used i n the analysis of major dams (Naylor a n d Pande 1981). A layer by layer construction procedure is incorporated in T A R A - 3 for the purpose of simulating the sequence of construction loading. T h e method of analysis is detailed in the following sections. Chapter 2 : 28 2.6.2 Method of Analysis T h e construction sequence is modelled by computing the incremental stresses, strains and deformations due to the placement of each new layer. T h e r e are several methods by which the layer by layer construction can be handled. T h e y all differ in the approach by which the stress dependent moduli are evaluated ( K u l h a w y et al 1969; Desai and Christian 1979). T h e r e are three cases possible: (1) T h e initial stress approach (2) T h e final stress approach (3) T h e average stress approach For b o t h the final stress and the average stress approach, one cycle of interation is necessary for each layer placement, so that the final stresses will be known for the evaluation of m o d u l i directly or to find the average stresses and for subsequent evaluation of m o d u l i . Studies carried out by Kulhawy et al (1969) showed that the average stress approach is much more accurate and efficient than the other two approaches. In T A R A - 3 , the average stress approach is adopted and therefore, placement of a layer is analysed twice. T h e first time analysis is carried out using the m o d u l i based on the stresses at the beginning of the increment and the second time using the m o d u l i based on the average stresses d u r i n g the increment. T h e changes in stresses, strains and displacements are added to the values at the beginning of the increment. A p a r t f r o m this option, there is also a provision to evaluate m o d u l i based on average strains, as in T A R A - 2 , rather than on average stresses. Since only one iteration is carried out for a layer placement, equilibrium may not Chapter 2 : necessarily to satisfy 29 be satisfied (Desai a n d A b e l 1972). Therefore, correction forces are employed the equilibrium condition. T h e correction forces corresponding to changes i n shear stresses are computed and applied as nodal forces at the next load increment. T h e procedure for obtaining nodal forces is outlined in A p p e n d i x I. T h e placement of a fresh layer is simulated by applying forces to represent the weight of the fresh layer. For freshly placed elements, m o d u l i are based on the estimated stresses. T h e vertical effective stress, a' , the horizontal effective stress, a' , and the shear stress, r , y x xy of a freshly placed element are estimated following the suggestion by Ozawa et al (1973), as o\ = l d a' = K a' s x r xy 0 (2.26) (2.27) y = 0.5 cr'y s i n a (2.28) 0 where, d = the depth of the centre of the element from the top surface, K 0 7 S a 0 = coefficient of earth pressure at rest, = appropriate unit weight of the soil depending on the submerged condition, = slope of the overlying surface. In the m e t h o d adopted here, it is assumed that the position of newly placed elements immediately after placement is the reference state for movements resulting from subsequent loadings. Therefore, the displacements at the top of a newly placed elements are set equal to zero. A l s o , the strains in the newly placed elements are set equal to zero. Earth structures are often built by placing layers on existing foundation. cases, the foundation should be treated as consisting of pre-existing elements. In these Provision is Chapter 2 : 30 included in T A R A - 3 to account for pre-existing elements, in which case the initial stress state of the elements is required to compute the m o d u l i for the subsequent analysis. CHAPTER 3 M E T H O D O F D Y N A M I C A N A L Y S I S IN TARA-3 3.1 Introduction The greatest challenge in developing a method of dynamic analysis of a soil structure system during earthquakes is the inclusion, in a realistic manner, of all the factors that have a strong influence on soil behavior. T h e major factors that must be included are: (1) in-situ stress states and corresponding m o d u l i , (2) stress strain variation during phases of initial loading, unloading and reloading, (3) seismically induced porewater pressures, (4) effective stress changes due to porewater pressure changes, (5) viscous and hysteretic d a m p i n g , (6) volume changes induced by shear. In order to incorporate these factors into any mathematical modelling process, the real behavior of a soil structure system has to be idealised. T h e dynamic method of analysis incorporated in T A R A - 3 includes all these factors. It is an extensively revised and greatly expanded version of an earlier program T A R A - 2 (Siddharthan and F i n n 1982). 31 T h e the- Chapter oretical foundations of this method of analysis and the assumptons S : 32 implied in relation to the dynamic analysis are presented in this chapter. 3.2 Equations of Motion The dynamic equilibrium equations for a linear finite element system subjected to earthquake ground motions can be expressed in the form [M] {X} in which {X}, {X} and {X} + [C] {X} + [K] {X} (3.1) = {P} are the vectors of relative nodal acceleration, velocity and displacement respectively and [M], [C] and [K\ are the mass, damping and stiffness matrices respectively. {P} is the inertia force vector. {P} in which {/} = - is a column vector of 1 and X b T h i s is defined as, [M] {/} X (3.2) b is the base acceleration. T h e base acceleration is assumed to be identical at every nodal point along the base and therefore {P} is strictly a function of time. D y n a m i c analysis of a linear system may be solved either by the mode superposition method or by direct step-by-step integration method (Clough and Penzien 1975). E a c h of these methods has its own advantages and disadvantages. T h e mode superposition method requires the evaluation of the vibration modes and frequencies. It essentially uncouples the response of the system and evaluates the response of each mode independently of others. T h e main advantage of this approach is that an adequate estimate of the dynamic response can often be obtained by considering only a few modes of vibration, even in systems that may have many degrees of freedom; thus the computational efforts may be reduced significantly. T h e main disadvantage is that it is not applicable to nonlinear systems. Chapter S : 33 O n the other h a n d , the direct step-by-step integration method which involves the direct numerical integration of the d y n a m i c equilibrium equations has the advantage that it can be applied to both linear and nonlinear systems. T h e nonlinear analysis is approximated as a sequence of analyses of successively changing linear systems. In other words, the response is calculated for a short time increment assuming a linear system having the properties determined at the start of the interval. Before proceeding with the next increment, properties are determined which are consistent with the state of deformation and stress at that time. In T A R A - 3 , the step-by-step m e t h o d is used so as to account for the nonlinear behavior of the soil structure system. T h e basic formulation for the step-by-step integration method employed in T A R A - 3 is given in the next section. 3.3 Incremental Equations of Motion A s described earlier, in order to account for the nonlinear behavior, it is neccessary to work with the incremental equations rather with the original equations in equation (3.1). Let t anf T be the times corresponding to the beginning and end of a short time interval At. T h a t is, T = t + At. E q u a t i o n (3.1) should hold at these two instants of time and therefore, [M] {X} + [C] {X} + [K} {X}t = {P} T T T T T T (3.3) and [M\ T {X} t + [C} T {X} + T [K] {X} T T = {P}T (3.4) where subscripts refer to the instant of time. T h e mass matrix is constant throughout the analysis. A lumped mass matrix is used in T A R A - 3 instead of the more accurate consistent mass matrix. T h e procedure for obtaining the lumped mass matrix along with the reasons for adopting the lumped mass approach are Chapter S : 34 discussed in section 3.5. T h e d a m p i n g and stiffness matrices in equations (3.3) and (3.4) are, however, dependent on the current responses owing to the nonlinear behavior of the soil. Therefore, approximations are required to solve these equations. O n e way would be to represent the d a m p i n g and stiffness matrices by an average d a m p i n g and stiffness matrices applicable to the time interval A i . T h i s would yield the incremental equation shown below. [M] { A l } + [ C U {AX} + [K} av {AX} (3.5) = {AP} where the subscript av refers to the average damping and stiffness matrices and AX, AX AX, and A P refer to the incremental values during the time interval At, defined as, {AX} = {X} T - {X} t {AX} = {X} T - {X} t {AX} = {X} T - {X} {AP} = {P} T - {P} (3.6) (3.7) (3.8) t and (3.9) t However, this approach will involve an iterative solution scheme and may become very expensive as iterations are required at every time increment. Therefore, in practice, tangent damping and tangent stiffness matrices which correspond to time t (at the beginning of the interval) are used. T h i s would produce a tendency for the computed stress-strain to deviate from the stress-strain approximated response relationship of the soil since the nonlinear behavior is by a series of linear steps. Appropriate corrections are made so that the stress-strain state at the end of the increment is on the stress-strain curve of the soil. T h e stress-strain relationship is described in section 3.4 and the formulation of the stiffness matrix at time t, [Kt]t, is given in section hysteretic 3.6. In TARA-3, tangent d a m p i n g other than is accounted through the use of Rayleigh d a m p i n g in which case the element Chapter S : 35 damping matrix is expressed as a linear combination of element mass and stiffness matrices. The procedure is described i n section 3.7. The dynamic incremental equilibrium equations can now be rewritten as, [M] (3.10) {Al} + [C] {AX} + [K ]t {AX} = {AP} t t where, [C]t = the global d a m p i n g matrix at time t. Equations (3.10) represent a set of second order differential equations and can be solved using numerical procedures developed by Newmark (1959) or Wilson et al (1973). 3.4 Dynamic Stress-Strain Behavior A s noted earlier, an incrementally elastic approach has been adopted to model nonlinear behavior of soils. In this approach, the soil behavior is assumed to be linear within each increment of the load. The soil is assumed to behave isotropically. required to represent and its behavior. Therefore only two elastic constants are A s in the case of static analyses, bulk modulus, G and Bt were selected as the required constants. t the tangent shear T h e stress strain relationship in shear and the volume change behavior assumed in T A R A - 3 for the dymanic analysis is described i n detail in the next section. 3.4.1 Dynamic Shear Stress-Shear Strain Behavior T h e seismic loading imposes irregular loading pulses which consist of loading, unloading and reloading. T h e soil exhibits different behavior in each of these above phases. Adequate Chapter S : modelling of each of these phases is essential in order to obtain the true dynamic of the soil system. hysteretic, 36 response In T A R A - 3 , the behavior of soil in shear is assumed to be nonlinear and exhibiting M a s i n g (1926) behavior during unloading and reloading. T h e relationship between shear stress, r, and shear strain, 7 , for the initial loading phase under either drained or undrained loading conditions is assumed to be hyperbolic and is given by - - G m a x (3.11) 7 or, r=f( ) (3.12) 1 in which, Gmax = the m a x i m u m shear modulus, Tmax — the appropriate shear strength. T h i s initial loading or skeleton curve is shown in F i g . 3.1(a). T h e unloading-reloading has been modelled using the M a s i n g criterion. T h i s implies that the equation for the unloading curve from a point (y , r ) at which the loading reverses direction is given by r r r - T G r max (7 - 7 ) / 2 r or 2 =fC~~) (3-14) T h e shape of the unloading-reloading curve is shown in F i g . 3.1(b). T h e M a s i n g criterion implied in equations (3.13) a n d (3.14) means that the unloading and reloading of a hysteretic loop are the same skeleton curve with the origin translated point and the scales for the stress and strain increased by a factor of two. branches to the reversal Chapter S : 37 Lee (1975) a n d F i n n et al (1976) proposed rules for extending the Masing concept for irregular loading. T h e y suggested that the unloading and reloading curves should follow the previous skeleton loading curves when the magnitude of the previous m a x i m u m shear strain is exceeded. In F i g . 3.2(a), the unloading curve beyond B becomes the extension of the initial loading i n the negative direction, i.e., B C . In the case of a general loading history, they assumed that when the current loading curve intersects a previous loading curve, the stress strain curve follows the previous loading curve. T w o typical examples are provided in F i g . 3.2(b).to illustrate these rules (Finn et al 1976). (1) If loading along path B C is continued, the loading path is assumed to be B C A M , where A M is the extension of O A ; (2) If unloading along path C P B is continued, then the unloading path will be A B P ' . T h e tangent shear modulus, G , needed in the formulation is the value of the tangent t to the stress strain curve at the stress strain point. For instance, if the point is on the skeleton curve given by equation (3.11), then the tangent shear modulus in terms of strain, 7, is given by ^ Alternatively, Tmax ' Gt c a n be expressed in terms of shear stress, r, as Gt = Gmax (1 - — T. ) 2 (3.16) max M e t h o d s of d y n a m i c analysis commonly used in practice start the analysis from the origin of the stress strain curve for all the elements. These methods ignore the static strains in the soil structure system even in those elements which carry high shear stresses. However, in T A R A - 3 , an option is provided so that the dynamic analysis c a n start from the static stress-strain condition. It is believed that this option permits a more realistic estimation of dynamic response a n d of residual or permanent deformations. Chapter (a) first unloading F i g . 3.2 (b) general reloading Hysteretic Characteristics S 38 Chapter S : 39 3.4.1.1 Computation of Hyperbolic Model Parameters T h e m a x i m u m shear modulus, Gmax for sands is calculated using the equations proposed either by H a r d i n a n d D r n e v i c h (1972) based on resonant column tests or by Seed and Idriss (1970). T h e H a r d i n and Drnevich (1972) equation is of the form Gmax = 320.8 P a ( 3 f ". (1 + e) 1 7 3 e ) 2 (OCR)" (3.17) P a in which e = void ratio, OCR = overconsolidation ratio, k = a constant dependent on the plasticity of the soil, P = atmospheric a pressure, <j' = current mean normal effective stress. m The equation suggested by Seed and Idriss (1970) takes the form Gmax = 1000 K (OV2 2max (i p f) n (3.18) S in which Klmax = a constant dependent on the type of soil a n d relative density Equation (3.18) has been modified to reflect D. r previous stress history by including a term with the overconsolidation ratio and also to allow its usage in any system of units by expressing it i n a similar form as in the Hardin and D r n e v i c h equation. Gmax = 21.7 Ki max The variation of K 2 P a [OCR) k (3.19) with shear strain and relative density for sands (Seed and Idriss Chapter S : 1970) is shown in F i g . 3.3. T h e constant K2 max (the value of K 2 40 at small strains) may be estimated using the approximation suggested by B y r n e (1981), i f w = 15 + 0.61 Z? r where D (3.20) is expressed in percentage. r For clays, the m a x i m u m shear modulus is calculated based on the undrained shear strength, S , using the equation, u in which, Kday ~ a constant for a given clay. T h e variation of GjS u with shear strain for saturated clays is shown in F i g . 3.4 (Seed and Idriss 1970). T y p i c a l values of K \ vary between 1000 and 3000. c ay T h e m a x i m u m shear strength, Tmax-, for soils is dependent on the current stress system, the way by which the soil element is brought to failure and the failure criterion. and Drnevich (1972) suggested that the value of r m < M Hardin calculated using the M o h r - C o u l o m b failure envelope defined by the static strength parameters such as c' (effective cohesion) and <f>' (internal angle of friction) is adequate for dynamic loadings. Therefore, the options for selecting the value of Tmax reported in section 2.3.1.1 are all retained in the case of dynamic analyses. It should be noted that there is also a provision in T A R A - 3 for both Gmax and Tmax to be specified directly by the user. field or laboratory tests directly. T h i s facilitates the input of values obtained from either Chapter t 80 icr icr 4 3 icr io-' 2 Sheor Strain -percent Fig. 3.3 Shear M o d u l i of Sands at Different Relative Densities (Adapted From Seed and Idriss •j 1970) S : 41 30,000 A Wilson ond Dietrich (I960) x Thiers (1965) A Idriss (1966) + Zeevoerl (1967) • Shonnon and Wilson (1967) Shannon ond Wilson (1967) v Thiers ond Seed (I960) O Kovacs (I960) a llordinond Olock (1968) ^—lAisiks ond Torshonsky (1968) HimSeed and Idriss (1970) ^ T s a i ond Mousner (1970) 10.000 3000 —w -B 1 •' •• — • • 'i 1000 300 100 30 10 10" 10 - 3 Sheor Fig. 3.4 Strain - percent Shear M o d u l i for Saturated Clays (Adapted From Seed and Idriss 1970) Chapter 8 : 43 3.4.2 Volume Change Behavior The response of the soil to uniform all round pressure elastic and dependent on the mean normal effective stress. neglected in this mode. is assumed Hysteretic to be nonlinearly behavior, if any, is T h e relationship between tangent bulk modulus, Bt, a n d mean normal effective stress, a' , is assumed to be in the form m B t = K h P a (3.22) & ) » "a in which, Kb = the bulk modulus constant, P a — the atmospheric pressure in units consistent with a' , m n = the bulk modulus exponent. For fully saturated deposits, Bt has to be of high value to simulate undrained conditions in the case of dynamic analysis. 3.5 Formulation of Mass Matrix T h e mass matrix in equation (3.10) can be obtained by two different methods. In the first method, the mass matrix is formulated so as to be consistent with the assumed displacement interpolation function. T h e resulting matrix is known as the consistent mass matrix. In the second method, the mass matrix is obtained through a lumped mass approximation, giving what is called a lumped mass matrix. T h e presence of the off diagonal terms i n the consistent mass matrix greatly increases the computational time required to solve the dynamic equilibrium equations. O n the other h a n d , the lumped mass matrix is simple to obtain and has only diagonal terms. T h e degree Chapter 3 : 44 of accuracy obtained through the use of l u m p e d mass approximation is considered to be good enough for typical geotechnical problems (Desai a n d Christian 1979). In T A R A - 3 , the l u m p e d mass approximation is used, in which one-fourth of the mass of each quadrilateral element a n d one-third of the mass of each triangular element are l u m p e d at respective nodes. T h e total mass at any one node is the summation of the contributions from all the elements c o m m o n to that particular node. 3.6 Formulation of Stiffness Matrix A s mentioned earlier, the analysis incorporated in T A R A - 3 assumes isotropic behavior of soil and further it is applicable to the restricted but practical case of plane strain. Under these conditions, the relationship between the incremental stresses {ACT} a n d incremental strains {Ae} in an element of soil, can be written as, {ACT} = [D] {Ae} (3.23) where [D] is the elasticity matrix w h i c h , in this case, is a function of any two elastic constants. In the present analysis, tangent shear and bulk moduli are selected to form the [D] matrix. A s shown i n A p p e n d i x I, [D] i n terms of G and B , is given as t B + [D) = B t 4/3 t 2/3 G G t t t B B + t 2/3 t 4/3 0 0 G G t 0 t 0 (3.24) G t T h i s could be rewritten as, 1 1 0 1 1 0 0 0 0 + G t 4/3 -2/3 0 -2/3 4/3 0 0 0 1 (3.25) Chapter 3 : 45 or [D] = B [Qi] where [Qi] and [Q ] are the constant matrices. (3.26) + G [Q ] t t 2 Now the expression for the element 2 tangent stiffness matrix [kt], as obtained i n A p p e n d i x I, can be written as [kt] = JJJ [BY[D][B]dV (3.27) v W h e n the expression for [D] in equation (3.26) is incorporated into equation (3.27), the resulting expression for [kt] can be written as, [kt] = B t ffj [B] [Q ] [5] 1 L dV + G t j j f [BY [Q ] [B] dV 2 (3.28) It should be noted that equation (3.28) is valid only if B and G are assumed constants for t an element. However, in the isoparametric t formulation adopted i n T A R A - 3 , the stresses and strains vary and consquently m o d u l i are not constant within the element. It is therefore assumed that the values of moduli computed using the stresses obtained at the centre of the element are the representative values for the element. U n d e r this assumption, equation (3.28) can be used. Therefore, [kt] c a n be written in the form, [kt] = B [Ri] + t G t (3.29) [R ] 2 where, [Ri] = UJ \ \ [<?i] [B] dV B v (3.30) l and [*a] =f f j v [BY [ f t ] [B] dV (3.31) [Ri] and [R ] will be constant matrices provided changes in the geometry of the elements 2 are not considered. In T A R A - 3 , changes i n the geometry of the elements are not taken into account. Therefore [R±] a n d [R ] are evaluated only once during the dynamic analysis. 2 Chapter T h e element tangent stiffness matrix 8 : [kt] can be updated merely by multiplying 46 matrices [Ri] and [R%] by the current Bt and Gt values respectively and adding them together. This procedure can save computing time as [kt] need not be re-formulated at every load step. The global tangent stiffness matrix matrices [Kt] can be assembled using element tangent stiffness [kt] following conventional procedures. 3.7 Formulation of Damping Matrix T h e types of damping that occur when the vibrational energy is transmitted a m e d i u m can be broadly divided into two categories: through viscous and hysteretic d a m p i n g . Viscous d a m p i n g depends on the velocity and is frequency dependent. O n the other hand, hysteretic d a m p i n g depends largely on the magnitude of the strain and is frequency independent. For linear analysis, the d a m p i n g must be introduced in the form of viscous d a m p i n g . However", in the true non-linear analysis, where the hysteretic stress strain law is used, the damping is already introduced in the form of hysteretic d a m p i n g and therefore viscous damping" may not be needed. inside the soil structure, However, to take into account of the effect of flow of water some viscous damping is required. Moreover, small amounts of viscous d a m p i n g may be needed to control any pseudo high frequency responses that are introduced by the numerical integration procedures. While the hysteretic d a m p i n g is inherent, the viscous d a m p i n g in T A R A - 3 is of the Rayleigh type. In this context, the element damping matrix is expressed as a linear combination of mass matrix [m] a n d tangent stiffness matrix - [kt] as shown below, [c] = a[m] + b [k ] t in which a and b are constants. (3.32) Chapter 3 : 47 T h e element tangent stiffness matrix [kt] varies with time during the dynamic analysis. Therefore whenever [kt] is changed, [c] matrix is also changed. However, T A R A - 3 has also an option whereby the [c] matrix is not varied according to the current stiffness matrix but kept constant based on [AfJ^o- Accordingly, [c] is expressed as [c] = a[m] + b [kt]t=o (3.33) T h e above formulation will give a damping ratio A for the rfi mode as, n 1 where ui is the 11 mode frequency. TH n Equation (3.34) implies that if a = and when 6 = 0 , 0 the d a m p i n g is proportional to the frequency the d a m p i n g is inversely proportional to the frequency. A l s o from equa- tions (3.32) and (3.33), if a = 0, the damping matrix contains only the mass proportional components a n d if 6 = 0, it contains the stiffness proportional component. In a typical soil strucure system only the first few modes of vibration govern the dynamic response and therefore it is unnecessary customary to compute 6 and, if necessary, to include the higher mode components. It is a using only the natural frequency of the system (Lee 1975). For instance, if it is desired to have stiffness proportional d a m p i n g (a = 0), 6 could be computed as, 6 = — (3.35) where A is the critical d a m p i n g ratio and w i is the fundamental natural frequency of the system. Chapter 8 : 48 3.8 Computation of Correction Force Vector A s mentioned earlier, i n T A R A - 3 the nonlinear behavior of soil is approximated by a series of linear steps. Therefore, at the end of a load increment, the computed strains and stresses for an element may not be compatible with the stress-strain In order to make t h e m compatible, correction forces are used. relation of the soil. T h e correction forces are calculated assuming that the computed strains are the true strains. However, the correction forces do not necessarily satisfy the equilibrium equations. Therefore, a condition of global equilibrium at each step of the analysis is imposed. In order to d o this, it is necessary to compute all components representing both the right and left hand sides of the equilibrium equation. A n y differences constitute the correction force vector, {P r}CO A m o n g the components of the left hand side of the equilibrium equation, the inertia and d a m p i n g terms at time t, {Fi}t and {Fr>}t respectively, can be calculated i n a straightforward manner as, {F^ = [M] {X} (3.36) t and {F } =[C} {X} D T h e spring force term, t t (3.37) t {Fs}t, is obtained by representing the element dynamic stresses, {o~d}, as nodal forces acting o n the nodes a n d summing the contributions from all the elements as shown below, {Fsh = iJIJ v W M. dV where, N is the total number of elements and \B\ is the transpose e l (3-38) of the displacement matrix [B] defined i n A p p e n d i x I. If the right h a n d side of the equation representing the external load, at time t, is {P}t, Chapter S : then {P r} CO c a n 49 be calculated as {Pcor} = {P}t- {F^t- {F } D {F } t s (3.39) t C o m b i n i n g equations (3.36) through (3.38) into equation (3.39) will yield, {Pcor} = {P}t~ [M] {X} t T h e correction [C] {X} t E t /// W W d V (3.40) force vector calculated above can be added to the right hand side of the incremental equation formulated at time t for solving the responses at time T, as [M] {AX} + [C] {AX} + [K ] {AX} = {AP} + t t t {P } (3.41) cor 3.9 Residual Porewater Pressure Model D u r i n g seismic shaking two kinds of porewater sands. T h e y are the transient are due to changes saturated pressures and residual porewater are generated pressures. in staurated T h e transient pressures i n the applied mean normal stresses d u r i n g seismic excitation. For sands, the transient changes i n porewater pressures are equal to changes in the mean normal stresses. Since they balance each other, the effective stress regime in the sand remains largely unchanged. Hence the stability and deformability of the sand are not seriously affected due to the transient pressures. are due to plastic deformation O n the other h a n d , the residual pressures in the sand skeleton. drainage or internal diffusion and therefore These persist until dissipated by they exert a major influence on the strength and stiffness of the sand skeleton. Changes in the total mean normal stresses also affect the post earthquake value of the residual pressures. Skempton's B value. ignored. These pressures c a n be calculated using In all studies in this thesis, these changes are small and hence are Chapter S : 50 In T A R A - 3 , the residual porewater pressures are generated using the M a r t i n - F i n n - S e e d model ( M a r t i n et al 1975). T h e transient pressures are not modelled. Therefore, computed porewater pressure time histories will show the steady accumulation of pressure with time but will not show the fluctuations in pressure caused by the transient changes in mean normal stresses. 