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The verification of relationships for effective stress method to evaluate liquefaction potential of saturated… Bhatia, Shobha K. 1982

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THE VERIFICATION OF RELATIONSHIPS FOR EFFECTIVE STRESS METHOD TO EVALUATE LIQUEFACTION POTENTIAL OF SATURATED SANDS b y SHOBHA K. BHATIA B.E., University of Roorkee, Roorkee, India, 1971 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY in the Department of C i v i l Engineering Faculty of Applied Science We accept this thesis as conforming to the required standards ( T ) Shobha K. Bhatia , 1982 THE UNIVERSITY OF BRITISH COLUMBIA November 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f CCw^ r^.v/vg\uvcexv The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 2-2 \\ Movj-euuta-u; t DE-6 (3/81) THE VERIFICATION OF RELATIONSHIPS FOR EFFECTIVE STRESS METHOD TO EVALUATE LIQUEFACTION POTENTIAL OF SATURATED SANDS ABSTRACT The constitutive relationships proposed by Finn, Lee and Martin (1977) for the effective stress analysis of saturated sands during earthquakes are studied. The basic assumptions of their porewater pres-sure model appears to be well founded. There is a strong verification of a unique relationship between volumetric strain in drained tests and pore-water pressures in undrained tests for both normally and overconsolidated sands. An important point to emerge from this study is that the rebound modulus used in converting the volumetric strains to porewater pressures should be measured under dynamic conditions. The porewater pressure model predicts successfully the porewater pressure response under undrained con-ditions for uniform and irregular cyclic strain and stress histories. When the porewater pressure model is coupled with a non-linear stress-strain relationship in effective stress analysis, i t predicts rea-l i s t i c porewater pressure response in undrained tests for cyclic stress histories representative of earthquake loading. Results suggest that strain-hardening effects do not occur unless the sand i s allowed to drain. A new porewater pressure model based on endochronic theory is presented in which the porewater pressures are directly related to dynamic response parameters. This approach bypasses the need for converting volu-metric strains to porewater pressures. The proposed formulation relates porewater pressure to a single monotonically increasing function of a damage parameter. This parameter allows the data from constant strain or stress cyclic loading tests to be applied directly to predict the porewater pressure generated in the f i e l d by irregular stress or strain histories due to earthquakes. This formulation is an extremely efficient way of represen-ting a large amount of data and can be easily coupled with dynamic response i i i i i analysis to perform effective stress analysis. This study is based on extensive experimental data on Ottawa sand, crystal s i l i c a sand and Toyoura sand. In total, one hundred and f i f t y tests were performed for this study. The tests were performed under cyclic simple shear conditions using Roscoe type simple shear appa-ratus. Dry sand was used for both the drained and constant volume tests conducted for this study. The tests were performed under both stress con-trolled and strain controlled conditions. W.D. Liam Finn, Thesis Supervisor. TABLE OF CONTENTS PAGE ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS ACKNOWLEDGEMENTS CHAPTER 1 - INTRODUCTION 1 1.1 Scope of t h i s Research 5 CHAPTER 2 - REVIEW OF CONSTITUTIVE RELATIONSHIPS USED 8 FOR VARIOUS EFFECTIVE STRESS ANALYSES 2.1 CRITICAL REVIEW OF CONSTITUTIVE RELATIONSHIPS 8 2.1.1 Stress Path Models 9 2.1.2 Volumetric S t r a i n Models 13 2.1.3 Endochronic Models 14 2.1.4 Kinematic Hardening Models 23 2.1.5 Empirical Models 24 2.2 DISCUSSION 24 CHAPTER 3 - CONSTITUTIVE RELATIONS FOR THE EFFECTIVE STRESS 25 MODEL OF FINN, LEE AND MARTIN 3.1 PORE PRESSURE MODEL 26 3.1.1 Volume Change C h a r a c t e r i s t i c s Under Drained 29 C y c l i c Loading 3.1.2 One-Dimensional Volumetric Unload ing 33 Ch a r a c t e r i s t i c s 3.2. STRESS-STRAIN RELATIONSHIP 36 3.2.1 I n i t i a l Loading 38 3.2.2 Unloading and Reloading 38 3.2.3 Influence of Hardening and Porewater 40 Pressure 3.3 VERIFICATION OF CONSTITUTIVE RELATIONSHIPS 45 3.4 DISCUSSION 47 iv i i v v i i v i i i x v i xix V PAGE CHAPTER 4 - VERIFICATION OF CONSTITUTIVE RELATIONSHIPS 49 FOR EFFECTIVE STRESS MODEL 4.1 VERIFICATION OF FUNDAMENTAL ASSUMPTIONS 50 4.2 EVALUATION OF THE PORE PRESSURE PREDICTIVE 54 CAPACITY OF THE PORE PRESSURE MODEL 4.2.1 Rebound C h a r a c t e r i s t i c s of Sand 58 4.2.1.1 Comparison between s t a t i c and dynamic 60 rebound modulus 4.2.2 Volume Change C h a r a c t e r i s t i c s of Sand 63 Under C y c l i c Loading Conditions 4.3 PORE PRESSURE PREDICTION 66 4.3.1 Pore Pressure P r e d i c t i o n for Irregular 71 St r a i n History 4.4 PORE PRESSURE PREDICTION FOR SAMPLES WITH 75 PREVIOUS STRAIN HISTORY 4.5 DISCUSSION 84 CHAPTER 5 - VERIFICATION OF CONSTITUTIVE RELATIONSHIPS 85 FOR OVERCONSOLIDATED SAND 5.1 COMPARISON BETWEEN NORMALLY AND OVER- 87 CONSOLIDATED SAND BEHAVIOUR 5.2 VERIFICATION OF THE CONSTITUTIVE 92 RELATIONSHIPS 5.2.1 Volume Change C h a r a c t e r i s t i c s 93 5.2.2 Dynamic Rebound Modulus 98 5.2.3 I n i t i a l Shear Modulus and Shear Strength 104 5.2.4 Pore Pressure P r e d i c t i o n 106 5.3 DISCUSSION 108 CHAPTER 6 - POREWATER PRESSURE MODEL BASED ON 109 ENDOCHRONIC THEORY 6.1 ENDOCHRONIC THEORY 110 6.2 ENDOCHRONIC FORMULATION OF PORE PRESSURE 112 DATA 6.2.1 Inverse Transformation 123 6.2.2 Endochronic Representation of Porewater 126 Pressure Data for Various Relative Densi-t i e s , Overconsolidation Ratios and Types of Sands 6.2.2.1 Various r e l a t i v e d e n s i t i e s 126 v i PAGE 6.2.2.2 Overconsolidation r a t i o s 130 6.2.2.3 Endochronic representation f o r other sands 130 6.2.3 V e r i f i c a t i o n of Endochronic Pore Pressure 137 Formulation 6.2.4 Endochronic Representation of Porewater 140 Pressure from Stress Controlled Undrained Tests 6.2.4.1 Pore pressure as a function of stress path 144 6.2.4.2 Pore pressure as a function of s t r a i n path 148 6.3 DISCUSSION 150 CHAPTER 7 - SUMMARY AND CONCLUSIONS 7.1 SUMMARY 152 7.2 CONCLUSIONS 152 7.3 SUGGESTIONS FOR FUTURE RESEARCH WORK 155 REFERENCES 157 APPENDIX I 163 APPENDIX II 175 APPENDIX III 179 LIST OF TABLES TABLE PAGE 4.1 Volume Change Constants for Ottawa Sand (C-109) 68 4.2 Increase i n Shear Modulus and Shear Strength 81 Due to S t r a i n History 5.1 Experimental k Values for Various OCR 90 o 5.2 Relationship Between k' for Overconsolidated 90 Sample to Normally Consolidated Sample 6.1 Endochronic Constants for Various Relative 129 Densities 6.2 Endochronic Constants for Various Types of Sands 136 6.3 Pore Pressure C a l c u l a t i o n f or Irregular S t r a i n 139 History Using the Endochronic Formulation, Ottawa Sand, D = 45% r 6.4 C a l c u l a t i o n of Porewater Ratio for Stress 143 Controlled Undrained Tests on Ottawa Sand at D = 45% at x/o' = 0.089 r vo I I - l Properties of Sands 178 v i i LIST OF FIGURES FIGURE 'PAGE 2.1 S t r e s s - S t r a i n Relationship for Sand 10 2.2 Y i e l d L o c i for Loose Sand (After Ishihara et a l , 1975) 10 2.3 Comparison of Two Kinds of Y i e l d L o c i 10 (After Ishihara et a l , 1974) 2.4 E l l i p t i c a l Stress Path for Undrained Loading Test 12 (After Ghaboussi and Dikmen, 1978) 2.5 Relationship Between Material Parameter X and 12 Relative Density (After Ghaboussi and Dikmen, 1978) 2.6 Volumetric S t r a i n vs. Number of Cycles for C r y s t a l 17 S i l i c a Sand (After Cuellar et a l , 1977) 2.7 E l a s t i c and I n e l a s t i c Stress Increment 17 (After Bazant and Krizek, 1976) 2.8 P r e d i c t i o n of the Hysteretic Loops for Crystal S i l i c a 19 Sand (After Cuellar et a l , 1977) 2.9 Pore Pressure vs. Number of Cycles i n Constant Stress 19 Undrained Test (After Bazant and Krizek, 197 6) 2.10 Volumetric S t r a i n vs. Length of S t r a i n Path 21 (After Zienkiewicz et a l , 1978) 2.11 Volumetric S t r a i n vs. Damage Parameter 21 (After Zienkiewicz et a l , 1978) 2.12 Idealized Y i e l d and P l a s t i c P o t e n t i a l Surface 22 (After Zienkiewicz et a l , 1978) 2.13 Two-Surface Model Showing Unloading and Isotropic 22 Consolidation (After Mroz et a l , 1979) 3.1 Schematic I l l u s t r a t i o n of Mechanism of Porewater 27 Pressure Generated During C y c l i c Loading (After Seed, 1976) 3.2 Volumetric S t r a i n vs. C y c l i c Shear S t r a i n Amplitude 31 (After Seed and S i l v e r , 1971) 3.3 Void Ratio Change vs. Frequency i n C y c l i c S t r a i n Tests 31 on Dry and Saturated Drained Samples (After Youd, 1972) v i i i ix FIGURE PAGE 3.4 Volumetric S t r a i n vs. Number of Cycles of Constant 32 St r a i n (After Martin et a l , 1975) 3.5 Incremental Volumetric S t r a i n vs. C y c l i c Shear 32 St r a i n for Various Volumetric S t r a i n (After Martin et a l , 1975) 3.6 Generalised One-Dimensional Unloading Curves 35 (After Martin et a l , 1975) 3.7 V e r t i c a l E f f e c t i v e Stress vs. Recoverable 35 Volumetric S t r a i n f o r Monterey Sand (After Seed et a l , 1973) 3.8 Increase i n Av. Shear Modulus with Various Number 37 of Cycles of Constant Shear S t r a i n (After Lee, 1975) 3.9 Str e s s - S t r a i n Relationship by Finn, Lee and Martin 39 (1977) 3.10 Av. Shear Modulus vs. Shear S t r a i n for Various 42 Volumetric Strains 3.11 Comparison Between Predicted and Measured Stress- 44 S t r a i n Curve 3.12 Volumetric S t r a i n V a r i a t i o n with C y c l i c Shear 46 St r a i n i n Drained Tests (After Finn et a l , 1980) 4.1(a) V e r t i c a l E f f e c t i v e Stress vs. Volumetric S t r a i n f o r 51 Str a i n Controlled Undrained Test 4.1(b) V e r t i c a l E f f e c t i v e Stress vs. Volumetric S t r a i n f o r 51 St r a i n Controlled Drained Test 4.2 Volumetric S t r a i n vs. Number of Cycles for Constant 53 Cy c l i c Shear S t r a i n Test on Loose Ottawa Sand 4.3 Relationship Between Volumetric Strains and 55 Porewater Pressures i n Constant S t r a i n C y c l i c Simple Shear Tests, D r = 45% 4.4 Relationship Between Volumetric Strains and 56 Porewater Pressures i n Constant S t r a i n C y c l i c Simple Shear Tests, D r = 60% 4.5 Relationship Between Volumetric Strains and 57 Porewater Pressures i n Constant S t r a i n C y c l i c Simple Shear Tests, 0 ^ o = 300 kN/m2 X FIGURE PAGE 4.6 V e r t i c a l E f f e c t i v e Stress vs. Volumetric S t r a i n 59 During Dynamic Unloading 4.7 V e r t i c a l E f f e c t i v e Stress vs. Recoverable 59 Volumetric S t r a i n During S t a t i c Unloading Conditions 4.8 Rebound of Ottawa Sand Under Various Load ing 61 Conditions 4.9 Ratio of Dynamic Recoverable S t r a i n to S t a t i c 64 Recoverable S t r a i n f o r Various Values of E f f e c t i v e V e r t i c a l Stress 4.10 Dynamic Unloading Curves from Three I n i t i a l 64 V e r t i c a l E f f e c t i v e Stress f o r Ottawa Sand 4.11 V e r t i c a l E f f e c t i v e Stress vs. Dynamic 65 Recoverable S t r a i n f o r Various Relative Densities 4.12 Incremental Volumetric S t r a i n vs. Shear S t r a i n 67 Amplitude f o r Various Levels of Cummulative Volumetric S t r a i n 4.13 Predicted and Measured Porewater Pressure i n 70 Constant Stress C y c l i c Simple Shear Tests, D r = 45% 4.14 Predicted and Measured Porewater Pressures i n 72 Constant Stress C y c l i c Simple Shear Tests, D r = 45% 4.15 Predicted and Measured Porewater Pressures i n 73 Constant Stress C y c l i c Simple Shear Tests, D r = 60% 4.16 Comparison of Calculated and A n a l y t i c a l Pore 74 Pressure Ratio f o r Irregular S t r a i n History 4.17 C y c l i c Stress Ratio vs. Number of Cycles to 7 6 Liquefaction for Samples with Previous S t r a i n History 4.18 Horizontal E f f e c t i v e Stress vs. V e r t i c a l 78 E f f e c t i v e Stress f o r C y c l i c Shear Controlled Undrained Test on Samples with Previous Shear S t r a i n History 4.19 Predicted and Measured Porewater Pressures i n (a&b) a Sand with Previous Loading History 80,83 x i FIGURE PAGE 5.1 C y c l i c S h e a r / i n i t i a l Mean E f f e c t i v e Stress vs. 86 Number of Cycles to I n i t i a l Liquefaction for Various k 0 Values (After Ishibashi and Sherif, 1974) 5.2 C y c l i c Shear S t r e s s / I n i t i a l Mean E f f e c t i v e Stress 86 vs. Number of Cycles to I n i t i a l Liquefaction f o r Various OCR (After Seed and Peacock, 1971) 5.3 V a r i a t i o n of E f f e c t i v e Horizontal and V e r t i c a l 89 Stresses During I n i t i a l Consolidation, S t a t i c Unloading and C y c l i c Loading for Samples with OCR = 1,2,3 and 4 5.4 C y c l i c Shear S t r e s s / I n i t i a l V e r t i c a l Confining 91 Stress vs. Number of Cycles to I n i t i a l L iquefaction for Various Values of OCR 5.5 C y c l i c Shear S t r e s s / i n i t i a l Mean Normal Stress vs. 92 Number of Cycles to I n i t i a l Liquefaction for Various Values of OCR 5.6 Decrease i n the Ratio of Horizontal to V e r t i c a l 94 E f f e c t i v e Stress vs. Number of Cycles f o r Various C y c l i c Shear S t r a i n Amplitude for Overconsolidated Sample 5.7 Volumetric S t r a i n Behaviour f o r F i r s t Two Cycles 96 of Shearing f o r an Overconsolidated Sample 5.8 Volumetric S t r a i n vs. Cycles of Constant Shear 96 S t r a i n Amplitude y - 0.10% for Various OCR 5.9 Incremental Volumetric S t r a i n i n F i r s t Cycle f o r 97 Ottawa Sand f or Various OCR Values 5.10(a) Incremental Volumetric S t r a i n vs. Shear S t r a i n 97 Amplitude f o r Various Levels of Volumetric Strains 5.10(b,c) Incremental Volumetric S t r a i n vs. Shear S t r a i n 99 Amplitudes f o r Various Values of Volumetric Strains 5.11(a) Relationship Between Volumetric