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Dynamic soil-structure interaction : theory and verification Yogendrakumar, Muthucumarasamy 1988-12-31

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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. 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D L i a m . , (1975),  *DESRA-1:  Program for D y n a m i c Effective  Stress Response Analysis of Soil Deposits including Liquefaction Evaluation , Soil M e c h a n ics Series, No.36, Dept.  of C i v i l Engineering, University of British C o l u m b i a , Vancouver,  B . C . , Canada.  Lee, M . K . W . and F i n n , W . D . L i a m (1978) " D E S R A - 2 : Analysis of Soil Deposits with Energy Liquefaction Potential"  Transmitting  Soil Mechanics Series Report  D y n a m i c Effective Stress Response Boundary  Including Assessment  of  N o . 38, Dept of C i v i l Engineering,  University of British C o l u m b i a , Vancouver, C a n a d a .  Lysmer,  J . , and Kuhlemeyer,  R . L . , (1969),  "Finite D y n a m i c M o d e l for Infinite  Media",  Journal of the Engineering Mechanics Division, A S C E , V o l . 95, E M 4 , Sept. p p . 859-877  Lysmer, J . , U d a k a , T . , T s a i , C . F . , and Seed, H . B . , (1975), " F L U S H : A Computer for Approximate 3 - D Analysis of Soil-Structure 75-30, Earthquake  Interaction Problems",  Program  Report N o . E E R C  Engineering Research Center, University of California, Berkeley,  fornia.  277  Cali-  Lysmer, J . , and Wass, G . , (1972), "Shear Waves in Plane Infinite Structures", Journal of Engineering Mechanics Division, A S C E , V o l . 98, E M I , F e b . , p p . 85-105.  M a r t i n , G . R . , F i n n , W . D . L i a m . , and Seed, H . B . , (1975), "Fundementals  of Liquefaction  U n d e r C y c l i c L o a d i n g " , Journal of the Geotechnical Engineering Division, A S C E , V o l . 101, G T 5 , M a y , p p 423-438.  M a s i n g , G . , (1926) "Eigenspannungen International  and Verfestigung B e i m M e s s i n g " , Proceedings, 2nd  Congress of A p p l i e d Mechanics, Zurich, Switzerland.  National Research Council of the United States (1982), Report  by Committee  on Earthquake  Engineering  "Earthquake  Research,  Engineering -1982",  National A c a d e m y  Press,  Washington, D . C .  National Research Council of the U n i t e d States (1985), "Liquefaction of Soils D u r i n g E a r t h quakes", Report by Committee on Earthquake Engineering, National A c a d e m y Press, Washington, D . C .  Naylor, D . J . , and Pande,  G . N . , (1981),  "Finite Elements  in Geotechnical  Engineering",  Rainbow - B r i d g a Book C o . L t d .  Newmark, N . M . , (1959), " A M e t h o d of C o m p u t a t i o n for Structural D y n a m i c s " , Journal of the Engineering Mechanics Division, A S C E , V o l . 85, E M 3 , July.  Newmark, N . M . , (1965), "Effects of Earthquake on Dams and E m b a n k m e n t s " , 5th Rankine Lecture, Geotechnique 15, No.2, p p . 139-160.  Newmark, N . M . , and Rosenblueth, E . , (1971), "Fundermentals of Earthquake  Engineering",  Prentice-Hall Inc., Englewood, Cliff, N . J . , p p . 162-163.  O z a w a , Y . , and D u n c a n , J . M . , (1973),  " I S B I L D : A Computer  Static Stresses and Movements in E m b a n k m e n t s " ,  Program  for Analysis of  Geotechnical Engineering Research Re-  port N o . T E - 7 3 - 4 , Department of C i v i l Engineering, University of California, Berkeley, D e c .  Prevost, J . H . , (1981), " D Y N A F L O W : A Nonlinear Transient Finite Element Analysis Program",  Department  of C i v i l Engineering,  Princeton  University,  Princeton,  New Jersey,  U.S.A.  Roesset, J . M . , and Ettouney, International  M . M . , (1977),  "Transmitting  Boundaries:  A Comparison",  Journal for Numerical and A n a l y t i c a l M e t h o d s in Geomechanics,  151 - 176.  278  V o l . 1, p p .  Robertson, P . K . , (1982), "In-situ Testing of Soils with Emphasis on Its A p p l i c a t i o n to L i q uefaction Assessment",  P h . D Thesis, Department of C i v i l Engineering, University of British  C o l u m b i a , Vancouver, C a n a d a .  Roscoe, K . H , (1968), "Soils and M o d e l Tests", Journal of Strain Analysis, N o . 3, p p . 57-64.  Schnabel, P . B . , Lysmer, J . , and Seed, H . B . , (1972), " S H A K E : A C o m p u t e r Program for Earthquake  Response Analysis of Horizontally Layered Sites",  Earthquake  Engineering Research Center, University of California Berkeley, Dec.  Schofield, A . N . , (1981), International  " D y n a m i c and Earthquake  Geotechnical  Conference on Recent Development in Earthquake  namics, Missouri, U . S . A . , A p r i l  Report  No. E E R C  Centrifuge  Modelling",  Engineering and Soil D y -  28-May2.  