3.9.1 Martin-Fmn-Seed Model T h e original M - F - S model applies only to level ground, so that there are no static shear stresses acting on horizontal planes prior to the seismic loading. T h e model was subsequently modified to include the effects of initial static shear stresses present in two dimensional analyses. T h e original model is briefly described in this section and the modifications in the subsequent section. In the model, the increments in porewater pressure A U that develop in a saturated sand under cyclic shear strains are related to the volumetric strain increments Ae„d that occur in the same sand under drained conditions with the same shear strain history. Consider a sample of saturated sand under a vertical effective stress, a' . v Let the increment in volumetric compaction strain due to grain slip caused by a cycle of shear strain, 7, d u r i n g a drained cyclic simple shear test be Ae . va Let the increment in porewater pressure caused by a cycle of shear strain, 7, during an undrained cyclic simple shear test starting with the same effective stress system be A U. It was shown by M a r t i n et al (1975) that for fully saturated sands and assuming that water to be incompressible, that AU and Ae - are related by va A U = E in which E r r Ae vd (3.42) is the one-dimensional rebound modulus of sand at a vertical effective stress a\. T h e y also showed that under simple shear conditions the volumetric strain increment, Chapter 8 : 51 Ae d, is a function of the total accumulated volumetric strain, e -, and the amplitude of the v va shear strain cycle, 7, and is given by A e ^ = C i (7 - C e ) 2 in which C\, C , 2 vd + ^f"* (7 + C C3 and C4 are volume change constants. e^J 4 (3.43) These constants depend on the sand type and relative density. A n analytical expression for the rebound modulus, E , T at any vertical effective stress level a' is given by M a r t i n et al (1975) as, v *°»M!O" in which a' vo (3 is the initial vertical effective stress and K , r ' 44) m and n are rebound constants. These are derived from rebound tests in a consolidation ring. T h e i n c r e m e n t in porewater pressure, A U, during a given loading cycle with a m a x i m u m shear strain amplitude, 7, can now be computed using equations (3.42), (3.43) and (3.44) given the volume change and rebound constants. T h e important assumption in the formulation of the M - F - S model is that there is a unique relationship between the volumetric strains in drained tests and porewater pressures in undrained tests for a given sand at the same effective stress system and subjected to the same strain histories. T h i s assumption has been verified to be valid through an extensive laboratory program involving drained and undrained tests on normally and overconsolidated sands ( B h a t i a 1982 and F i n n 1981). B h a t i a (1982) found out that when the M - F - S model is coupled with the stress strain model reported in section 3.4, it can satisfactorily predict b o t h the rate of porewater pressure generation and liquefaction strength curve in undrained tests for cyclic stress histories representative of earthquake loading. Chapter S : 52 3.9.2 Extension Of M-F-S Model to 2-D Conditions In the 2 - D analysis of isotropic soil, the permanent volume changes due to shearing action are related to the cyclic shear stresses on horizontal planes because the seismic input motions are usually assumed to be shear waves propagating vertically. Therefore, in T A R A 3, for computation of Ae„d i n equation (3.43), the shear strain on the horizontal plane, is substituted in place of 7. Also, er' a n d a' v respectively, where a' and a y yo v0 y , xy i n equation (3.44) are replaced by <j' a n d a' y y0 are the current a n d initial vertical effective stresses. Static shear stresses are present on horizontal planes i n 2 - D problems. T h e presence of initial static shear stresses may significantly affect the cyclic behavior of sands depending on and the relative density of the sand and the level of the initial static shear stress F i n n 1978; V a i d and C h e r n 1983). In saturated (Vaid sands, the rate of development of porewater pressures, the level to which they may rise and the liquefaction potential curve are all dependent on the static shear stress level. in the porewater These effects are taken into account pressure model by specifying model constants such that they produce a reasonable match for the liquefcation potential curves and the rates of porewater pressure generation observed in laboratory samples with different initial static shear stress ratios. 3.10 Evaluation of Current Effective Stress System The global system of equations that relate the incremental nodal forces { A P } and incremental displacements {A} is given by (see A p p e n d i x I) {AP) = [K \ {A} + [K*} { A t / } t in w h i c h , [Kt] = the global tangent stiffness matrix, [K*] — the matrix associated with porewater pressures, (3.45) Chapter {AC/} = the incremental porewater S : 53 pressures. T h i s equation is used to evaluate the changes in effective stresses resulting from the changes in residual porewater pressures by setting { A P } = 0. T h e incremental displace- ments, strains and stresses given by this procedure constitute the response of the deposit to softening of the elements. T h e incremental stresses give rise to the new effective stress system which can now be used to modify soil properties as described in the next section. T h e incremental strains are components of the permanent strains. 3.10.1 Modification of Soil Properties The m a x i m u m shear modulus, Gmax, and the shear strength, Tmax, in the hyperbolic stress strain relationship are dependent on effective stresses. porewater pressure increases, A s the seismically induced and reduces the effective stress, the modulus and strength must be adjusted to be compatible with the current effective stress system. In T A R A - 3 , the m a x i m u m shear modulus is assumed to be proportional to (cr^) ^. 1 Therefore, the m a x i m u m shear modulus, G ax for the current cycle of loading is obtained m by (3.46) in which (Gm^o is the m a x i m u m shear modulus corresponding to the initial effective stress system defined by (r' . mo T h e computation of T^^X compatible with the current effective stress system is already outlined in section 3.4.1.1. Chapter 3 : 54 3.10.2 Estimation of Maximum Residual Porewater Pressure Laboratory investigations of samples with initial static shear stress on potential failure planes (Chern 1981) reveal that there is a limit to which the residual porewater pressures can rise. For triaxial conditions, the limiting residual porewater pressure, Umax, has been found to be given by (Chern 1981; C h a n g 1982) Uma, = o' 3c [1 - fJ - I sin <p 1) 3c (3-47) in which a\ and a' are the major a n d minor principal consolidation stresses respectively c 3c and <p' is the angle of internal friction. Equation (3.47) implies that the limiting value of the residual porewater pressure depends on the static shear stress level that existed after the end of consolidation. T h e direct application of equation (7 (3.47) to estimate mai based on the field stress conditions will not be correct since loading from earthquakes resembles simple shear rather than triaxial conditions. Therefore, equation (3.47) should be modified to reflect simple shear conditions. T h e modification takes the form, i ^1 r Umax = CT's* 1 " ~ cr 1) — - sin <t>'. . , J , 3.48 I sin <p' 3+ in which a' and a' ^ are the applied major and minor principal stresses in a triaxial sample u 3 that would produce a stress condition on a plane inclined at an angle (45 + <p'/2) horizontal, the same as on the horizontal plane in the field with initial stresses (o , r ). xy y T h e condition is clearly illustrated in F i g . 3.5. F r o m the M o h r circle i n F i g . 3.6, a' u <j' 5it to the and can be calculated as, i » (1 + sin <t>') r~ cos <p i (1 i * = °> ~ r « _ sin d>') c o s / ( 3 5 0 ) Chapter * r e= triaxial condition simple shear Fig. 3.5 S Simple Shear and Triaxial Stress Conditions Fig. 3.6 M o h r Circle Construction (45 + 4? ft) Chapter 8 : 56 It should be noted that the above computation can be equally applied in the case of the level ground conditions, where the limit on residual porewater pressure will be equal to the initial vertical effective stress. One of the options included in T A R A - 3 regarding the porewater pressure limit is the one described above. However, there are other options available including the option that would terminate porewater pressure generation in an element which has reached failure according to M o h r - C o u l o m b failure criterion. 3.11 Interface Representation In the conventional finite element approach, the relative displacement at the interface between two finite elements is not modelled. structure B u t , in practice, interface, realtive movements do occur. particularly at the soil- Therefore, in situations where relative motions are anticipated, a m o d e l that incorporates the relative movement at the interface is indeed necessary for a realistic solution of the problem. In T A R A - 3 , the relative movement at the interface between two finite elements is m o d elled using the two-dimensional slip elements presented by G o o d m a n et al (1968). The element is of zero thickness and capable of allowing relative movement in both sliding and rocking modes during the earthquake excitation. T h e slip element formulation is presented in the subsequent section. 3.11.1 Slip Element Formulation T h e slip element incorporated in the method of analysis is a two-dimensional element with four nodes and eight degrees of freedom. T h e horizontal and vertical displacements at each node are the degrees of freedom. F i g . 3.7 shows a slip element with the global (z, y) and local element (s, n) axes. Since the element thickness is zero, nodes P and Q will have Chapter Fig. 3.7 Definition of Slip Element S Chapter 8 : 58 the same [x, y) coordinates as that of R and S respectively. T h e force displacement relationship at any point within the slip element is assumed to be of the form, K 0 s 0 w (3.51) 3 K n in which, /„ = shear force per unit area of the element, / „ = normal force per unit area of the element, K s K n = unit shear stiffness in the direction of the element, = unit normal stiffness in the direction normal to the element, w = shear displacement at the point of interest and, 3 w n = normal displacement at the point of interest. A linear variation of displacement in the slip element is assumed. matrix K m T h e n the stiffness in terms of local co-ordinates as derived in A p p e n d i x II, is • 2K L K 2K K, 0 0 - 6 0 3 0 K s 0 K 2K, 0 n 0 n 0 K - 0 . 0 3 0 2K 3 0 - 2K n - - - 2K n 3 - 3 2K n 0 0 2K K. 0 0 2K K K 0 n n n - n K n (3.52) 0 s 0 2K 0 s K " K 2K n 0 2K 0 K 0 2K, 0 n 0 K. 0 s - 0 - 0 K 0 n 2K 2K, 0 - 0 K 3 n 0 2K n . in which L— the length of the slip element. T h e assumed linear displacement variation is consistent with the variation in the isoparametric quadrilateral finite element along a side. T h e stiffness matrix in terms of the global co-ordinates can be obtained using the transformation matrix consisting of the d i - Chapter 3 : 59 rection cosines. 3.11.2 Analysis Procedure In the incremental analysis, the values of K ans K are kept constant until yield is indi3 n cated. Therefore, the incremental stresses A / and A / „ are obtained using the incremental a force displacement relationship, 3 K 0 A/„ 0 K A/ s Aw n (3.53) n in which, Aw 3 Aw n = incremental shear displacement, = incremental normal displacement. Because of the linear displacement field, the stresses vary from point to point within the slip element. T h e average stresses are assumed to be representative stresses of the element. The average incremental stresses Af | Af I A/ 3av 1 3av and Af Aw = J n a v are calculated using the relationship, nav K 0 Aw n 3 (3.54) n in which, Aw sav = (Au R •+ Au )/2 - {Au + - {Av s P + Au )/2 (3.55) + Av )/2 (3.56) Q and Aw nav = {Av R Av )/2 s P Q Here R and S are the top nodes and P and Q are the bottom nodes defining the slip element (Fig. 3.7). Chapter S : 60 T h e total stresses /„ and / „ are computed by adding the incremental values from all load steps. T h e stress displacement relationship along the direction of the slip element is assumed to be elastic-perfectly plastic, while along the normal direction it is assumed to be elastic. T h e plastic region is defined by the M o h r - C o u l o m b yield criterion. Slip is assumed to occur when the shear stress exceeds the shear strength, /max, given by fmax = c, + f n tan (3.57) <f>'s in which, c = cohesion, 3 (f>'3 = friction angle. W h e n slip is indicated, the shear stiffness K s K n is set equal to zero, b u t the normal stiffness is kept at its current value. T h e separation is also indicated when the normal stress / „ reaches a negative value. U n d e r this circumstance, both K a n d K s T h e parameters These parameters K, K, 3 n c and 3 <f>'3 n are set to a small value. adequately define the behavior of the slip element. depend on many factors such as surface roughness a n d shape and char- acteristics of the asperities. Estimates of the parameters can be obtained from direct shear tests ( G o o d m a n et al 1968; Tatsuoka et al 1985), simple shear tests (Uesugi et al 1986), ring torsion tests (Yoshimi et al 1981) a n d rod shear tests (Felio et al 1987). 3.12 Computation of Deformation Pattern There are basically three components of deformation that occur in a soil system as a result of earthquake loading. structure T h e first component is the dynamic residual deformation that occurs at the end of the earthquake as a result of the hysteretic strain response. stress In order to compute this, an earthquake record with enough trailing zeros Chapter S : 61 should be used so that the free damped vibration response of the system can be included in the analysis. T h e second component increasing porewater is the deformation of the system that occurs as a result of pressures during the dynamic analysis. T h i s occurs because of the gravity acting on the softening soil. T h i s is mostly of the constant volume type of deformation in the saturated regions of the soil structure. T h e third component is the deformation of the system that occurs after the earthquake due to consolidation as the seismically induced residual porewater pressures dissipate. A l l three components are computed in T A R A - 3 analysis. are computed directly in a straightforward T h e first two components manner. T h e deformation due to dissipation of residual porewater pressures can be obtained by treating the problem as a two-dimensional consolidation problem in which the deformations are obtained at discrete time intervals as porewater pressures dissipate. T h e post consolidation deformations can also be obtained using t h « volumetric strains computed by the porewater pressure model. T h e computed volumetric strains are distributed to form a strain field depending on the degree of freedom of the nodes forming the element. T h e obtained strain field is used to compute nodal forces, which are then applied to the nodes to obtain the deformation field. T h i s procedure is carried out at several equal steps, each time only a portion of the total accumulated volumetric strains is used. B o t h of these options are available in T A R A - 3 . T h e final post earthquake deformation computed by T A R A - 3 is the sum of all three components described in this section. CHAPTER 4 ENERGY TRANSMITTING BOUNDARY 4.1 Introduction N u m e r i c a l techniques for dynamic analysis of a c o n t i n u u m require a finite domain with well denned boundaries. These boundaries often do not exist naturally and therefore must be artificially imposed on the computational model. In d y n a m i c analysis involving earthquake excitations, two different types of artificial boundaries are imposed when a semi-infinite m e d i u m is modeled by a finite domain, namely, the b o t t o m boundary (base) and lateral boundaries. problem involving earthquake excitations, In a typical soil-structure interaction it is common practice to apply the input excita- tion along the base of the finite element mesh and to assume vertical propagation of waves through the soil. T h e incident waves that are produced by the earthquake excitation and any waves reflected downward from the surface or any structures in the region pass through the b o t t o m boundary. T h e lateral boundary divides the core region from the free field. A n y waves other than those that pass through the b o t t o m boundary pass through the lateral boundary. For a realistic computation of dynamic responses, the conditions imposed on these boundaries must be such that they reproduce the physical behavior of the actual p r o b l e m being analyzed. 62 Chapter 4 •' 63 Often boundaries are represented by elementary boundaries on which either forces (free boundary) or displacements (fixed boundary) or combination of forces and displacements (roller boundary) mentary are specified depending on the problem. T h e major problem with ele- boundaries is that the energy that is transmitted out of the finite domain does not correspond to what is transmitted in the field. For example, either acceleration or velocity or displacement is often specified as the condition on the b o t t o m boundary. Such an assignment implies that the underlying m e d i u m is rigid. Therefore, no energy is allowed to radiate out of the system into the underlying m e d i u m . The use of elementary appropriate boundaries for the lateral boundaries in dynamic analysis is only in cases where the boundaries are located far enough from the zone of interest so that either the reflected waves will not reach the zone of interest within the time period under consideration or they will be removed before they reach the zone of interest by internal d a m p i n g . If the boundaries are located far away from the zone of interest, the finite element mesh will become large and therefore the computing time and cost will increase. Hence, elementary boundaries may not be practical i n some cases. Boundaries that account for the radiation of energy out of the finite domain are desirable for the proper evaluation of the dynamic response. S u c h boundaries are termed as energy transmitting or energy absorbing boundaries. These boundaries are achieved by precribing a set of normal and tangential stresses in such a way that the continued effect of these stresses and the stresses due to any incident waves will reflect the proper amount of energy back into the finite domain. O v e r the span of the last 20 years, many types of transmitting boundaries have been proposed for use in dynamic analyses involving wave propagation. However, many of them are not applicable to true nonlinear systems and most importantly they cannot be accommodated within the framework of time domain analysis. T h e next section describes briefly Chapter 4 •' 64 the review of the possible transmitting boundaries that can be adopted for implementation in computer program TARA-3. 4.2 Review of Possible Transmitting Boundaries O n e of the simplest accommodated and most effective energy transmitting boundary that could be in time domain analyses is the viscous boundary. In concept, this boundary is achieved by connecting viscous dashpots with appropriate constant properties along the nodes of the boundary. T h e properties of the viscous dashpots are based on the specific type of wave. T h e earliest solution for the viscous boundary was proposed by Lysmer and Kuhlemeyer (1969) for two-dimensional plane strain problems. the nodal dashpots were assumed to be m e d i u m and V and V 3 b are constants. p apV 3 and In their formulation the properties of bpV , p where p is the mass density of the are the shear (S) a n d compression (P) wave velocities and a a n d In their evaluation of this boundary, they showed that for any choice of a a n d 6, the effectiveness of the boundary i n absorbing energy depends on the Poisson's ratio. T h e case with a =1 and b =1 was found to be most efficient in absorbing plane b o d y waves a n d was termed the standard viscous boundary. T h e i r study and subsequent studies by White et al (1977) indicated that the standard viscous boundary is efficient in absorbing plane b o d y waves for Poisson's ratio ranging from 0.0 to 0.40. A n o t h e r possibilty is the transmitting boundary known as the superposition boundary. T h e technique for the superposition boundary was first introduced by Smith (1974). It is a method where the complete solutions of two independent boundary value problems using N e u m a n n (free) a n d Dirichlet (fixed) boundary conditions are superimposed so as to cancel out single boundary reflections. T h e formulation is independent of frequency and incident angles a n d very effective for both b o d y and surface waves. It requires 2 n complete dynamic Chapter 4 •' 65 solutions if n reflections occur during the time span of interest. However, the method fails when a given wave is reflected at the same boundary more than once. There have been refinements proposed to the original superposition boundary. The notable refinement is the one proposed by K u n a r a n d M a r t i (1981), in which the boundary conditions are changed from fixed and free to constant velocity and constant stress. T h e reflected waves are eliminated as they occur in the boundaries. A c c o r d i n g to K u n a r and M a r t i (1981), this refinement has the advantage that it avoids multiple reflections and the need for 2 n complete solutions as required in the original superposition boundary formulation. Between both of these boundaries, the viscous boundary was selected to be incorporated in T A R A - 3 for the simple reason that it is easy to implement. R a n d o l p h (1986), who conducted a comparative In fact, Simons a n d study of the standard viscous boundary and the superposition boundary of K u n a r and M a r t i , concluded that while the superposition boundary is found to be an effective absorber, the improvement in results obtained by the more rigorous superposition boundary formulation in preference to a simple viscous b o u n d ary formulation does not appear to warrant the increased computational effort required for the superposition formulation. Roesset et al (1977) have conducted parametric studies to compare the effect of different boundaries using single frequency oscillation input. T h e y have shown for the examples considered, that the responses (transfer functions) depend strongly o n the distance from the boundary to the structure and that satisfactory results can be obtained if elementary a n d viscous boundaries are located at an appropriate distance from the structure. T h e y recommended a distance of 10B to 20B for the cases with moderate values of internal d a m p i n g and a distance of 5 B for cases with high values of internal damping, where B is the width of the structure. T h e y have also shown that both roller and viscous boundaries are effective and the differences resulting from the use of these boundaries are not significant provided Chapter 4 •' 66 boundaries are located at an appropriate distance away from the edge of the structure. These studies were restricted to linear systems. Consequently, it is not known whether an improvement could be achieved by incorporating the viscous boundary for nonlinear problems with earthquake type of excitations. In order to investigate this, the viscous b o u n d a r y formulation is incorporated in T A R A - 3 a n d the effectiveness of the boundary is evaluated through simple examples. 4.3 Energy Transmitting Boundaries in TARA-3 T h e transmittimg base in T A R A - 3 is modeled by viscous dashpots with constant properties as used in the 1-D nonlinear program D E S R A - 2 (Lee and F i n n , 1978). T h e dashpots are similar to the ones proposed by Joyner a n d C h e n (1975) which are extensions of the viscous dashpots proposed by Lysmer a n d Kuhlemeyer (1969) to allow for incident waves from excitations outside the model to come into the model. T h e viscous dashpots placed along the lateral boundary are very similar to the ones proposed by Lysmer and K u h l e m e y e r (1969). However, the formulation is such that the properties of the dashpots placed along the lateral boundary c a n be either constant or varying. In the case of constant properties, the boundary is identical to the standard viscous boundary. T h e formulation is such that the lateral boundaries have to be vertical. the program. T h i s places a limitation on the capability of However, this does not seem to be a serious limitation in the case of the soil-structure interaction problem involving earthquake excitations. 4.4 Finite Element Formulation For Transmitting Base Consider a system of horizontal soil deposit bounded above by free surface a n d below by a semi-infinite m e d i u m . In the m e t h o d proposed by Joyner a n d C h e n (1975), the finite rigidity of the underlying m e d i u m is taken into account by including the stresses transmitted Chapter 4 •' 67 across the boundary between the soil deposit and the underlying m e d i u m into the l u m p e d mass system. In order to evaluate the stresses at the boundary, the following assumptions are implied. T h e underlying m e d i u m is elastic and the propagating shear and compression waves are plane waves travelling vertically. If U is the horizontal displacement of a particle in the underlying m e d i u m located at a depth z, then the shear stress r is given by, r = d_U (4.1) G where, G = shear modulus of the underlying m e d i u m . If Vi and Vj are the displacement and velocity components due to the incident wave and UR and VR are the displacement a n d velocity components due to the reflected waves, then Ui = UR = U,{z UR(Z + - V, t) V t) S (4.2) (4.3) where V, = shear wave velocity in the underlying m e d i u m , t = time. Now from equation (4.1), the shear stress at any point in the m e d i u m is given by, (4.4) Chapter 4 •' 68 F r o m equations (4.2) and (4.3), dUl _ V! (4.5) and ~J7=~T <-> 4 3 6 therefore, ( 4 . 7 ) 3 V Supposing VJB a n d VRB are the velocity components of the incident and reflected waves at the boundary, then the shear stress at the boundary, r £ , ' i s given by, T B = G { V l B - V R B (4.8) ) '3 T h e particle velocity at the boundary, Vg, is given by, V B (4.9) = VJB + V RB F r o m equations (4.8) and (4.9), TB can be rewritten as, (2V TB IB = — G ^ - IB V) ^ B (4.10) now, G and V are related by, 3 G = p V? (4.11) where, p = mass density of the underlying m e d i u m . C o m b i n i n g equations (4.10) a n d (4.11) will yield, T B = pV 3 (2V IB - V) B (4.12) Chapter 4 •' 69 T h i s is the expression for the shear stress transmitted across the b o u n d a r y between the soil deposit and the underlying m e d i u m . T h i s shear stress can be included i n the l u m p e d mass system by considering the equilibrium of the mass on the boundary. Consider a discrete mass q at node Q on the bottom boundary shown in F i g . 4.1. Let x and y be the horizontal a n d vertical directions respectively, and the b o u n d a r y stresses on segment S T in the x a n d y direction be T and a, respectively. In the case of transmitting boundaries, the input base motion is interpreted as the "control outcrop m o t i o n " . T h i s is simply the surface motion expected at the outcrop of the base material. Supposing the velocity of the motion in the horizontal direction expected at the outcrop of the base material is x , then equation (4.12) can be rewritten as b T = p V (x - x ) s b q (4.13) where, x = velocity of the mass q in the horizontal direction. q Similar arguments give the expression for normal stress as * = pV {y - y) p b q (4.14) where, y = velocity of the motion i n the vertical direction expected b material, y = velocity of the mass q in the vertical direction, q V p = compression wave velocity in the underlying m e d i u m . at the outcrop of the base Chapter .