Strains and Pore- 100 water Pressures i n Constant S t r a i n C y c l i c Simple Shear Tests, OCR = 2 5.11(b) Relationship Between Volumetric Strains and Pore- 102 water Pressures i n Constant S t r a i n C y c l i c Simple Shear Tests, OCR = 3 5.11(c) Relationship Between Volumetric Strains and Pore- 103 water Pressures i n Constant S t r a i n C y c l i c Simple Shear Tests, OCR = 4 x i i FIGURE PAGE 5.12 Average Value of Shear Modulus vs. Mean P a r t i c l e 105 Size for S o i l s with OCR = 1.33 to 2 (After A f i f i and Richart, 1973) 5.13 Av. Shear Modulus vs. Shear S t r a i n f or F i r s t 105 Cycle of Shearing at Various OCR 5.14 C y c l i c Stress Ratio vs. Number of Cycles for 107 I n i t i a l L i q u e f a c t i o n f o r Various OCR Ratios 6.1 Porewater Pressure Ratio vs. S t r a i n Cycles 113 6.2 Porewater Pressure Ratio vs. Natural Logarithm 115 of Length of S t r a i n Path 6.3 Various Values of Transformation Factor, A 118 6.4 Pore Pressure Ratio vs. Natural Logarithm of 119 Damage Parameter 6.5 Porewater Pressure Ratio vs. Damage Parameter 121 6.6 Pore Pressure Ratio vs. Number of S t r a i n Cycles 122 at Various Confining Stresses 6.7 Comparison of Computed and Experimental Pore- 124 water Pressure i n £-plot 6.8 Comparison of Computed and Measured Porewater 125 Pressures i n N-plot 6.9 Porewater Pressure Ratio vs. S t r a i n Cycles of 127 0.20% at Various Relative Densities 6.10 Porewater Pressure Ratio vs. Natural Logarithm 127 of Damage Parameter at Various Relative Densities 6.11 Pore Pressure Ratio vs. Ln (Length of S t r a i n Path) 131 for Various OCR 6.12 Porewater Pressure Ratio vs. Ln (Damage Parameter) 132 for Overconsolidated Sands 6.13 Porewater Pressure Ratio vs. Ln (Damage Parameter) 134 for Various Types of Sands 6.14 Porewater Pressure Ratio vs. S t r a i n Cycles of 135 Y = 0.20% for Various Types of Sands x i i i FIGURE PAGE 6.15 Comparison Between Calculated and Experimental 138 Porewater Pressure Ratios for Irregular S t r a i n History 6.16 Predicted and Measured Porewater Pressure i n 141 Constant Stress C y c l i c Simple Shear Tests, D r = 45% 6.17 Predicted and Measured Porewater Pressure i n 142 Constant Stress C y c l i c Simple Shear Tests, D r = 60% 6.18 Porewater Pressure Ratio vs. Number of Cycles 145 for Various C y c l i c Shear Stress Ratios 6.19 Porewater Pressure Ratio vs. Natural Logarithm 147 of Damage Parameter 6.20 Pore Pressure Ratio vs. C y c l i c Shear S t r a i n 149 During Stress Controlled Undrained Test 6.21 Porewater Pressure Ratio vs. Ln (Damage) 149 Parameter) I - l Constant Volume C y c l i c Simple Shear 166 Apparatus at U.B.C. I- 2 Vibrations Applied to Sand Sample 170 I I - l Grain Size D i s t r i b u t i o n Curves 177 I I I - l Volumetric S t r a i n vs. S t r a i n Cycles for 179 Ottawa Sand III-2 Volumetric S t r a i n vs. Natural Logarithm of 179 Length of S t r a i n Path for Ottawa Sand III-3 Volumetric S t r a i n vs. Ln (Damage Parameter) 181 for Ottawa Sand III-4 Volumetric S t r a i n vs. Damage Parameter for 181 Ottawa Sand III-5 Comparison of Computed and Experimental 182 Volumetric S t r a i n i n £-plot III-6 Comparison of Computed and Experimental 182 Volumetric S t r a i n in N-plot x i v FIGURE PAGE III-7 Av. Shear Stress vs. Number of Cycles for 183 Constant Stress Drained Test III-8 Comparison of Calculated and Experimental 183 Volumetric Strains LIST OF PLATES PLATE PAGE 1-1 C y c l i c Simple Shear Apparatus 164 1-2 Constant Volume Simple Shear Setup 167 1-3 Membrane Stretched Out 169 1-4 Placing Top Plate on Sand Sample 169 I- 5 C y c l i c Simple Shear Apparatus with Recording 172 Equipment I I - l Grain Shapes 176 xv LIST OF SYMBOLS a,b,c = constants a , b , c , d , e , = constants used by Bazant and Krizek (1976) A,B = endochronic constants for pore pressure A,B,C,D = endochronic constants f o r pore pressure A^,A2,A2 = hardening constants A^,B^,C^,D^ = endochronic constants f o r pore pressure B ^ J B ^ J B ^ = hardening constants C^,C2,C^,C^ = volume change constants = bulk compressibility of s o i l skeleton 1/C = secant constrained modulus c D = r e l a t i v e density r e = void r a t i o de. . = d e v i a t o r i c shear s t r a i n (E ), . = dynamic rebound modulus r dynamic E^ = tangent modulus of the one-dimensional unloading curve G = secant shear modulus G = i n i t i a l shear modulus mo G = maximum shear modulus i n n1-*1 c y c l e mm H^JH^JH^JH^ = hardening constants Hz = cycles per second k2,m,n = rebound modulus constants k = r a t i o of l a t e r a l e f f e c t i v e stress to v e r t i c a l e f f e c t i v e stress k = bulk modulus of water w L(a' -a') = c o r r e c t i o n c o e f f i c i e n t of Bazant and Krizek (1976) vo n = porosity x v i x v i i N = number of cycles OCR = overconsolidation r a t i o p 1 = mean e f f e c t i v e normal stress PI = p l a s t i c i t y index q = d e v i a t o r i c stress R = shape factor u = porewater pressure Au = incremental pore pressure u/a' = porewater pressure r a t i o vo a = constant Y = shear s t r a i n Y Q = shear s t r a i n amplitude Y = maximum shear s t r a i n max P = p l a s t i c shear s t r a i n Y Y^ = shear s t r a i n at r e v e r s a l point e , = volumetric s t r a i n / cummulative volumetric s t r a i n vd e = t o t a l recoverable s t r a i n due to unloading vro e = non-recoverable s t r a i n due to i n t e r p a r t i c l e s l i p vso As j = incremental volumetric s t r a i n vd Ae = incremental recoverable s t r a i n vr £ = d e n s i f i c a t i o n v a r i a b l e of Cuellar et a l (1977) n = d i s t o r t i o n v a r i a b l e dn = incremental length of stress path K = damage parameter X = material parameter of Ghaboussi and Dikmen (1978) X = transformation f a c t o r 5 = rearrangement measure or length of s t r a i n path x v i i i d£ = incremental length of strain path 6.. = deviatoric stress a' = vertical effective stress v vo = i n i t i a l vertical effective stress a = effective stress z T = shear stress T / O ' = ratio of cyclic shear stress to i n i t i a l mean mo effective stress T / O ' = shear stress ratio vo T Q = shear stress amplitude x, = shear stress hv x = maximum shear stress mo x = maximum shear stress in cycle mn x = shear stress at reversal point r <j)' = effective angle of shearing resistance 3 / 3 = constrained strain level w z ACKNOWLEDGEMENTS I wish to record my deepest appreciation of the guidance and encouragement offered by Professor W.D. Liam Finn throughout this research, and to his many helpful suggestions in writing this manu-script. I am also grateful to Dr. P.M. Byrne, Dr. R.G. Campanella and Dr. N.D. Nathan for their encouragement and advice and to Dr. Y.P. Vaid and Dr. M.K. Lee for their generous help and invaluable discussions. During my studies Mr. Y. Koga and Mr. B i l l Deacon took personal interest in my progress and were always ready with their unselfish help. Ms. Desiree Cheung not only typed the manuscript but was a best friend to me in time of need. Ms. Monica Gutierrez showed everlasting patience while drafting the figures. Mr. Fred Zurkichen and Mr. Wolfram Schmitt from the C i v i l Engineering workshop provided invaluable assistance with the equipment. Finally, I would like to express my very special thanks to my husband for his support and constant encouragement. The financial support by the Canadian Commonwealth Scholarship and Fellowship Committee is acknowledged. xix CHAPTER 1 INTRODUCTION In 1964 a v i o l e n t earthquake h i t Niigata and Yamagata prefectures i n Japan i n f l i c t i n g damage on the c i t y of Niigata f a r out of proportion of the magnitude (7.5) of the earthquake. In Niigata c i t y , where sand deposits i n the lowland areas are widespread, the damage was pr i m a r i l y associated with l i q u e f a c t i o n of loose sand deposits. B u i l d -ings not embedded deep i n fi r m s t r a t a sank and t i l t e d . Underground structures, such as manholes, sewage conduits and septic and storage tanks f l o a t e d up a meter or two above ground l e v e l . On l e v e l ground, sand flows and mud volcanoes ejected water and sand 2 to 3 minutes a f t e r the earthquake. Sand deposits 20-30 cm thick covered the en t i r e c i t y as i f the whole area had been devastated by a sand flood (Kawasumi, 1964). In the same year, an earthquake of magnitude 8.3 occurred i n southern Alaska. This c i t y i s located on a d e l t a composed of s i l t and f i n e sand occurring as beds and st r i n g e r s within coarser sand and gravel deposits which l i q u e f i e d and resulted i n a massive s l i d e involving approximately 18,000,000 cubic yards of material extending inland above 500 f t from the coast l i n e and destroying the harbour and nearshore f a c i l i t i e s . Several s i m i l a r case h i s t o r i e s where l i q u e f a c t i o n has been the c e n t r a l factor i n large lan d s l i d e s have been documented by Seed (1968) . In 1936, A. Casagrande explained f or the f i r s t time the phenomenon of l i q u e f a c t i o n induced by s t a t i c loading, enabling engineers to understand the massive Fort Peck Dam s l i d e of 1936. F l o r i n and Ivanov 2 (1961) presented r e s u l t s obtained i n shaking table experiments where the increase of porewater. pressure due to the c y c l i c loading i n saturated sands was demonstrated. But i t was not u n t i l 1964 and the event at Niigata and Alaska that the process of l i q u e f a c t i o n due to c y c l i c loading gained widespread atte n t i o n . Since t h i s time engineers' spon-taneous i n t e r e s t i n the process has been increased by the need to assess the safety of foundations of c r i t i c a l structures such as nuclear power plants and p i p e l i n e s which are t y p i c a l l y located i n s o i l s which are susceptible to l i q u e f a c t i o n . Study of the l i q u e f a c t i o n process has yielded several methods of evaluating l i q u e f a c t i o n p o t e n t i a l . These f a l l into two groups: the empirical methods and the a n a l y t i c a l methods. Empirical methods are based on observations of the performance of sand deposits i n locations where earthquakes have taken place. For example, r e l a t i o n s h i p s l i k e those proposed by Kishida (1966) and Ohsaki (1966) are based on the Standard Penetration Resistance of the sand deposits i n the Niigata area. Such r e l a t i o n s h i p s , however, cannot be applied to other s i t e s , where shaking i n t e n s i t y or water tables may be at d i f f e r e n t depths than that i n the Niigata area. More recently, Seed et a l . (1975) have presented a c o r r e l a t i o n where the values of shear stress r a t i o known to be asosciated with l i q u e f a c t i o n or no l i q u e f a c t i o n i n the f i e l d are plotted as a function of the corrected average Standard Penetration Resistance of the s o i l deposit involved. The shear stresses at any depth produced during earthquakes are estimated by the method proposed by Seed and I d r i s s (1967) . Observation of the conditions under which l i q u e f a c t i o n has occurred i n previous earthquakes w i l l always be valuable when predicting the probable performance of a saturated sand deposit during earthquakes. However, i t i s l i m i t e d i n three ways: 1. Very meagre information i s a v a i l a b l e on l i q u e f a c t i o n occurring during any earthquake; 2. The method of observation cannot accommodate fa c t o r s such as the duration of the earthquake or the p o s s i b i l i t y of drainage and r e d i s t r i b u t i o n of pore water pressure; and 3. The Standard Penetration Resistance of s o i l i s not always r e l i a b l y determined i n the f i e l d . I t s values may vary due to boring and sampling conditions. Available a n a l y t i c a l methods can be grouped into t o t a l stress methods and e f f e c t i v e stress methods. The important d i f f e r e n c e between the two types of methods i s that t o t a l stress methods do not take e x p l i c i t account of change i n porewater pressure. The t o t a l stress analysis per-formed to evaluate l i q u e f a c t i o n involves two independent determinations: 1) an evaluation of the c y c l i c stresses induced i n the f i e l d i n the sand deposit by the earthquake, and 2) a laboratory i n v e s t i g a t i o n to determine the c y c l i c stress r a t i o which w i l l cause l i q u e f a c t i o n . Evaluation of l i q u e f a c t i o n p o t e n t i a l i s based on the comparison of c y c l i c stresses induced i n the f i e l d with the c y c l i c stresses required to cause l i q u e -f a c t i o n i n the laboratory. In 1967, Seed and I d r i s s presented a method to perform dynamic response analysis f o r c a l c u l a t i n g c y c l i c stresses occurring i n the f i e l d . Schnabel et a l . (1972), i n t h e i r method for response analysis treated s o i l 4 as an equivalent l i n e a r e l a s t i c material and t h i s method i s used to c a l c u l a t e c y c l i c stresses induced i n the f i e l d . The i r r e g u l a r c y c l i c h i s t o r y obtained by dynamic response analysis at a given depth i s conver-ted to an equivalent number, Neq, of uniform shear stress, x a v e . The converted uniform shear stress, x & v e , i s compared with uniform c y c l i c shear stress, x or required to cause l i q u e f a c t i o n of the sample i n c y c l i c t r i a x i a l conditions, where the sample'is consolidated under the same all-round confining pressure as i n the f i e l d , i n Neq c y c l e s . However, i t should be noted that the stress conditions i n c y c l i c t r i a x i a l are d i f f e r e n t than those e x i s t i n g i n the f i e l d during an earthquake. There are two l i m i t a t i o n s to t h i s method. F i r s t , t r eating s o i l as an equivalent l i n e a