Scott, R . F , (1978), "Summary Specialty Session 7 - M o d e l i n g " Proceedings, ference on Earthquake  72-12,  Engineering and Soil D y n a m i c s , A S C E ,  Specialty C o n -  V o l . I l l , Pasadena, C A ,  June 19-22,  Seed, H . B . , (1979a),  "Considerations  in the Earthquake-Resistant  Design of E a r t h and  Rockfill D a m s " , 19th Rankine Lecture, Geotechnique 29, N o . 3, p p . 215-263. Seed, H . B . , (1979b), "Soil Liquefaction and C y c l i c Mobility Evaluation for Level G r o u n d D u r i n g Earthquakes",  Journal of Geotechnical Engineering Division, A S C E , V o l . 105, N o .  G T 2 , p p . 201-255. Seed, H . B . , and Idriss, I . M . , (1970), "Soil M o d u l i and D a m p i n g Factors for D y n a m i c R e sponse A n a l y s i s " , Report N o . E E R C 70-10, Earthquake Engineering Research Center, U n i v . of California, Berkeley,  December.  Seed, H . B . , a n d Lee, K . L . , (1966),  Liquefaction of Saturated Sands D u r i n g Cyclic L o a d -  i n g " , Journal of the Soil Mechanics and Foundation Engineering Division, A S C E , V o l . 92, N o , S M 6 , November. Serff, N . , Seed, H . B . , M a k d i s i , F . I . , and C h a n g , C . Y . , (1976), formations  of E a r t h  D a m s " , Report  No. E E R C  R . , (1984),  76-4, Earthquake  Induced D e -  Engineering  Research  " A Two-Dimensional Nonlinear Static and D y n a m i c  Response  Center, University California, Berkeley, Siddharthan,  "Earthquake  Sept.  Analysis of Structures", P h . D Thesis, Department of C i v i l Engineering, University of British C o l u m b i a , Vancouver, C a n a d a .  Siddharthan, R . , and F i n n , W . D . L i a m . , (1982),  "TARA-2:  Static and D y n a m i c Response A n a l y s i s " , Department  279  T w o Dimensional Nonlinear  of C i v i l Engineering, University of  British C o l u m b i a , Vancouver, C a n a d a .  Simons, H . A . , and R a n d o l p h , M . F . , (1986), "Short C o m m u n i c a t i o n :  Comparison of Trans-  mitting Boundaries in D y n a m i c Finite Element Analyses using Explicit T i m e Integration", International  Journal for Numerical and A n a l y t i c a l M e t h o d s in Geomechanics, V o l . 10, p p .  329-342.  Steedman,  R . S . , (1985),  "Seismically  Induced Settlements in Soils:  trifuge M o d e l Tests, RSS90 and R S S 9 1 " , Engineering Department,  D a t a Report of C e n Cambridge University,  Cambridge, E n g l a n d .  Steedman,  R . S . , (1986), " E m b e d d e d Structure on Sand Foundation:  D a t a Report of C e n -  trifuge M o d e l Tests, RSS110 and R S S 1 1 1 " , Engineering Department, Cambridge University, Cambridge, E n g l a n d .  Tatsuoka,  F . , and Haibara, O . , (1985),  Lubricated Surfaces",  "Shear  Resistance Between  Sand and Smooth or  Soils and Foundations, V o l . 25, N o . 1, p p . 89-98.  Uesugi, M . , and K i s h i d a , H . , (1986), "Influential Factors of Friction Between Steel and D r y Sands", Soils and Foundations, V o l . 26, N o . 2, p p . 33-46. V a i d , Y . P . , and C h e r n , J . C . , (1981),  "Effect  of Static Shear  on resistance to Liquefac-  t i o n " , Soil Mechanics Series N o . 51, Department of C i v i l Engineering, University of British C o l u m b i a , Vancouver, C a n a d a . V a i d , Y . P . , a n d F i n n , W . D . L i a m . , (1979), "Effect of Static Shear on Liquefaction  Poten-  t i a l " , Journal of the Geotechnical Engineering Division, A S C E , V o l . 105, G T 1 0 , O C T . , p p . 1233-1246. Vaziri-Zanjani, H . H . , (1986), "Nonlinear Temperature  and Consolidation Analysis of Gassy  Soils", P h . D Thesis, Department of C i v i l Engineering, University of British C o l u m b i a , V a n couver,  Canada.  Wedge, N . E . , (1977), "Problems in Nonlinear Analysis of Movements in Soils", M . A . S c T h e sis, Department  of C i v i l Engineering, University of British C o l u m b i a , Vancouver, C a n a d a .  Wilson, E . L . , Farhomand,  I., a n d Bathe, K . J . , (1973), " N o n - L i n e a r D y n a m i c Analysis of  Complex Structures", International  Journal of Earthquake Engineering and Structural D y -  namics, V o l . 1, p p 241-252.  Yogendrakumar, Simple Shear  M . and F i n n , W . D . L i a m . , (1984),  Tests on D r y and Saturated Sands"  British C o l u m b i a , C a n a d a . 280  "SIMCYC:  Simulation of the Cyclic  Soil Dynamics G r o u p , University of  Yogendrakumar, M . , and F i n n , W . D . L i a m . , (1986), " C - P R O : A Program for Evaluating the Constants in the M a r t i n - F i n n - S e e d Porewater Pressure M o d e l " Soil Dynamics G r o u p , D e partment of C i v i l Engineering, University of British C o l u m b i a , Vancouver, C a n a d a , November.  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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