{ : core region bottom boundary P -> Fig. 4.1 s T x B o u n d a r y Stresses on a Discrete M a s s on Horizontal B o t t o m Boundary 70 Chapter 4 •' T h e corresponding boundary forces S a n d S x 71 are then given by y S = p V Al (x - x ) (4.15) S = p V Al(y -y ) (4.16) x 3 s b p b q q where, Al = length of segment S T , which is the s u m of 1/2 of the distance between nodes P and Q a n d 1/2 of the distance between nodes Q a n d R (Fig. 4.1). Now the d y n a m i c equilibrium of the discrete mass q in the horizontal direction gives the equation i n the form, m q x + c (x - x -i) q q q q + k (x - x -i) q q = q S (4-17) x where m , c , a n d k are the mass, d a m p i n g a n d stiffness terms associated q q with mass q. q Subscript "q — 1" refers to responses of the mass connected to mass q. Substituting for S from equation (4.15) into equation (4.17) a n d rearranging will yield x m q x + (c + p V Al) x - c x -i + k (x - x - ) q q s q q q q q q X = p V Al x s b (4-18) Equation (4.18) indicates that in order to account for the b o t t o m transmitting base, it is necessary to increase the diagonal components of the [c] matrix associated with the nodes on the b o t t o m b o u n d a r y by p V Al and introduce a term p V Al x on the right hand side s 3 b of the equilibrium equation. Similar arguments would lead to the conclusion that for the vertical degree of freedom, the diagonal components of the [cj matrix associated with nodes on the bottom boundary should be increased by p V Al and a term p V Al y be introduced to the right hand side p p b Chapter 4 •' 72 of the equilibrium equation. Therefore, the increase i n the d a m p i n g matrix coemcents [c] , mc and the term on the right hand side of the equation {F} dd, a associated with a node on the b o t t o m transmitting boundary are given by Cm p V Al 0 0 p V Al s (4.19) p and <«<» 4.5 Finite Element Formulation For Lateral Viscous Boundary In the standard viscous boundary proposed by Lysmer a n d Kuhlemeyer (1969), the boundary stresses on a vertical boundary are expressed as, a = p V u (4.21) r = p V w (4.22) p s where a a n d r are the normal and shear stresses, respectively, a n d u and iv are the normal and tangential velocities, respectively. However, i n seismic soil-structure interaction problems where the input is applied along the base of finite element mesh, it is important to formulate the lateral energy transmitting boundary i n such a way that it reacts only to waves radiating away from the structure rather than to motion resulting from the propagation of the seismic input. T h i s can be achieved by having a formulation that permits the lateral viscous boundary to react only to any response different from the free field response, i.e., the differential velocity field. In order to impose this condition, let consider a discrete mass n at node B on the vertical lateral boundary. Let x and y be the horizontal a n d vertical directions as shown i n Chapter 4 •' 73 F i g . 4.2. Let the boundary stresses on the segment D E in the x and y directions be a and T respectively. These are now defined as, <r = pV (x p an - x ) (4.23) m - y ) (4.24) af r = P V [y 3 where p is the mass density of the soil, V P af and V are the compression and shear wave s velocities in the free field and subscript " a n " refers to absolute velocities of discrete mass n and subscript tt af refers to absolute velocities of the free field at the location of node B . T h e boundary forces F x and F y corresponding to the boundary stresses expressed in equations (4.22) a n d (4.23) are given by F = pV F = p V Al {y x y Al (x p a an - x ) (4.24) an - y ) (4.25) af af where, Al is the length of segment D E , which is the s u m of 1/2 of the distance between nodes A and B and 1/2 of the distance between nodes B a n d C ( F i g . 4.2). Equations (4.24) and (4.25) can be rewritten in terms of quantities relative to the base as F = p V Al ( i t F y p = p V Al ( a yrn m - Xr,) (4.26) - M (4.27) where, subscript " r " refers to the velocities relative to the base. T h e dynamic equilibrium in the horizontal direction of the discrete mass n on the lateral Chapter 4 lateral b o u n d a r y . D 6 rt core region >x Fig. 4.2 B o u n d a r y Stresses on a Discrete M a s s on Vertical Lateral Viscous Boundary 74 Chapter 4 •' 75 boundary without consideration of the boundary forces yields a typical equation of the form m 'i n m + C (x n m - Xm-i) + k (x n - x -i) m m = - m n x (4-28) b where, m, c , a n d k are the mass, damping and stiffness terms associated with mass n and n n n the base input acceleration. is Subscript " n — 1" refers to responses of the mass connected to mass n . If the boundary force given by equation (4.26) is introduced, then equation (4.21) should be rewritten as, m x n m + c n (x m - x -i) m + k n (x m - z TO _i) = - m n i& - p V Al (x p m - xj) (4.29) Here, the force on the segment D E is assumed to be applied at the node B . Rearrangement of equation (4.29) yields m x n m + ( c + p V Al) km - c Comparison n p n of equations xm-i + k (x n m - a; -i) = - m x m n b - p V Al p (4.30) (4.28) and (4.30) indicates that in order to account for lateral viscous boundary, it is necessary to increase the diagonal components of the damping matrix associated w i t h the nodes on the boundary by p V p Al and introduce an additional term — p V Al Xyj on the right hand side of the equilibrium equation. T h e relative velocity of the p free field at location of the node B , i ^ , has to be determined by a separate site amplification study. It should be noted here that the finite element discretization for the separate free field response study should be consistent with the discretization of the lateral boundary. T h e free field response study may be conducted using T A R A - 3 in the one-dimensional mode. Similar arguments will indicate that for the vertical degree of freedom, the increase in the diagonal components of the d a m p i n g matrix associated with nodes on the lateral Chapter 4 •' 76 boundary is p V Al and the additional t e r m on the right hand side of the equilibrium 3 equation is - p V Al y,f. However, i n the cases where the earthquake input at the base is s assumed to be of horizontal shear waves propagating in the vertical direction, this additional term will be zero because is zero. Therefore, for general cases the increase i n the d a m p i n g matrix, [ c ] term on the right hand side of the equation, {F}^, mc and the additional associated with a node on the lateral boundary, such as node B , is given by p V Al p 0 0 (4.31) p V Al s and 4.6 Effectiveness of the Transmitting Base The effectiveness of the transmitting base is evaluated by analysing a horizontally lay- ered soil-column, 58m deep, using both a rigid a n d energy transmitting base. In the latter case, the underlying m e d i u m is assumed to have the same m a x i m u m shear modulus as that of the soil layer above the boundary. The soil column is similar to that at Station 7 of the E l Centro Strong M o t i o n A r r a y in Imperial Valley, California. T h e soil is assumed to behave nonlinearly and the variation of shear modulus and shear strength are as shown in F i g . 4.3. Further details regarding the site can be found in C h e n (1985). T h e selection of a one-dimensional deposit eliminates any influences that might arise from the inclusion of lateral boundaries. The horizontal input motion for the T A R A - 3 analysis is the reversed spike with a duration of 3.0 seconds ( F i g . 4.4). T h e input motion consists of two parts; the first part is Chapter a' (MPa) G v max Vertical Effective Stress (MPa) w Shear M o d u l u s F i g . 4.3 (MPa) Shear Strength Soil Property Profile Input Motion 10.00 A •V 0 -10.00 i F i g . 4.4 I • 2 Time (sec) •, 3 " • Reversed Spike Input M o t i o n > .5 Chapter 4 •' 78 the reversed spike scaled to a peak acceleration value of 10.0%g with duration of 0.4 seconds and the peaks of the spike occurring at times 0.1 and 0.3 seconds; the second part consists of zero input from time 0.4 seconds up to 3.0 seconds. The soil column was analysed using T A R A - 3 with the nonlinear analysis option in the total stress mode. T h e results for the case of the rigid base are shown in F i g . 4.5, which shows the computed surface acceleration response and the input motion. T h e input acceleration is amplified on passage to the surface by a factor of 1.37. reflection from the rigid base are clearly evident in F i g . 4.5. T h e effects of wave Three distinctly different parts can be identified in the surface acceleration response. First, there is a time lag of about seconds. to 0.78 Second, the big cyclic pulse starting at about time 0.18 0.18 seconds and extending seconds which corresponds to the reversed spike of the input motion. there is the considerable surface response in the time range from about 0.78 Finally, seconds to 3.0 seconds during which the input motion is zero. T h i s response can be attributed to the effect of multiple reflections from the rigid boundary of incident waves reflected f r o m the free surface. A t successive reflections, the wave amplitudes are being attenuated slowly by viscous and hysteretic damping in the soil a n d , as a result, the surface response decays with time. T h e soil column was also analysed using an energy transmitting base. T h e results are shown in F i g . 4.6. Results for the rigid base are shown for the purpose of comparison. In constrast to the rigid base response, the surface motions in this case diminish rapidly with time after the input motions ceases. T h i s clearly indicates that very little wave reflection from the base occurs in the case of the transmitting base. T h e little reflections found in the case of transmitting base are due to the fact that there is constrast in rigidity between the soil layers within the deposit. T h i s example shows that the energy transmitting base incorporated in T A R A - 3 is very Chaptc Max.Val. Input Motion Time Fig- 4.5 (sec) Surface Acceleration Response W i t h Rigid Base Chapter 4 Max.Val. Elastic Base c V u V a. c o 10 7.26 0 a t. 1 o u -8.16 -10 -< Time (sec) F i g . 4.6 Surface Acceleration Responses W i t h Rigid and Elastic Bases Chapter 4 •' 81 effective in simulating the energy radiation into the underlying m e d i u m . 4.7 Effectiveness of the Lateral Viscous Boundary T h e soil-structure interaction problem shown in F i g . 4.7, involving a stiff elastic ture on a dry sand foundation, was selected to demonstrate viscous boundary. the effectiveness struc- of the lateral T h e material properties of the structure a n d the sand foundation are given in Table 4.1. Table 4.1 Properties Selected for the Example Problem Property Structure Foundation Soil U n i t Weight (pcf) 400.0 120.0 Shear M o d u l u s (psf) 1.6 x 10 9 B u l k M o d u l u s (psf) 3.5 x 10 9 Bulk Modulus Exponent - 0.40 Poisson R a t i o 0.30 - A n g l e of Internal Friction - 35.0 Cohesion - 0.0 D a m p i n g Coefficient, a 0.0 0.0 D a m p i n g Coefficient, 8 0.005 0.005 ^2max ~ K 800.0 b = 51.0 For T A R A - 3 analysis, a horizontal computational boundary is imposed at a depth 5 B below the base of the structure, where B is the width of the structure. T h e base was assumed to be rigid. Lateral boundaries are placed at various distances from the structure. Chapter J D — Stiff Elastic Structure T Lateral Boundary 20 _L_ 100 Sand Foundation Scale in feet Fig. 4.7 rigid base Soil-Structure Interaction Problem Chapter 4 •' 83 T h e T A R A - 3 analyses were conducted assuming b o t h linear and nonlinear soil response. The horizontal input motions applied at the base correspond to the first 3 seconds of the 1940 E l C e n t r o , S 0 0 E horizontal acceleration record, scaled to 10.0%g peak acceleration. The free field relative velocities required for the lateral viscous boundary were computed using T A R A - 3 with a finite element discretization in the vertical direction consistent with that of the soil-structure problem. 4.7.1 Linear Analysis T h e peak free field accelerations computed by T A R A - 3 for the case of linear analysis is shown in Table 4.2. Table 4.2 Linear Analysis: Free Field Peak Accelerations The Depth (ft) Acceleration 0.00 26 20.0 22 40.0 16 60.0 13 80.0 11 100.0 10 (%S) distribution of peak accelerations when horizontal roller boundaries are placed at distance D = 2 0 B , where D is the distance between the b o u n d a r y and the edge of structure, is shown in F i g . 4.8. T h e values at the grid intersection are the peak horizontal accelerations Linear Analysis 50 50 26 26 26 26 26 26 26 26 26 26 27 27 27 2B 2B 28 29 28 27 37 33 33 22 22 22 22 21 21 21 21 21 21 21 22 22 22 23 23 23 23 24 24 29 24 16 16 16 16 16 16 16 16 16 16 15 15 15 14 14 15 15 16 16 19 16 16 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 14 14 14 14 14 14 14 11 ] I 11 11 11 11 1 i 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 RIGID BASE GEO.SCALE ACCELERATION Fig. 4.8 o ' io ' 80 feet percent g Linear Analysis - Distribution of Accelerations When Roller Boundaries are at D=20B Chapter 4 • in % g . 85 It should be noted that the results are quoted only for the region on the left h a n d side of the centerline of the model. T h e accelerations edge of the structure from the structure. at locations far away from the are close to those of free field a n d do not vary much with distances T h i s indicates that true free field conditions are achieved in the wider region bounded by the boundary and the vertical grid at a distance around 10B. Therefore, these results can be assumed to be "correct" responses and consequently be used to assess the effectiveness of other boundary conditions. F i g . 4.9 shows the acceleration distribution for the case when horizontal roller b o u n d aries are placed at distance D = 1 0 B . Accelerations at the boundary and at locations near to the boundary are close to those of free field given in T a b l e 4.2. Further, the accelerations at. locations on and closer to the structure are still similar to the corresponding accelerations when the boundaries were at distance D = 2 0 B . T h e differences are within a few percent. For instance, at top center point of the structure, the acceleration is only 2% different when the boundary is at D = 1 0 B . T h e results for the case when horizontal roller boundaries are situated at distance D = 4 B are shown in F i g . 4.10. These results are significantly different from the "correct" response. T h e deviations in acceleration, particularly at locations on and closer to the structure, are higher than the corresponding deviations when the boundaries were located at D = 1 0 B . For instance, the difference in acceleration at top center point of the structure is now around 10%. T h e results clearly indicate that the responses are strongly dependent on distance D . A s D is changed, the natural periods that contribute strongly to the resposne are changed resulting in quite different responses. For a given problem, the choice of D depends on the degree of accuracy desired. In this case, for practical purposes, the boundary could be placed at distances not less than 4 B . Linear Analysis 26 26 26 27 27 27 27 27 26 37 32 32 21 21 21 21 22 22 21 21 21 24 29 15 15 14 14 14 14 15 15 16 18 16 13 13 13 13 13 13 14 14 14 14 |4 12 12 12 12 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 10 10 14 RIGID B A S E GEO.SCALE ACCELERATION Fig. 4.9 o 40 80 percent g Linear Analysis - Distribution of Accelerations When Roller Boundaries are at D=I0B tetl Linear Analysis 29 28 34 30 30 23 23 23 22 25 21 16 17 16 17 ie 17 15 15 15 14 15 15 12 13 12 12 12 12 10 10 10 15 10 10 GEO.SCALE ACCELERATION 4.10 <15 29 RIGID Fig. <J5 BASE o ' *o ' 80 '«t percent g Linear Analysis - Distribution of Accelerations When Roller Boundaries are at D = 4 B "5 5 OO -I Chapter 4 : Fig. 88 4.11 shows horizontal acceleration distribution when viscous boundaries with con- stant dashpot properties are placed at distance D = 2 0 B . T h e accelerations away from the edge of the structure accelerations at locations far are close to those of the free field. at locations on and closer to the structure The computed are similar to the corresponding responses when horizontal roller boundaries are in place. However, acceleration as shown in F i g . response particularly 4.12, when the viscous boundaries are at D = 1 0 B , at locations on and closer to the structure are the quite different than the corresponding response when D = 2 0 B . A t top center point of the structure, the acceleration is underestimated as m u c h as 14%. M a r k e d differences are noticeable when the viscous boundaries are located at distance D = 4 B as shown in F i g . 4.13. Structural responses are underestimated. For instance, at top center point on the structure, the horizontal acceleration is underestimated as m u c h as 16%. Similar differences are also noticeable in the case of responses at locations closer to the structure. Therefore, for elastic analysis the roller boundary seems preferable than the viscous boundary. 4.7.2 Nonlinear Analysis The peak free field acceleration presented in Table 4.3. responses assuming the nonlinear soil behaviour are T h e y are slightly less than the values in Table 4.2. T h i s is due to the fact that additional inherent hysteretic d a m p i n g is present in the nonlinear case. Linear Analysis S! 51 27 24 25 24 25 25 25 26 26 26 27 27 27 26 26 28 28 28 26 33 31 31 25 21 20 20 20 20 20 21 21 21 22 22 22 22 22 23 22 22 24 24 27 23 20 16 15 15 14 14 14 14 14 14 14 14 14 15 15 15 15 15 16 17 16 16 13 )3 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 RIGID BASE GEO .SCALE ACCELERATION Fig. 4.11 o 40 BO f«t percent g Linear Analysis - Distribution of Accelerations When Viscous Boundaries are at D=20B Linear Analysis 44 44 26 25 25 24 25 26 28 2B 27 33 31 22 20 20 20 21 21 22 22 23 23 26 16 10 14 15 15 15 16 16 17 16 |7 13 13 13 13 14 14 14 14 14 15 |4 11 11 11 12 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 10 10 RIGID BASE GEO .SCALE o ACCELERATION Fig. 4.12 '"" 40 percent ' so fe«t e Linear Analysis - Distribution of Accelerations When Viscous Boundaries are at D = 1 0 B S "5 Linear Analysis 43 43 27 22 21 26 ?5 25 22 18 18 20 20 16 16 15 15 16 15 16 13 13 14 13 14 13 11 12 11 12 II 12 10 10 10 10 10 10 RIGID B A S E GEO.SCALE o ACCELERATION Fig. 4.13 ' So ' BO feet percent g Linear Analysis - Distribution of Accelerations When Viscous Boundaries are at D = 4 B Chapter 4 : 92 Table 4.3 Nonlinear Analysis: Free Field Peak Accelerations Depth (ft) Acceleration 0.00 22 20.0 19 40.0 15 60.0 14 80.0 12 100.0 10 (%g) F i g . 4.14 shows the horizontal acceleration responses when horizontal roller boundaries are placed at distance D = 2 0 B . A s in the case of linear analysis, the response computed at locations far away from the structure are close to those of the free field. F i g . 4.15 shows the acceleration response when the horizontal roller boundaries are located at distance D = 1 0 B . It is clearly seen that at locations close to the boundary free field conditions are achieved. A l s o , the structural response is similar to those when D = 2 0 B . However, as may be seen form F i g . 4.16, are somewhat underestimated. For instance, the structural responses for the case D = 4 B at top center point on the structure, the acceleration is 10% smaller than the corresponding value when D = 2 0 B . F i g . 4.17 shows results obtained when viscous boundaries with constant dashpot properties are placed at distance D = 2 0 B instead of roller boundaries. T h e structural responses in b o t h cases are within very few percent. A s seen from F i g . 4.18, when the viscous boundaries are at D = 1 0 B , structural response is close to that when D = 2 0 B . However, the acceleration values c o m p u t e d at locations on Nonlinear Analysis 40 40 22 22 22 22 22 22 22 22 22 22 22 22 22 22 21 21 21 20 20 25 26 26 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18 18 18 19 21 23 20 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 16 17 19 17 17 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 15 14 15 14 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 1? 12 12 12 12 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 RIGID BASE GEO.SCALE ACCELERATION Fig. 4.14 o ' « ' so f«t percent g Nonlinear Analysis - Distribution of Accelerations When Roller Boundaries are at D=20B Nonlinear Analysis 39 21 21 21 21 21 21 20 20 20 26 25 18 18 18 18 18 18 n 18 18 21 23 14 14 14 15 15 15 15 16 17 17 |7 13 13 13 13 13 13 14 14 14 14 |4 11 11 11 11 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 10 10 39 RIGID B A S E GEO.SCALE ACCELERATION Fig. 4.15 o ' 40 1 oo percent g Nonlinear Analysis - Distribution of Accelerations When Roller Boundaries are at D = 1 0 B Nonlinear Anatysis 36 36 20 19 19 24 ?? 22 17 17 17 18 22 18 16 16 16 16 15 15 14 |4 14 13 13 13 12 12 12 12 12 12 10 10 10 10 10 10 RIGID BASE GEO.SCALE ACCELERATION Fig. 4.16 t 0 40 80 percent g Nonlinear Analysis - Distribution of Accelerations When Roller Boundaries are at D = 4 B feet Nonlinear Analysis 39 71 70 70 70 70 71 71 77 73 72 22 22 22 21 71 20 20 24 25 19 18 17 17 17 17 17 18 18 18 19 19 19 18 18 17 17 19 20 21 17 16 15 16 15 15 15 15 15 15 15 15 15 15 15 15 16 16 16 17 16 15 14 13 13 14 14 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 12 12 12 12 12 12 12 II 11 11 11 11 11 11 12 12 12 12 12 12 12 10 10 10 10 10 10 10 . 10 10 10 ID 10 10 10 10 10 10 10 10 10 10 7K 71 22 ' RIGID BASE GEO.SCALE ACCELERATION Fig. 4.17 o ' 40 ' 80 f«t percent g Nonlinear Analysis - Distribution of Accelerations When Viscous Boundaries are at D = 2 0 B 39 Nonlinear Analysis 39 27 21 20 20 19 19 19 19 19 24 23 23 19 18 17 17 17 17 17 19 20 22 17 15 15 15 15 16 16 16 17 18 17 17 15 13 14 14 14 14 14 14 15 14 15 14 12 12 12 12 12 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 10 10 RIGID BASE GEO.SCALE ACCELERATION Fig. 39 4.18 o ' 40— 80 percent g Nonlinear Analysis - Distribution of Accelerations When Viscous Boundaries are at D = 1 0 B p Chapter 4 : 98 the boundary are greater than that of the corresponding free field response values. T h i s may be due to disturbances caused by incomplete absorption of the surface waves and to some extent the body waves. A s the depth increases, the acceleration values become closer to the corresponding free field values. T h e distribution of accelerations when viscous boundaries are placed at distance D = 4 B is shown in F i g . 4.19. T h e structural response in this case shows that the difference in acceleration at top center point on the structure is now only 5.1%. 4.7.3 Discussion T h e results in both linear and nonlinear analysis clearly reveal that the responses of the soil-structure system depend on the distance D and the type of boundary conditions. However, the effect of boundary distance is m u c h more significant in the linear than in the nonlinear case because of the greater d a m p i n g in the latter case. results show that satisfactory In both cases, the results can be obtained using viscous or roller boundaries provided that they are located at an appropriate distance from the edge of the structure. T h e m i n i m u m distance for the nonlinear case seem to be 4 B and for the linear case a m i n i m u m distance somewhat greater than 4 B seems to be appropriate. In the linear case, the results reveal that the roller boundaries perform better than the viscous boundaries with respect to structural response. A l s o , in the nonlinear case, except for the case when D = 4 B , the roller boundary performs better than the viscous boundary. Therefore, the use of roller boundary is preferable. T h e roller boundary not only performs more efficiently but also requires less effort in data preparation and computer cost. Nonlinear Analysis 37 37 30 18 18 23 24 24 24 16 17 20 21 18 18 15 16 17 16 16 15 13 14 13 13 13 12 12 11 11 II 12 1<3 10 19 10 19 10 RIGID BASE GEO.SCALE ACCELERATION Fig. 4.19 o ' So - ' 80 percent g Nonlinear Analysis - Distribution of Accelerations When Viscous Boundaries are at D = 4 B f«t CHAPTER 5 SIMULATED SEISMIC TESTS ON CENTRIFUGE 5.1 Introduction A t present, only simulated seismic tests on centrifuge models can provide the flexibility and cost effectiveness necessary to provide a data base against which concepts of response to loading and methods of seismic analyses can be checked. tests on centrifuge models of simple 1-D system (Abghari 1983; and pile foundations ( F i n n and G o h l 1987) of numerical analyses. D a t a from simulated seismic L a m b e and W h i t m a n 1985) have been used successfully to verify methods T h i s chapter deals with the important aspects of the simulated seis- mic tests that were conducted on various centrifuged models to generate data to explore the capacity of T A R A - 3 to model soil structure and soil-structure These models include both dry and saturated embankments, interaction problems. and surface and embedded gravity structures on both dry and saturated sand foundations. A l l tests were conducted on the Cambridge University Geotechnical Centrifuge in the U n i t e d K i n g d o m by D e a n and Lee (1984) and Steedman Schofield. (1985 and 1986) under the general direction of Professor A . N . T h e tests were sponsored by the U n i t e d States Nuclear Regulatory C o m m i s - sion through the U n i t e d States A r m y C o r p s of Engineers ( U S A E ) and were monitored by Professor W . D . L i a m F i n n on behalf of the U . S A r m y Corps of Engineers. 100 T h e tests were Chapter 5 : 101 designed jointly with the collabaration of the University of British C o l u m b i a , the Cambridge University and the U S A E to ensure the rigorous evolutionary testing of the capability of TARA-3. T h e subsequent sections describe briefly a review of centrifuge testing and test procedures in Cambridge Geotechnical Centrifuge. Detailed descriptions can be found elsewhere (Schofield, 1981). 5.2 Centrifuge Testing In a centrifuge, the same unit stresses that exist in a full-scale structure can be reproduced at corresponding points in a small scale model by rotating the model around the axis of the centrifuge to create an artificial gravity field, Ng, where g is the acceleration due to the earth's gravity and 1/iV is the linear scale of the model. T h e ability to create prototype stresses in the model is important in studies of soil-structure interaction since many soil properties are dependent on effective stresses. For this reason, seismic tests on a centrifuge are superior to those conducted on a shaking table in l g environments. Since all stresses at each point in a centrifuged model can, in theory, be made the same at the corresponding point in the prototype, each element of soil can be expected to undergo the same response to loading as corresponding elements in the prototype (Barton, 1982). Since each model is of finite size, different parts of the model are at different radii from the rotational of the centrifuge. axis Therefore, at any given speed of the centrifuge arm, different parts of the model will be subjected to different gravitational intensities. T h i s results difference at corresponding points in the model and the prototype. in a stress T h e stress difference will be small if the space that the model occupies in the direction of the centrifuge arm is small compared to the radius of the centrifuge arm. For example (as illustrated by Schofield 1981) for a model that extends for a radial distance of one tenth of the centrifuge radius, Chapter 5 : 102 the error in vertical pressure within the model in the Cambridge Geotechnical Centrifuge is typically around ± 2%. Errors of this magnitude are certainly within the acceptable range of accuracy in the engineering profession. 5.3 Scaling Laws Scaling laws for the centrifuged models have been reported for granular media by many researchers (Roscoe, 1968 and Scott, 1978). A summary (Scott 1978) is given in Table 5.1. Table 5.1 Scaling Relations Quantity Full Scale M o d e l at N Linear Dimension 1 l/N Area 1 l/N 2 Volume 1 l/N 3 Stress 1 1 Strain 1 1 Force 1 1/iV Acceleration 1 N Velocity 1 1 T i m e - In D y n a m i c T e r m s 1 l/N T i m e - In Diffusion Cases 1 l/N Frequency in D y n a m i c Problems g's 2 2 N 1 In a l/N linear scale m o d e l , excess porewater pressures dissipate N 2 times faster in the Chapter 5 : 103 model than in the prototype if the same fluid is used in both. T h e rate of loading by seismic excitation will be only N times faster. Therefore, to model prototype drainage conditions during the earthquake, a pore fluid with a viscosity N times the prototype viscosity must be used. C o m m e r c i a l silicon oil blended to the appropriate viscosity is often used as pore fluid in centrifuge model tests. Saturated tests of centrifuged models for the verification study of T A R A - 3 were carried out using silicon oil as pore fluid (Dean and L e e , 1984 a n d Steedman, Triaxial tests by E y t o n (1982) showed that the stress-strain 1985 and 1986). behavior of fine sand was not changed when the silicon oil was substituted for water as pore fluid. Centrifuge model tests conducted at different linear scale ratios (40 and 80) also indicated that the responses were not changed when silicon oil was used as pore fluid. 5.4 Earthquake Simulation in Cambridge Geotechnical Centrifuge T h e Cambridge centrifuge has a 10m long rotor a r m driven by a 225kW motor. T h e effective radius of the centrifuge is around 4m. T h e centrifuge is housed i n a reinforced concrete chamber of diameter slightly larger than 10m. In general, earthquake simulation i n a centrifuged model is accomplished through the use of some form of a shaking system. There are many designs of shaking systems available, each of which has its own advantages and disadvantages ( A r u l a n a n d a n et al, 1984). T h e system that is currently adopted in the Cambridge centrifuge is a mechanical type. Seismic excitations are generated by a wheel linked to the model container travelling on a track with precisely machined sinusoidal undulations attached to the wall of the centrifuge The track extends over one third of the circumference system is known as the b u m p y road. of the centrifuge A model earthquake chamber. chamber. The involves a single pass of the actuating wheel along the b u m p y road track. T h e intensity of model shaking is controlled Chapter by adjusting the linkage between the wheel and model container. 