r e l a s t i c material may overestimate the seismic r e s -ponse due to pseudo-resonance at periods corresponding to the s t r a i n compatible s t i f f n e s s e s used i n the e l a s t i c dynamic response analysis as shown by Finn et a l . (1978). Second, the conversion of non-uniform stress cycles to equivalent uniform stress cycles as suggested by Seed et a l . (1975) and Lee and Chan (1976), i s computationally i n e f f i c i e n t and the development of porewater pressures at l e v e l s below l i q u e f a c t i o n may not be predicted with s u f f i c i e n t accuracy as pointed by Finn (1980), since the equivalence can be defined at only one point on the porewater pressure curve. In addi-t i o n , t o t a l stress methods cannot predict permanent deformation which occur during seismic loading. The l i m i t a t i o n s as outlined above for the t o t a l stress method, treating s o i l as an equivalent l i n e a r e l a s t i c material, are.overcome by development of e f f e c t i v e stress method used i n conjunction with nonlinear s t r e s s - s t r a i n laws. Streeter et a l . (1974) proposed the f i r s t true non-l i n e a r a n alysis where a Ramberg-Osgood representation of the s t r e s s - s t r a i n 5 behaviour of s o i l was used. But, i t was not u n t i l 1975 when Martin et a l . presented a fundamental model f or predicting the porewater pressure i n c y c l i c loading, coupled l a t e r with dynamic response analysis (Lee, 1975, Finn et a l . , 1977) that a r e a l i s t i c nonlinear e f f e c t i v e stress method emerged. This dynamic e f f e c t i v e stress analysis i s able to include the nonlinear s t r e s s - s t r a i n law along with pore pressure generation and d i s s i -pation, while allowing for continuous modification of s o i l properties with increasing pore pressure. In recent years, many other methods both e f f e c t i v e and t o t a l stress dynamic analysis have been developed. Some of them are quite noticeable because of t h e i r p o t e n t i a l for two- or three-dimensional analyses such as proposed by Zienkiewicz et a l . (1978) . Aspects of some of these methods are discussed i n Chapter I I . 1.1 Scope of t h i s Research The pore pressure model presented by Martin et a l . (1975) l a t e r coupled with appropriate c o n s t i t u t i v e r e l a t i o n s h i p s for methods of dynamic analysis by Finn et a l . (1977) accounts f o r the most e s s e n t i a l elements of e f f e c t i v e stress analysis, such as the nonlinear h y s t e r e t i c behaviour of s o i l under c y c l i c loading, concurrent generation and d i s s i p a t i o n of pore-water pressure, and continuous modification of s o i l properties with increas-ing porewater pressure. However, at the time the model was proposed i t was based on very l i m i t e d experimental data and contained some c r u c i a l assump-tions that had not been v e r i f i e d . This thesis undertakes to v e r i f y the most important assumptions made i n the model to check whether the model can make any us e f u l pore pressure predictions f o r general loading conditions. In the e f f e c t i v e stress analysis by Finn et a l . (1977) volumetric s t r a i n s are rela t e d with dynamic response parameters and by r e l a t i n g the 6 unloading characteristics of sand with volumetric strain, the porewater pressure are calcualted. However, a more efficient way w i l l be to directly relate porewater pressure with dynamic response parameters. In this thesis, a simple formulation based on the endochronic theory is presented where the porewater pressure are related with dynamic response parameters. This simple formulation is a very efficient way of representing large amounts of porewater pressure data and can easily be coupled with dynamic analysis to perform effective stress analysis. In Chapter II the constitutive relationships proposed by various investigators for their effective stress analysis are briefly discussed. In Chapter III the pore pressure model proposed by Martin et a l . (1975) is discussed emphasizing the most crucial assumptions. Chapter IV undertakes the verification of basic assumptions, providing adequate and experimentally based answers to several questions raised in Chapter III. Verification of the model in Chapter IV involves measurement of each variable in the model, prediction of pore pressure by these measured v a r i -ables and comparison of computed pore pressure with experimental results. Detailed investigation of stress-strain relations is also performed and the pore pressure prediction capacity of the model is checked for sands with stress histories. In Chapter V the predictive capacity of the model is checked for overconsolidated sands. The volumetric strains and rebound modulus parameters used for the model are discussed with respective comparison to normally consolidated sand. Lastly, in Chapter VI, the endochronic theory is invoked to relate porewater pressure with dynamic response parameters. A simple formulation is presented where porewater pressures are presented as a function of a single variable. This approach bypasses 7 the measurement of s o i l parameters (volume change constants, rebound modulus constants) as required by Martin et a l . (1975). The main advan-tages of the endochronic formulation are discussed with suggestions for incorporating i t in dynamic effective stress analysis. This research is based on extensive experimental data obtained in cyclic simple shear apparatus on Ottawa sand under strain and stress controlled drained and undrained conditions. In Chapter 6 experi-mental data for Toyoura and Crystal S i l i c a sand is also presented. Chapter 7 includes a brief summary of each chapter with suggestions for future research work. CHAPTER 2 REVIEW OF CONSTITUTIVE RELATIONSHIPS USED FOR VARIOUS  EFFECTIVE STRESS ANALYSIS E f f e c t i v e stress methods of evaluating the l i q u e f a c t i o n of sand deposits require r e a l i s t i c c o n s t i t u t i v e r e l a t i o n s h i p s to describe nonlinear h y s t e r e t i c behaviour, porewater pressure generation and d i s s i -pation and continuous modification of s o i l properties with increasing pore pressure i n a r e a l i s t i c manner. A method cannot be seen as r e l i a b l e unless the set of c o n s t i t u t i v e r e l a t i o n s h i p s used i n the model are thoroughly v e r i f i e d . This thesis undertakes to v e r i f y the c o n s t i t u t i v e r e l a t i o n s h i p s used for the e f f e c t i v e stress model of Finn et a l . (1977, 1978). By way of introduction, several recent methods of e f f e c t i v e stress analysis are discussed and a c r i t i c a l review of the c o n s t i t u t i v e r e l a t i o n -ships used i n each model i s presented. Recently a v a i l a b l e , they are broadly c l a s s i f i e d into groups based on t h e i r most prominent c h a r a c t e r i s t i c s and one model from each section i s discussed i n such .a way that the most e s s e n t i a l features of the c o n s t i t u t i v e laws used for models i n that category are highlighted. 2.1 CRITICAL REVIEW OF CONSTITUTIVE RELATIONSHIPS Several models developed since 1974 to predict pore pressure development for saturated sand under c y c l i c loading can broadly be c l a s s i f i e d as follows: 1. Stress Path Models - Ishihara et a l . (1975), Ghaboussi and Dikmen (1978) . 8 9 2. Volumetric S t r a i n Models - Martin et a l . (1975), Liou et a l . (1977) . 3. Endochronic Models - Bazant and Krizek (1976), Zienkiewicz et a l . (1978). 4. Kinematic Hardening Models - Mroz et a l . (1979). 5. Empirical Models - Ishibashi e t - a l . (1977) Martin and Seed (1978), etc. Sali e n t features of the c o n s t i t u t i v e laws for a model i n each of these categories w i l l be c r i t i c a l l y discussed. 2.1.1 Stress Path Models Ishihara et a l . (1975) presented the f i r s t stress path model, l a t e r adopted and s l i g h t l y modified by Ghaboussi and Dikmen (1978) and coupled with dynamic response a n a l y s i s . The model used by Ghaboussi and Dikmen (1978) i s based on the assumption that the e f f e c t i v e stress path during the unloading condition can be established with s u f f i c i e n t accuracy. In t h i s model, the r e s i d u a l pore pressures are developed due to p l a s t i c deformations which occur whenever a stress path penetrates a current y i e l d locus established by previous loading. The s o i l i s treated as a two-phase s o l i d - f l u i d media where the nonlinear s t r e s s - s t r a i n behaviour of the s o l i d phase i s i d e a l i s e d by a hyperbolic s t r e s s - s t r a i n r e l a t i o n s h i p ( F i g . 2.1A) for loading and unloading i s assumed to be e l a s t i c with an i n i t i a l shear modulus, G Q ( F i g . 2.IB). However, the shape of the hyperbolic form i s not changed with increasing pore pressure, which i s at variance with experimental observations by Hardin and Drnevich (1972). I t i s assumed that the shear 10 (a) Initial loading FIG. 2.1 St r e s s - S t r a i n Relationship for Sand. Hyperbolic (b) Loading and unloading 1200 cr in a> loose sample e = 0.72-0.78 0 400 800 Effective mean prjncipal stress p' ( k N / m 2 ) Critical state line Proposed yield locus B Yield locus of Granta-gravel FIG. 2.2 Y i e l d L o c i f o r Loose Sand (After Ishihara et a l , 1975) . FIG. 2.3 Comparison of Two Kinds of Y i e l d L o c i (After Ishihara et a l , 1974). 11 y i e l d l o c i take the form of str a i g h t l i n e s r a d i a t i n g from the o r i g i n i n the q-p' plane, where q i s the d e v i a t o r i c shear stress and p' i s the mean normal confining pressure. These shear y i e l d l o c i are also considered to be l i n e s of equal shear s t r a i n . This observation i s based on the experimental data of Poorooshasb et a l . (1971) and Tatsuoka and Ishihara (1974). A t y p i c a l set of y i e l d l o c i established by Ishihara et a l . (1975) i s shown i n F i g . 2.2 for F u j i r i v e r sand. The plotted curves i n F i g . 2.2 c l e a r l y show that the assumption of y i e l d l o c i as straight l i n e s r a d i a t i n g from the o r i g i n holds good only f or mean e f f e c t i v e stresses below 2.0 kN/m At higher values of e f f e c t i v e stress the y i e l d l o c i approach a state p a r a l l e l to p'-axis. Moreover, as Tatsuoka and Ishihara (1974) point out, the assumption of y i e l d l o c i as r a d i a t i n g l i n e s from the o r i g i n i n the p-q' plane i s i n d i r e c t c o n f l i c t with the c r i t i c a l state model presented by Schofield and Wroth (1968) for Granta Gravel. Two concepts of y i e l d i n g are shown i n F i g . 2.3 and d i r e c t c o n t r a d i c t i o n between them i s most c l e a r l y demonstrated by stress paths AB and AC. Ghaboussi and Dikmen (1978) considered the e f f e c t i v e s t r e s s -path for undrained loading to be a quarter of an e l l i p s e ( F ig. 2.4) with the r a t i o of the major to minor axes of the e l l i p s e being a function of r e l a t i v e density and sand grain shape. In addition, they indicated that A, which governs the porewater response, i s higher for angular p a r t i c l e sand than round p a r t i c l e sand for the same r e l a t i v e density as shown in F i g . 2.5. From F i g . 2.5, the pore pressure response of angular sand at a r e l a t i v e density of 20% would correspond to the pore pressure response of round p a r t i c l e sand at a r e l a t i v e density of 79%. This suggests that the p a r t i c l e shape has a s i g n i f i c a n t influence on the development of pore pressure, hence, l i q u e f a c t i o n . 12 f I = q - p'tah0' , f 2 = ( p'-pf ) 2 + —2 q 2 - (pj, - p 2 ) = 0 X X = pf tan0'/pQ-pf Complete liquefaction state (Critical state line ) ^Initial liquefaction state Shear yield loci (Lines of constant shear strains) 1 — Stress path , f 2 = 0 Pf FIG. 2.4 E l l i p t i c a l Stress Path for Undrained Loading Test (After Ghaboussi and Dikmen, 1978). 2! 3 •Z 2 ° P o • ' O O k N / m 2 ) • P; = 4 0 0 - 1 0 0 0 k N / m 2 J C° S' r 0 ( , 9 6 9 ) a K h o s l o a n d W u ( 1 9 7 6 ) a T o t s u o k a a n d I s h i h a r a ( 1 9 7 5 ) V e r y a n g u l a r P Q » 1 0 0 k N / m 2 / o J R o u n d to s u b a n g u l a r P 0 * 1 0 0 - 1 0 0 0 k N / m 2 2 0 4 0 6 0 8 0 R e l a t i v e D e n s i t y , D r % 100 FIG. 2.5 Relationship Between Material Parameter A and Relative Density (After Ghaboussi and Dikmen, 1978). The most c r i t i c a l handicap of the present stress path models as discussed by Finn (1979) i s the assumption of y i e l d l o c i and l i n e s of constant shear s t r a i n r a d i a t i n g from the o r i g i n . This assumption implies that the stress path during a s t r a i n c o n t r o l l e d c y c l i c loading test (except for the f i r s t half c y c l e i n contraction and extension) w i l l e s t a b l i s h the two l i m i t s of y i e l d l o c i , while further a p p l i c a t i o n of cycles of constant s t r a i n would move the e f f e c t i v e stress path i n a v e r t i -c a l d i r e c t i o n without any a d d i t i o n a l pore pressure generation. This behaviour of the model i s caused by the assumption that i s o t r o p i c hardening behaviour i s appli c a b l e for sand under c y c l i c loading. Since experimental r e s u l t s contradict t h i s p r e d i c t i o n of the model, the assumption on which the model i s based i s i n need of a complete review. 