5 : 104 For a given b u m p y road configuration, the frequency of oscillation is governed by the angular velocity of the rotor arm. Ideally, the b u m p y road should generate a model earthquake that is sinusoidal in na- ture with a constant period. However, the actual motion is much more complicated mainly due to resonances, mechanical linkage clearances and other factors, a n d as a result it has a broader frequency range. A typical model earthquake consists of three important compo- nents (Dean and Lee, 1984): (1) Small "wheel-on" accelerations associated with initial contact of the wheel with the track; (2) the model earthquake proper consisting of roughly sinusoidal pulses; (3) small "wheel-off" accelerations associated with the wheel leaving the track. In the b u m p y road system, it is difficult to obtain precisely the earthquake motions one wants. Often the linkage adjustments earthquakes of very small amplitudes. between the wheel and the model container produce Therefore, in order to obtain earthquake motions of significant amplitudes, a series of earthquakes is initiated and each time the linkage is adjusted so as to produce earthquakes of greater amplitudes. 5.5 Model Construction T h e models were constructed in a container whose exterior dimensions are 902mm long, 481mm wide and 225mm deep. Overflow troughs are provided to take excess soil should a failure occur. Leighton B u z z a r d sand was used in the construction of all centrifuged models. For most tests, sand passing through British Standard Sieve N o . 120 and retained by B.S.S N o . 200 Chapter 5 : 105 (B.S.S 120/200) was used. T h e aperture sizes of these two sieves are 0.125mm and 0.075mm respectively. F o r the remaining tests, Leighton B u z z a r d B . S . S 52/100 sand was used. T h e properties of each of these sands are given in Table 5.2. T h e standard sand was chosen to minimize the potential variability of model properties and it is not intended to model any real in-situ conditions. used to simulate specific real type prototypes of the program T h e model tests are not being but to provide data for the direct verification TARA-3. Table 5.2 Properties of Model Sand G Sand & min s Cmax Type Mean Grain Size (mm) B . S . S 52/100 2.65 0.585 0.928 0.225 B . S . S 120/200 2.65 0.650 1.025 0.100 5.5.1 Dry Model Construction T h e dry models were constructed a fixed height. construction. to a uniform density by allowing sand to fall through A l u m i n u m formworks were first fitted inside the model container to guide A hopper, fitted with a nozzle, containing a known weight of dry sand was suspended at an appropriate height above the base of the container. T h e nozzle and the height of drop required to give a specified relative density was determined by calibration tests in advance of the construction. T h e hopper valve was opened a n d the nozzle was moved slowly over the area of construction same rate over the entire area. so that the sand surface rose roughly at the A s the surface level rose, the hopper was raised so as Chapter 5 : to keep its height above the sand surface was temporarily approximately stopped whenever transducers constant. were installed. 106 Pouring of the sand Leads were carefully laid and attached to the side of the container in such a manner to avoid tensioning or jerking of leads d u r i n g the flight. Before placement of the structure, the top sand surface was levelled by vacuuming. For embedded structures, pouring of the sand continued around the structure to the required design profile. T h e transducers were then mounted at appropriate places on the structure. Once pouring was complete, the top sand surface was levelled and measurements were taken to define the actual surface. T h e formworks were removed and the roof of the container was then bolted on. 5.5.2 Saturated Model Construction T w o different techniques of saturated model construction were employed. T h e first method ( M e t h o d 1) involves pluviating de-aired s a n d / o i l mixture through de-aired silicon oil. U n d e r these conditions, it was difficult, to maintain uniform density, to relative density and to specify accurately the transducer locations. struction technique ( M e t h o d 2) was adopted in the later tests. sand dry as described above and then saturating the methods is discussed briefly in the subsequent 5.5.2.1 Method determine Therefore, a new con- T h i s involves placing the it slowly under a high vacuum. E a c h of sections. 1 A sufficient quantity of silicon oil at appropriate viscosity was de-aired under a vacuum of 27-30 inches of mercury for a period of 24 hours. T h e model container with the a l u m i n u m formworks in place was then filled with the de-aired silicon oil. Sufficient mass of dry sand was weighed and placed in a small dessicator. Silicon oil was then added to cover the sand surface and was thoroughly mixed with the sand. The Chapter 5 : mixture was placed under vacuum for 1/2 hour. the model container using a small beaker. 107 T h e s a n d / o i l mixture was tranferred to T h e beaker was inverted beneath the surface of the oil in the container to prevent the entrapment of air a n d the sand was allowed to pluviate through the oil. O n removal from the model container, a beaker full of silicon oil was transferred back to the dessicator. D u r i n g the pouring process, the beaker was moved slowly over the area of construction in order to achieve equal rate of rise of surface at all points. A t appropriate levels, transducers were placed. T h e porewater pressure transducers were also placed under the vacuum. T h e accelerometers were coated with a thin layer of silicon rubber as a seal. Once pouring was complete, the formworks were removed. T h e soil profile was surveyed and the roof of the container was bolted o n . D u r i n g the deposition process, the silicon oil in the container became very murky as some sand remained in suspension. T h i s made it difficult to see how the model was progressing. A l s o the sand surface was very soft which posed problems for the installing transducers. 5.5.2.2 Method 2 In this technique, the model is constructed first using d r y sand as described in section 5.5.1. O n c e the dry model construction was completed, the model container was sealed. T h e container was then evacuated to a v a c u u m of 28-30 inches of mercury. Silicon o i l , de-aired under a similar vacuum, was slowly introduced at b o t h ends of the model. T h e v a c u u m was maintained until the oil was u p to the desired level. T h e n , the v a c u u m was slowly released and the model container was unsealed. 5.6 Relative Density Estimation Estimates of average relative density of each model were made from estimates of the volume a n d mass of sand in the model. T h e void ratio e a n d relative density D r (in percentage) Chapter 5 : 108 of the models were then calculated from : e = Gs (V/M) - D = 100.0 [{e r max - e) / ( 1 (5.1) CmM - e^] (5.2) where G is the specific gravity of the sand, V is the model volume, M is the mass of the s sand, e max is m a x i m u m void ratio and e m m is m i n i m u m void ratio. Errors in the calculation of void ratio arise from inaccuracies of the balance used to weigh the model container and errors in volume measurement. from these two sources is of the order of ± 2%. T h e compounded error T h i s leads to a possible error of up to ± 10% in relative density (Dean and Lee, 1984). However, for saturated models constructed using M e t h o d 1, the error can be much greater especially because of migration of sand into the overflow troughs during construction. A n u n k n o w n amount of fines also remained in suspension in the oil. 5.7 Instrumentation and Accuracy T h e models were instrumented with accelerometers, porewater pressure transducers a n d linearly variable displacement transducers designated A C C , P P T and L V D T respectively. T h e number of transducers used in a test was limited by the number of channels available in the data acquisition system and the size of the model. 5.7.1 Accelerometers M i n i a t u r e piezo-electric D J B A 2 3 type accelerometers supplied by D . J . Birchall L t d . , C h e l t e n h a m , E n g l a n d , were used in the model tests. T h e frequency response is flat to above 10 k H z . T h e accuracy of calibration is about ± 4% of the measured values (Dean and Lee, Chapter 5 : 109 1984). Besides calibration accuracy, a number of operating factors also affects the accuracy of measured response. T h e piezo-electric accelerometers respond sharply to sudden increases in tension in leads giving the appearance of spiky high frequency response. In order to minimize the effect of lead tension, leads were laid perpendicular to the direction of shaking as shown in F i g . 5.1. Since accelerometers are capacitive devices, any lead bending may affect the capacitance of the leads and consequently alter the measurement. A poor earth connection can cause the signals to "float" about its base line. Often, it is possible to correct data using simple digital techniques. However, even if corrected, some error is likely to remain. Taking all these factors into account, D e a n and Lee (1984) concluded that the overall accuracy of the accelerometer is believed to be i n the order of ± 5% of the measured values. 5.7.2 Porewater Pressure Transducers Porewater pressures were measured using P D C K 81 type porewater pressure transducers, supplied by Druck L t d . , Leicester, E n g l a n d . A silicon integrated pressure sensor forms the diaphragm of the device. T h e calibration accuracy for these transducers is about ± 5% of the measured values (Dean and L e e , 1984). In order to register pressure, the transducer requires a small but finite volume of fluid to flow into and out of it. T h i s volume has to be provided by the surrounding soil. K u t t e r (1983) has found that i n saturated clays the required flow causes negligible measurement inaccuracies and has a negligible effect on model behavior. Dean and Lee (1984) concluded that in fine sands the effects were also negligible. Occasionally drainage channels may be introduced along the path of the leads. Such Chapter Fig. 5.1 Layout of the A c c e l e r o m e t e r Leads 5 : 110 Chapter 5 : 111 an event can be detected by the fall off in measured porewater pressures with respect to measured pressures by adjacent transducers. If tension is suddenly applied to the lead, the transducer may move relative to the surrounding soil and a sudden decrease in the porewater pressure will be measured. Therefore, careful study is necessary to determine whether sharp drops in porewater pressures are due to this effect or dilations due to shearing. T h e overall accuracy of the porewater pressure transducers is estimated to be of the order ± 10% of the measured values (Dean a n d Lee, 1984). 5.7.3 Linearly Variable Displacement Transducers (LVDT's) T h e L V D T ' s were used mostly to measure vertical settlements a n d were attached to the gantry spanning the box. Because of the poor dynamic response of these devices a n d limitations of available channels the L V D T ' s were read only at discrete times- for instance, during swing up and at the beginning a n d end of earthquakes to give complete settlement increments during the tests. T h e accuracy of these devices is about ± 2% for static readings (Steedman, 1985). 5.8 Data Acquisition and Digitisation Signals from the model were recorded on a 14 track R A C A L tape recorder. These analogue signals were processed and digitised at a suitable time increment using the software package, F L Y - 1 4 , developed by D e a n (1984). T h e raw digitised data was smoothed once using a three point smoothing scheme as suggested by Dean (1984). A c c o r d i n g to this scheme, the current value at any time is replaced by the sum of 1/2 of the current value plus 1/4 of the previous value a n d 1/4 of the next value. T h e smoothing function is symmetric and therefore does not introduce phase shift. T h e smoothing was necessary to filter out Chapter 5 : 112 very high frequency electrical noise which contained negligible energy. T h i s type of noise is unavoidable in dynamic centrifuge tests as it originates as a result of ambient sources such as container vibrations etc. T h e accuracy of digitisation from analogue magnetic tape is dependent on the magnitude of the signal. In general, a strong signal is digitised with an accuracy of better t h a n ± 0.1%. For a weak signal, the error in digitisation may exceed ± 2% (Steedman, 1985). These cases are identified with a code P A P standing for "Possible A c c u r a c y P r o b l e m " in the time history plot. 5.9 Centrifuge Flight T h e container is first secured on the centrifuge. A s the centrifuge speed is increased, the box swings up and encounters end stops which prevent the box from swinging further. A t this point, the base of the container is vertical. Further increase in centrifuge speed will make the radial acceleration field more dominant. T h e centrifuge acceleration is increased in steps of 20g until the desired g level is reached. A t every 20g steps, readings from porewater pressure transducers ( P P T ) and displacement transducers are recorded. A f t e r the centrifuge has reached the desired g level, sufficient time is allowed for porewater pressures to come into equilibrium before the model is subjected to earthquake loading. D u r i n g each earthquake, the transducer data are recorded by the high speed analogue tape recorder. RACAL A b o u t 15 minutues is allowed between earthquakes i n a sequence to allow the model to drain a n d porewater pressure and L V D T transducers to stabilize. L V D T measurements are taken at the beginning and the end of each earthquake. A f t e r the test series, the centrifuge is brought to a stop and the model container is removed from the centrifuge. T h e post-test site profile is measured and the final locations of the transducers are determined during careful excavation of the model. Chapter 5 : 113 5.10 Typical Test Data A centrifuge model of a gravity structure and foundation is illustrated in F i g . 5.2. T h e foundation layer is 110mm thick and the width perpendicular to the plane of the figure is 480mm. T h e gravity structure is modelled by an a l u m i n u m cylinder 150mm in diameter and 100mm high, embedded 30mm in the foundation soil. T h e centrifugal acceleration was nominally 80g. T h e model, therefore, simulated a structure approximately 8 m high and 12m in diameter embedded to a depth of 2.4m in the foundation soil. T h e average contact pressure of the structure on the soil was 200kPa. T h e model was instrumented by accelerometers, displacement transducers. porewater pressure transducers and T h e locations of these instruments are shown i n F i g . 5.2. T h e typical output of smoothed d a t a from F L Y - 1 4 is shown F i g . 5.3. It should be noted that there are wide variations in the scales of the various records and the apparently quite different forms of some of the records are due primarily scales are model scales. T h e accelerations to differences in the scale. A l l are expressed as percentages of the centrifuge acceleration. Porewater pressures are those actually measured. Equivalent prototype times are given by multiplying measured times by the linear scale factor. T h e accelerations expressed as percentage of model gravity and porewater pressures are the same in model and prototype. T h e peak acceleration of the input motion as measured by A C C 2036 is 0.16g. T h e peak acceleration transmitted to the soil near the base ( A C C 1487) is almost the same. T h e peak horizontal acceleration recorded on the structure by A C C 2033 is 0.26g. T h e porewater pressures increase steadily during the shaking. T h e porewater presure transducers far away from the structure on the right h a n d side of the model ( P P T 2338, 2335, 2251 and 2511) show a relatively smooth development of porewater pressure with none of the large oscillations usually associated with dilatant behavior or rocking of the Chapter 1 5 0 mm ACC 728 20?3 ACC 1 ACC 734 WEST EAST Accelerometer Porewater 7 P mm pressure transducer 3 0 mm 1 1 0 mm -»-»ACC . • PPT 2631 PPT 2561 PPT ACC Fig. 5.2 2338 PPT 2335 « PPT 2626 •*— ACC PPT 1225 68 PPT 2251 p p 1487 2036 Instrumentation of a Centrifuged M o d e l T 2 ni Chapter too SO • ACC731 20.0 X/div - ACC2033 SO.O X/div •• ACC728 20.0 X/div •• ACCU87 50.0 X/div is s 26 -- i -23.9 -. 11.0 -10.1 38.0 kPa -too 16.9 X -11.7 13.3 '-• ^ X 4- -16.2 •• ACC2036 SO.O X/div 16.0 X -IS. 3 50 100 miUisecs Scatis Fig. 5.3 : Model Typical Test Data on Seismic Response of the Model . • ) : 1 1 Chapter 5 : structure. 116 It seems that cyclic shear strains in the free field area are not sufficient to cause significant dilation. O n the contrary, the porewater pressure transducers beneath the structure ( P P T 2631, 2626 and 68) show large swings in the recorded porewater pressure with cycles of loading indicating that the effects of rocking a n d the cyclic shear strains under the structure are sufficiently large to induce significant dilation. However, despite the oscillations, there is a steady increase in residual porewater pressure under the structure. T h e effects of increasing porewater pressure on the rocking mode are clearly evident. T h e rocking is portrayed by the vertical acceleration records A C C 728 a n d A C C 734 at opposite ends of the diameter of the structure i n the plane of excitation. are 180 degrees out of phase. These records W h e n A C C 728 indicates an upward acceleration, A C C 734 indicates a downward acceleration. T h e input motion, except for r a n d o m effects, is primarily a horizontal acceleration, and in the initial stages of shaking the recorded vertical accelerations are very small, showing insignificant rocking, which is not surprising in such a squat structure. However, as porewater pressure increases, the vertical accelerations become quite large, upto 0.16g at A C C 734 and O . l l g at A C C 728. T h e -amplitude of the input motion to the base of the structure ( A C C 1225) increases slightly with duration and it may be thought that the sharp increase i n rocking may be due to this. However, it should be noted that despite significant horizontal acceleration (at A C C 2033) at the level of A C C 734 and A C C 728, in the early stages of shaking there is very little rocking evident from the records despite the fact the scale of the vertical accelerations is 2.5 times that of the horizontal accelerations. T h e d a t a presented i n this section are a typical sample of the kind of information obtained d u r i n g a centrifuge model test. T h e description of the d a t a is intended to be a guide to the reader i n interpreting similar data for the tests to be discussed later. T h i s will help to avoid tiresome repetition in the presentation of the data. Chapter 5 : 117 5.11 Centrifuge Tests Used in the Verification Study Six different centrifuge tests, one from each series, were used in the T A R A - 3 verification study. T h e y involve two-dimensional (2-D) plane strain and three-dimensional (3-D) models simulating a variety of structures and soil-structure interaction systems. the simple embankments to surface and embedded structures These range from on both dry and saturated sand foundations. T h e surface structures are modeled by mild steel plates and the embedded structures are modeled by a solid piece of a l u m i n u m alloy. A summary of the test series is given in Table 5.3. Detail descriptions of each of the models are presented in chapter 7 and chapter 8 along with the T A R A - 3 analyses. Table 5.3 Centrifuge Test Summary Series Model Description Foundation LDOl 2-D Embankment Dry LD02 2-D Surface Structure Dry LD04 2-D Surface Structure Saturated R S S 110 2-D E m b e d d e d Structure Dry R S S 111 2-D E m b e d d e d Structure Saturated R S S 90 3-D E m b e d d e d Structure Dry CHAPTER 6 SOIL PROPERTIES FOR TARA-3 ANALYSES 6.1 Introduction T h e centrifuge model tests used in the verification of T A R A - 3 were conducted over a three year period from 1983 to 1986. In 1983 the technology for conducting seismic tests on large scale models was in its infancy and techniques were not available for measuring the in-situ properties of the sand models in flight. Not until 1987 (Finn and G o h l , 1987) was a technique developed for measuring reliably the in-situ shear modulus. T h i s technique involves measuring shear wave velocities using piezoceramic bender elements in the sand model while the model is in flight. Therefore, the soil properties required for the T A R A - 3 using other procedures. three robust parameters, analyses have to be derived It is fortunate that the constitutive model in T A R A - 3 is based on shear modulus, bulk modulus and shear strength which can be related to the relative density and effective stresses in the model. Hence the required soil properties were estimated on the basis of the relative density of the model. A s outlined in section 5.6, the gross density of a model was determined from its geometry and weight and the relative density was then calculated from a knowledge of the density at m i n i m u m and m a x i m u m void ratios of the sand. T h i s procedure worked well for dry models which could be constructed to defined geometry. 118 A l l sand placed in the model stayed within Chapter 6 : 119 the boundaries of the model thus ensuring that an accurate model weight could be obtained. However, in the case of saturated models where the sand-oil mixture was pluviated through silicon oil this procedure was less accurate. the oil resulting in an overestimation mixture sometimes Some fines remained in suspension in of model weight. D u r i n g construction the sand-oil migrated outside the boundary of the model proper and ended up in the overflow trough a n d other areas of the container. In these circumstances it was difficult to calculate accurate densities. A s the test series progressed, model construction technique for the construction of saturated improved with experience. A new models, referred to as M e t h o d 2 in this thesis, was developed in 1985 and in later tests such as the R S S 111 series the relative density can be determined as accurately as in dry tests. T h e technology of model construction h a d important implications also for the homogeneity of the model. very homogeneous. Test d a t a show that the models constructed using M e t h o d 2 were T h e earlier models show evidence of non-homogeneity. T h i s does not appear to affect very much parameters such as acceleration which depend strongly on average global properties but can have a marked effect on porewater pressures which are very strongly affected by purely local conditions. These effects are discussed fully later when reviewing the test data. 6.2 Shear and Bulk Moduli Parameters A s mentioned previously, the initial in-situ shear modulus is related directly to the relative density a n d effective stresses. by Seed and Idriss T h i s was calculated using the expression proposed (1970) as given in equation (2.4). T h e value of shear modulus pa- rameter, Kimaxi was obtained using the expression proposed by Byrne (1981) as shown in equation (2.5). F i n n and G o h l (1987) showed that the correlations proposed by Seed and Chapter 6 : 120 Idriss (1970) and H a r d i n and Drnevich (1972) give very good estimates of shear moduli for centrifuge modeling in flight by comparing estimates by these procedures with moduli measured directly in-situ using their new technique. T h e bulk modulus parameter, K , for the static analysis was obtained using the expresb sion reported by Byrne and C h e u n g (1984). T h i s takes the form 19 0.0655 - (6.1) 0.0535 log where, D r = relative density expressed in percentage. For dynamic analysis, a value of K five times the value given by equation (6.1) was used b for saturated portions. T h e higher value is necessary to simulate the undrained conditions during the earthquake loadings. Parametric studies with different higher values of Kt, some as high as twenty times of that given by equation (6.1), indicate that the responses not affected significantly. T h e bulk modulus exponent, were m , was selected to be equal to 0.40. T h e effective angle of internal friction of the Leighton B u z z a r d sand was determined by both triaxial tests (Eyton 1982) and simple shear tests and over the range of density used in the model tests was taken to be around 35 degrees. 6.3 Liquefaction Resistance Curve T h e liquefaction resistance of the Leighton B u z z a r d sand was determined using the University of British C o l u m b i a simple shear device. T h e liquefaction resistance curve de- termined for a relative density of D = 65% is shown in F i g . 6.1. Resistance r at other Chapter F i g . 6.1 Liquefaction Resistance Curve 0 Chapter 6 : 122 relative densities were estimated on the assumption of a linear dependence on relative density as shown by Seed and Lee (1966). T h e volume change constants C\ to C and the rebound constants in the M a r t i n - F i n n 4 Seed porewater (Yogendrakumar pressure model were determined by regression and F i n n analysis using SIMCYC-2 1984) to result i n a close fit between the measured dicted liquefaction resistance curves. a n d pre- Table 6.1 gives the set of volume change and rebound constants for different relative densities used in the tests. Table 6.1 Porewater Pressure Model Constants Constants D r = 75% D r = 64% D r = 52% 0.820 0.960 1.00 0.790 0.430 0.40 0.450 0.161 0.161 0.730 0.376 0.376 m 0.430 0.430 0.430 n 0.620 0.620 0.620 K 0.006 0.007 0.007 Ci c 2 c r 4 Chapter 6 : 123 6.4 Structural Properties T h e structural response is assumed to be linearly elastic in the analyses and therefore the structure was modeled using linear elastic elements. T h e assumption of linear elastic behavior is justifiable, because of the very small strains that develop in the structure during the earthquake. The properties selected for a l u m i n u m alloy (Dural) and mild steel are shown in Table 6.2. Table 6.2 Structural Properties. Property Aluminum M i l d Steel Specific Gravity 2.83 7.80 27.8 76.5 U n i t Weight (kN/m ) 3 Shear M o d u l u s (kPa) 2.4 x 1 0 7 7.6 x 10 7 Bulk M o d u l u s (kPa) 6.7 x 1 0 7 1.7 x 10 8 Poisson 0.34 0.30 D a m p i n g Coefficient, a 0.0 0.0 D a m p i n g Coefficient, /? 0.005 0.005 Ratio 6.5 Slip Element Properties Experimental studies by many researchers (Tatsuoka el al 1985; Uesugi et al 1986; Uesugi et al 1987) on the behavior of sand-structure interface under cyclic loading reveal that the interface behavior is essentially of the rigid-perfectly plastic type. Therefore, the Chapter 6 : 124 high unit stiffnesses deduced from the test results involving sand and steel surfaces (Tatsuoka et al 1985) were used in the T A R A - 3 analyses. These values are considered appropriate for the steel structures used in the centrifuge studies. T h e properties for the slip element are tabulated in Table 6.3. Table 6.3 Slip Element Properties. Property Slip Element U n i t N o r m a l Stiffness U n i t Shear Stiffness Friction Angle, 4>' Cohesion, c s s (kPa/m) (kPa/m) 6.3 X 10 5 6.3 x 10 5 10.0 0.0 CHAPTER 7 VERIFICATION BASED ON DRY MODEL TESTS 7.1 Verification Study Based on Test Series L D O l 7.1.1 Centrifuge Model in Test Series L D O l A schematic view of a 2 - D plane strain model embankment is shown in F i g . 7.1. T h e embankment is 116mm high and has a flat crest 239mm wide and a base 732mm wide. T h e length of the model in the direction perpendicular to the plane of shaking is 481 m m . The model was constructed using Leighton B u z z a r d B . S . S 120/200 sand by the dry construction method outlined in section 5.5.1. T h e estimated relative density of the sand is about 50 ± 10%. The model was shaken by an earthquake, E Q 1 , while under a nominal centrifugal acceleration of 80g. T h e model, therefore, corresponds to a prototype embankment 9.2m high with a crest and base width of 18.9m and 58.5m respectively. The responses of the model embankment to the simulated earthquake were measured by the instruments located in the model as shown in F i g . 