2.1.2 Volumetric S t r a i n Models Martin et a l . (1975) presented the f i r s t fundamental porewater pressure model, l a t e r substantiated by Lee (1975), that accounts for the nonlinear h y s t e r e t i c behaviour of sand during c y c l i c loading. This model w i l l be discussed i n d e t a i l i n Chapter I I I . Liou et a l . (1977) proposed a model where s o i l i s again treated as a two phase material. The nonlinear, s t r a i n dependent hystere-t i c behaviour of s o i l i s accounted for by a shear wave sub-model. This shear wave sub-model uses the modified Ramberg-Osgood s t r e s s - s t r a i n r e l a -tionship, where i n i t i a l shear modulus and maximum shear strength i s given by Hardin and Drnevich (1972). The pressure wave sub-model r e l a t e s pore pressure to the bulk compressibility of water, secant constrained modulus of s o l i d skeleton and porosity. The pore pressure model i s given by where ~ - = G + (2.2) i n which a_ i s e f f e c t i v e stress, 77- i s secant constrained modulus of the s o i l skeleton, - r — i s the constrained s t r a i n l e v e l , G i s the secant shear dz modulus and C^ , i s the bulk c o m p r e s s i b i l i t y of the s o i l skeleton. Further, i t i s assumed that C^ remains constant during shearing. Thus, any change i n C c i s d i r e c t l y a t t r i b u t e d to a change i n G. Since the secant shear modulus decreases with increasing shearing amplitude, C c would increase with increasing shear s t r a i n and any change i n C c w i l l c a l c u l a t e the change i n e f f e c t i v e stress by equation (2.1). The pore pressure p r e d i c t i o n by t h i s model, i n i t s simplest form, depends on the r e l a t i o n s h i p (equation 2.2) which i s true i n the range of s t r a i n where s o i l behaves as an e l a s t i c material. However, beyond t h i s range of s t r a i n the a p p l i c a t i o n of t h i s r e l a t i o n s h i p i s questionable. In addition, Finn (1979) points out that changes i n G with s t r a i n are recoverable on stress r e v e r s a l , whereas changes i n C c are assumed to be permanent and cumulative which suggests that the r e l a t i o n -ship (equation 2.2) should not be assumed to hold for s o i l . 2.1.3 Endochronic Models The endochronic theory was proposed by Valanis (1971) for metals to describe nonlinear s t r e s s - s t r a i n law, s t r a i n hardening behaviour and contraction of hysteresis loops with c y c l i c s t r a i n i n g . The p o t e n t i a l of t h i s theory for saturated sands was f i r s t r e a l i s e d by Bazant and Krizek (1976). In the c o n s t i t u t i v e laws proposed for the pore pressure model of Bazant and Krizek, s o i l i s treated as a two-phase porous media where i n e l a s t i c and e l a s t i c behaviour are considered separately. The c o n s t i t u t i v e equations proposed by Biot (1956,1957) for e l a s t i c behaviour 15 of the two-phase media were extended to include inelastic volumetric strain in 1975 and used for a pore pressure model. Bazant and Krizek assumed that the deviatoric stress-strain relationships for the inelastic isotropic two-phase media have the same form as that of a solid phase alone (i.e., dry sand). In addition, they,assumed that in the undrained condition the presence of porewater in porespace produces viscous and ine r t i a l forces which oppose the relative movement of sand particles. They further postulated that intergranular stresses, a', serve the purpose of overcoming the microscopic viscous and in e r t i a l forces. Hence, the actual extent of densification which is realised under undrained condi-tions is a function of intergranular stress a'. Based on this concept, the actual densification which is responsible for pore pressure generation under undrained conditions is related with densification in drained conditions for the same strain history as given by Bazant and Krizek. ^ Evd^Potential Volumetric L^ avo ° ^ Evd^Drained (2.3) Strain in Undrained where L ( a ' -a') is a correction coefficient whose value is arbit r a r i l y assigned. vo is the i n i t i a l vertical confining stress and a' is the intergranular stress at a particular instant during cyclic loading. The proposed porewater pressure model under undrained condi-tions and constant overburden stress, o' , is given as vo (e L ( a ' -a') . , vd Drained vo / r i da' = - (2.4) cb in which da' is the change in effective stress and is bulk compressi-b i l i t y of drained sand. It is not clear from the literature whether is assumed constant or a function of intergranular stress. To describe densification and hysteretic behaviour of drained sand under cyclic loading, the endochronic theory was applied. The only independent variable for these formulations is a scalar which characterizes the accumulation of total shear strain in time which is called the rearrangement measure Since both inelastic volumetric strains (densification) and inelastic shear strains are mainly due to rearrangement of sand particles under cyclic loading, they are considered to be related to £. For this purpose the following two variables are proposed by Bazant and Krizek. 1) Densification variable (5) 1 which relates densification, e v d with £. 2) Distorsion variable (ri) which relates inelastic shear strain, yP with The following relationship is proposed by Bazant and Krizek for densifi-cation under cyclic loading which accounts for strain hardening and softening behaviour e . = - L (1 + a?) (2.5) vd a n and for constant strain cyclic shear condition r, = Y Q q N (2.6) where y Q is shear strain amplitude, N is the cycles of constant shear strain and a and q are s o i l constants which depend on the relative density. In Fig. 2.6, volumetric strain (densification) is plotted against the number of cycles of constant shear strain amplitude for Dr = 45% and 65%, where the solid curve represents experimental data and the dotted curve shows the analytical data. Figure 2.6 shows that the analytical and For the same variable Bazant and Krizek (1976) used the symbol, K. 17 0.001 001 010 0.001 I 2 4 10 30 Number of cycles,N "O > VI/ c o a> E o > 0.01 0.10 2 4 10 30 Number of cycles , N FIG. 2.6 Volumetric S t r a i n vs. Number of Cycles f o r Cry s t a l S i l i c a Sand (After Cuellar et a l , 1977). FIG. 2.7 E l a s t i c and I n e l a s t i c Stress Increment (After Bazant and Krizek, 1976). 18 experimental data do not match very w e l l . The a n a l y t i c a l curves (double dotted) overpredicts the volumetric s t r a i n . In Chapter 6, a better presentation i s proposed for the volumetric s t r a i n s with endochronic v a r i a b l e s . The nonlinear s t r e s s - s t r a i n law for one-dimensional shearing where the i n e l a s t i c shear s t r a i n increment (Fig. 2.7) i s related with shear modulus and shear stress through the following equation: dx = (a')'"2 x (c + —-4- r ) x d Y - f \ r (2.7) e + <W> T d ( i + ± 0 h where a,b,c,d,e,x^ are s o i l constants which depend on s o i l - t y p e and r e l a t i v e density, dy i s t o t a l shear s t r a i n , G i s the shear modulus, and x i s the shear stress. By using equation (2.7) Cuellar et a l . (1977) were able to describe nonlinear h y s t e r e t i c behaviour of sand as shown i n F i g . 2.8. However, the number of constants required f or the s t r e s s -s t r a i n r e l a t i o n s h i p i s extremely large and i t i s not clear how such constants can be obtained. The s t r e s s - s t r a i n r e l a t i o n s h i p proposed for general stress conditions i s extremely d i f f i c u l t to understand. Bazant and Krizek (1976) checked the p r e d i c t i v e capacity of the pore pressure model f o r stress controlled undrained test data as shown i n F i g . 2.9, and by varying the value of the c o r r e c t i o n f a c t o r a r b i t r a r i l y , a good c o r r e l a t i o n between experimentally observed pore pressure and a n a l y t i c a l r e s u l t s was obtained. However, i t i s not necessary that t h i s convection f a c t o r w i l l give r e s u l t s for samples with other c y c l i c stress r a t i o s . In addition, i t i s not clear from the l i t e r a t u r e what values of C^ and G are used f o r the a n a l y t i c a l r e s u l t s presented. In conclusion, the formulation of the problem presented by 19 FIG. 2.8 P r e d i c t i o n of the Hysteretic Loops for C r y s t a l S i l i c a Sand (After Cuellar, 1977). FIG. 2.9 Pore Pressure vs. Number of Cycles i n Constant Stress Undrained Test (After Bazant arid Krizek, 1976). 20 Bazant and Krizek (1976) is impressive but requires detailed verification for its pore pressure predictive capacity without which i t s performance is really unknown. Moreover, the constitutive relationships have not yet been verified for multi-directional loading. In the model of Zienkiewicz et a l . (1978) the volumetric strain which accounts for pore pressure rise was related to the length of the strain path, £ (Fig. 2.10). Endochronic variables were used to relate volumetric strain (autogenous volumetric strain) with a parameter called the damage parameter. An experimental transformation was proposed to transform the length of strain path to damage parameter through the following relationship: dK = exp(c • x/a m Q)d5 (2.8) where T / O ^ Q is the stress ratio and c is the constant. Data shown in Fig. 2.11 shows volumetric strain against damage parameter, where points corresponding to various stress ratios (Fig. 2.10) collapse into one curve (Fig. 2.11). To calculate pore pressure, these volumetric strains were multiplied by a constant tangent bulk modulus of sand. (Martin et a l . (1975) have shown that tangential bulk modulus is a function of effective stress.) Since volumetric strains are explicitly calculated, a non-associated theory of plasticity which combines a Mohr-Coulomb yield surface (Fig. 2.12) with a Tresca-type potential surface with zero dilatancy was used. However, as Finn (1979) points out, "to what extent a potential surface selected on the basis of dilatancy characteristics only may be capable of representing the remaining strain is yet unknown". This model is based on very limited data and i t s performance under idealised conditions (stress controlled undrained test) is yet to be examined. 21 L e n g t h of s t r a i n p a t h , FIG. 2.10. Volumetric S t r a i n vs. Length of Strain, Path (After Zienkiewicz et a l , 1978). FIG. 2.11 Volumetric S t r a i n vs. Damage Parameter (After Zienkiewicz et a l , 1978). 22 with Ztro Oi lo loncy . 2.12 Idealized Y i e l d and P l a s t i c P o t e n t i a l Surface (After Zienkiewicz et a l , 1978). Cr i t ica l state line State sur face -Yield sur face ( i n i t i a l pos i t i on ) . - Y i e l d s u r f a c e (Current p o s i t i o n ) FIG. 2.13 Two-Surface Model Showing Unloading and Isotrop Consolidation (After Mroz et a l , 1979). i c 23 2.1.4 Kinematic Hardening Models Mroz et a l . (1978) proposed a model for c y c l i c behaviour of s o i l based on the concept of kinematic hardening. The model, i n i t s simplest form, assumes that a state boundary surface established by con-s o l i d a t i o n stresses, (F=0), i t s Roscoe-Burland surface as shown i n F i g . 2.13, and the e l a s t i c domain enclosed by the y i e l d surface, f o=0, i s much smaller than (F=0). [F=0 i s considered as the y i e l d surface used for the i s o t r o p i c hardening model.] For i n e l a s t i c material response i t i s assumed that the y i e l d surface, f o=0 may t r a n s l a t e , expand or contract i n the stress space but cannot i n t e r s e c t the e x i s t i n g boundary surface. An asso-ciated flow r u l e governing the p l a s t i c flow and hardening r u l e describes the v a r i a t i o n of the y i e l d surface and state boundary surface, together with the v a r i a t i o n of the hardening modules along any t r a j e c t o r y , although with changing y i e l d surface the flow r u l e r e t a i n s i t s usual associated flow. T r a n s l a t i o n of the surface i s constrained by conditions presented by Mroz (1967). For determination of the c y c l i c behaviour of s o i l , instead of using a nest of y i e l d surfaces with associated p l a s t i c moduli between y i e l d i n g and bounding, the two-surface model uses an i n t e r p o l a t i n g r u l e . The i n t e r p o l a t i n g r u l e used i n the model follows the work of Dafalias and Popov (1976). Recently, Finn and Martin (1980) c r i t i c a l l y reviewed the main points of the model. In F i g . 2.13, the two-surface model for the case of unloading and reloading from i s o t r o p i c consolidation i s shown. Since t h i s theory permits the occurrence of volumetric changes on unloading, i t has p o t e n t i a l for i m p l i c i t determination of v o l u -metric s t r a i n and dynamic analysis of porewater pressure. But the formu-l a t i o n s of the theory for c y c l i c loading are for the pseudo-static case only. The model has not been extended to include earthquake loading which 24 requires i n e r t i a e f f e c t s . 2.1.5 Empirical Models Since 1975, along with the development of models based on the theory of p l a s t i c i t y and endochronic theory, several empirical models have also been proposed, such as those of Ishibashi et a l . (1977), Martin and Seed (1978), and others. These methods are, no doubt, of p r a c t i c a l importance and represent d e f i n i t e progress. However, the a p p l i c a t i o n of such models to general conditions i s always l i m i t e d . 