7.2. A l l accelerometers horizontal acceleration responses. measured Accelerometers A C C 1544 a n d A C C 1486 were not 125 Chapter Fig. 7.1 Schematic of a Model Embankment 7: 126 { of box 1 Rough concrete base Fig. 7.2 A>.i.L + L fixed to concrete Instrumented Model Embankment in Test Series L D O l •a Chapter 7 : 128 activated for this particular test. 7.1.2 Model Response in Test L D O l T h e model responses recorded during the test are shown in F i g . 7.3 at model scale. A C C 1244 fixed to the concrete base measured the acceleration input to the model. T h e peak amplitude of the input is 10.1% of the centrifuge acceleration and it occurs at approximately 50.0 milliseconds. A C C 1932 the test (Dean a n d Lee, 1984) was reported to have been functioning incorrectly and will therefore be ignored. during A C C 734 malfunctioned during this and subsequent test series a n d data from it are not used (Dean and Lee 1984). Accelerometers A C C 988, A C C 1225, A C C 1908, A C C 1928 and A C C 2036 show responses that are distinctly different in frequency content from the other accelerometer the input motion. responses and These transducers were located in the upper part of the embankment and therefore they responded differently from those located in the lower part. A C C 1225 and A C C 988 were located at the same elevation (Z=90 mm) but at different positions in the direction perpendicular to the plane of shaking. A C C 1225 was near the centre section (Y=10 mm) a n d A C C 988 was near the rear window ( Y = -200 mm). T h e y both show responses that are somewhat different in peak amplitudes and in frequency content. A C C 988 was close to the window and end effects might have distorted the response and hence the record has to be interpreted with caution. T h e input motion as measured by A C C 1244 is shown in F i g . 7.4 along with the baseline corrected motion at prototype scale. T h e baseline corrected A C C 1244 record was used as the input motion for the T A R A - 3 analysis. It has about 10 roughly sinusoidal pulses of horizon al base shaking with a predominant frequency of about 1.50 H z . It consists of 5 cycles of more-or-less constant amplitude shaking followed by 2 big cycles of shaking. T h e amplitude gradually decreases in the last three 3 cycles and significant shaking ceases at Chapter 7: 129 millisecs 0 10. 7 X -9.97 -9.97 = 11.2 " 73 K.O , A A A A A. A A, A. A wy v y V/ y v v ^ V - A AA VV v A A A, A. A J. X , 150 - u A A A A .VV TV A JV ^AL IA. Ar»~M — ty" * w > W " y v V^jv™» . v - "*r\r* ^y* Y ^yi' ~ X 100 1 ! ~ 11.3 X 50 v A v A ~ - - - - 20.0 - X/div - ACC 1258 20.0 X/div V .A j r ACC1187 _ - .. • .... 20.0 ^ , p f l i .^v. . V ) A/UIftfViAAI ftM.nnrtft/Vi fl^li ft*i **f* v\Ju v y v •tyv^yv VUV l / v v\(V "\Jy V T v y v r r '<>•«• - • • ACC2033 i - n X/div - ACC2036 50.0 ACC 1928 -16.9 t •* •• - 13.0 " • -11.7 ^ - X/div 10.9 " - ACC1908 -11.2 =, w X X/div 20.0 X 13. 1 X vyv* »\rv fywyw ntf »\A/"yv 20.0 vyjv™»v««-»» - ~ — — . . — " - X/div - ACC 1225 20.0 -U.2 X/div ^ 10.5 -10. 1 •• - i " . LHJ'-'Al. W *r V\f \ - i/V V l l i/li A. «A,1 v V^v Yy* yiiv vy UT y "If _/lA r *jYUl.f. V — . . . — — • • * * • • - ACC988 20.0 - X/div ACC 1932 5.97 " • -s.y; = - X/div • ACC 1938 5>. i9 X 10.0 y \ A. A, A. A A. A A /v A r- - 20.0 X/div ro. ? X X. 93 X -C36 «— \/ v v- *V V \/ v - A / \ / \ / \ / ^ / \ f \ [ \ f \ / \ / \ " „ -9.50 = , J\ A A A A A \ A A. r vv vv y y yv w - A —i0 1 50 • 1 100 millisecs Scales Fig. 7.3 : - X/div - ACC73i 10.0 = - ACC 1583 20.0 ^ - 10. 1 X - ^•^ A A A . A A A A A Model Model Response in Test L D O l / E Q l - - X/div - ACC12U 20.0 - 150 X/div Chapter 7: Max.Val. 1 1 1 1 1 1 1 1 1 Not Corrected 10.2 , A A H A A IV itV VJLVA ~ Vv / VVV V • A f\ 1 A ft I v 1 1 1 1—: 1 1 1 1 -8.48 1 Baseline Corrected VA A A A A A v v vw v y v v v 1 ,r A fi II 9.94 k I IL -8.69 0 i ]0 i 2.0 i i i t i i 3.0 4.0 5.0 6.0 7.0 8.0 Time Fig. 7.4 i 9.0 (sec) Input Motion for Test LDOl/EQl 1 0 .0 Chapter around 7.50 seconds. 7 : 131 T h e relative density of the sand was taken as 50% for the T A R A - 3 analysis. 7.1.3 Comparison of Acceleration Responses of Test L D O l / E Q l T h e computed and measured accelerations near the base at the locations of A C C 1583, ACC 1258, A C C 1938 and A C C 2033 are shown in Figs. 7.5 to 7.8 respectively. In each of these locations, the responses are very similar i n frequency content, each corresponding to the frequency of the input motion. T h e peak amplitudes and the variation of ampli- tudes with time agree very closely. T h e computed and measured peak amplitudes at these locations are tabulated in Table 7.1 and they differ only by a few percent. A C C 1487 and A C C 1908 were located at half way between the crest a n d base with A C C 1487 closer to the left hand side slope. T h e comparison of accelerations is shown in Figs. accelerations 7.9 a n d 7.10. T h e agreement at these locations between the measured and computed in terms of frequency content a n d amplitude variation with time is good at both locations. Figs. 7.11 to 7.14 show comparison of acceleration responses i n the upper part of the embankment at the locations of A C C 1928, A C C 2036, A C C 988 and A C C 1225 respectively. T h e overall agreement is good except at the locations of A C C 988 and A C C 2036. A s pointed out earlier, A C C 988 may have been affected by end-effects because of its proximity to the end a n d therefore it is not surprising to see differences between the measured and computed responses. Chapter 7 Max.Val. •r 1 i •— 11 r" T~ i • m T —r- * • • Recorded Response - k A « 9.29 A f\ ft ft „ „ A A A /I A A A A A „ . . J V v v'Vv / VVVV ~ - • A 1 ' v i i \i y i v i "i -8.03 i i i i i , Computed Response - l\ A 9.95 M-A A A A A A V v v/V i/WW •• " A IV A K y 1 .0 1.0 I A 1 f If " ..1 2.0 1 3.0 y • M y 1 4.0 -8.62 1 1 5.0 Time 7.5 n H 6.0 • 7.0 1 8.0 1 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1583 in Test LDOl/EQl Chapter 7 : Max.Val. 20 c o 10 u <v Cu c 0 o 0 1 1 1 • r T i 1 T Recorded Response 0 A AA l i n n An AA7\ A \ft/I A A 0 it 1 u -10 0 o -20 0 20 0 11.2 i * VV^VV /yyvy ' ' ii if. V r *J CO • T A If --- - r- -9.97 CJ <o i i i i i i i i i , _ cCU o Computed Response 10.0 10 0 —^ p< v C 0 0 O V AV VA VA YA VA \\ A \A AVAV V w CO i~ cu -10 0 -20 0 0 .0 " r V v > -8.54 cu o <o 1 1 1 1 1 I 1 1 1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Time Fig. 7.6 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1258 in Test LDOl/EQl 133 Chapter Max.Val. 20 -1 1 1 1 CJ }-> 1 r- Recorded Response *J C OJ 7 : 9.50 10 OJ ^—' c 0 _o '<-> cu -8.42 -10 % o o -20 < 20 OJ o Computed Response 10.3 10 OJ a, G o CO OJ 0 -9.14 -10 »—1 OJ CJ o -20 < .0 Fig. 1.0 7.7 2.0 3.0 4.0 5.0 6.0 Time (sec) 7.0 8.0 9.0 10.0 C o m p u t e d and Measured Accelerations at the Location of A C C 1938 in Test L D O l / E Q l 134 Chapter 7.8 7 : C o m p u t e d and Measured Accelerations at the Location of A C C 2033 in Test LDOl/EQl Chapter 7 : Max.Val. W> 20.0 W> 20.0 a OJ o 10.0 c o 0.0 03 •10.0 OJ o o -20 < .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.9 6.0 7.0 8.0 C o m p u t e d and Measured Accelerations of A C C 1487 in Test 9.0 10.0 (sec) LDOl/EQl at the Location 136 Chapter 7 : Max.Va). Mi 20.0 W) 20.0 O 10 a o 0.0 *•—* a u •10 <v a> o -20 .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.10 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1908 in Test LDOl/EQl 137 Chapter 7 : Max.Val. <aO 20.0 Recorded Response 13.0 11.7 M 20.0 V o 10.0 a 0.0 u <u o • I—I CO 10.0 U. ' OJ o o .-20 .0 < F i g . 7.11 1.0 2.0 3.0 4.0 5.0 Time 6.0 7.0 8.0 C o m p u t e d and Measured Accelerations of A C C 1928 in Test 9.0 10.0 (sec) LDOl/EQl at the Location 138 Chapter 7 : Max.Val. W> 20.0 W> 20.0 1.0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.12 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 2036 in Test LDOl/EQl 139 Chapter 7 Max.Val. 20 0 •*-> C <v 10 0 o S-, OJ 0, 0 0 c o rt-10 0 OJ OJ o CJ < -20 0 „ be 20 0 OJ o 10 0 OJ c 0 .0 1.0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.13 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations of A C C 988 in Test LDOl/EQl at the Location Chapter 7: Max.Val. .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.14 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1225 in Test LDOl/EQl Chapter 7 : 142 Table 7.1 Comparison of Peak Acceleration in Test L D O l / E Q l Transducer No. Measured Computed (%g) (%g) A C C 1583 9.3 9.9 A C C 1258 11.2 10.0 A C C 1938 9.5 10.3 A C C 2033 11.2 10.8 A C C 1487 10.7 12.1 A C C 1908 11.3 11.1 A C C 1928 13.0 12.7 A C C 2036 16.9 12.8 A C C 988 10.5 13.0 A C C 1225 14.2 13.2 The stress strain response at two locations near A C C 1583 and near A C C 1932 are shown in F i g . 7.15 and F i g . 7.16 respectively. T h e y are drawn to the same scale and hence they offer direct comparison of stress strain response at representative locations in the lower and upper part of the embankment. T h e responses are not strongly nonlinear. However, the hysteretic behavior at location near A C C 1583 is somewhat more pronounced than at location A C C 1932. 7.1.4 Comparison of Settlements in Test L D O l / E Q l T h e measured and computed settlements at the locations of L V D T 46999 a n d L V D T 13893 are tabulated in Table 7.2. T h e values quoted in the table are at prototype scale. A t Chapter o O ™ t -0.020 Shear Fig. 7.15 0.000 Strain 0.020 (percent) C o m p u t e d Shear Stress-Strain Response Near the Location of A C C 1583 in Test L D O l / E Q l 7 : 143 Chapter Fig. 7.16 7 : C o m p u t e d Shear Stress-Strain Response Near the Location of A C C 1932 in Test L D O l / E Q l 144 Chapter 7 145 both locations, the settlements are predicted satisfactorily. Table 7.2 Comparison of Settlements in Test L D O l / E Q l LVDT Measured Computed No. (mm) (mm) 46999 10.2 9.0 13893 10.8 9.4 The vertical settlements of the embankment are also shown in F i g . 7.17. T h e dotted lines show the initial shape a n d the solid lines show the computed post-earthquake taking only the vertical settlements into account. shape T h e circular points indicate the locations of the tips of L V D T s and the triangular points show the final positions. It is clear that the agreement between the measured a n d computed vertical settlements is very good. Vertical settlements could not be measured satisfactorily on the slopes of the embank- ment due to the sliding of material during shaking, the effects of wind erosion a n d the difficulties in setting up the L V D T properly on the slope. 7.2 Verification Study Based on Test Series L D 0 2 7.2.1 Centrifuge Model in Test Series L D 0 2 A schematic view of a 2 - D plane strain soil-structure model is shown in F i g . 7.18. T h e embankment was constructed by dry method described in section 5.5.1 using Leighton B u z z a r d B . S . S 120/200 sand. T h e estimated relative density of the sand is 71 ± 8%. T h e embankment is 105mm high and has a flat crest 230mm wide a n d a base 720mm wide. T h e LVDT 46999 LVDT 13893 GEO.SCALE o DISPLACEMENT Settlement Pattern in Test 0 ' — LDOl/EQl Chapter 7 Surface Structure D i r e c t i o n of S h a k i Scale in m m F i g . 7.18 Schematic of a M o d e l E m b a n k m e n t W i t h Surface Structure Chapter 7 : 148 length of the embankment perpendicular to the direction of shaking is 480 m m . T h e surface structure consisted of three mild steel plates, each of which is 15mm thick and 65mm wide. T h e steel plates were placed end to end along the centerline of the crest. T h e two end pieces were each 4 0 m m long and the central piece was 385mm long. T h e model experienced a nominal centrifugal acceleration of 80g. T h e model, therefore, simulated a prototype embankment approximately 8.8m high w i t h crest width and base width of 18.4m and 57.6m respectively and a structure approximately 1.2m high and 5.2m wide. T h e complete instrumentation of the model is shown in F i g . 7.19. T h e transduc- ers are distributed in the model in order to obtain a comprehensive picture of the model responses. A C C 1932 and A C C 1938 measured vertical accelerations while other accelerom- eters measured horizontal accelerations. A C C 1544 mounted on the concrete base recorded the acceleration input to the model. 7.2.2 Model Response in Test L D 0 2 For the first three earthquakes ( E Q 1 to E Q 3 ) , A C C 1544 was not working. earthquake motions are of small amplitudes with peak values less than 5%g. These T h e response to these earthquakes was not analysed. O n l y the response to the fourth earthquake (EQ4) which has a peak amplitude of 12.4%g was analysed using T A R A - 3 . T h e output of smoothed data for test L D 0 2 / E Q 4 is shown in F i g . 7.20. T h e number of channels in the d a t a acquistion system was limited and less t h a n the number of transducers. Therefore, not all transducers could be recorded in each test in the sequence. this particular test, only the accelerometers activated. whose responses were given in F i g . 7.20 For were 5 •^3 CO Chapter 7 : SO 150 millisecs 12.5 -11.6 -- X/div - ACC 1932 10.0 i/div 50.0 16.8 T ACC 1225 20.0 X X/div -15.0 8.09 X 70.0 -7.57 X/div 16.6 - -12.9 + 20.0 20.0 X j- -11.9 X/div ^ACC2033 16.2 -j j X -12.5 -17.1. X/div ACC1928 16.9 19.1 ACC 1908 v v v w V V V V v v •j-j - 13.3 20.0 X/tf/Y ACC 1258 50.0 20.0 X X/div -10.5 - - /CC748<f 11.3 X 20.0 -10.1 X/div i 12. X -10.8 vvVwvVVvv^ 50 20.0 100 millisecs Scales Fig. 7.20 Model : Model Response in Test L D 0 2 / E Q 4 Chapter 7 : 151 A s noted earlier in section 7.1.2, A C C 734 malfunctioned during this test series and data from it is ignored. T h e records A C C 1225, A C C 1258, A C C 1932 and A C C 1938 should also be viewed w i t h caution as they show considerable high frequency response during shaking and even after 90 milliseconds when the earthquake motion had already ceased. A possible explanation for this noisy response is given in the next section. T h e peak horizontal acceleration 12.4%g. of the input motion as recorded by A C C 1544 is F i g . 7.21 shows the input motion along w i t h the base corrected motion. B o t h of these records are smoothed once and are shown at prototype scale. T h e y show no apparent differences. T h e Fourier spectrum of the base corrected A C C 1544 record is shown in F i g . 7.22. It has a predominant frequency of 1.5 H z . It also contains relatively small energy at higher frequencies, for instance, at 4.5 H z a n d 7.5 H z . Except for a small drop in peak values, the acceleration transmitted to the soil near the base as given by A C C 1486 is similar to that of the input motion. There is an increase in peak acceleration values as the structure is approached. Close to the base of the surface structure, the peak acceleration recorded by A C C 2033 is 16.9%g. measured at the top of the structure ( A C C 1583) is 18.7%g. T h e peak These indicate that there is a steady amplification of the response as the motion is transmitted model to the top of the surface ACC from the base of the structure. 1932 and A C C 1938 were placed to measure edges of the steel plate. acceleration vertical accelerations T h e y show quite a different type of response content higher than that of the other records. at opposite with frequency T h e reason for this is explained later. A C C 2033 and A C C 1928 were located at same elevation (Z = 90mm) but in different vertical planes, 60mm (model scale) apart. These records are almost identical. This observation suggests that the model behaved i n a plane strain mode. T h e prototype of the model was analysed using T A R A - 3 with base corrected A C C 1544 Chapter 7 : Max.Val. 'bio 20 —> C OJ o 10 1) a, c 0 ra •o —< OJ -10 0) o CJ < -20 20 o 10 OJ c 0 o —' CO -10 OJ 0) o o -20 < .0 2.0 3.0 4.0 5.0 Time F i g . 7.21 6.0 7.0 8.0 9.0 (sec) Input M o t i o n for Test L D 0 2 / E Q 4 10.0 152 2000 £ a. E -< 1000[ ~i—i—i—i—|—i—i—r~i—|—i—i—i—i—p 5 10 15 T—i—pr—i—II 20 [ l TT 25 i^ 30 T T " r I—|—I—I—I—I—|—I—I—i—i—|—i—i—i—r 35 40 45 Frequency (Hz) Fig. 7.22 Fourier Spectrum of A C C 1544 50 "a Record in Test L D 0 2 / E Q 4 Cn CO Chapter 7 : 154 record as the input. T h e relative density of the sand was taken at 71%. T w o analyses were conducted: one with slip elements between the soil and structure and the other without slip elements. C o m p u t e d responses were compared with the corresponding measured responses at prototype scale in the following section. 7.2.3 Comparison of Acceleration Responses of Test LD02/EQ4 Comparison between the measured and computed horizontal acceleration responses at locations of A C C Fig. 1486, A C C 1487, A C C 2033 and A C C 1928 is shown in F i g . 7.23 through 7.26 respectively. These accelerometers are located in the foundation soil along the centerline of the model, with A C C 1486 near the concrete base, a n d A C C 1487 midway between the base of the surface structure and the concrete base, and A C C 2033 and A C C 1928 near the base of the surface structure. T h e magnitude a n d the frequency content of the computed responses are similar to the corresponding measured responses. T h e comparisons in terms of peak acceleration values shown in Table 7.3 are quite good. Table 7.3 indicates that the computed responses with a n d without slip elements show little differences. However, predictions are generally better when the slip elements are used. It appears, however, that very little slip occurs during shaking. Figs. 7.27 to 7.29 show the comparison of measured acceleration responses with the computed responses at the locations A C C 1908, A C C 1258 and A C C 1225 respectively. A t location A C C 1908, agreement between the measured and computed responses in terms of magnitude and frequency content, as shown in F i g . 7.27, is good. T h e difference i n peak acceleration values with and without slip elements is not very significant although again prediction with slip elements is better. Chapter 7 Max.Val. M 20.0 M ?0.0 3.0 4.0 5.0 Time F i g . 7.23 6.0 7.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1486 in Test L D 0 2 / E Q 4 Chapter 7 : Max.Val. 20.0 M C D O U 10.0 OJ Q. a o 0.0 CO 10.0 "3 o o < •20.0 20.0 W> c o u <u a. c o re a; QJ O O < 10.0 0.0 10.0 •20. F i g . 7.24 C o m p u t e d and Measured Accelerations of A C C 1487 in Test L D 0 2 / E Q 4 at the Location 156 Chapter 7 Max.Val. w> 20.0 C y 10.0 a. c o 0.0 -10.0 u o -20.0 < W> 20.0 a CD CJ I* <U a. a 10.0 0.0 o CO 1. -10.0 "53 o o < 20.0 W> c 1) o 1- a. 20.0 10.0 1) c 0.0 « u 10.0 V o o < -20.t Fig. 7.25 C o m p u t e d and Measured Accelerations of A C C 2033 i n T e s t LD02/EQ4 at t h e L o c a t i o n Chapter M 7 : Max.Val. 20.0 fi 0) o 10.0 C o S-lo.o oi o W -20.0 M 20.0 C o 10.0 c 0.0 o *-> CO 10.0 o o 20.0 V < F i g . 7.26 C o m p u t e d and Measured Accelerations of A C C 1928 in Test LDQ2/EQ4 at, the Location 158 Chapter Max.Val. 20.0 M Recorded C o 10.0 a 0.0 7 : Response 16.8 C O u- •12.7 V 10.0 «—t V o o . <: 20.0 M c 10.0 cu O t. 0J a c 20.0 0.0 o C O >-, 0) o o < W> -10.0 -20.0 20.0 c u o a c o CS u V o u < 10.0 0 0 10.0 -20. F i g . 7.27 C o m p u t e d and Measured Accelerations at the Location of A C C 1908 in Test LDQ2/EQ4 159 Chapter 7 : Max.Val. be 20 0 — J C V V u V 10 0 a. c _o ZCO> >- 0 0 -10 0 u -20 0 < f , w 20 0 c o 10 0 u V a c 0 o o —' CD S- -10 l> 0 V O o -20 0 < bD 20 0 <v V 10 .0 u. o a c 0 .0 v —' CO L. -10 With Slip Element Computed Response C 0 V V u o -20 n .0 < a F i g . 7.28 y V» V V 1.0 2.0 3.0 4.0 f if 5.0 y v 13.1 V •12.4 6.0 7.0 8.0 9.0 10.0 Time (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1258 in Test L D 0 2 / E Q 4 Chapter 7 : M Max.Val. 20.0 C QJ U U V c 10.0 0.0 o C O 10.0 0) I—. o o •< -20.0 W> 20.0 C CO 10.0 u u v O. c 0.0 o cs u -10.0 "a! o o < •20.0 M 20.0 c v »V a a. o 0) F i g . 7.29 C o m p u t e d and Measured Accelerations of A C C 1225 in Test L D 0 2 / E Q 4 at the Location Chapter 7 162 Table 7.3 Comparison of Peak Acceleration in Test LD02/EQ4 Transducer No. Measured Computed Computed (%g) (%g) Without slip elements (%g) With slip elements ACC 1486 11.3 12.4 12.5 ACC 1487 13.6 12.8 13.1 A C C 2033 16.6 14.9 15.3 ACC 1928 17.1 14.9 15.3 ACC 1908 16.8 14.9 15.2 ACC 1583 18.4 16.3 17.0 Measured acceleration histories at A C C 1258 and A C C 1225 show higher peak values and more high frequency noise than the computed responses. A C C 1487 together with ACC 1908 may provide some indication as to whether A C C 1258 record is anomalous or not. A C C 1487 is at the same elevation as A C C 1258 distance away from the centerline as A C C 1258. high frequency characteristics the A C C 1258 and A C C 1908 is at the same Clearly both of them do not show the as seen in the A C C 1258 record. record are in excess of those in A C C 1487. Also the peak values in Therefore it is apparent that the A C C 1258 record contains responses other than the motions resulting from shear wave transmission from the base. T h e fact that A C C 1258 has recorded significant responses after the earthquake supports the aforementioned notion. T h e same conclusion may be extended to the A C C 1225 record. In centrifuge tests, the measured acceleration responses may usually have components other than those resulting from shear wave transmission f r o m the base. These are motions due to container vibrations and are transmitted to soil Chapter 7 : through the side walls and the top of the container. 163 These motions are usually of the high frequency type and contain negligible energy. O n e of the other possible sources for the spiky high frequecy response is the tension in the transducer However, T A R A - 3 analysis takes into account input. leads as discussed in section 5.7.1. only the motions resulting from the base Hence, it is not surprising to see differences between the computed and measured responses. Despite this, the comparison at location A C C 1225 is good. Figs 7.30 to 7.32 show the comparison of measured acceleration responses to that of the computed responses at locations A C C 1583, A C C 1932 and A C C 1938 respectively. accelerometers are mounted on the structure These in such a way that A C C 1583 measures the horizontal acceleration at the middle of the structure and A C C 1932 and A C C 1938 measure the vertical (rocking) accelerations at opposite edges of the structure. It is apparent from the measured acceleration responses that the frequency content of the vertical accelerations is very different from that of the horizontal acceleration at the same level in the structure. T h e frequency content of the horizontal acceleration ( A C C 1583) is similar to that of the input motion while the frequency content of vertical accelerations ( A C C 1932 a n d A C C 1938) is m u c h higher than that of the input motion ( A C C 1544). T h i s phenomenon is reproduced in the corresponding computed acceleration responses. accelerations T h e high frequency content i n vertical is due to the fact that the foundation soils are much stiffer under the normal compressive stresses due to rocking than under the shear stresses induced by the horizontal accelerations. A s shown in F i g . 7.30 and Table 7.3, the acceleration response at the location A C C 1583 is predicted satisfactorily. A s noted earlier, both A C C 1932 a n d A C C 1938 have recorded significant responses even after the earthquake motion ceased. A s i n the case of A C C 1258 and A C C 1225, this casts doubts as to whether or not b o t h A C C 1932 and A C C 1938 were measuring only the motions resulting from the base input. T h e vertical accelerations appear to be relatively Chapter 7 : M Max.Val. 2 0 . 0 C u 10.0 cu a c C O u 10.0 cu "cu u o •20.0 < M 2 0 . 0 C cu o 10.0 u cu a. c 0.0 o CO- 1 0 . 0 u V o V 20.0 < F i g . 7.30 C o m p u t e d and Measured Accelerations at the Location of A C C 1583 in Test LDQ2/EQ4 Chapter 7 Mi Max.Val. 20.0 Recorded Response a v u u 10.0 3: 0.0 OJ c o at I, -10.0 o o < •20.0 M 20.0 OJ C 0J u Computed Response Without Slip Element Computed Response With Slip Element 10.0 OJ a. 0.0 e o -10.0 - rH a) OJ "3 •20.0 <W> 20.0 CJ O c CJ CJ 10.0 u OJ 0.0 fy*>> c o CO 10.0 u• CJ »—I 20. CJ u o . Fis. .0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 T i m e (sec) 7.31 Computed and Measured Accelerations at the Location of A C C 1932 in Test L D Q 2 / E Q 4 Chapter 7 : M 20.0 C 01 u u 10.0 Max.Val. OJ a. c o 0.0 CS u -10.0 V % V o •20.0 < M c 20.0 o 10 0 a c 0 0 OJ Computed Response Computed Response Without Slip Element o —' CO-10 L. CD "v O CJ -20 < ^ 0cJ 0 0 20 0 O «-, 10 0 c 0 0 a > a. C O t. -10 u o -20. < With Slip Element 0 .0 F i g . 7.32 1.0 2.0 3.0 4.0 5.0 6.0 T i m e (sec) 7.0 6.0 9.0 10.0 C o m p u t e d and Measured Accelerations at the Location of A C C 1938 in Test L D Q 2 / E Q 4 Chapter 7 : more sensitive to the presence of high frequency noise than the horizontal 167 accelerations. A C C 1932 a n d A C C 1938, which are located symmetrically about the centerline, are supposed to record almost similar histories showing a phase lag of 180 degrees. both accelerometers are measuring very different peak values. Fig. It is clear that A s seen in F i g . 7.31 and 7.32, the peak values measured by A C C 1932 a n d A C C 1938 are 7.55%g a n d 12.5%g respectively. Recall that the input motion has high energy at 1.5 H z and relatively low energy at 4.5 and 7.5 H z . F i g . 7.33 and F i g . 7.34 show the Fourier spectrum of A C C 1932 and A C C 1938 records. It is seen that both have significant energy at frequencies higher than 7.5 H z , which may be primarily due to noise. Therefore, in an attempt to isolate the noise, these records were passed through a low pass 8.0 H z filter whereby at frequencies higher t h a n 8.0 H z were removed. filtered responses with c o m p u t e d responses respectively. responses F i g . 7.35 a n d F i g . 7.36 compare the at the locations of A C C 1932 a n d A C C 1938 T h e comparison in terms of frequecy contents is fairly good at b o t h locations but the peak values are somewhat different to each other. 7.2.4 Comparison of Settlements in Test LD02/EQ4 The computed a n d measured vertical settlements at the locations of L V D T 48406, L V D T 48407 and L V D T 46997 are tabulated at prototype scale in Table 7.4. T h e computed values are for the analysis with slip elements. L V D T 48406 and L V D T 48407 were mounted on opposite edges of the structure, and L V D T 46997 was located on the flat crest of the sand berm. A t all three locations the comparison is good between computed a n d measured settlements. T h e complete settlement pattern as computed by T A R A - 3 is shown in F i g . 7.37. T h e dotted lines show the initial shape a n d the solid lines show the computed shape taking only vertical settlements into account. post-earthquake T h e circular points indicate the initial 800 oo Fig. 7.34 Fourier Spectrum of A C C 1938 Record in Test L D Q 2 / E Q 4 Chapter 7 M v u u y a c o CO l- Max.Val. 20.0 Filtered Response 10.0 NriM/'lfinAAAfinNiyMMM^' 0.0 «v 10.0 0> "3 y •20.0 y <: M c o u 20.0 Computed Response Without Slip Element Computed Response With Slip Element 10.0 V a c 0.0 o cs - 1 0 . 0 1M 1) "3 y y < M c V o u y a CO 20.C 20.0 10.0 o.c -10.0 V "3 y y < •20. .0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Time (sec) Fig. 7.35 C o m p u t e d and Filtered Accelerations at the Location of A C C 1932 in Test L D Q 2 / E Q 4 Chapter 7 M Max.Val. 20.0 e OJ o In Filtered Response 6.41 10.0 CI c c 0.0 o CO cj -7.39 10.0 "S o CJ •20.0 < M 20.0 CJ CJ u Without Slip Element Computed Response C 10.0 3.15 a. CJ 0.0 CO -3.29 u -10.0 CJ ~ a o 20.0 < M c OJ 20.0 With Slip Element Computed Response 10.0 o «. 4.52 CJ & C o 0.0 C8 - 1 0 . 0 0J -4.39 o CJ -20. .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.36 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Filtered Accelerations at the Location of A C C 1938 in Test L D Q 2 / E Q 4 ^1 to Chapter 7 : 173 locations of the tips of the L V D T s and the triangular points show the final positions. It is clear that that the agreement between the computed a n d the measured vertical settlements is good. Table 7.4 Comparison of Settlements in Test LD02/EQ4 Transducer No. Measured (mm) Computed (mm) 46997 4.8 5.0 48407 5.3 6.9 48406 5.3 6.3 A s noted earlier, the vertical settlements could not be measured satisfactorily on the slopes due to sliding of materials during shaking, the effects of wind erosion and the difficulties in setting up the L V D T properly on the slopes. 7.3 Verification Study Based on Test Series RSS110 7.3.1 Centrifuge Model in Test Series RSS110 A schematic view of a 2 - D plane strain model in which the structure is embedded in the soil is shown in F i g . 7.38. T h e embankment was constructed by dry method described in section 5.5.1 using Leighton B u z z a r d B . S . S 52/100 sand. T h e estimated relative density of the sand is 64%. T h e sand foundation is 110mm high a n d has a base 900mm wide. T h e side slopes are at 2.2:1. shaking is 480mm. T h e length of sand foundation perpendicular to the direction of Chapter 7 : Embedded F i g . 7.38 Schematic of a Model Embankment W i t h E m b e d d e d Structure 174 Chapter 7 : 175 T h e heavy structure is made from a solid piece of aluminum alloy and has dimensions 105mm wide by 108mm high in the plane of shaking. plane of shaking is 470mm. foundation. T h e structure T h e length perpendicular to the is embedded to a depth of 25mm in the sand Coarse sand was glued to the base of the structure to prevent slip between structure and sand. D u r i n g the test the model experienced a nominal centrifugal acceleration of 80g. The model, therefore, simulated a structure approximately 8.64m in height, 12m in width and embedded to a depth 2m in sand foundation. T h e average contact pressure between the structure and sand foundation was approximately 240 k P a . T h e complete instrumentation of the m o d e l is shown in F i g . 7.39. was measured by A C C 3441 ACC 1552 and A C C 1572 mounted to the concrete base. measured vertical accelerations T h e input motion Accelerometers while the other A C C 1925, accelerometers measured horizontal accelerations. 7.3.2 Model Response in Test RSS110 T h e model response to a simulated earthquake E Q 1 is shown in F i g . 7.40. A C C 1925 and A C C 1552, which were located in the sand foundation, show large baseline shifts and they were not used in the study. These shifts may be due to drifts caused by poor earth connection (Steedman 1985). It is also probable that the gauges rotated measure a mixture of vertical and horizontal accelerations. so that they A C C 1572 is also very highly suspect because of the large baseline shift a n d the very noisy response. A l l accelerometer responses contain high frequecy noises a n d therefore they were filtered using a 10 H z low pass filter. T h e input motion measured by A C C 3441 is shown in F i g . 7.41 along with the baseline corrected motion at prototype scale. T h e baseline corrected motion was used as the input IVOTHU) IVDT2US7 : •E3 Scale in i I mm 3Z XCCIfH I l~*jH ACCH7I Fig. 7.39 Instrumented Model in Test Series RSS110 ^4 05 Chapter 7 : millisecs 0 50 i i 100 i 6.69 • ACC1572 - X/d/v - ACC1938 - X/d/v - ACC3i36 X 10.0 -6.68 j 9.89 J\ X -7.71 •> 7.76 " X S.80 V ^ 11.5 - X -11.5 - ; - 8.50 - •A A X -8.93 •< 9.47 X - -8.95 •> 10.9 - -««r «/V v A J\ A / A A A^"\ ^/ \fv vv w v-v Vv ^ r'\ . v y y y v/ V - - - v . v ^ w 20.0 " " ,,yV i/, yV i A IAI«A IAI,,A ' " 1 AVr^S-urA M. A\. >*V X/c//v - ACC 1225 - X/div • ACC3i57 A rV . A A - 20.0 - nft A ; ^ y V V V V V • ACC3i77 20.0 - w ^ - . ^ - — • — »rJ An 1^.1 , X/div ACC 1552 20.0 r • X/div • ACC3i78 20.0 ~ ^ t n - - X/div - 8. 18 -8.30 • - /-v 70.0 .- 20.0 r-v J/ A A A A yV A J — \ y r r v * " N X % ^ \ y r "fyr y ^ " " - u V A - / * ^ X -9.40 U J\ A I A . J A I A iA IA »A IA,LAII u ^ ^ ^ . "S'r y Y | T F V \ | Vyf ¥ V W F!T X -ii.i /A M y\ vy w ^ i/VV ^ ^ v • X/div • ACC3U66 :- 10.0 X/div - - ACC 1925 -5.61 - - X/div 6.81 " • A CC3i 79 ^ - X/div 10. 4 - - ACC3H1 X -8.07 - - X/div 11.7 X 20.0 A A A X -6.83 A A H u / \ ^ _ ._ _ J 10.0 20.0 0 50 100 millisecs Scales Fig. 7.40 M o d e l Response : Model in Test RSS110/EQ1 177 Chapter 7 Max.Val. <M> 2 0 . 0 -i r- 1 1 Not Corrected a r- OJ O 10.3 10.0 oj (X 0.0 c o 2 -IO.O -7.99 OJ "OJ cj 3 -20.0 20.0 W> Baseline Corrected a 0J o u 10.7 10.0 0J 0.0 C o *-> 10.0 CO ^ •7.50 • 0J ID o o < -20. 1.0 1.0 2.0 3.0 4.0 5.0 Time Fig. 7.41 6.0 7.0 8.0 9.0 (sec) Input Motion for Test RSS110/EQ1 10.0 Chapter 7 : 179 for the T A R A - 3 analysis. 7.3.3 Comparison of Acceleration Responses of Test RSS110/EQ1 Figs. 7.42 to 7.44 compare the measured and computed acceleration responses at locations A C C 3479, A C C 3466 a n d A C C 3477 respectively. T h e comparison i n terms frequency content and variation of amplitudes with time is good. T h e comparison of peak accelerations as shown in Table 7.5 is good at these locations. Table 7.5 Comparison of Peak Accelerations in Test RSS110/EQ1 Transducer Measured Computed No. (%g) (%g) ACC 3479 6.41 6.21 ACC 3466 7.10 6.50 ACC 3477 7.06 6.50 ACC 3478 10.6 7.42 ACC 3457 10.5 6.95 ACC 1225 11.6 6.88 ACC 1938 10.1 8.89 ACC 1572 3.79 3.76 ACC 3478 and A C C 3457 were located outside the edge of the structure a n d were placed symmetrically opposite about the centerline of the model. T h e comparison at these locations is shown in F i g . 7.45 and F i g . 7.46 respectively. Except for minor differences, the measured responses at these locations are similar. T h e measured responses contain higher F i g . 7.42 C o m p u t e d and Measured Accelerations of A C C 3479 in T e s t RSS110/EQ1 at the Location Max.Val. 1 1 1 1 1 1 1 1 1 Recorded Response 7.10 J w y v ^ v /\ y V V \ / \ r\, i\ / "l / " / v \ /VI / \n 1 \ v ^ V ^ -7.64 > i i i i i • i Computed i Response 6.50 y\AKhj\f\hi\/\t\ " V v ~ v VV V V V V v V V . ~ \f ~ -6.27 1 0 1 1 2 1 3 1 1 1 5 1 6 1 7 1 8 9 i 10 T i m e (sec) Fig. 7.43 Computed and Measured Accelerations at the Location of A C C 3466 in Test RSS110/EQ1 Max.Val. 1 r 1 ~ " r~ 1 1 r 1 r Recorded Response 6.68 " A A A / I A A A A A A - — -7.06 i i • r i i i i r Computed Response 6.86 ' A / A A wA A A A A A A V 1/ ^w - - . ^ y w Vy v V w I 0 1 i1 2 i 1 3 i 1 4 i I 5 i I 6 i I 7 i I 8 " ^ -6.50 i I 9 i1 l_ 10 Time (sec) F i g . 7.44 Computed and Measured Accelerations of A C C 3477 i n Test RSS110/EQ1 at the L o c a t i o n Max.Val. -1 1 1 1 1 1 1 Recorded Response 10.6 -8.94 _J L_ _l l_ -1 I I— Computed Response 7.42 -6.95 Time Fig. 7.45 (sec) Computed and Measured Accelerations at the Location of A C C 3478 in Test R S S 1 1 0 / E Q 1 , 1 , r 1 —i 1 1 Recorded Response 1 9.04 ' A I\AA Ah y if v v y y i 1 i M AA / \ r i if V wr ^ i i i i A A v w -10.5 i i C o m p u t e d Response 7.42 A / \ / \ A • 1 1 / I A / 1 A A . / i I 2 i I 3 1 i I i 1 5 \ *, .. V wr ~ ^ vV VV V ;w V 0 A i 1 6 i I 7 i I 8 i1 9 ' -6.95 10 Time (sec) Fig. 7.46 Computed and Measured Accelerations at the Location of A C C 3457 in Test R S S 1 1 0 / E Q 1 Chapter 7 : frequency contents than the computed responses 185 and the measured peak amplitudes are consistently higher than the computed values. A similar order of difference is also observed at location A C C 1225 as shown in F i g . 7.47 a n d Table 7.5. A C C 1938 was mounted on top of the structure to measure horizontal accelerations and ACC 1572 near the right hand edge to measure vertical accelerations. c o m p u t e d accelerations T h e measured and at location A C C 1938 are compared in F i g . 7.48. similar i n frequency content. T h e peak accelerations T h e y are very tabulated i n Table 7.5 agree fairly closely. T h e vertical acceleration due to rocking as recorded by A C C 1572 and those computed are shown in F i g . 7.49. acceleration A g a i n , the computed accelerations closely match the recorded in both frequency contents and peak values. 7.3.4 Comparison of Settlement in Test RSS110/EQ1 T h e computed and measured settlements are tabulated in Table 7.6 at prototype scale. The comparison at locations on top of the structure ( L V D T 81648 and L V D T 77452) is excellent with very little difference between the measured and computed values. However, at locations on the crest of the sand foundation ( L V D T 48411 and L V D T c o m p u t e d values are consistently higher than the measured value. 92032), the Max.Val. —'— — T" " ~ 1 I 1 ' 1 1 ~1 1 ~ 1 1 1 Recorded Response 'AhAAh 10.7 -11.6 i i i i i i i i 1 Computed Response 7.47 ' w -A /A M A A A *\ A * - ; V V V y V v V V ur ~ ~ w W 0 i 1 2 Fig. 7.-17 i i 3 i 4 1 5 1 6 i 7 8 i 9 1 10 Time (sec) Computed and Measured Accelerations at the Location of A C C 1225 in Test R S S 1 1 0 / E Q 1 -6.88 Max.Val. -1 r~ Recorded Response 10.1 -7.65 C o m p u t e d Response 8.89 / \ /1 /I A f\ -~ A -8.49 1 1 1 1 1 5 1 6 I I I 10 T i m e (sec) Fig. 7.48 Computed and Measured Accelerations at the Location of A C C 1938 in Test R S S 1 1 0 / E Q 1 Max.Val. 201 1 1 1—• 1 1 1 1 r- 1 Recorded Response io| 1 3.79 fe.iol 1 -3.08 g o V a. es v o o •< -20 ' 1 1 1 1- C o m p u t e d Response 3.76 ,,.«A AA n A> « « I\ A / I A A A M /( AA- A/\ <\ ^ N « • - - - i ~ v V M y u y v v y * v w ^ v / v/y v v 1/1/ ^ v " v*' v " y v ^ ^ " ^ -v^ • w v 1 -3.60 3 1 • 2 F i g . 7.49 i 3 • i 1 i i 5 6 Time (sec) 7 i 8 Computed and Measured Accelerations of A C C 1572 in Test RSS110/EQl • 9 at the i 10 Location Chapter 7 : 189 Table 7.6 Comparison of Settlements in Test RSS110/EQ1 Transducer No. Measured (mm) Computed (mm) 48411 2.4 3.6 81648 3.2 3.1 77452 3.2 3.5 92032 2.4 4.4 Part of this is due to the difficulty of making accurate measurements with the L V D T s in sand, especially when the dry sand is subject to mobilization by wind d u r i n g flight. The complete computed settlement pattern is shown in F i g . 7.50 along w i t h the mea- sured values. 300. It should be noted that the settlements are plotted with a magnification of T h e notations are same to those used in sections 7.1.4 and 7.2.4. It is clear that the measured and computed settlements lie closely at the locations on the top of the structure. 7.4 Verification Study Based on Test Series RSS90 7.4.1 Centrifuge Model in Test Series RSS90 A schematic view of a 3 - D soil-structure model is shown in F i g . 7.51. T h e model was constructed by dry method as described in section 5.5.1 using Leighton B u z z a r d B . S . S 120/200 sand. T h e estimated relative density of the sand is 64%. T h e sand foundation was 110mm high, 900mm wide at the base and has side slopes of 2.2:1. T h e length of the sand foundation in the direction perpendicular to the plane of shaking is 480mm. L V D T 81648 o LVDT LVDT o 48411 77452 LVDT GEO.SCALE DISPLACEMENT F i g . 7.50 92032 o 0 ' ' o"o3r S e t t l e m e n t P a t t e r n i n Test R S S U O E Q l Chapter 7 : 192 T h e embedded structure is a solid cylindrical block of a l u m i n u m alloy (Dural) 150mm in diameter and 100mm high. T h e block was embedded to a depth of 30 m m i n the foundation soil. D u r i n g the test the model experienced a nominal centrifugal acceleration of 80g. Therefore, the model simulated a structure embedded to a d e p t h 2.4m. approximately 8 m in height a n d 12m in diameter T h e average contact pressure of the structure on the soil was 220 k P a . T h e complete instrumentation of the model is shown in F i g . 7.52. 728 and A C C 734 accelerations. measured vertical accelerations Accelerometers A C C while the others measured horizontal A C C 2036 mounted on the concrete base measured the input motion to the model. 7.4.2 Model Response in Test RSS90 T h e first earthquake of this test series has a peak amplitude of the order of 5%g. The response to this earthquake was not analysed. O n l y the response to the second earthquake ( E Q 2 ) which has a peak amplitude of 21.0%g was analysed. T h e model response to the second earthquake (EQ2) is shown in F i g . 7.53. and A C C 1258 study. A C C 1244 records show a large bias in one direction and they were not used in the A s mentioned earlier, the shifts may have been caused by poor earth connection which make the signals float above the baseline. Except for the vertical records ( A C C 734 and A C C 728), motion. all other records show frequency characteristics similar to that of the input A C C 734 a n d A C C 728 both show frequency characteristics typical of a vertical acceleration record. T h e r e is very little rocking evident in the early stages of shaking, i.e., up to the time around 50 milliseconds. Sharp increases in rocking are evident after time 50 milliseconds. T h i s is due to the fact that there is an increase in input to the base of the LVDT72$75IVDT2W1 and ACOlt II r-JLiwr"'" || mi) Jj Scale in m m *«»«t=|)d i1 LVDTlllll |)m L*. A C a . 1 1 c O c 1 | 1 z» ACC125I ^\*CC203t Fig. 7.52 Instrumented M o d e l in Test Series RSS90 Chapter 7 : 21. 4 ACC 1932 50.0 X -215 X/div ACC15U 23. l 50.0 X -22. 194 X/div i ACCU87 26.3 SO.O X X/div -23.7 ACC 1583 17.8 50.0 X X/div -16.1 ACC 12 25 25.6 50.0 X X/div -27.0 ACC988 18.3 50.0 X X/div -17.9 43.4 ACC 1258 50.0 X X/div -20.6 ACC 30.5 HU- SO. 0 X X/div -5.4 7 ACC2036 20.9 50.0 X X/div -21.0 50 mitlisecs Scales F i g . 7.53 : 100 Model M o d e l Response in Test R S S 9 0 / E Q 2 Chapter structure as shown in A C C 1583 and A C C 1487 It is interesting to note that accelerometers 7 ; 195 records. in sand foundation on the right hand side of the centerline of the model measure peak values much higher than those of the counterparts on the left hand side. T h i s suggests that the model may not be uniform and homogeneous in its properties about the centerline. For instance, A C C 1583 and A C C 1487, which were located under the structure and symmetrically opposite about the centerline, measure peak acceleration values which differ by 8.5%g. Further, A C C 1486 is located on the right hand side at a distance from the centerline approximately the same as the average distance of A C C 1932 measures ACC and A C C 1544 peak acceleration 1544 respectively. which are located on the left hand side. value 9.9%g and 8.3%g Yet A C C 1486 higher than those of A C C 1932 and T h e differences are too high and therefore there is certainly local inhomogeneity in the properties of the model. The motion. input motion is shown in F i g . 7.54 It has a peak acceleration at prototype scale along with base corrected value of 21.0% g. It consists of 5 cycles of low level shaking followed by another 5 | cycles of high level shaking. T h e total duration of input motion is around 10 seconds with the last 2.0 seconds of input representing wheel-off accelerations. T h e 3-D prototype was analysed as 2-D plane strain soil-structure system with the foundation soil assumed to be homogeneous with a relative density of 64%. C o m p u t e d and corresponding measured responses are compared at prototype scale in the following sections. 7.4.3 Comparison of Acceleration Responses of Test RSS90/EQ2 Figs. 7.55 to 7.58 show comparison between the measured and computed at locations of A C C 988, A C C 1225, A C C 1583 and A C C 1487. responses These are located in foundation soil with A C C 988 and A C C 1225 in the free field away from the structure and Chapter 7 : Max.Val. <=X) 30.0 -i 1 1 1 1 Not 1 1 Corrected 20.9 -21 .0 30 Or en <a0 20 0 C J i~ OJ 10 0 d. c o 0 0 *J -10 co 0 u OJ -20 0 *0J u CJ -30 0 < 4.0 0. 5.0 Time Fig. 7.54 6.0 (sec) Input Motion for Test RSS90/EQ2 196 Chapter 7 : 30 en Max.Val. bo 20 o u OJ C o 10 0 -10 CO k. OJ - 2 0 OJ CJ o < -30 30 Computed Response - en • 20 CJ OJ p. —— 10 A s C 0 O *J -10 CO s-, OJ - 2 0 A A A • Vv V v II ft1 ft f A 1 20 vv\ /\ -20 .8 .7 'C3 J CJ - 3 0 < D.O 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.55 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 988 in Test R S S 9 0 / E Q 2 Chapter 7 : Max.Val. W> OJ CJ CJ < 30.0 0.0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.56 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1225 in Test RSS90/EQ2 198 Chapter 7 Max.Val. W> 30.0 OJ CJ CJ _30 < n 0.0 1 1 1 1.0 2.0 1 3.0 ' 4.0 1 5.0 Time Fig. 7.57 1 6.0 1 1 7.0 1 1 8.0 9.0 — 10.0 (sec) Computed and Measured Accelerations at the Location of A C C 1583 in Test RSS90/EQ2 Chapter 7 : ^ W) 3 0 . 0 i CJ < on n l 0.0 Max.Val. 1 1 1.0 1 1 2.0 1 1 3.0 1 1 1 4.0 1 1 5.0 Time F i g . 7.58 1 6.0 1 1 1 7.0 1 8.0 1 1 1 1 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1487 in Test RSS90/EQ2 200 Chapter ACC 1583 and A C C 1487 under the structure. 7 : 201 A l l responses show the same trend as the input motion; that is. they show a low amplitude response for the first 3.7 seconds followed by a high amplitude response for the next 4.0 seconds. Responses are very similar in frequency content, each corresponding to the frequency of the input motion. A t locations of A C C 988 and A C C 1225 the agreement accelerations is quite satisfactory. acceleration between the measured and computed peak A t locations of A C C 1583 and A C C 1487, measured peak values are 17.8%g and 26.3%g respectively, while computed values are both 21.2%g. For T A R A - 3 analysis, the model was assumed to be homogeneous and therefore it is not surprising to see the same computed peak values at these two locations. Figs. 7.59 to 7.61 compare measured and computed responses at locations A C C 1544, A C C 1932 and A C C 1486 respectively. A t locations of A C C 1544 and A C C 1932, comparison is good both in terms of magnitude and frequency contents. values at locations of A C C 1544 Measured and computed peak and A C C 1932 differ only slightly. Measured response at A C C 1486 has a peak value of 31.4%g, whereas computed has 23.7%g. E v e n though there is a large difference in peak values, frequency contents are very similar. Comparison the structure between measured and computed acceleration are shown in Figs. 7.62 to 7.64. A C C 728, mounted on top of the structure as shown in F i g . 7.52. responses at locations on A C C 2033 and A C C 734 were A C C 728 and A C C 734 were placed to measure vertical accelerations due to rocking while A C C 2033 was placed in the middle of the structure to measure horizontal accelerations. A t location A C C 2033, measured and computed accelerations closely match in both peak values and frequency content. Measured and computed peaks are 26.1%g and 26.6%g respectively. istics very similar to that of the input motion. B o t h responses show character- T h a t is, they both show 5 cycles of low amplitude response followed by 5 | cycles of high amplitude response. T h i s observation is true for vertical acceleration responses, even though it is not as distinct as in the case Chapter 7 Max.Val. 2.0 3.0 4.0 5.0 Time Fig. 7.59 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1544 in Test R S S 9 0 / E Q 2 Chapter 7 Max.Val. W> 30.0 30 Or 20 0 en —> o uJ O p. G o *J CO CJ OJ 1—* o a <C 10 0 0 0 -10 0 -20 0 -30 Q 0 .0 1.0 F i g . 7.60 2.0 3.0 4.0 5.0 Time 6.0 7.0 1.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1932 in Test RSS90/EQ2 Chapter 7 Max.Val. bo 40 Or C 30 o• o 20 0 - 10 o• _> OJ u OJ - • cu c 0 o -10 4-> CO t, 0 0 -20 0 *0J - 3 0 0 OJ o CJ < -40 3.0 4.0 5.0 Time Fig. 7.61 6.0 7.0 9.0 10.0 (sec) Computed and Measured Accelerations at the Location of A C C 1486 in Test RSS90/EQ2 Chapter 7 Max.Val. 10.0 0 .0 10.0 Computed Response 10.0 - 7.58 0.0 -5.72 10.0 0.0 _J 1.0 I_ 2.0 1 3.0 I 4.0 Time F i g . 7.62 I 1_ 5.0 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 728 in Test RSS90/EQ2 Chapter 7 : Max.Val. Recorded Response • '0.0 1.0 2.0 1 3.0 4.0 5.0 Time F i g . 7.63 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 2033 in Test RSS90/EQ2 206 Chapter 7 : Max.Val. 0.0 1.0 2.0 3.0 4.0 5.0 Time F i g . 7.64 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 734 in Test R S S 9 0 / E Q 2 207 Chapter 7 : of horizontal acceleration and response. 208 Both measured and computed responses at A C C 728 A C C 734 have frequency content much higher than that of the horizontal acceleration response at the same level in the structure ( A C C 2033) and that of the input motion A C C 2036. T h e reason for this has already been given in section 7.2.3. Unlike symmetrical pairs of accelerometers and A C C 734 measure peak acceleration in the foundation soil, the pair A C C 728 values very close to each other. T h e measured peaks at A C C 728 and A C C 734 are 7.5%g and 8.3%g respectively. T h e computed peak for both case is 7.6%g. T h e differences are very small. T h e computed responses at locations A C C 728 and A C C 734 are such that they show a phase lag of 180 degrees. T h i s indicates that rocking is accounted correctly in the computations. There are two major factors contributing to discrepancies between measured and computed accelerations at some locations in this test. First, as observed earlier, the model is not homogeneous in its properties. Secondly, in T A R A - 3 analysis the responses were computed assuming plane strain behavior of the model. However, the model is a 3 - D model. Hence it is not surprising that some discrepancies may exist between computed and measured accelerations. 7.4.4 Comparison of Settlement in Test RSS90/EQ2 The comparison between measured and computed vertical settlements at locations of L V D T 82280, L V D T 72875, L V D T 72873, L V D T 48411 and L V D T 82273 is given in Table 7.7. T h e values quoted in the table are at prototype scale. Settlements computed at L V D T 72875, L V D T 72873 and L V D T 48411, which were mounted on top of the structure, show remarkable agreement with measured values. L V D T 82280 was placed on top surface of the sand b e r m approximately half way between the shoulder of the b e r m and the edge of the structure. A t this location, the comparison is very good with computed settlement 7.7% Chapter 7 : 209 higher than measured. L V D T 82273 is located close to right hand side shoulder of the berm. A s seen from Table 7.7. measured value at location L V D T 82273 is very much higher than computed. Part of this is due to the effects of wind erosion during the centrifuge The vertical settlement is also compared in F i g . 7.65 where the recorded settlements are indicated by the triangles. and flight. It can be seen that the agreement measured settlements is very good. Table 7.7 Comparison of Settlements in Test RSS90/EQ2 LVDT No Measured (mm) Computed (mm) 82280 15.4 14.2 72875 12.7 12.6 72873 12.4 12.6 48411 12.0 12.6 82273 110.6 11.0 between the computed Fig. 7.65 Settlement Pattern in Test R S S 9 0 / E Q 2 CHAPTER 8 VERIFICATION BASED ON SATURATED M O D E L TESTS 8.1 Verification Study Based on Test Series L D 0 4 8.1.1 Centrifuge Model in Test Series L D 0 4 A schematic view of a 2 - D plane strain saturated soil-structure model is shown in F i g . 7.18. T h e embankment was constructed by M e t h o d 1 described in section 5.5.2.1 using Leighton B u z z a r d B . S . S 120/200 sand. T h e estimated relative density of the sand is 91 ± 17%. T h e embankment is 110mm high and has a flat crest 230mm wide and a base 720mm wide. T h e length of the embankment perpendicular to the direction of shaking is 480mm. The and surface structure consisted of three mild steel plates, each of which is 15mm thick 65mm wide. T h e steel plates were placed end to end along the centerline of the crest as depicted in F i g . 7.18. T h e two end pieces were each 40mm long and the central piece was 385mm long. T h e model experienced a nominal centrifugal acceleration of 80g. T h e model, therefore, simulated a prototype embankment approximately 8.8m high with crest width and base width of 18.4m and 57.6m respectively and a structure approximately 1.2m high and 5.2m wide. The instrumentation of the model is shown in F i g . 8.1. A l l accelerometers 211 measured N3 Chapter horizontal accelerations. 8 : 213 A C C 1932 mounted on the base of the model container recorded the acceleration input to the model. 8.1.2 Model Response in Test L D 0 4 ACC 1932 was not working during the first earthquake of the test series. Only the response to the second earthquake, E Q 2 , was analysed using T A R A - 3 . T h e smoothed data from all acceleration and porewater shown in F i g . 8.2 at model scale. pressure transducers from the test L D 0 4 / E Q 2 are T h e input motion measured by A C C 1932 has a peak- amplitude of 16.4% of the centrifugal acceleration and has 10 complete cycles of significant shaking in the range 10 to 100 milliseconds. A l l acceleration records were filtered to remove frequencies above 10Hz at prototype scale. Also A C C 2033 located near the base shows baseline distortion in the form of a small drift in the negative direction and hence this record has to be baseline corrected. Transducer P P T 2330 shows a record with negative porewater pressures in the entire time span. It is probable that the signs were switched around and hence it is assumed that the correct record is the exact opposite of that shown in F i g . 8.2. P P T 2332 record is anomalous as it does not show any accumulation of porewater pressures during shaking. T h i s is not consistent with the input motion or with other tranducers located at similar location such as P P T 2331. Therefore, this record is ignored in the study. A l l other porewater pressure transducer records, except for P P T 2255 record, are very consistent with the input motion. T h e y all show a rapid accumulation of porewater pressure during the first two cycles of strong shaking. During the next two cycles of weak shaking, the accumulation is shown to be very slow. However, during the subsequent two to three cycles of strongest shaking, rapid accumulation along with large swings of transient porewater pressures are shown in the records. Contrary to these observations, P P T 2255 record shows Chapter 8 : 214 millisecs PPT2330 20.0 •15.8 kPa/div " 11.9 4- -3.71 kPa/div PPT 6 8 11.2 10.0 kPa -1.6i PPT2255 20. 0 kPa kPa/div ^S^YVV^TV^^VSJ* PPT2332 10.0 20.0 kPa kPa/div -10.3 ACC 15 a 50.0 •1i.7 - X/div ACC1908 13.8 20.0 X X/div -11.1 ACC?3i 8. 78 20. C X X/div -8.39 ACCI928 12.5 20.0 X X/div -9.96 ACC 1258 18.8 SCO X X/div -18.5 ACC2033 15.5 50. 0 X X/div -15.5 ACC 1932 16.1 50.0 X X/div -12.0 50 100 millisecs Scales Fig. 8.2 : Model M o d e l Response in Test L D 0 4 / E Q 2 Chapter 8 : 215 a large decrease in porewater pressure at the time of strongest shaking. T h i s behavior at the location of P P T 2255 and also the behavior at the location of P P T 2332 may be due entirely to localised effects such as drainage along the cable leading to the transducers or tension on the transducer leads due to lateral displacements. Hence, data from these transducers have to be interpreted cautiously. T h e effect of soil-structure interaction on porewater pressure responses can be clearly- identified by comparing the records of P P T 2335 and P P T 2331 with those of P P T 2252 and P P T 68. P P T 2335 and P P T 2331 records show larger cycles of oscillations in pressures about the residual level than P P T 2252 and P P T 68 records. fluctuations These oscilations are due to in mean normal stresses caused by rocking of the structure. P P T 68 were located under the structure P P T 2252 and on the centerline of the model and hence they were not subject to large normal stress fluctuations. O n the other h a n d , P P T 2331 and P P T 2335 were located close to edge of the structure and hence they were subject to larger normal stress fluctuations. Therefore, it is not surprising to see larger and more pronounced oscillations at locations P P T 2331 and P P T 2335 than at P P T 2252 and P P T 68. T h e input motion of the earthquake E Q 2 is shown in F i g . 8.3 along with the baseline corrected motion at prototype scale. T h e significant shaking starts around 1.0 seconds and ceases around 7.7 seconds. T h e peak acceleration of 16.3%g occurs at around 4.47 seconds. T h e predominant period of shaking is 0.67 seconds. T h e prototype was analysed as a 2 - D plane strain soil-structure system using T A R A - 3 . T h e sand foundation was assumed to be homogeneous and uniform with a relative density of 75%. T h i s value is within the range of values quoted for the model. T h e baseline corrected A C C 1932 record, shown in F i g . 8.3, was the input for the T A R A - 3 analysis. Slip elements were introduced at the interface between the structure and sand foundation to model slippage between them. T h e computed respones are compared with corresponding Chapter 8 : Max.Val. W> 2 0 . 0 1 1 . c OJ O OJ a. o 2 1 o .0 fry y w -IO.O y T r OJ I 1 Not Corrected 1 1 k 10.0 1 16.4 y 11 | r y y ^1 %/ -12.0 OJ cj Si - 2 0 .0 W> 2 0 . 0 .0 1.0 2.0 3.0 4.0 5.0 Time Fig. 8.3 6.0 7.0 8.0 9.0 (sec) Input Motion for Test LD04/EQ2 10.0 216 Chapter 8 : 217 measured responses at prototype scale in the next section. 8.1.3 Comparison of Acceleration Responses in Test LD04/EQ2 The baseline distortion and the high frequency noises found in the measured A C C 2033 record is highlighted in F i g . 8.4. corrected records. A s seen in the figure, the corrected a n d / o r high frequency noises. Fig. It also shows the comparison between the original and T h e corrected record has no baseline distortion and computed responses are compared in 8.5. T h e comparison in terms of frequency content, peak amplitudes and variation of amplitude with time is excellent. Both responses show characteristics very similar to that of the input motion. T h e peak amplitudes both in the positive and negative directions are fairly close to the corresponding values of the input. T h i s could be expected because A C C 2033 was located near the base. T h e measured and computed peak acceleration values are 15.8%g and 15.9%g respectively and the difference is very small. ACC 1258 was located on the centerline of the model approximately half way between the base of the structure and the base of the model. T h e measured response is compared with the computed response in F i g . 