2.2 DISCUSSION I t i s quite c l e a r from the above disc u s s i o n that none of the c o n s t i t u t i v e r e l a t i o n s h i p s considered are complete. Work i n t h i s area i s s t i l l at the stage when modification and v e r i f i c a t i o n i s required for each of the models. However, the development of a l l these models i s so recent that any p r a c t i c a l experience with them i s very l i m i t e d . In t h i s t h e s i s , i t i s undertaken to v e r i f y the performance of one of the f i r s t models discussed, that of Martin et a l . (1975). This f i r s t model, to predict pore pressure under c y c l i c loading, considered for the f i r s t time the fundamental mechanism of pore pressure development under c y c l i c loading. Lee (1975) extended t h i s model to perform e f f e c t i v e dynamic stress a n a l y s i s . This model i s very simple and r e a l i s t i c a l l y p redicts the behaviour of s o i l . The method has been used i n evaluation of l i q u e f a c t i o n p o t e n t i a l of offshore s i t e s and i n conjunction with a s o i l p i l e i n t e r a c t i o n program to predict the performance of p i l e founda-tion s . Recently, i t has been used i n Japan to evaluate the l i q u e f a c t i o n performance of s i t e s i n the Off-Tokachi earthquake, 1978. I t i s considered e s s e n t i a l to v e r i f y the model and check i t s performance under general conditions. CHAPTER 3 CONSTITUTIVE RELATIONS FOR THE EFFECTIVE STRESS MODEL OF FINN, LEE AND MARTIN Finn, Lee and Martin (1977) presented a complete method of dynamic e f f e c t i v e stress response analysis f or the evaluation of l i q u e -f a c t i o n p o t e n t i a l i n h o r i z o n t a l l y layered saturated sand deposits. This method i s based on a set of c o n s t i t u t i v e r e l a t i o n s h i p s which take into account the important f a c t o r s known to influence the response of a saturated sand layer i n a given earthquake: the i n s i t u shear modulus, the v a r i a t i o n of shear modulus with shear s t r a i n and mean normal e f f e c t i v e stress, the contemporaneous generation and d i s s i p a t i o n of porewater pres-sure, h y s t e r e t i c and viscous damping and s t r a i n hardening. The pore pressure model adopted in t h i s method i s proposed by Martin et a l . (1975). This model i s the f i r s t to explain the fundamental mechanism of porewater pressure generation during c y c l i c loading. Lee (1975) proposes nonlinear s t r e s s - s t r a i n laws for sand under c y c l i c loading and presents a nonlinear e f f e c t i v e stress model which consists of f i v e basic elements: 1. A method for solving the nonlinear equation of motion; 2. A s t r e s s - s t r a i n law for the nonlinear behaviour of sand; 3. A procedure for c a l c u l a t i n g p o t e n t i a l volume changes caused by c y c l i c shear; 4. A r e l a t i o n s h i p between p o t e n t i a l volume changes and r e s i d u a l porewater pressure; 5. A procedure for c a l c u l a t i n g the d i s s i p a t i o n of excess porewater pressure. 25 In this chapter, only the constitutive relationships used for the effective stress model w i l l be discussed. Broadly, the constitutive relations used in the analysis for the effective stress model can be divided into two categories: 1. Relations required for the porewater pressure generation model; 2. Relations required for the representation of the nonlinear stress-strain behaviour of sand including strain hardening and softening. 3.1 PORE PRESSURE MODEL In the pore pressure model proposed by Martin et a l (1975) the behaviour of sand under undrained cyclic loading conditions is com-puted from the knowledge of volume change characteristics under drained cyclic simple shear conditions and the rebound characteristics under effective stress unloading. A schematic representation of the model is shown in Fig. 3.1. It is considered that in a saturated sand sample subjected to one cycle loading under simple shear conditions, the net volumetric strain increment occurring due to interparticle slip under drained condi-tions is A e v < j . If an identical saturated sample is being subjected to the same shear strain amplitude but under undrained conditions i t is assumed that sl i p at the grain contacts resulting in volumetric strain, A e V ( j , again must occur provided that the magnitude of the residual pore-water pressure increase occurring during the cycle is small relative to the i n i t i a l effective stress. However, in the undrained condition slip deformation Ae V (j must transfer some of the vertical stress previously carried by intergranular 27 Vertical effective stress, <xv FIG. 3.1 Schematic Illustration of Mechanism of Porewater Pressure Generated During Cyclic Loading (After Seed; 1976) . 28 forces to the l e s s compressible water and, as porewater pressure increases, the corresponding decrease i n e f f e c t i v e stress w i l l r e s u l t i n a release of recoverable s t r a i n , Ae v r, stored at grain contacts due to e l a s t i c deformation. Hence, from the volumetric com p a t i b i l i t y at the end of the load cycle, Martin et a l (1975) derived the following r e l a t i o n s h i p -Au • + f ] = A e v d (3.1a) w r or Au = A e v d / (.f- + j~) (3.1b) w r i n which Ae y (j i s the volumetric s t r a i n due to s l i p deformation for the cycle, n i s the porosity of the sample, 1^ i s the bulk modulus of water, and E r i s the tangent modulus of the one-dimensional unloading curve at a point corresponding to the i n i t i a l v e r t i c a l e f f e c t i v e stress. In saturated samples, 1^ i s approximately two orders of magni-tude greater than E r for the usual v e r t i c a l stress interested from the l i q u e f a c t i o n point of view, hence i t can be assumed that the compressibi-l i t y of the porewater pressure becomes n e g l i g i b l e . I t i s usually assumed that undrained t e s t s are constant volume t e s t s . The consequence of t h i s assumption i s that volumetric s t r a i n , e v c j , due to i n t e r p a r t i c l e s l i p ( p l a s t i c strain) must be equal and opposite to recoverable s t r a i n , A e v r ( e l a s t i c s t r a i n ) , released due to the decrease i n e f f e c t i v e s t r e s s . This, for example, i s a fundamental assumption of the c r i t i c a l state theory as applied to undrained loading by Schofield and Wroth (1968). Hence, for the constant volume condition equation (3.1b) implies: Au = E Ae , (3.2) r vd v ' The key to the practical application of the theory rests in the fact that the value of Aey(j is independent of vertical effective stress. The theory in i t s simplest form implies that i f "saturated sand loaded to an i n i t i a l vertical effective stress of a v o has a recoverable volumetric strain of £ v r o , then liquefaction 1 w i l l occur under an applied cyclic strain history that produces a volumetric st r a i n 2 , e V (j = £ v r o , under drained conditions", Martin et a l (1975). The porewater pressure model of equation (3.2) gives rise immediately to the important question: how are Ae V (j and E r to be obtained? In the model i t is assumed that the plastic volumetric strain, Ae V (j, which develops during one cycle of uniform shear strain, y> 1 1 1 a n undrained simple shear test w i l l be the same as the volumetric strain in a drained simple shear test. A question arises - can i t be proven? Therefore, a fundamental assumption of the pore pressure model is that there is a unique relationship between volumetric strain in drained tests and porewater pressure in undrained tests for samples of a given sand with corresponding strain histories. The following sections present the volume change characteristics under drained cyclic shear conditions used to obtain Ae V (j, and rebound characteristics under static unloading to calculate E r. 3.1.1 Volume Change Characteristics Under Drained Cyclic Loading Consideration of volume change in drained cyclic tests is res-tricted to simple shear test conditions, which best simulate f i e l d defor-mation induced in horizontal sand deposits by earthquake excitation. The liquefaction is the state where porewater pressure has become equal to i n i t i a l v e rtical confining stress. 2In the later part of this thesis, volumetric strain under undrained conditions w i l l be referred to as potential volumetric strain. 30 volumetric s t r a i n c h a r a c t e r i s t i c s of sand under c y c l i c simple shear have been given i n studies by S i l v e r and Seed (1971), Youd (1972,1975), Pyke (1973) and Martin et a l (1975). A few important observations from these studies are: 1. Due to the a p p l i c a t i o n of c y c l i c shear stress or s t r a i n i n the drained condition, i n t e r p a r t i c l e s l i p r e s u l t s i n volumetric 3 s t r a i n ( p l a s t i c ) which i s proportional to shear s t r a i n amplitudes (y<0.3%) ( S i l v e r and Seed, 1971). 2. Volumetric s t r a i n s are independent of v e r t i c a l confining stress as shown i n F i g . 3.2 ( S i l v e r and Seed, 1971). 3. Volume change c h a r a c t e r i s t i c s are the same for both dry and saturated samples subjected to the same c y c l i c shear s t r a i n amplitude (Fig. 3.3) and are independent of frequency i n the range 0.2 Hz to 2 Hz (Youd, 1971). With increasing volumetric s t r a i n s during c y c l i c loading i n the drained condition, both shear modulus and increase (Pyke (1973), Youd (1975)) Based on these observations, Martin et a l (1975) suggest determining A e v d from experimental data on dry sand under c y c l i c s t r a i n conditions as shown i n F i g . 3.4. For the a p p l i c a t i o n of data shown i n Fi g . 3.4 to i r r e g u l a r or random c y c l i c shear s t r a i n h i s t o r y , data i s replotted i n terms of incremental volumetric s t r a i n , Ae vd, against shear s t r a i n amplitude, y, for constant accumulative volumetric s t r a i n , e v c j . Such a representation i s shown i n F i g . 3.5 for C r y s t a l S i l i c a Sand, for which a function of the following form has been f i t t e d : r r. 2 l _ ' r ' " ^ 2 ^ ' y+C.e , 4 vd A e v d = C l ( Y - C 9 e „ H > + v+r g OA) Since l a t e r a l s t r a i n s i n the simple shear condition are zero, v e r t i c a l s t r a i n s are equal to volumetric s t r a i n s , ^ko i s the r a t i o of l a t e r a l e f f e c t i v e stress to v e r t i c a l e f f e c t i v e stress i n a l a t e r a l l y confined condition. 31 001 -a > 14 6 o > 01 10 i — r — . Crystol silica sand » 6 0 % , Cycles «10 • i \ e Legend : • 500 a 2000 * 4000 1 _ i 01 .10 Cyclic shear strain amplitude ,y% FIG. 3.2 Volumetric S t r a i n vs. C y c l i c Shear S t r a i n Amplitude (After Seed and S i l v e r , 1971). 0 200 0160 0120 o 0080 0040 a Soturoted drained 0 5 5 < e 0 < 0 56 ° D r Y 2 ! % < / < : 2 3 % CTi = 500 psi Ns 10,000 " U O v 100 10 20 40 60 80 100 Frequency in cycles per minutes 120 FIG. 3.3 Void Ratio Change vs. Frequency i n C y c l i c S t r a i n Test on Dry and Saturated Drained Sample (After Youd, 1972). L e s 32 Number of cycles , N FIG. 3.4 Volumetric S t r a i n vs. Number of Cycles of Constant S t r a i n (After Martin et a l , 1975). 0 01 0.2 03 Cyclic shear strain amplitude, y% FIG. 3.5 Incremental Volumetric S t r a i n vs. C y c l i c Shear S t r a i n f o r Various Volumetric S t r a i n (After Martin et a l , 1975) (1 psf =47.9 N/m2). 33 where Ae v (j i s the volumetric s t r a i n 5 per cyc l e , £ y (j i s the accumulated volumetric s t r a i n , y i s the shear s t r a i n 5 amplitude and Cj^C^jC-jjC^ are volume change constants which depend on r e l a t i v e density and s o i l type. Finn and Byrne (1976) point out that the volumetric s t r a i n increment, Ae V (j, corresponding to r e l a t i v e density, Dr^, for which the volume change constants have been evaluated can be rela t e d to the volumetric s t r a i n increment at another r e l a t i v e density, D r2, by the r e l a t i o n : ( A £ v d > D r 2 = R<*Evd>Drl ( 3 ' 5 ) i n which R i s a shape parameter that v a r i e s with r e l a t i v e density. The proposed r e l a t i o n (equation 3.5) interpolates volume change c h a r a c t e r i s t i c s at r e l a t i v e d e n s i t i e s other than for those which volume change constants are known. 3.1.2 One-Dimensional Volumetric Unloading C h a r a c t e r i s t i c s The pore pressure model requires a knowledge of the recover-able deformation c h a r a c t e r i s t i c s of sand during one-dimensional unloading from a given i n i t i a l v e r t i c a l e f f e c t i v e stress. Martin et a l (1975) explain the behaviour of sand i n one-dimensional loading and unloading. The volumetric s t r a i n on loading with a v e r t i c a l confining stress can be subdivided into two components: 1. Non-recoverable s t r a i n due to i n t e r p a r t i c l e s l i p , evso5 a n d 2. Recoverable s t r a i n due to e l a s t i c deformation of grain contacts, e v r o -For the pore pressure model, a r e l a t i o n s h i p between v e r t i c a l e f f e c t i v e stress and the recoverable component of volumetric s t r a i n i s required. Martin et a l . (1974) conducted a few tes t s on Crystal S i l i c a bShear s t r a i n and volumetric s t r a i n are expressed i n percentages. 34 sand i n an NGI 6 type simple shear device to obtain the unloading charac-t e r i s t i c s of sand. Their experimental data shown i n F i g . 3.6 ind i c a t e that: 1. The t o t a l recoverable s t r a i n , e v r o , stored at grain contact points increases with increasing confining stress, o v o ( c f . F i g . 3.6) according to the following r e l a t i o n s h i p : e = k 9 ( a ' ) n (3.5) vro 2 vo 2. Unloading curves from d i f f e r e n t confining stresses are geometrically s i m i l a r i n shape (cf. F i g . 3.6) and can be related with each other by: = (7?-) (3-6) vro vo From the above two observations Martin et a l . (1975) obtained a r e l a t i o n s h i p f or the unloading modulus, E r , ax any stress, given by: E r = ( a ; ) 1 _ m / m k 2 ( a ; o ) n " m (3.7) where m, n are constants for a given sand and density, a^Q i s the i n i t i a l confining stress and a v i s the e f f e c t i v e stress at a p a r t i c u l a r instant. Seed et a l . (1975) studied the recoverable c h a r a c t e r i s t i c s of sand and determined that the t o t a l volumetric s t r a i n recovered due to unloading from a c e r t a i n confining stress i s increased by 20% due to c y c l i c shear p r i o r to unloading. I t i s further observed that most of the increase i n recoverable s t r a i n i s i n the l a s t one-third of the unloading curve as shown i n F i g . 3.7. Hence, there i s a p o s s i b i l i t y that due to c y c l i c shearing the rebound modulus E r might change in dry sand. Any change i n E r i s c r u c i a l f o r pore pressure p r e d i c t i o n , so the Norwegian Geotechnical I n s t i t u t e . 35 5 Recoverable volumetric »lroin , € y | . % FIG. 3.6 Generalised One-dimensional Unloading Curves (After Martin et a l , 1975), (1 psf =47.9 N/m2). I OO E z a> in a> > o 4) O u > Sand type > Monterey No 0 Relative density • 60 % 0 75 I-*7*0 050 025 Static unloading Static unloading after it has been subjected to cyclic loading 0 010 0 20 0 30 0 40 Recoverable volumetric s t ra in , « v r % V e r t i c a l E f f e c t i v e Stress vs. Recoverable Volumetric S t r a i n f o r Monterey Sand -(After Seed et a l , 1973). " observations made by Seed et a l . (1975) should be further investigated. The most crucial point about the measurement of E r is under which condition i t should be measured. Rebound occurs during undrained conditions under cyclic loading but direct measurement of E r under such conditions is not possible. Therefore, E r is measured during static rebound in the simple shear apparatus. However, the basic question arises: can E r measured in static conditions (consolidation ring) be used for the pore pressure model? 3.2 STRESS-STRAIN RELATIONSHIP For the range of strains expected during earthquake loading, sand manifests a hysteretic stress-strain relationship which indicates both a nonlinear behaviour and the capacity of dissipating a considerable amount of energy especially at high levels of shear strain. The extent to which this behaviour is manifested depends on shear strain amplitude and the relative density of the s o i l . Also, as a consequence of continuous application of cyclic shear stress or strain, i t is observed that shear modulus increases with increasing number of cycles for a constant shear strain amplitude. This behaviour is reported by Pyke (1975) for simple shear as shown in Fig. 3.8 for Monterey Sand. In addition, Hardin and Drnevich (1972) are the f i r s t to observe that shear modulus is a function of mean effective stress. Under undrained conditions i t gradually decreases with increasing pore pressure. Hence, in any generalised stress-strain relationship to describe the nonlinear hysteretic behaviour of sand for irregular loading under drained and undrained conditions, the shear modulus has to be continuously modified for the hardening and softening behaviour of sand. Constitutive laws used .02 05 10 20 50 1.0 Cyclic shear strain amplitude , / % FIG. 3.3 Increase in Av. Shear Modulus with Various Number of Cycles of Constant Shear Strain (After Lee, 1975). 38 for the e f f e c t i v e stress analysis by Finn et a l . (1977) are those proposed by Lee (1975) which include s t r a i n hardening and softening behaviour. The dif f e r e n c e i n behaviour of sand under i n i t i a l loading, unloading and reloading i s recognised by Lee (1975) and behaviour i n each phase i s treated separately. 3.2.1 I n i t i a l Loading Up to the f i r s t r e v e r s a l i n loading, i t i s assumed that the response of the sand i s defined by the hyperbolic s t r e s s - s t r a i n r e l a -tionship formulated by Konder and Zelasko (1963) and i l l u s t r a t e d i n Fig . 3.9: G y x = G y / (1 + -S£_) (3.8) mo x mo in which x i s the shear stress corresponding to a shear s t r a i n amplitude Y, G m o i s i n i t i a l shear modulus, and x m o i s the maximum shear stress which can be applied to sand i n the i n i t i a l state without f a i l u r e . Finn et a l . (1977) suggest evaluating the values of G Q and x m o from the equations proposed by Hardin and Drnevich (1972): G = 14760 ( 2 - 9 7 3 - e ) 2 ( 3 > 9 ) mo 1+e 3 v 1+k 1-k l Tmo = (~T~^  sin<n - (-y-V (3.10) where e i s the void r a t i o with a maximum value of 2.0, i s v e r t i c a l e f f e c t i v e stress i n psf., k Q i s the c o e f f i c i e n t of earth pressure at res t , and <j>' i s e f f e c t i v e angle of shearing resistance. 3.2.2 Unloading and Reloading While a hyperbolic s t r e s s - s t r a i n r e l a t i o n s h i p i s considered sui t a b l e to describe the i n i t i a l loading curve, the hysteresis loop that 39 FIG. 3.9 S t r e s s - S t r a i n Relationship by Finn, Lee and Martin (1977). AO appears to simulate the actual behaviour of sand during c y c l i c loading can be constructured by the r u l e suggested by Masing (1936). The r u l e i s gene-r a l l y stated as: the shape of the unloading and reloading curve i s the same as that of the i n i t i a l loading curve except that the scale i s enlarged by a factor of two. If the i n i t i a l loading curve i s described by a function, x=f(Y), and loading r e v e r s a l occurs at ( y r , x r ) , then the s t r e s s - s t r a i n curve f o r subsequent unloading and reloading from a re v e r s a l point i s x-x Y~Y = f (-y^) (3.11) Such behaviour i s shown i n F i g . 3.9(b). Lee (1976) proposes the following r u l e s f o r general s t r e s s -s t r a i n conditions: 1. The unloading and reloading curves should follow the i n i t i a l loading curve i f the previous maximum shear s t r a i n i s exceeded. In F i g . 3.9(c) the unloading path beyond point B w i l l l i e on the extended skeleton curve. 2. If the current loading or unloading curve i n t e r s e c t s the curve described by the previous loading or unloading curve, the s t r e s s - s t r a i n r e l a t i o n w i l l follow the previous curve. Newmark and Rosenblueth (1971) suggest the same behaviour. (In F i g . 3.9(d) at point B, a f t e r path 4 the curve w i l l follow path BA'.) The above two rules require that the model re f e r to the coordinates of the greatest excursion i n either d i r e c t i o n . The coordinates are a lso required f o r descending sequences of s t r a i n s . 3.2.3 Influence of Hardening and Porewater Pressure During the process of c y c l i c shear stress or s t r a i n on dry 41 sand or saturated sand under drained conditions, i n t e r p a r t i c l e s l i p r e s u l t s i n volumetric s t r a i n and shear modulus gradually increases with increasing volumetric s t r a i n as shown i n F i g . 3.10. Martin et a l . (1974) suggest incorporating the e f f e c t of hardening i n the s t r e s s - s t r a i n r e l a -tionship. I t i s proposed that the s t r e s s - s t r a i n r e l a t i o n s h i p be a function of cummulative volumetric s t r a i n . For a given volumetric s t r a i n , e V ( j , the smoothed shear stress and s t r a i n curve can be approximated by hyperbola, and v . vd :u = —TC- where a = A. - — — (3.12) hv a+by 1 A„+A„e , 2 J vd 1 W v d and A^,A2>A3, B^,]^ and B3 are constant f o r a given sand at a given r e l a t i v e density, i s e f f e c t i v e v e r t i c a l stress and T and y are the shear stress and shear s t r a i n . Lee (1975) suggests a simpler and more e f f i c i e n t way to r e l a t e shear modulus, shear stress and cummulative volumetric s t r a i n as follows: e , G = G [1 + „ ] (3.13) mn mo IL+H-e , 1 2 vd and e , x = T [1 + „ ^ u v a ] (3.14) mn mo H„+H.e , 3 4 vd where G m n i s maximum shear modulus i n the n*-*1 cycle, x m n i s maximum shear stress i n the n cycle, e V (j i s volumetric s t r a i n up to the n*-^  cycle, and Hi,H.2,H.3,H4 are hardening constants. The s t r e s s - s t r a i n behaviour i s now completely defined by equations (3.10),(3.11),(3.13) and (3.14) for the c y c l i c drained condition. .02 .05 .1 .2 .5 S h e a r s t r a i n , f ( % ) FIG. 3.10 Av. Shear Modulus vs. Shear Strain for Various Volumetric Strains. 43 Using the proposed s t r e s s - s t r a i n law, Finn et a l . (1977) c a l c u l a t e the s t r e s s - s t r a i n behaviour of C r y s t a l S i l i c a Sand i n a s t r a i n c o n t r o l l e d drained condition and compare i t with experimental data. This comparison i s shown i n F i g . 3.11 for the 2nd and 4th cycles of loading where the t h e o r e t i c a l s t r e s s - s t r a i n loops appear to be a good approximation of the measured s t r e s s - s t r a i n loops. This suggests that the assumptions that the skeleton curve i s hyperbolic, unloading and reloading curves are Masing, and shear modulus, shear stress are functions of cummulative volumetric s t r a i n are r e a l i s t i c . To incorporate the e f f e c t of porewater pressure, i t i s suggested that equations (3.17) and (3.18) be modified to read: G = G (1 + „ J V d ) (r^-)h (3.15) mn mo H , + H 0 E , a 1 2 vd vo e a' x = x (1 + „ ) -4- (3.16) mn mo H » + H , e , a J 4 vd vo where a v 0 i s the i n i t i a l v e r t i c a l stress and i s the v e r t i c a l e f f e c t i v e stress at the beginning of the n t n cycle. The generalised s t r e s s - s t r a i n r e l a t i o n s h i p s which account for hardening and porewater pressure increase are given by equations (3.10), (3.11), (3.15) and (3.16). These s t r e s s - s t r a i n r e l a t i o n s are incorporated i n the dynamic e f f e c t i v e stress method of Finn et a l . (1977). Can the s t r a i n hardening e f f e c t occur during undrained condi-tions i n which the p o t e n t i a l volumetric st r a i n s ( p l a s t i c ) are absorbed by rebound, or i s i t postponed u n t i l the volumetric s t r a i n s develop a f t e r drainage? In the c r i t i c a l state theory of Schofield and Wroth (1967) the s t r a i n hardening i s considered to occur i n clays i n undrained shear and, therefore, the area enclosed by the y i e l d surface increases. Finn 44 FIG. 3.11 Comparison Between Predicted and Measured Stress-Strain Curve. 45 et a l . (1977) suggest that, since the major contribution to s t r a i n hardening comes from the generation of a stable structure due to i n t e r -p a r t i c l e s l i p , rather than from increased density, hardening should occur under undrained conditions. I t i s further assumed that s t r a i n hardening under drained and undrained c y c l i c shear should be the same, although i n applying t h e i r method i n engineering p r a c t i c e they were not including s t r a i n hardening e f f e c t s f or undrained: conditions. I t i s f e l t that i t i s s t i l l not s e t t l e d whether s t r a i n hardening should be included i n the undrained condition. Correction for hardening and porewater pressure are made only during the unloading phases as shown in F i g . 3.9(c). This i s based on the observation i n laboratory c y c l i c simple shear t e s t s , that most of the volumetric s t r a i n i n dry sand and porewater pressure i n undrained saturated sand occurs during the unloading portion of the load cycle. However, recently Finn et a l . (1980) have showed that i n c y c l i c drained t e s t s , volumetric strane takes place during loading and unloading phase for the f i r s t few cycles, and l a t e r on most of the volumetric s t r a i n takes place during the unloading phase as shown i n F i g . 3.12. Correction of the s t r e s s - s t r a i n curve should be made continuously over the loading and unloading phases. 3.3 VERIFICATION OF CONSTITUTIVE RELATIONSHIPS The porewater pressure model and s t r e s s - s t r a i n r e l a t i o n s are v e r i f i e d by Finn et a l . (1977) f o r the c y c l i c undrained stress c o n t r o l l e d t e s t . For porewater pressure computation, both s t r a i n hardening and softening behaviour i s considered and computed r e s u l t s agree w e l l with the experimental data (Finn et a l . , 1977). However, the performance of the model i s only v e r i f i e d by a l i q u e f a c t i o n strength curve. Moreover, 46 FIG. 3.12 Volumetric Strain Variation with Cyclic Shear Strain in Drained Tests (After Finn et a l , 1980). 47 the simple shear equipment employed to perform the undrained test had a considerable shear equipment employed to perform the undrained test had a considerable amount of compliance and as Finn and Vaid (1977) show, the presence of compliance considerably a f f e c t s the pore pressure 7 response. Hence, i t i s important to v e r i f y the pore pressure p r e d i c t i v e capacity against experimental data obtained under constant volume conditions. With the modification of c y c l i c simple apparatus at the University of B r i t i s h Columbia by Finn and Vaid (1977) to perform drained constant volume c y c l i c simple shear, i t i s possible to carry out a constant volume test and make t h i s comparison. I t i s further important to check whether t h i s model can accurately predict the h i s t o r y of development of porewater pressure i n undrained stress c o n t r o l l e d tests and not j u s t the l i q u e f a c t i o n strength curve. From a p r a c t i c a l point of view, i t i s very important to check whether t h i s model can make any u s e f u l predictions for overconsolidated sand. 3.4 DISCUSSION analysis by Finn et a l . (1977) require the v e r i f i c a t i o n of the basic assumptions of the model. This would consist of providing adequate and experimentally based answers to the following questions: The c o n s t i t u t i v e r e l a t i o n s h i p s used f o r e f f e c t i v e stress 1. Are.volumetric s t r a i n s i n undrained tests the same as that of drained t e s t s when both samples are i d e n t i c a l and subjected to the same shear s t r a i n history? 2. In constant s t r a i n c y c l i c loading t e s t s , i s there a unique r e l a t i o n s h i p between volume changes i n drained tests and porewater pressure i n undrained tests? 3. Can E r be measured s t a t i c a l l y ? Refer to Appendix I for d e t a i l s . 