8.6. T h e comparison in the time range 0.0 to 3.5 seconds is good. However, in the range 3.5 to 5.2 seconds, the computed accelerations are somewhat lower than the measured accelerations. Fig. 1928. B u t the overall agreement is good. 8.7 compares the measured and computed accelerations at the location of A C C T h e y are very similar except for one large peak in the computed response. A C C 1928 was located just outside the edge of the structure at the same elevation as A C C 1258. Yet the differences between measured accelerations at these two locations are very high, whereas the difference between computed accelerations are small, and for a uniform and homogeneous model as assumed in the analysis, this small difference seems to be reasonable. Therefore, the large difference between the measured and computed accelerations at these Chapter 8 : Max.Val. be 20.0 W> 20.0 C cu CJ 10.0 IH CJ ^o. 0.0 c o •ZJ 10.0 co i* • cu -20. f.O "v 1.0 CJ CJ < F i g . 8.4 2.0 3.0 4.0 5.0 Time 6.0 7.0 8.0 9.0 10.0 (sec) Original and Corrected Accelerations at the Location of A C C 2033 in Test L D 0 4 / E Q 2 218 Chapter 8 : Max.Val. <ai) 20.0 W) 20.0 .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 8.5 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 2033 in Test L D Q 4 / E Q 2 219 Chapter 8 Max.Val. 20 0 10 0 0 0 -10 0 -20 0 20 0 O 0) ra c o OJ "OJ o CJ < r . tut Computed *-> fi OJ CJ t* 14.1 10 . 0 AA OJ Q, C Response 0 0 "vyi/vVy < 0.0 1.0 i 2.0 i 3.0 -11.4 i « 4.0 5.0 Time Fig. 8.6 A; i 6.0 i 7.0 t 8.0 i 9.0 10.0 (sec) Computed and Measured Accelerations of A C C 1258 i n T e s t L D 0 4 / E Q 2 at t h e L o c a t i o n Chapter 8 : Max.Val. <a© 20 Recorded »J C OJ o u OJ Response 10.4 10 D. C o »-> cfl 0 -9.28 -10 OJ OJ o < -20 ^ _^ b£) 20 »-> c OJ CJ 10 OJ c o 0. 0 CO t- -10 CJ "OJ a o -20 < .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 8.7 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1928 in Test L D 0 4 / E Q 2 221 Chapter 8 : 222 locations might primarily be due either to local inhomogeneity of the model in density or due to measurement error of the transducers. T h e measured and computed accelerations at the location of A C C 1908 in the zone directly beneath the right hand shoulder of the model are shown in F i g . 8.8. T h e comparison in terms of frequency content, peak amplitude and distribution of amplitude with time is excellent. T h e measured peak value is 13.4%g and the computed peak value is 14.5%g. A C C 1544 was mounted on the top of the structure, as shown in F i g . 8.1, to measure horizontal accelerations. T h e measured accelerations are compared with those computed by T A R A - 3 in F i g . 8.9. Except for the thin peak in the computed response, the peak values and frequency content agree very closely. T h e measured and computed peak accelerations are 14.7%g and 16.3%g respectively. 8.1.4 Comparison of Porewater Pressures in Test LD04/EQ2 T h e measured porewater pressures near the base of the model at the location of P P T 2252 is shown in F i g . 8.10 along with those computed by T A R A - 3 . B o t h the rate of development and peak residual porewater pressure are predicted very well. computed peak residual porewater pressure ratio, u/a' , yo T h e measured and are 23.0% and 22.0% respectively. T h e variation of amplitude in the input is clearly reflected in both measured and computed responses. For instance, during the strong shakings in the time ranges 1.0 to 2.2 seconds and 3.5 to 5.5 seconds, the accumulation of porewater pressures are rapid and during the weak shakings in the ranges 2.2 to 3.5 seconds and 5.5 to 9.8 seconds, the accumulation is very slow. T h e comparison between the measured and computed porewater pressures at the location of P P T 2335 is shown in F i g . 8.11. T h e computed pressures are consistently lower than the measured pressures. T h e measured peak residual porewater pressure ratio is 46.0% and Chapter F i g . 8.8 8 C o m p u t e d and Measured Accelerations at the Location of A C C 1908 in Test L D Q 4 / E Q 2 Chapter 8 : Max.Val. fee 20.0 fee 20.0 C cu o 10.0 u CU c _o «-> cc 0.0 •10.0 p- • cu "a. •20 o o. .0 1.0 2.0 3.0 4.0 5.0 Time <: F i g . 8.9 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations at the Location of A C C 1544 in Test L D Q 4 / E Q 2 224 Chapter to 8 : 225 Recorded Computed Cu I) U 10.0 F i g . 8.10 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2252 in Test L D 0 4 / E Q 2 to ^ 10.0 Time F i g . 8.11 (sees) C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2335 in Test L D Q 4 / E Q 2 Chapter 8 : 226 the computed ratio is 31.0%. A s indicated by the designation m a x . O T L L in F i g . 8.2. the m a x i m u m measured values of porewater pressures at this location are outside the guaranteed linear range of the tape recorder. Therefore, measured values have to be viewed with some skepticism. Transducer P P T 2255 was located in the upper part of the sand foundation as shown in F i g . 8.1. T h e comparison of porewater pressures shown in F i g . 8.12 indicates that, the computed and measured porewater pressures agree closely for the first 4.0 seconds of the record and then deviate sharply. A s discussed in the previous section, the measured response is somewhat dubious. It shows a sudden decrease in porewater pressures at around 4.5 seconds when the strongest shaking occurs. In constrast to this, the computed response shows a steady build up of porewater pressure in response to the strong shaking. Hence it is postulated that during the strongest shaking either the transducer moved in relation to the surrounding soil and thereby caused an apparent decrease in the measurement or drainage occurred along the cable leading to the transducer. Fig. 2331. 8.13 shows comparison of porewater pressure responses at the location of P P T T h e computed pressures are less than the measured pressures in the early stages of the shaking. However, after 4.0 seconds, the computed pressures build up rapidly and match the measured pressures in the later stages of shaking. T h e peak residual porewater pressure is predicted satisfactorily. T h e measured and computed peak porewater pressure ratios are 45.0% and 46.0% respectively. Tranducer P P T 2330 was located under the structure as shown in F i g . 8.1. The porewater pressures at this location are compared in F i g . 8.14. T h e measured and computed pressures agree very closely for the first 5.5 seconds of the record. In subsequent stages, in constrast to the little development shown in the computed response, the measured response shows a steady increase upto 7.0 seconds and thereafter shows a steady decrease in pressures. Chapter S 227 CO CL, Recorded Computed CD 0J L. 3 V) V Cu u. o 1- a. 1 cO l& o 0) u o Cu 0 2.0 4.0 6.0 8.0 Time (sees) Fig. 8.12 Computed and Measured Porewater Pressures at the Location of P P T 2255 in Test L D 0 4 / E Q 2 Fig. C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2331 in Test L D Q 4 / E Q 2 8.13 10.0 Chapter The decrease in pressures estimate is due to drainage after the quake ceased. 8 : 228 A fairly reliable of peak residual pressure is given by the record around 7.0 seconds. T h e peak residual porewater pressure ratio of the measured and computed responses are 18.0% and 14.8% respectively. The measured porewater pressures at the location of P P T 68 are compared with the computed pressures in F i g . 8.15. A s seen in Fig.8.1, P P T 68 was located directly beneath the structure on the centerline of the model. T h e measured porewater than the computed pressures throughout the shaking. in the range 3.5 to 6.0 seconds. pressures are less However, differences appear only T h e reason is that the rapid development exhibited in the computed pressures in response to the strongest shaking in the time range 3.5 to 5.5 seconds is absent in the measured response. A p a r t from this, the overall agreement is quite satisfactory. 13.0% T h e measured and computed porewater pressure ratios at this location are and 15.0% respectively. 8.1.5 Comparison of settlements in Test LD04/EQ2 The measured vertical settlements at the locations of L V D T 82280 and L V D T 46997 are compared with corresponding computed values in Table 8.1. T h e values are reported at prototype scale. B o t h L V D T s were located symmetrically opposite about the centerline at top of the structure. T h e measured values are higher than the computed values. Chapter 8 : 229 Recorded Computed 0.0 2.0 4.0 6.0 8.0 10.0 T i m e (sees) F i g . 8.14 CL, o f\j C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2330 in Test L D Q 4 / E Q 2 Recorded Computed QJ U 3 Vi re cn o Cu i~ QJ J-J CO ft QJ o u O Cu 0 2.0 4.0 6.0 8.0 T i m e (sees) F i g . 8.15 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 68 in Test L D 0 4 / E Q 2 10.0 Chapter 8 : 230 Table 8.1 Comparison of Settlements in Test L D 0 4 / E Q 2 Transducer No. Measured (mm) Computed (mm) 82280 16.1 8.5 46997 16.9 8.3 8.2 Verification Study Based on Test Series R S S l l l 8.2.1 Centrifuge Model in Test Series R S S l l l A schematic view of a 2 - D plane strain model structure embedded in a saturated foundation is shown in F i g . 7.38. T h e model was constructed 5.5.2.2 using Leighton Buzzard B.S.S 52/100 sand. possible at a nominal relative density estimated by M e t h o d 2 described in section T h e sand was placed as uniformly as to be about 52%. T h e sand foundation is 110mm high and has a base 900mm wide. T h e side slopes are at 2.2:1. T h e length of sand foundation perpendicular to the plane of shaking is 480mm. T h e heavy structure is made from a solid piece of aluminum alloy and has dimensions 105mm wide by 108mm high in the plane of shaking. plane of shaking is 470mm. foundation. T h e length perpendicular to the T h e structure is embedded to a depth of 25mm in the sand Coarse sand was glued to the base of the structure to prevent slip between structure and sand. D u r i n g the test the model experienced a nominal centrifugal acceleration model, therefore, simulated a structure approximately embedded to a depth 2m in sand foundation. of 80g. T h e 8.6m in height, 12m in width and T h e average contact pressure between the Chapter structure 8 : 231 and L V D T s are and sand foundation was approximately 240 k P a . T h e locations of the accelerometers, shown in F i g . acceleration 8.16. A C C 3441 porewater pressure transducers mounted on the base of the model container defined the input to the model. In this test, as may be seen from F i g . 8.16, the porewater pressure transducers were duplicated at corresponding locations on both sides of the centerline of the model except for P P T 2255 and P P T 1111. T h e purpose of the duplication was to check the reliability of the recorded data. T h i s was not done in earlier tests and in some cases it was difficult to decide whether differences between measured and computed responses were due to instrumentation problems, lack of homogeneity in the sand foundation or deficiencies in the method of analysis. If the model was homogeneous and the instrumentation was perfect, then theoretically responses measured at pairs of locations should yield very similar responses. T h e extent to which the records for corresponding locations agree with each other is an indication of reliability and homogeneity. 8.2.2 Model Response in Test RSS111 T h e smoothed data from all tranducers for the earthquake 8.17 and F i g . 14.3% 8.18. T h e input motion measured by A C C 3441 of the centrifugal acceleration Accelerometers A C C 1552, A C C 1925, ( E Q l ) are shown in F i g . has a peak amplitude of and has 10 complete cycles of significant shaking. A C C 1900 and A C C 1572 measured vertical accel- erations and other accelerometers measured horizontal accelerations. 1552 ( F i g . 8.17) A C C 3457 and A C C records have to be viewed with caution as they both show a large bias in one direction. Therefore, they have to be corrected for baseline distortion before making comparisons. Besides the drifts, A C C 1552 shows a response primarily at a frequency similar to that of the input motion right from the beginning of shaking. T h i s is quite unusual ACCI572 '00 i i E3 ACCM4 o PPUI1I o PPT22SS -ait* 1 — o PP12IU o PPT285S F i g . 8.16 o 1 o PPT243I o «>rj*« >PTWI o o PPT2H0 PPT2}3t 1 1 ACCISS2 o PPTltSI DDTUli Instrumented Model in Test Series R S S l l l to to Chapter S : .0858 233 LVDTU57 mm . 1000 -. 0520 mm/div max.OTRR LVDT16i8 ma . 1000 -.157 mm/div 5.90 -6.39 ~ ACC 1900 *T X/div 8. 17 4- ACC 1572 10.0 -7.37 _+ 10.5 -- X/div -r ACC 1925 20.0 •11.2 X/div -• ACC3166 20.0 -11.8 11.9 •f 1 X/div X -13.6 ~~13~1 " i -H -13.0 20.0 4 in XCC3457 X 20.0 -20. 1 X/div 17. i ACC 1552 X -11.5 11.6 20.0 i X/div 1 ACC3U1 20.0 X -U.3 J" X/div 50 millisecs Scales Fig. 8.17 100 : Model M o d e l Response in Test RSSlll/EQl Chapter 8 : 234 millisecs 50 100 —+~ —\- 16.1 T ACC 1938 SO.O X X/div -16.9 100 50 tillisecs Scales Fig. 8.18 : Model Model Response in Test R S S l l l / E Q l Chapter for a vertical acceleration S : 235 record at a location in the middle of the sand foundation. It is probable that the transducer rotated so that it measures a mixture of vertical and horizontal accelerations. located Because of the uncertainty. A C C 1552 was not used in the study. A C C 1925, adjacent to the edge of the structure, shows significant response milliseconds when the significant motion of earthquake had already ceased. may be suspect and therefore is not used in the study. even after 95 T h i s record A C C 1900 and A C C 1572 were placed at opposite edges of the structure symmetrically about the centerline of the model. Since the model embankment was constructed to be homogeneous, both these should record similar forms of response. Yet both records show quite different forms of responses. A C C 1572 has a lot noise compared to the much cleaner record of A C C 1900. T h e porewater pressure data, shown in F i g . 8.18, show the sum of the transient and residual porewater pressures. T h e peak residual porewater pressures were attained when the earthquake excitations ceased at about 95 milliseconds. After this, most of the records show significant decreases in pressures due to drainage. T h e pressures recorded by the symmetric pairs P P T 2631 and P P T 2338, P P T 2626 and P P T 2848, P P T 2628 and P P T 2851, and P P T 2855 and P P T 2846 are quite similar although there are obviously minor differences in the levels of both transient and residual porewater pressures. Therefore it can be assumed that the sand foundation is remarkably symmetrical in its properties about the centerline of the model. P P T 2631 and P P T 2338 records show large oscillations about the residual porewater pressure levels. These are due to soil-structure interaction. T h e transducers were located directly underneath the structure and therefore they were subjected to large cycles of normal stresses due to rocking of the structure. fluctuations that the The fluctuations in mean normal stress and hence in porewater fluctuations in stress resulted in similar pressure. It is also apparent in these records are almost 180 degrees out of phase. For instance, Chapter 8 : 236 at time 50 milliseconds. P P T 2338 records a pressure below the steady residual component while P P T 2631 records a pressure above it. T h e phase shift results from the fact that the cyclic normal stresses caused by rocking of the structure are 180 degrees out of the phase at these locations. A s free field is approached, it is evident that the influence of soil-structure interaction decreases. For instance, all other pairs show records that contain somewhat smaller oscil- lations than those contained in the pair P P T 2631 and P P T 2338. However, the pair P P T 2846 and P P T 2855 show somewhat larger oscillations than those recorded in the free field. T h e locations of P P T 2846 and P P T 2855 are close enough to the structure to be affected by the cyclic normal stresses caused by rocking and therefore it is not surprising to see small oscillations present in the records. P P T 2842 is located on the centerline of the model approximately midway between the base of the model and the base of the structure. T h i s location is not subjected to large normal stress fluctuations due to rocking and therefore the porewater pressure record does not oscillate much about the residual porewater pressure. However, P P T 2842 record is not consistent with other porewater pressure records or with the input motion. T h e strongest shaking occurs between time 50 and 75 milliseconds and strong shaking persists up to 90 milliseconds. Y e t P P T 2842 shows significant drainage from time 60 milliseconds which is not evident in any other records. It is probable that drainage occurred along the lead of the transducer. D u r i n g strong shaking, P P T 1111 record show large negative porewater pressures. structure. fluctuations in pressures causing P P T 1111 was located near the surface and adjacent to the Hence, due to rocking of the structure, this was subjected to large shear strains. T h i s , along with low confining pressure at this location led to the strong dilatant behavior. T h e input motion measured by A C C 3441 is shown in F i g . 8.19 at prototype scale. Chapter 8 : Max.Val. W> o o < 20.0 .0 1.0 2.0 3.0 4.0 5.0 Time Fig. 8.19 6.0 7.0 8.0 9.0 10.0 (sec) I n p u t M o t i o n for T e s t RSSlll/EQl 237 Chapter It also include the baseline corrected motion. 8 : 238 It can be seen that the uncorrected and corrected motions are identical. T h e total duration of the earthquake is around 10.0 seconds and significant shaking ceases around 7.5 seconds. T h e peak acceleration of 14.3%g occurs at 4.17 seconds. The prototype was analysed as a 2-D plane strain problem using T A R A - 3 . T h e foun- dation sand was assumed to be symmetrical in its properties about the centerline. In the centrifugal acceleration field of 80g, the heavy structure underwent consolidation settlement, which led to an increase in density under the structure compared to that in the free field. For the analysis, the soil density under the structure was adjusted to be 64% based on the consolidation settlements. 8.2.3 Comparison of Acceleration Responses in Test R S S l l l / E Q l Figs. 8.20 to 8.22 show comparison between measured and computed responses at locations of A C C 3479, A C C 3466 and A C C 3478 respectively. A C C 3479 was located near the base, A C C 3466 near the surface in the free field and A C C 3478 near the edge of the structure. Measured and computed responses at the location A C C 3479 (Fig. 8.20) are similar to that of the input motion. T h i s is expected because A C C 3479 was located very close to the base. The C o m p u t e d peak amplitudes closely agree with those of measured ones. measured and computed peaks are 14.4%g and 13.3%g respectively. Comparison in terms of frequency content is also good. A t location of A C C 3466. the comparison shown in F i g . 8.21 is generally good both in terms of peak values and frequency content. However, the computed peak ordinates between time 4.0 and 6.0 seconds are somewhat less than the measured values. T h e peak acceleration values for measured and computed responses are 14.4%g and 11.0%g respectively. Comparison at the location of A C C 3478 in F i g . 8.22 is good with computed peak Chapter 8 : Max.Val. 20.0 to 20.0 .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 8.20 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Measured Accelerations of A C C 3479 in Test RSSlll/EQl at the Location 239 Chapter 8 Max.Val. bC 20 0 c OJ 10 0 o u OJ Q. 0 0 ra C o -10 0 -20 0 20 0 OJ OJ CJ CJ < bo Computed Response <-> d OJ o 10 .0 OJ a. a 0 .0 1 1 .0 AA A J ra o -10 .0 V OJ OJ CJ o < -20 • 0 0 .0 1 1 0 2.0 1 3.0 Uljvvy t 1.0 1 5.0 Time Fig. 8.21 1 6.0 -12.1 1 7.0 1 8.0 ! 9.0 10 .0 (sec) C o m p u t e d and VIeasured Accelerations at the Location of A C C 3466 in Test R S S l l l / E Q l Chapter 8 Max.Val. tuO 20 0 *-> C <u cj u QJ 10 0 a, c 0 0 ra o OJ OJ CJ CJ < tuj -10 0 -20 0 20 0 Computed Response *J QJ O U QJ 10 0 &, ra C o QJ QJ CJ CJ < 0 0 -10 0 -20 0 0 0 /\ A A A A Al A A ti i 1.0 F i g . 8.22 • 2.0 « 3.0 i 4.0 12.1 A V i 5.0 Time /u*" -10.9 i 6.0 i 7.0 i 8.0 i 9.0 10 .0 (sec) Computed and Measured Accelerations of A C C 3478 in Test RSSlll/EQl at the Location Chapter 8 : ordinates matching closely with those of the measured. 242 T h e peak values in measured and computed responses are 13.5%g and 10.9%g respectively. T h e measured and computed horizontal accelerations at the top of the structure at the location of A C C 1938 are shown in F i g . 8.23. T h e y are very similar in frequency content, each corresponding to the frequency of the input motion given by A C C 3441 (Fig. 8.19). T h e peak accelerations agree fairly closely. T h e measured and computed peak values are 16.9%g and 16.3%g respectively. T h e computed and measured vertical accelerations shown in F i g . 8.24. at the location of A C C 1900 are T h e computed response closely matches the recorded response in both peak values and frequency content. A s seen in F i g . 8.17, high frequency noises are present in A C C 1572 record and therefore frequency components higher than 10.0 H z were removed by a low pass filter. T h e original and filtered responses are shown in F i g . 8.25. T h e filtered response does not have the noises anymore and moreover it is now similar in frequency to A C C 1900 record. F i g . 8.26 shows the comparison of the filtered and computed responses. T h e agreement in both frequency content and peak values is excellent. T h e measured and computed peaks at location of A C C 1572 are 7.22%g and 6.86%g while at A C C 1900 they are 6.32%g and 6.86%g respectively. T h e measured and computed accelerations location of A C C 3436 are shown in F i g . 8.27. A C C 3436 was located on the vertical edge of the structure that lies parallel to the plane of shaking as shown in F i g . 8.16. accelerations at the T h e peak and frequency content agree fairly closely. A s may be seen from F i g . 8.17. A C C 3457 record shows a large shift in one direction. T h e original (uncorrected) and the baseline corrected records are compared in F i g . 8.28. T h e baseline distortion is not present in the corrected record. F i g . 8.29 shows that comparison between corrected and computed responses is good both in terms of frequency content and peak values. T h e measured and computed peaks are 12.7%g and 11.7%g respectively. Chapter 8 Max.Val. W> 20.0 W> 20.0 F i g . 8.23 C o m p u t e d and Measured Accelerations of A C C 1938 in Test R S S l l l / E Q l at the Location Chapter 8 : Max.Val. 20.0 1 -i -I v r- C QJ O (-( Recorded Response 5.88 10.0 QJ a c o 0.0 2 -IO.O •6.35 QJ % o u -20.0 be 20.0 C QJ CJ u 10.0 •< QJ Pi c o Z • » Computed Response 6.96 ./i.Aft 0.0 A A . A I I «IVAA. A» III A ll .A...IVI A /\ A r<\ -6.32 -io.o QJ o « -20 '0.0 Fig. 1.0 8.24 2.0 3.0 4.0 5.0 6.0 T i m e (sec) 7.0 8.0 9.0 10.0 C o m p u t e d and Measured Accelerations at the Location of A C C 1900 in Test RSSlll/EQl Chapter 8 : Max.Val. W> 20.0 — I r~ C <v o OJ Recorded Response 8.16 10.0 cx c 0.0 o £ -7.36 -IO.O CJ 2 -20.0 to 20.0 cu JJ 10.0 o 2 Corrected Response 7.22 0.0 -4.80 -10.0 QJ CJ -20. .0 1.0 2.0 3.0 4.0 5.0 Time F i g . 8.25 _i 6.0 i_ 7.0 8.0 9.0 10.0 (sec) Original and Corrected Accelerations of A C C 1572 i n T e s t R S S l l l / E Q l at t h e L o c a t i o n 245 Chapter 8 : Max.Val. fee 2 0 . 0 -i r- ( c 1 Corrected Response c OJ v CJ a 1 7.22 10.0 0.0 o £ OJ -IO.O -4.80 % a 2 fet -20.0 -j 1 i_ _j 20.0 i_ Computed Response OJ o 6.32 10.0 OJ CS 0.0 o AM/^A.A/1M K - i iM J l l Ann ^ l/v w vY ^ 1 1 ^ nr A/H A A Jl/U f%A- r-\ r -6.96 £ - 1 0 . 0 OJ OJ o o •20 '8.0 1 1.0 ' 2.0 3.0 4.0 5.0 Time F i g . 8.26 6.0 7.0 8.0 9.0 10.0 (sec) C o m p u t e d and Corrected Accelerations at the Location of A C C 1572 in Test RSSlll/EQl 246 Chapter 8 Max.Val. bo 20 , —; C 0) 10 CJ u c o 0 ra a -10 OJ QJ O CJ < ! , M Art/ -20 , * i i 1 \ T r 13.1 / 1 V VYYV i_ . i i i bo 20 ~" '• -13.0 i i i Computed Response 10.6 10 P. C o .«H 0 CO -10 A AA A A A A A A lyl/Vv/ <-> OJ — i \ ft i —> C CJ o 1 Recorded Response A v u H -10.7 OJ o o -20 < 0 i 1.0 F i g . 8.27 i 2.0 i 3.0 i i i 4.0 5.0 6.0 T i m e (sec) i 7.0 i 8.0 i 9.0 10 .0 C o m p u t e d and Measured Accelerations of A C C 3436 in Test R S S l l l / E Q l at the Location Chapter 8 : Max.Val. W> 20.0 w> 20.0 '0.0 Fig. 1.0 8.28 2.0 3.0 4.0 5.0 6.0 T i m e (sec) 7.0 8.0 9.0 10.0 Original and Corrected Accelerations at the Location of A C C 3457 in Test RSSlll/EQl 248 Chapter 8 Max.Val. 20 G CU cj 10 cu —-' G 0 O *J ra S cu "cu -10 CJ o -20 < bo 20 Computed Response G cu CJ u cu G 11 .2 10 A A A A U 0 O ca cu / -10 y v y \r "CJ o CJ < -20 .0 i 1.0 1 2.0 1. 3.0 i 4.0 Time F i g . 8.29 i 5.0 i 6.0 i 7.0 v w v s . -11.7 • 8.0 i 9.0 30.0 (sec) C o m p u t e d and Corrected Accelerations at the Location of A C C 3457 in Test RSSlll/EQl Chapter 8 : 250 8.2.4 Comparison of Porewater Pressure Response in Test R S S l l l / E Q l T h e measured and computed porewater pressures at locations of P P T 2338 and P P T 2631 are shown in F i g . 8.30 and F i g . 8.31 respectively. These transducers were located directly beneath the structure and symmetric about the centerline. T h e measured responses have two types of oscillations superimposed on steady accumulating residual porewater pressures. T h e first type is the large oscillations with cycles of loading which are of low frequency and second type is the higher frequency peaks superimposed on the large oscillations. T h e low frequency oscillations are due to fluctuations in mean normal stresses caused by rocking of the structure and the higher frequency peaks are due to dilations caused by shear strains. However, the computed responses do not have any of these oscillations because only residual porewater pressures are computed by T A R A - 3 . T h e computed rate of porewater pressure development at both locations matches fairly well with that of the measured response. T h e m a x i m u m residual porewater pressure is observed between 7.0 and 7.5 seconds just after the strong shaking has ceased and before significant drainage has time to occur. T h e measured and computed residual porewater pressure, as given in Table 8.2, agree very well at both locations. T h e computed m a x i m u m residual porewater pressure at both locations is 16.0% of the initial effective vertical stress. It is also clearly evident that both measured responses show significant drainage starting at time 7.5 seconds immediately after the strong shaking has ceased. T h e pair P P T 2848 and P P T 2626 were located symmetrically about the centerline, outside the edge of the structure 2631, at the same elevation as the pair P P T 2338 and P P T and the comparisons are shown in F i g . 8.32 and F i g . 8.33 respectively. T h e pressures measured at these locations show somewhat smaller oscillations than those recorded under the structure. T h i s is due to the fact that the effect of rocking on mean normal stresses at these locations is less than at locations under the structure. In these cases, the computed Chapter CS Cu 8 : 251 Recorded Computed JX o OJ st/3 to OJ s~ CU o !* OJ CO is o OJ Vo 2.0 0 6.0 4.0 Time F i g . 8.30 8.0 10.0 (sees) C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2338 in Test R S S l l l / E Q l <0 Recorded Computed C u o \r OJ u C/3 re 73 C u o r\j 0J CO is o OJ o C u 0 2.0 6.0 4.0 Time F i g . 8.31 8.0 (sees) C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2631 in Test RSSlll/EQl 10.0 Chapter 8 : 252 Recorded Computed 4.0 6.0 Time F i g . 8.32 10.0 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2848 in Test F i g . 8.33 8.0 (sees) RSSlll/EQl C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2626 in Test RSSlll/EQl Chapter 8 : 253 residual porewater pressures are less than the measured ones (Table 8.2) but the overall agreement is quite satisfactory. Table 8.2 Comparison of Peak Residual Porewater Pressures in Test R S S l l l / E Q l Transducer No. Measured (kPa) Computed (kPa) PPT 2338 33.5 33.5 PPT 2631 33.0 33.5 PPT 2848 24.5 18.0 PPT 2626 24.0 18.0 PPT 2851 24.3 26.6 PPT 2628 23.7 26.6 PPT 2846 38.1 38.0 PPT 2855 36.0 38.0 PPT 2342 - 72.0 PPT 2255 37.0 38.0 PPT 1111 4.0 2.9 It is interesting to note that measured P P T 2848 response shows a slight increase in pressures in the range 7.5 to 8.2 seconds before showing a decrease in pressures. This in- crease is thought to have occurred due to migration of porewater pressures from surrounding areas of high porewater pressure such as the location of P P T 2338. However, unlike P P T 2848, P P T 2626 record shows decrease in pressures after 7.5 seconds. Since the drainage and internal redistribution are not modeled in T A R A - 3 analysis during shaking, differences Chapter 8 between the measured and computed responses 254 could occur especially after 7.5 seconds when drainage begins to dominate. T h e pair P P T 2851 and P P T 2628 were located out in the free field at the same elevation as the pair P P T 2338 and P P T 2631 and the responses at these locations are compared in F i g . 