48 4. Can the model accurately predict the hi s t o r y of development of porewater pressures i n constant stress tests and not j u s t the l i q u e f a c t i o n strength curve? 5. Does s t r a i n hardening occur during undrained tests? 6. Can the model predict the e f f e c t of s t r a i n h i s t o r y under previous loading? 7. Can the model make us e f u l predictions for over-consolidated sand under stress c o n t r o l conditions? 8. F i n a l l y , can the model make us e f u l predictions of porewater pressures under general loading and drainage condition i n simple shear? CHAPTER 4 VERIFICATION OF CONSTITUTIVE RELATIONSHIPS FOR  EFFECTIVE STRESS MODEL This chapter undertakes the v e r i f i c a t i o n of the fundamental assumptions of the model by Martin et a l . (1975). F i r s t , that the p l a s t i c volumetric s t r a i n s which develop during an undrained simple shear test are the same on the volumetric s t r a i n s which would develop i n a drained simple shear test and, second, that there i s a unique r e l a t i o n s h i p between volu-metric s t r a i n s i n drained tests and porewater pressures i n undrained tests when tests are performed on a given sand and use the same shear s t r a i n h i s t o r i e s . In 1976, Finn and Vaid showed that i n a constant volume c y c l i c simple shear t e s t , the reduction i n e f f e c t i v e pressure i s equivalent to the increase i n porewater pressure i n the corresponding undrained t e s t . They also observed that the constant volume test i s almost free of compliance and has an e x t r a o r d i n a r i l y high degree of r e p r o d u c i b i l i t y and consistency. In t h i s study, Martin's model i s v e r i f i e d using the e f f e c t i v e stress data from constant volume te s t instead of porewater pressure data from undrained t e s t . For these constant volume (undrained) t e s t s , the change in e f f e c t i v e stress i s r eferred to as an increase i n porewater pressure. To v e r i f y the performance of the pore pressure model, required sand c h a r a c t e r i s t i c s such as volume change behaviour during c y c l i c shearing in the drained condition and rebound c h a r a c t e r i s t i c s are measured. Rebound c h a r a c t e r i s t i c s of sand are measured.under both s t a t i c unloading conditions, as suggested by Martin et a l . (1975), and dynamic conditions. A comparison of dynamic and s t a t i c rebound c h a r a c t e r i s t i c s i s made and a c o r r e c t i o n factor proposed by which the s t a t i c rebound modulus can be modified so that i t i s 49 50 suitable for pore pressure effective stress prediction in the undrained constant volume cyclic loading condition. The effective stress predictive capacity of the model is evaluated for stress controlled undrained tests and for tests with irregular cyclic loading histories representative of earthquake loading. The constitutive relationships for the nonlinear, hysteretic stress-strain behaviour of sand as proposed by Lee (1975), when coupled with the porewater pressure model, are used to predict pore pressure response in stress controlled undrained tests. Moreover, the performance of the pore pressure model and stress-strain relationship is evaluated for samples with previous strain histories. 4.1 VERIFICATION OF FUNDAMENTAL ASSUMPTIONS To verify that the plastic volumetric strains are the same under undrained and drained cyclic loading conditions, a series of experiments has been performed where plastic volumetric strains occurring in undrained and drained samples were measured. One typical set of these experiments is shown in Fig. 4.1(a) and (b). Two samples prepared in exactly the same 2 manner are consolidated to a vertical confining stress, a' = 210.0 kN/m . vo The resulting volumetric strains during i n i t i a l consolidation are shown in Fig. 4.1(a) and (b) by curve A. The f i r s t sample, tested under undrained (constant volume) condi-tions, is subjected to 21 cycles of y = 0.198%. Due to cyclic shearing the 2 2 vertical effective stress decreases from 209.0 kN/m to 15 kN/m , shown in Fig. 4.1(b) as curve B. The second sample is also subjected to 21 cycles of Y = 0.198% but under drained conditions, which results in a plastic volume-t r i c strain of the order of 0.975%. Since both drained and undrained samples are subjected to exactly the same cyclic shear strain history, i t is 51 £ v r =0.827% Volumetric strain , € v % FIG. 4.1(a) V e r t i c a l E f f e c t i v e Stress vs. Volumetric S t r a i n f o r S t r a i n Controlled Undrained Tests. FIG. 4.1(b) V e r t i c a l E f f e c t i v e Stress vs. Volumetric S t r a i n f o r S t r a i n Controlled Drained Tests. . 52 conceivable that the porewater pressure developed i n the undrained test i s a consequence of a p l a s t i c volumetric s t r a i n of 0.975% (as occurs i n the drained condition). In order to measure the p l a s t i c volumetric s t r a i n occurring i n the undrained t e s t , the sample i s allowed to drain a f t e r c y c l i c shearing and the r e s u l t i n g volumetric s t r a i n s measured (see curve C i n Fig. 4.1(b)). The p l a s t i c volumetric s t r a i n measured i n the undrained test i s 0.847% which i s 85% of that measured i n the drained t e s t . This lack of complete correspondance may be due to one or more of the following f a c t o r s : 1. The process during which the p l a s t i c volumetric s t r a i n s are recorded i s d i f f e r e n t i n the drained and undrained t e s t s . During the drained te s t , volumetric s t r a i n s are measured when sand p a r t i c l e s are under-going c y c l i c shearing, whereas i n the undrained condition the p l a s t i c volumetric s t r a i n s are measured a f t e r c y c l i c shearing has ceased. 2. F r i c t i o n a l forces acting at the sides of the simple shear equipment may r e s i s t the recovery of p l a s t i c volumetric s t r a i n s i n the undrained t e s t . 3. The sample used for the undrained test i s not exactly s i m i l a r to that used f o r the drained t e s t . It can be noted from F i g . 4.1(a) and (b) that the volumetric s t r a i n s due to i n i t i a l consolidation are d i f f e r e n t . However, the experimental data obtained i n the drained and undrained conditions when both samples are subjected to 1,3,5 and 10 cycles of y = 0.198% show that the p l a s t i c volumetric s t r a i n s i n the undrained condi-t i o n are generally 80 to 85% of those observed i n the drained condition. It i s f e l t that the 20 to 15% differ e n c e i n volumetric s t r a i n s i n drained and undrained conditions may be due to the three reasons mentioned above and i t i s j u s t i f i e d to conclude that p l a s t i c volumetric s t r a i n s i n undrained condi-tions are the same as those i n the drained conditions when both the drained and undrained samples are subjected to the same shear h i s t o r i e s . To v e r i f y the existence of a unique r e l a t i o n s h i p between v o l u -metric change under drained conditions and porewater pressures under undrained 53 1 3* > o o E o > 2.0 •c I.O 0.5 0 l l l l - Sand type : Ottawa sand (C ~! 1 1 - I 0 9 ) ^ K ^ " -Relative density =45% y = 0 . 2 5 % -/T 0 .245% -/ X °"v'0 = 2 0 0 k N / m2 1 * a™ =i i i i 300 k N / m 2 i i 0 10 20 30 40 50 Number of cycles , N 60 FIG. 4.2 Plot Between Volumetric Strain vs. Number of Cycle for Constant Cyclic Shear Strain Test on Loose Ottawa Sand (C-109). 54 conditions f o r samples with the same s t r a i n h i s t o r y , s t r a i n c o n t r o l l e d t e s t s i n both drained and undrained conditions have been performed. In drained tests on Ottawa sand, volumetric s t r a i n are measured while sample c o n s o l i -2 2 dated to 200 kN/m and 300 kN/m . The r e s u l t s presented i n F i g . 4.2, show that the r e s u l t i n g volumetric s t r a i n s are independent of confining stress. Experiments f o r the undrained s t r a i n c o n t r o l l e d condition have been performed on sand specimens i d e n t i c a l to those used i n the drained condition. Volumetric s t r a i n s , £ V (j% are plotted against the porewater pressure r a t i o , u/a' , i n Figs. 4.3 and 4.4 for Ottawa sand at D = 45% and 60%, r e s p e c t i v e l y , where each point represents the corresponding value of these v a r i a b l e s f o r a given number of cycles of constant s t r a i n amplitude. For example, i n F i g . 4.2 point A represents the state of the volumetric s t r a i n , e v { j % , due to 2 cycles of shear s t r a i n amplitude of y = 0.10%, and the pore-water pressure r a t i o , U/O^q, developed due to 2 cycles of shear s t r a i n Y = 0.10%. From F i g . 4.3, i t can also be seen that there i s a s l i g h t devia-t i o n i n shear s t r a i n l e v e l amplitudes used f o r drained and undrained t e s t s , the d i f f e r e n c e ranging from 0% to 5% but t h i s i s not considered s i g n i f i c a n t . This scatter i s due to the s l i g h t d i f f e r e n c e i n r e l a t i v e d e n s i t i e s f o r the drained and undrained t e s t s . Experimental data shown i n Figs. 4.3 and 4.4 c l e a r l y i n d i c a t e the existence of a unique r e l a t i o n s h i p between volumetric s t r a i n i n the drained condition and porewater pressure i n the undrained condition. In addition, t h i s assumption i s tested f o r samples consolidated to a v e r t i c a l 2 confining s t r e s s , =300 kN/m as shown i n F i g . 4.5. 4.2 EVALUATION OF THE PREDICTIVE CAPACITY OF THE PORE PRESSURE MODEL To v e r i f y the p r e d i c t i v e capacity of the porewater pressure o - > i i i 1 1 r- 1 1 1 D v. Sand type : Ottawa sand (C -109) *• in o-JQ= 200 k N / m 2 , Relative density = 4 5 % Ires 1.00 -nfining si 0.75 X p X • J— £ A o u Legend O Shear strain amplitudes E 0.50 Drained Undrained -ire/ "Jo A o 0.056 % 0.056 % pressi 0.25 / ' • 0 .100% 0. 100 % x 0 . 2 0 0 % 0,210 % -o A 0.314 % 0.300 % 0 1 1 i ! i 1 1 i • 1 1 ' 1 1 1 I I 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Volumetric strain in percent, € v d % FIG. 4.3 Relationship Between Volumetric Strains and Porewater Pressures in Constant Strain Cyclic Simple Shear Tests, Dr = 45%. b tn tn c 1.00 r-~ 0.75 h o u I 0.50 c E 0.25 a. o 0_ 0 ! 1 i • i . . . Sand type = Ottawa sand ( C - 109 ) a ' = 200 k N / m 2 , Relative density = 6 0 % vo ' O -Legend Shear strain amplitudes Drained Undrained • 0 . 1 0 3 % 0 .1245% x 0 . 207% 0.203 % i * 0 . 2 8 5 % 0.270 % 0.2 0.4 0.6 0.8 1.0 \ 12 . 1.4 Volumetric strain in percent , € y ( j % 1.6 1.8 FIG. 4.4 Relationship Between Volumetric Strains and Porewater Pressures in Constant Strain Cyclic Simple Shear Tests. Volumetric strain , e. FIG. 4.5 Relationship Between Volumetric Strains and Porewater Pressures in Constant Strain Cyclic Simple Shear Tests. 58 model, we need the following two physical properties of the s o i l skeleton: (a) volumetric change c h a r a c t e r i s t i c s during a c y c l i c loading test; and (b) rebound c h a r a c t e r i s t i c s of sand. 4.2.1 Rebound C h a r a c t e r i s t i c s of Sand The porewater pressure model i n i t s simplest form implies that under undrained (constant volume) conditions the p l a s t i c volumetric s t r a i n , A e V ( j , i s equal and opposite to the e l a s t i c or recoverable volumetric s t r a i n , A e v r . The p l a s t i c volumetric s t r a i n increment i n the undrained con-d i t i o n i s the same as that i n the drained condition as shown i n section 4.1, hence, the p l a s t i c volumetric s t r a i n i n the undrained condition i s a known quantity. From the undrained s t r a i n c o n t r o l l e d t e s t , the decrease i n e f f e c t i v e v e r t i c a l stress, Ao\^, caused by p l a s t i c volumetric s t r a i n , A e V ( j , i s also known. Therefore, the rebound modulus, E r , corresponding to the increment of e f f e c t i v e stress, A a y , i s Aa' Aa' A E = — = - 1 - = — ( 4 . 1 ) r Ae Ae , Ae , vr vd vd Hence, to measure the rebound modulus, the increase i n porewater pressure measured during a s t r a i n c o n t r o l l e d undrained test and volumetric s t r a i n s during a s t r a i n c o n t r o l l e d drained t e s t are required. A t y p i c a l set of such data i s plotted i n F i g . 4.6, f o r samples subjected to a c y c l i c shear s t r a i n amplitude, y = 0.10%, i n both drained and undrained s t r a i n c o n t r o l l e d t e s t s . The curve shown i s c a l l e d the dynamic unloading curve. Since there i s a unique r e l a t i o n s h i p between pore pressure increase i n the undrained condition and volumetric s t r a i n i n the drained for a l l l e v e l s of shear s t r a i n amplitude (as shown i n Figs. 4.3 and 4.4), 59 i r I 25 Recoverable volumetric strain , e v r % FIG. 4 . 6 Vertical Effective Stress vs. Volumetric' Strain During Dynamic Unloading. 250 200 150 100 c v 0 =200 KN/m2 Sand type Ottowa sand (C-109) Relative density = 44 2 % Static unloading , e v r 0 = 0 6 9 6 % Static unloading [Subjected to 20 cycles of 0 2 % shear strain] « v r 0 - 1 . 0 9 9 % --ic-jr-fr-0.2 0 4 0 6 0.8 1.0 Recoverable volumetric strain , c v r % 1 2 FIG. 4.7 Vertical Effective Stress a v vs. Recoverable Volumetric Strain During Static Unloading Conditions. 60 the dynamic unloading curve i s also a unique curve for a l l l e v e l s of shear s t r a i n amplitude. The rebound modulus, E r , i s the incremental slope of the curve shown i n F i g . 4.6. Since t h i s modulus i s determined from data produced under c y c l i c loading conditions, i t w i l l be c a l l e d the dynamic rebound modulus ((E r)dynamic^• Martin et a l . (1975) suggest that the rebound modulus of sand under one-dimensional, l a t e r a l l y confined conditions should be determined from s t a t i c unloading curves. In F i g . 