8.34 and F i g . 8.35 respectively. T h e measured peak residual porewater pressure is slightly less than the computed one but the overall agreement is good. A s seen from Table 8.2, the differences in measured and computed peak residual values are small. It is also interesting to note that at these locations little drainage takes place even though they are close to drainage boundaries. areas of high porewater T h i s is again due to migration of porewater pressures from pressures. F i g . 8.36 and F i g . 8.37 show comparison of porewater pressure responses at the locations of P P T 2846 and P P T 2855 respectively. In both cases, the comparison is excellent both in terms of the rate of development and peak residual value. T h e measured and computed peak residual porewater pressures, shown in Table 8.2, agree closely. A s expected, significant differences appear only in the time range 7.5 to 10.0 seconds owing to drainage and diffusion. T h e large low frequency oscillations observed in the P P T 2338 and P P T 2631 responses are absent indicating that the influence of soil-structure interaction is not prominent at these locations. P P T 2842 was located on the centerline midway between the base of the model and base of the structure. C o m p u t e d and measured porewater pressures shown in F i g 8.38 agree closely for the first 5.0 seconds of the record and then deviate sharply. A s discussed in section 8.2.2, the measured pressures are not compatible with all other records or the input m o t i o n . T h e record shows significant drainage from time 5.0 seconds. possible reason for such drainage is that during the strongest T h e only shaking in the range 4.0 to 6.0 seconds, a drainage path developed along the cable to the tranducer P P T 2842. T h e Chapter F i g . 8.34 8 : C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2851 in Test RSSlll/EQl T i m e (sees) Fig. 8.35 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2628 in Test RSSlll/EQl 255 Chapter 0.0 2.0 4.0 6.0 8 : 8.0 256 10.0 T i m e (sees) F i g . 8.36 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2846 in Test 1 ccj Cu 1 1 1 Recorded Computed o RSSlll/EQl 1 A A, 1 — i 1 1 i / l A .. i j i : L* 3 re <f) V) Cu o CM »/ 0J cs •s O OJ o Cu 0 2.0 F i g . 8.37 4.0 6.0 8.0 T i m e (sees) C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2855 in Test RSSlll/EQl 10.0 Chapter 8 : T i m e (sees) F i g . 8.38 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2842 in Test RSSlll/EQl 257 Chapter 8 : 258 computed pressures show a steady increase in the range 4.0 to 6.0 seconds consistent with the input. T h e demonstrated homogeneity of the model about the centerline and the close agreement between measured and computed porewater pressures for all other transducers support the notion that the behavior of P P T 2842 is anomalous. Fig. 8.39 compares responses at the locations of P P T 2255 which was located out in the free field directly below P P T 2628. C o m p u t e d and measured pressures at this location agree very well for the first 7.0 seconds and then show differences. T h e measured response shows significant drainage after time 7.0 seconds a n d therefore it is not strange to see discrepancies between them after 7.0 seconds. However, the measured and computed peak residual pressures differ only by a few percent. The contours of peak residual porewater in F i g . 8.40. pressures computed by T A R A - 3 are shown T h e integers are the contour values in the unit k P a . T h e triangles show the locations where the porewater pressures were measured and the numbers with the decimal points indicate values of measured peak residual pressures. T h e figure demonstrates the overall agreement between the measured and computed values. It also illustrates the symmetric nature of the contours. T h e contours also support the notion that the movement of water during drainage and diffusion is from areas under the structure to outside towards the sloping and top horizontal boundaries of the sand foundation. 8.2.5 Stress-Strain Behavior C o m p u t e d shear stress-strain responses at selected locations are presented in this section to illustrate the effect of soil-structure interaction and porewater pressures on stress-strain responses. F i g . 8.41 and F i g . 8.42 show stress-strain responses at the locations of P P T 2338 a n d P P T 2842 respectively. A t these locations, hysteretic behavior is evident b u t the response for the most part is only mildly nonlinear. T h i s is not surprising as the initial Chapter 8 : Fig. 8.39 C o m p u t e d and Measured Porewater Pressures at the Location of P P T 2255 in Test RSSlll/EQl 259 Chapter i -0.040 F i g . 8.41 -0.020 0.000 0.020 Shear Strain (percent) Shear Stress-Strain Response at the Location of P P T 2338 in Test Fig. 8.42 0.040 RSSlll/EQl Shear Stress-Strain Response at the Location of P P T 2842 i n Test R S S l l l / E Q l Chapter 8 : 262 stresses under the structure are high and the porewater pressure ratio, u/a' , defined as the y0 ratio between porewater pressure, u, and the initial effective vertical stress, a' , reached a yo level of only 16% and 24% at the locations of P P T 2338 and P P T 2842 respectively. Such low porewater pressure in relation to the initial effective vertical stress does not cause significant reduction in either shear modulus or shear strength; hence hysteretic loops remain narrow and stiff. A s the free field is approached, strong nonlinear behavior is evident. Particularly, the response in the free field at the location of P P T 2851 ( F i g . 8.43) is strongly nonlinear with large hysteresis loops. T h i s indicates considerable softening due to high porewater pressures and shear strains. A t this location, the porewater pressure ratio reached about 80%. T h e stiffer loops found in the response are associated where very low porewater pressure are generated. porewater with the initial stages of the shaking However, as the shaking continues, high pressures are generated and as a result shear modulus a n d shear strength are reduced giving rise to the softer and flatter hysteretic loops. At the location of P P T 2848, even though the response as shown in F i g . nonlinear, it is not as strongly nonlinear as at the location of P P T 2851. 8.44 is T h e porewater pressure ratio reached a level of about 66% at this location. A t the location of P P T 2846, where the porewater pressure ratio reached a level of 65%, the stress-strain response shown in F i g . 8.45, has the same trend as at the location of P P T 2848. 8.2.6 Comparison of Displacements in Test R S S l l l / E Q l T h e displacement time histories shown in F i g . 8.16 were not considered for comparison as L V D T s used in this test series have poor dynamic response characteristics 1986). T h a t is, the response of the L V D T is frequency dependent. (Steedman, Therefore, unless the measured cyclic displacements are corrected appropriately for the frequency dependency Chapter 8 : CD -0.20 Shear Fig. 8.43 0.00 Strain 0.20 (percent) Shear Stress-Strain Response at the Location of P P T 2851 in Test R S S l l l / E Q l 263 Chapter 8 : | ' 1 -0.100 ' 1 0.000 L_ , _ J 0.100 Shear Strain (percent) Fig. 8.44 Shear Stress-Strain Response at the Location of P P T 2848 i n Test R S S l l l / E Q l Fig. 8.45 Shear Stress-Strain Response at the Location of P P T 2846 i n Test R S S l l l / E Q l 264 Chapter 8 : 265 of the L V D T s , they cannot be used for comparison. In order to illustrate the influence of frequency dependence of L V D T on cyclic displacements, the measured A C C 1938 acceleration record and the L V D T 4457 record are plotted together at prototype scale in F i g . 8.46. A C C 1938 was mounted on top of the structure and L V D T 4457 on the top left hand edge of the structure to measure horizontal displacements. Therefore, one should expect the horizontal acceleration response of A C C 1938 to be almost in phase with the horizontal displacement record of L V D T 4457. B u t it is evident as indicated in the figure that the displacement cycle lags behind the acceleration cycle by almost 50 degrees. T h i s phase lag cannot be entirely due to dynamic response but primarily due to L V D T response. ture. Problems of this nature have already been reported in the litera- L a m b e and W h i t m a n (1985) reported a similar phase lag between acceleration and displacement cycles in their centrifuge tests. T h e y have also conducted calibration tests to study the frequency dependence of L V D T s used to measure transient displacements in their centrifuge tests. F i g . 8.47 shows a typical result obtained in their study. T h e circles and crosses show the results measured for two different L V D T s . T h e figure clearly shows that the amplitude ratio is a function of the cyclic frequency and it depends on the particular L V D T used. Therefore, improvements must be made in methods employed for measur- ing transient displacements. characteristics Ideally, one should use transducers that have flat frequency in the range of frequencies contributing to the transient displacement time history. However, for static readings, L V D T s used in this test series are often adequate. The final displacements produced by the earthquake are compared at the locations of L V D T 1648 and L V D T 4457 in Table 8.3. T h e values quoted are at prototype scale. L V D T 1648 was mounted at the left hand top edge of the structure so as to measure vertical settlement while L V D T 4457 was located around the same place to measure horizontal displacement. Max.Val. LVDT C \\ A 5 s O a 6.73 11 0 (0 5 * 4457 Record -4.12 -5 C o N o. 10, l l l l 5 1 6 I 7 I 8 I 9 10 T i m e (sec) Fig. 8.46 Measured cyclic displacement and accelerations at the Locations of L V D T 4457 and A C C 1938 in Test R S S l l l / E Q l to (31 (35 Chapter Fig. 8.47 Frequency Dependent Characteristics of L V D T s Chapter 8 : It can be seen that the computed vertical settlement tlement. 268 is 66% more than measured set- T h e computed horizontal displacement is very much higher than the measured value. T h e final deformation pattern as computed by T A R A - 3 is shown in F i g . 8.48. T h e discontinuous line shows the undeformed shape and the solid line shows the deformed shape. It should be noted that for the purpose of clear illustration the deformations are magnified about 10 times. T h e top surface of the sand foundation settles more than the structure. Also, at the lower end of the sloping faces, the sand bulges out on both sides. T h i s is close to a constant volume type of deformation as often found in fully saturated Table 8.3 Comparison of Displacements in Test R S S l l l / E Q l Transducer No. Measured (m) Computed (m) Direction L V D T 1648 0.012 0.020 Vertical L V D T 4457 0.0016 0.006 Horizontal cases. !<L--.. V GEO.SCALE DISPLA C E M E N T Fig. 8.48 / 1 __J —-J-—^ ;> t a" Computed Deformation Pattern in Test R S S l l l / E Q l ^ 00 to CHAPTER 9 SUMMARY AND CONCLUSIONS 9.1 Summary A nonlinear effective stress method of analysis for determining the static and dynamic response of 2-D embankments and soil-structure interaction systems is presented. method of analysis has been incorporated into the computer program T A R A - 3 . The It is a re- vised and extensively modified version of an earlier program T A R A - 2 and has more efficient algorithms and additional features including energy transmitting boundaries. A n extensive verification of the capability of T A R A - 3 to model the dynamic response of structures using comprehensive data from a series of simulated earthquake tests on centrifuged model is presented. T h e models simulated a variety of structures ranging from simple embankments to soil-structure interaction systems which included surface and embedded structures on both dry and saturated sand foundation. The centrifuge model tests used in the verification of T A R A - 3 were conducted over a three year period from 1983 to 1986. In the earlier period, the technology of model construction and as well as the technology for conducting seismic tests on large scale models was in its infancy. Consequently, the earlier model construction techniques led to rather inhomogeneous models with wide variations in density as evident from data in tests such 270 Chapter as the L D 0 4 series. 9 : 271 A t some locations in these models, it was difficult to decide whether differences between the computed and measured responses were due to instrumentation problems, lack of homogeneity in the sand foundation or deficiencies in the method of analysis. A s the test series progressed, model construction improved with experience and a new technique that produced homogeneous models was developed. Further, in order to obtain an unambiguous data base, the instruments were duplicated at corresponding locations on both sides of the centerline of the model. T h e extent to which the records at corresponding locations agree is an indication of the reliability and homogeneity. T h e model in test series RSS111 was constructed in this new approach and the data indicated that the model was very homogeneous. T h e differences between the computed and measured responses in this model were found to be very small and within the acceptable accuracy for engineering purposes. T h i s indicates that T A R A - 3 is capable of conducting dynamic response analysis of soil structure systems with acceptable accuracy for engineering purposes. 9.2 Conclusions T h e study described in this thesis led to the following conclusions: 1) T h i s study clearly demonstrated the utility of centrifuge modeling in providing comprehensive data base for validating methods of seismic response analyses. a In no other way can such complete data coverage be obtained when required and at such a low cost. 2) T h e centrifuge tests clearly demonstrated key aspects of soil-structure interaction, namely, the high frequency rocking response, the effects of rocking on porewater pressure patterns and the distortion of free-field motions and porewater pressures by the presence of a structure. Chapter 3) T h e comparison between measured and computed responses 9 : 272 for the various cen- trifuged models demonstrated the wide ranging capability of T A R A - 3 for performing complex effective stress soil-structure interaction analysis with acceptable accuracy for engineering purposes. Seismically induced residual porewater pressures are satisfactorily even when there are significant effects of soil-structure interaction. predicted C o m p u t e d accelerations agree in magnitude, frequency content and distribution of peaks with those recorded. particular, In the program was able to model the high frequency rocking vibrations of the model structures. soil-structure T h i s is an especially difficult test of the ability of the program to model interaction effects. C o m p u t e d settlements also agree reasonably well with those measured. 4) It is necessary to incorporate an energy transmitting base to account properly for the energy transmitted into the underlying m e d i u m . T h e usual rigid base assumption may result in overestimation of the the dynamic response of the soil deposit. 5) Appropriate lateral boundaries for the model are also necessary to avoid feedback to the structure from the sides. Satisfactory results can be obtained when lateral boundaries are located at an appropriate distance from the edge of the structure. For both linear and nonlinear problems, the simple roller boundary proved as efficient and more economical than the other types of lateral boundaries. 9.3 Recommendations For Further Study 1) T h e capability of the method of analysis may be extended problems. for the analysis 3-D Chapter 9 : 273 2) A d d i t i o n a l validation studies may be carried out to verify the predictive capability of T A R A - 3 to model the dynamic response of other geotechnical soil structures such as retaining walls and anchored bulkheads. 3) T h e program has been validated for models with homogeneous sand foundations. However, the method is also applicable to more heterogeneous conditions of real sites. It is obviously highly desirable when field become available to test the capability of T A R A - 3 under these variable conditions. Such a study is planned for later in 1988 when seismic data from the L o T u n g Reactor in Taiwan becomes available. 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Y o s h i m i , Y . , a n d K i s h i d a , T . , (1981), " A R i n g Torsion A p p a r a t u s for Evaluating Friction Between Soil and M e t a l Surfaces", Geotechnical Testing Journal, G T J O D J , V o l . 4, N o . 4, pp. 145-152. Zienkiewicz, O . C . , and C h e u n g , Y . K . , (1967), "Finite Element M e t h o d in Structural and C o n t i n u u m M e c h a n i c s " , M c G r a w Hill Book C o m p a n y . 281 APPENDIX I STIFFNESS MATRLX IN TERMS OF EFFECTIVE STRESSES The strain vector, {e}, is related to the nodal displacement vector, {8}, {*} = [B] {8} as follows: (Al.l) in which, [B] = strain displacement matrix which depends on the element geometry. The effective stress vector, {cr }, is related to the strain vector by 1 W) = \D\ {e} (A1.2) where, [D] — elasticity matrix. {a } and [D] for 2-D plane strain problems are given by 1 {*'} =<*'.} U1.8) and D] ~B + B - 4/3 G 2/3 G 0 B - B + 2/3 G 4/3 G 0 0 0 (A1A) G where, B = bulk modulus and G = shear modulus. For equilibrium, the principle of virtual work requires that the work done by the virtual displacement, {8}, must equal the work done by the internal stresses. 282 Supposing virtual strains due to the virtual displacement, {6}, be {e}, then the internal work done, W{ , is given by n W = in jjj {e) {a} dV T v (A1.5) where, {a} — total stress vector. Now by effective stress principles, {a} = {a'} + {u} (A1.6) where, {«} = porewater pressure vector which is defined as, {«} = { u \ I (Al.l) 0 oJ in which, u 0 — porewater pressure i n the element. Substituting equation (.41.6) into equation (A1.5) yields, W = in JIj {e} [ {a'} + {u} } dV (Al.S) [ [D] {e} + {u} } dV (A1.9) T v Further from relationship in equation (A 1.2), W = in jjJ {e} v T Using equation ( A l . l ) , the above expression can be rewritten as, W = in IJj {6} v T [ [B} [D] [B] {6} T 288 + [B] {u} } dV T (ALIO) Supposing the external load vector is {p}, then the external work done, W , is ex W„ = {6} {p} (ALU) T Now by principle of virtual work, W = ex W (A in 1.12) or {5} T {p} = //J {S} v T [ [B\ [D] [B\ {6} + [B] {u} } dV T T (A1.13) or {p} = j/J[B] [D] [B] dV {6}+ IIl \B\ dV {u} T T v (ALU) or {P} = [*] {*} + [*1 {«} (A1.15) in which, [k] = element stiffness matrix, [k~] = element porewater pressure matrix. They are denned as, W = / / l \ B ] [D\ [B\ dV T v IH [B} [kl = v T dV (AIM) (Al.lt) Nonlinear problems are solved using incremental elastic approach. Therefore, the displacements, stresses, strains and moduli values are replaced by incremental displacements, incremental stresses, incremental strains and tangent moduli respectively. The global incremental equation can then be written as, {AP}= [K \ t {A} + [IT] {AU} 284 (A1.18) where, {AP} = incremental global load vector, [Kt] = global tangent stiffness matrix, [K*] = global porewater pressure matrix, {A} = incremental global displacement vector, {At/} = incremental global porewater pressure vector. It is often required to express nodal forces in an element in terms of stresses and strains. The following expressions give nodal forces in terms of stresses and strains respectively. (A1.20) and 285 APPENDIX II STIFFNESS MATRIX FOR SLIP ELEMENT T h e force displacement relationship at any point within the slip element shown in F i g . A2.1 is given by (A2.1) or, {/} = [*] {»} (A2.2) in which, f = shear force per unit area of the element, s / „ = normal force per unit area of the element, Kg = unit shear stiffness in the direction of the element, = unit normal stiffness in the direction normal to the element, K n w = shear displacement at the point of interest a n d , s w n = normal displacement at the point of interest. Let up, UQ, UR and us be the nodal displacements in the direction of the slip element of nodes P, Q , R and S respectively. be linear, then the displacement, Uf o p , Since the variation in displacement is assumed to in the direction of the slip element at any point on segment R S at a distance / from S, is given as "top = 7 u + (1 - -) u R s (A2.3) or utof = Ni u R 286 + N 2 u s (A2.4) in which, JV, = - (A2.5) (A2.6) Similarly, the displacement in the direction normal to the slip element, Ub t, at any 0 point on segment PQ at a distance / from P, is given by Hot = N\ UQ Now, the shear displacement, w s + N 2 {A2.1) up at that point is given by, W s = Ufop u - (A2.8) bot or ( up } w = [- N s Uq - Ni Ni N } < 2 2 (A2.9) U R \ u s Similarly, the normal displacement, w = [ - JV 2 n C o m b i n i n g equations / w , can be shown as, n - Ni Ni N } < 2 (A2.10) VR (A2.9) and (A2.10) will yield, ' Up x Vp UQ {«,} = - 7Y 2 0 - 0 - N 2 TYi 0 0 - ^ Ni 0 0 Ni ^ 0 0 VQ N 2 UR VR US VS 287 (A2.ll) T h i s takes the form, {w} = [B ] {8} (2.12) s Now, the elastic energy stored in the slip element due to the applied forces, {/}, is given by te=\ M {/} f T Using the relationships in equations (A2.12) a n d <t>E=\ [B ) j\s} T S dl (A2.13) (A2.2), [k] T <pE can be expressed as, [B }{6} s dl (A2.U) {6} (A2.15) T h i s can be arranged as, f [B ] a Jo [k] T [B ] t dl Therefore, the stiffness matrix, [K ], of the slip element can be deduced as, m [K ] = m f [B } L s Jo [k] [B ] dl T (42.16) 3 T h a t is, \K,^} = (42.17) f [K] L Jo where, [~N 2 0 " 0 -N 0 -Ni 2 0 \K] = Ni 0 0 iVi 0 N 2 . 0 K 0 o K n n -N 0 0 -N 2 N . 2 2 -tfi 0 Ni 0 N 0 -Ni 0 Ni 0 2 0 N 2 (A2.18) Equations (42.17) and (42.18) indicate that the following integrals have to be evaluated in order to define terms in [K ], m 288 / L 0 Nf dl, # Ni N dl and ft 7Y dl. 2 2 2 Now, ( Nidi JO JQ L NiN dl = 6 (A2.20) 2 o Nidi L (A2.19) 3 1 L [\l-L)*dl = JO JO Li L Nidi ' (A2.17) through an m K 0 0 2K 0 K K 0 2K, 0 0 K 0 2K 0 0 2K 0 a n n 0 - K, 0 . can be shown as, [K ] 0 3 - (A2.21), 2K t \K ] = (A2.21) 3 Jo Using equations Li 2 L Nidi = - L o f f\h dl = L - 2K S n - 2K n - 0 - - s 2K n K n 289 - 0 2K S - a 0 - 2K n 0 2K - A; - 2A„ 0 K. 0 - K n 0 0 2A„ A, 0 K, 0 2K An 0 0 K 0 2K S 0 K, 0 K 0 n 0 K 0 2K, 0 - n a n n (A2.22) Fig. A2 • 1 Definition of Slip Element 290 Publications 1. F i n n , W.D. Liam, Yogendrakumar, M. and Y o s h i d a , N., "Dynamic N o n l i n e a r Hysteretic E f f e c t i v e Stress Analysis i n Geotechnical Engineering", I n v i t e d S t a t e - o f - t h e - A r t Address, S i x t h I n t e r n a t i o n a l Conference on Numerical Methods i n Geomechanics t o be h e l d i n Innsbruck, A u s t r i a , A p r i l 1988. 2. F i n n , W.D. Liam and Yogendrakumar, M., " S e i s m i c S o i l - S t r u c t u r e I n t e r a c t i o n " , Proceedings o f the P a c i f i c Conference on Earthquake E n g i n e e r i n g , Auckland, New Zealand, August 1987. 3. F i n n , W.D. Liam and Yogendrakumar, M., " C e n t r i f u g a l M o d e l l i n g and A n a l y s i s o f S o i l - S t r u c t u r e I n t e r a c t i o n " , Proceedings o f the F i f t h Canadian Conference on Earthquake E n g i n e e r i n g , Ottawa, Canada, J u l y 1987. 6-8, 4. F i n n , W.D. Liam, Yogendrakumar, M., Y o s h i d a , N. and Y o s h i d a , H., " A n a l y s i s o f Pore P r e s s u r e s i n S e i s m i c C e n t r i f u g e T e s t s " , Proceedings o f the T h i r d I n t e r n a t i o n a l Conference on S o i l Dynamics and Earthquake E n g i n e e r i n g , P r i n c e t o n , N.J., U.S.A., June 22-24, 1987. 5. F i n n , W.D. Liam, Yogendrakumar, M. and N i c h o l s , A., " S e i s m i c Response A n a l y s i s : P r e d i c t i o n and Performance", I n v i t e d S t a t e - o f - t h e - A r t Address, Proceedings o f the I n t e r n a t i o n a l Symposium on P r e d i c t i o n and Performance i n G e o t e c h n i c a l E n g i n e e r i n g , C a l g a r y , Canada, 1987, E d i t o r s : R.C. J o s h i and F . J . G r i f f i t h s . 6. F i n n , W.D. Liam, Yogendrakumar, M., Y o s h i d a , N. and Y o s h i d a , H., " V e r i f i c a t i o n o f Dynamic S o i l - S t r u c t u r e I n t e r a c t i o n A n a l y s i s " , Proceedings o f the Seventh Japan Earthquake E n g i n e e r i n g Symposium, Tokyo, Japan, December 1986. 7. F i n n , W.D. Liam, Steedman, R.S., Yogendrakumar, M. and L e d b e t t e r , R.H., "Seismic Response o f G r a v i t y S t r u c t u r e s i n a C e n t r i f u g e " , Proceedings o f the Seventeenth O f f s h o r e Tech. Conference, OTC paper #4885, Houston, Texas, U.S.A., May 1985. 8. F i n n , W.D. Liam, S i d d h a r t h a n , R. and Yogendrakumar, M., "Response o f C a i s s o n R e t a i n e d and Tanker I s l a n d s t o Waves and Earthquakes", • Proceedings o f the 36th Canadian G e o t e c h n i c a l Conference, Vancouver, Canada, June 22, 1983.
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Dynamic soil-structure interaction : theory and verification Yogendrakumar, Muthucumarasamy 1988
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Title | Dynamic soil-structure interaction : theory and verification |
Creator |
Yogendrakumar, Muthucumarasamy |
Publisher | University of British Columbia |
Date | 1988 |
Date Issued | 2010-10-15 |
Description | A nonlinear effective stress method of analysis for determining the static and dynamic response of 2-D embankments and soil-structure interaction systems is presented. The method of analysis is incorporated in the computer program TARA-3. The constitutive model in TARA-3 is expressed as a sum of a shear stress model and a normal stress model. The behavior in shear is assumed to be nonlinear and hysteretic, exhibiting Masing behavior under unloading and reloading. The response of the soil to uniform all round pressure is assumed to nonlinearly elastic and dependent on the mean normal effective stresses. The porewater pressures required in the dynamic effective stress method of analysis are obtained by the Martin-Finn-Seed porewater pressure generation model modified to include the effect of initial static shear. During dynamic analysis, the effective stress regime and consequently the soil properties are modified for the effect of seismically induced porewater pressures. A very attractive feature of TARA-3 is that all the parameters required for an analysis may be obtained from conventional geotechnical engineering tests either in-situ or in laboratory. A novel feature of the program is that the dynamic analysis can be conducted starting from the static stress-strain condition which leads to accumulating permanent deformations in the direction of the smallest residual resistance to deformation. The program can also start the dynamic analysis from a zero stress-zero strain condition as is done conventionally in engineering practice. The program includes an energy transmitting base and lateral energy transmitting boundaries to simulate the radiation of energy which occurs in the field. The program predicts accelerations, porewater pressures, instantaneous dynamic deformations, permanent deformations due to the hysteretic stress-strain response, deformations due to gravity acting on the softening soil and deformations due to consolidation as the seismic porewater pressures dissipate. The capability of TARA-3 to model the response of soil structures and soil-structure interaction systems during earthquakes has been validated using data from simulated earthquake tests on a variety of centrifuged models conducted on the large geotechnical centrifuge at Cambridge University in the United Kingdom. The data base includes acceleration time histories, porewater pressure time histories and deformations at many locations within the models. The program was able to successfully simulate acceleration and porewater pressure time histories and residual deformations in the models. The validation program suggests that TARA-3 is an efficient and reliable program for the nonlinear effective stress analysis of many important problems in geotechnical engineering for which 2-D plane strain representation is adequate. |
Subject |
Soil Structure Embankments |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-10-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062504 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/29222 |
Aggregated Source Repository | DSpace |
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