4.7, the s t a t i c unloading curve for Ottawa sand at D r = 44.2% consolidated to a v e r t i c a l e f f e c t i v e stress of 200 kN/m2 i s shown as a s o l i d l i n e . From t h i s curve, i t i s observed that about 80% of the t o t a l recoverable s t r a i n i s measured i n the l a s t 15% of unloading. To evaluate the increase i n recoverable volumetric s t r a i n due to c y c l i c shear p r i o r to unloading, as discussed i n section 3.1.2, a c y c l i c shear of 20 cycles at y = 0.20% i s applied to another sand sample before unloading. Results of t h i s test are also plotted i n F i g . 4.7 as a dotted curve. In F i g . 4.7 r e s u l t s from samples with no shear s t r a i n h i s t o r y show a rebound modulus about 5% to 20% higher than that f or the strained samples, with a d i f f e r e n c e for the l a s t 20% unloading becoming quite marked. However, i t should be noted that by measuring rebound character-i s t i c s of sand i n t h i s manner, the sand p a r t i c l e s are not under c y c l i c shearing conditions. 4.2.1.1 Comparison between s t a t i c and dynamic rebound modulus A comparison has been made between s t a t i c and dynamic unloading curves from i n i t i a l v e r t i c a l e f f e c t i v e stress 200 kN/m for Ottawa sand at D r = 45%. One dynamic and two s t a t i c unloading curves are plotted i n Fi g . 4.8. This f i g u r e c l e a r l y shows that i n most of the range of v e r t i c a l 61 Sand type : Ottawa sand ( C -109 ) c j 0 * 200 kN / m 2 , Relative density* 4 5 % Obtained under cyclic loading conditions in simple shear Obtained by static unloading in simple shear apparatus Obtained by static unloading in consolidation ring 10 30 50 .70 90 Recoverable strain . e I 3 I 5 vr Rebound of Ottawa Sand Under Various Loading Conditions. 62 effective stress, recoverable strains obtained by static unloading even with the strain history effect, are considerably lower than dynamic recoverable strain. Figure 4.9 contains a comparison between static recoverable strain without shear strain history and dynamic recoverable strain at various levels of vertical effective stress. It can be noted that, in most of the range of vertical effective stress, static recover-able strains are about 3 to 5 times smaller than dynamic strains. In other words, the static rebound modulus is about 3 to 5 times higher than dynamic rebound modulus. The static rebound modulus is also measured by the consolida-tion equipment where the error due to lateral f l e x i b i l i t y of the equipment does not influence the measurement of recoverable strains. These tests performed on loose Ottawa sand resulted in rebound modulus values 10% to 15% higher than those for identical samples in the simple shear equipment. In this case, the static rebound modulus is 5 to 5.5 times higher than the dynamic rebound modulus. To use the pore pressure model to predict porewater pressures in stress controlled undrained tests, i t is necessary to use the rebound modulus and volumetric strain for the sand under consideration. In the latter part of this chapter i t is shown that pore pressure response pre-dicted using the dynamic rebound modulus is close to experimentally deter-mined pore pressures. From the remarks above i t w i l l be obvious that predictions using the static rebound modulus cannot be r e a l i s t i c . In order to use the pore pressure model, i t is necessary to evaluate the dynamic rebound modulus at various levels of vertical effec-tive stress. For that purpose dynamic unloading curves from various levels of i n i t i a l vertical effective stress are obtained. In Fig. 4.10, 63 dynamic unloading curves from three different i n i t i a l vertical effective stresses are shown. Although the curves look geometrically similar, they are not quite algebraically similar. However, by assuming one algebraic equation for a l l these curves, the error involved is not significant. Hence, the generalised form of the equation for dynamic rebound modulus can be of the same form as given in equation (3.7). Experimental data similar to that shown in Fig. 4.10 can be used to calculate dynamic rebound constants m,n and k2*. It should be noted that for porewater pressure verification, rather than using a gene-ralised form similar to equation (3.7), the dynamic rebound modulus at a particular effective stress a' is obtained by the slope of the dynamic unloading curve for a given i n i t i a l vertical confining stress, o y o . Since the porewater pressure model requires verification of i t s predictive capacity for sand under various relative densities, i t is necessary to obtain dynamic unloading characteristics of Ottawa sand for various relative densities. Experimental data plotted for relative densi-ties of 45,54,60 and 68%, shown in Fig. 4.11, indicate that dynamic rebound characteristics of Ottawa sand are independent of relative density. This finding is significant since this obviates the need for measuring dynamic unloading characteristics at different relative densities. 4.2.2 Volume Change Characteristics of Sand under Cyclic Loading  Conditions In order to verify the porewater pressure model, apart from the dynamic rebound characteristics, volume change characteristics under cyclic loading in the drained condition are also required. For this purpose, •^Values of these constants w i l l be different for static rebound modulus. 64 H o 3 80 ~- 6.0 > - _ 4.0 E o c T3 C o in c o o 20 0 1 1 1 Sand type • Ottawa sand (C-109) o-v'0 = 200 kN/m 2 , Relative density =45% App. average factor = 4.2 200 150 100 50 0 Vertical effective stress, cr', kN/m 2 FIG. 4 . 9 , Ratio of Dynamic Recoverable Strain to Static Recoverable Strain for Various Values of Effective Vertical Stress. Sand type = Ottawa sand (C-109) Recoverable volumetric strain , « % Dynamic Unloading Curves from Three I n i t i a l Vertical Effective Stress for Ottawa Sand. 200 > Sand type : Ottawa sand (C -109 ) Legend = Shear strain amp. , / % Relative density Drained Undrained • Dr = 4 5 % 0.100 0.100 x Dr = 54 % 0.130 0 126 A Dr =60% 01030 0.1245 • Dr =68% 0.122 0.109 Dynamic unloading curve 0.25 0.50 0.75 1.00 1.25 Recoverable volumetric stra in , € v r % ON Ul 1.50 FIG. 4.^J> Vertical Effective Stress vs. Dynamic Recoverable Strain for Various Relative Densities. 6 6 cyclic strain controlled tests have been performed on Ottawa sand at Dr = 4 5 % . Experimental data from cyclic drained tests on Ottawa sand (D r = 4 5 % ) are replotted in terms of incremental volumetric strain, Ae V (j, vs. cyclic shear strain amplitude, y, for a given value of cumulative volumetric strain, E V ( J % , as shown in Fig. 4 . 1 2 . The analytical function as given in equation ( 3 . 4 ) is fitted with a family of curves and four volume change constants obtained which are: C± = 0 . 9 1 3 , C 2 = 0 . 4 6 2 , C3 = 0 . 1 6 1 and C4 = 0 . 3 7 6 . The set of constants obtained give an almost exact f i t with the experimental data. However, cyclic drained tests also performed at three other relative densities 5 4 % , 6 0 % and 6 8 % yielded the volume change constants given in Table 4 . 1 . During experiments on dense sand i t was observed that some dilation was present within a cycle, however, the net effect of cyclic shear is a reduction in volumetric strain. 4 . 3 PORE PRESSURE PREDICTION At this stage i t seems best to f i r s t check the porewater pres-sure model before evaluating the validity of the stress-strain law. Hence, the porewater pressure response is calculated in stress controlled undrained tests by the following two procedures: 1 . The shear strains measured during a stress controlled undrained test are used to calculate the incremental plastic volumetric strain, Ae v cj, and hence pore pressure. Thus, the validity of the pore pressure model is checked without involving the stress-strain relationship. 2 . The generated strain under stress controlled undrained conditions is calculated using the proposed constitu-tive relationship for stress-strain, and from that the porewater pressure is predicted. 67 F I G . A . 12 Incremental Volumetric Strain vs. Shear Strain Amplitude for Various Levels of Cummulative Volumetric Strain. 68 TABLE 4.1 VOLUME CHANGE CONSTANTS FOR OTTAWA SAND (C-109) D % C. C_ C„ C, r 1 2 3 4 45 0.913 0.462 0.1612 0.376 54 0.626 0.525 0.100 0.258 60 0.467 0.658 0.100 0.235 68 0.357 1.060 0.1100 0.05 69 The pore pressure response is calculated for stress controlled undrained (constant volume) tests performed on Ottawa sand at Dr = 45%. During these tests, in addition to the measurement of porewater pressures, shear strains were a l so monitored down to a level of 0.02%. These measured shear strains are used to calculate volumetric strain from equation (3.4) with the constants given in Table 4.1 for Dr = 45%, and the dynamic rebound modulus is calculated from the slope of the dynamic unloading curve as shown in Fig. 4.3. A comparison between predicted and measured porewater pressures is shown in Fig. 4.13 for Dr = 45%. The predictive capability of the pore pressure model appears to be very good. Hence, i t can be safely concluded that the pore pressure model can predict accurate pore pressure response provided the dynamic rebound modulus is used in conjunction with incremen-tal volumetric strains. Porewater pressures have also been calculated for two tests from strains calculated by the following equation: T - T G ( y - y ) G ( y - y ) r mn r , r i mn r // o\ — 2 — = 2 / U + — 2 ^ J (4.2) mn where G m n and x m n are given by equations (3.15) and (3.16) and the defini-tion of x r and y r is the same as that given in equation (3.11). Since G m Q could not be measured, both i t and T M Q are computed for void ratios e = 0.676 (D r = 45%) and e = 0.628 (D r = 60%) using the well-known Hardin-Drnevich equations (3.9 ). The magnitude of G m n and x m n in equation (4.2) are modified during the strain calculations for the effect of increasing porewater pressure by the equations: G /G = (a'/a' ) H (4.3) mn mo v vo and Sand type : Ottawa sand (C-109) a v ' 0 = 200 k N / m 2 , Relative density =45% o Experimental curve Analytical curve o (Predictions based on measured strains) 6 10 30 Number of cycles , N 60 100 FIG. 4.13 Predicted and Measured Porewater Pressure in Constant Stress Cyclic Simple Shear Tests, Dr = 45%. o 200 71 x / T = o-'/a' (4.4) mn mo v vo These expressions neglect the s t r a i n hardening functions i n equations (3.15) and (3.16). Neglecting s t r a i n hardening ( s t r a i n history) e f f e c t s for undrained tests appears j u s t i f i e d on the evidence of Finn et a l . (1970) and Seed et a l . (1977) that increased resistance to pore pressure develop-ment as a r e s u l t of previous loading or s t r a i n h i s t o r y i s achieved only i f d i s s i p a t i o n of the porewater pressures caused by the previous loading i s allowed. Later on, i n section 4.4, the correctness of t h i s approach w i l l be demonstrated by comparisons between predicted and measured porewater pressures as shown i n F i g . 4.14 for D r = 45% and i n F i g . 4.15 for D r = 60%. The comparisons are quite good, although not as good as when measured shear s t r a i n s have been used i n the porewater pressure model. This i s not unexpected since the actual i n i t i a l i n - s i t u moduli were not measured but calculated by the Hardin-Drnevich equations. I t can be concluded that the c o n s t i t u t i v e r e l a t i o n s h i p s for the nonlinear s t r e s s - s t r a i n behaviour of sand c a l c u l a t e r e a l i s t i c shear s t r a i n h i s t o r y f o r the undrained condition, provided hardening i s not included. Moreover, the c o n s t i t u t i v e r e l a t i o n -ships f o r the pore pressure model predict accurate pore pressures i n the stress c o n t r o l l e d undrained conditions provided instead of s t a t i c , dynamic rebound modulus i s used. 4.3.1 Pore Pressure P r e d i c t i o n f o r Irregular S t r a i n History A more severe test of the p r e d i c t i v e c a p a b i l i t y of the pore pressure model i s provided by the i r r e g u l a r s t r a i n h i s t o r y shown i n F i g . 4.15(a). The computed porewater pressure response to t h i s s t r a i n pattern i s shown i n F i g . 4.15(b) and i s quite s a t i s f a c t o r y . Some of the di f f e r e n c e between experimental and a n a l y t i c a l pore pressure curves i s probably due to.the f a c t that the i r r e g u l a r s t r a i n pattern has been imposed by manual Sand type : Ottawa sand ( C - 1 0 9 ) c r v ' 0 = 2 0 0 k N / m 2 .Re la t i ve density = 4 5 % T/cr' =0.074 T / c r '=0.065 Number of cycles , N FIG. 4.14 Predicted and Measured Porewater Pressures in Constant Stress Cyclic Simple Shear Tests, Dr = 45%. to CO to c 3 (O CO <L> o. o 0. .00 Sand type = Ottawa sand (C -109 ) °"v'o = 2 0 0 k N / m 2 , Relative density = 6 0 % ~ 0.75 c o u :i o.5o c T/^i / o 8 5 0 .097 T/O"' = 0.126 0.25 0 ° Experimental curve Analytical curve Predictions based on calculated strains using eqns. ( I ) to (5) 6 10 30 Number of cyc les , N 60 100 200 FIG. 4.15 Predicted and Measured Porewater Pressures in Constant Stress Cyclic Simple Shear Tests, Dr = 60%. 74 en o h_ .30 Q . C 20 <B 3 .JO Q . E o 0 c I D J J -.10 UJ k . o -.20 a> x: CO - 3 0 + 1 o V 1.00 u> (0 a> k. </> cn c .75 . c C o o o .50 c \ a> 25 01 05 <D. k. a. (a). o CL 9 0 % pore pressure {/ o Experimentol curve • - — A n a l y t i c a l curve 1?" Sand type ; Ottawa sand ( C - 1 0 9 ) <Ty0 = 2 0 0 kN/m2,Relative density =45% _L _l_ (b) 10 20 30 Number of cycles , N 4 0 FIG. 4.£f$ Comparison of Calculated and Analytical Pore Pressure Ratios for Irregular Strain History. c o n t r o l and not by a programmed automatic c o n t r o l . Thus, the measured shear s t r a i n h i s t o r y may not be accurate enough. Hence, the pore pressure p r e d i c t i v e capacity of the model i s quite good for i r r e g u l a r s t r a i n h i s t o r y as generated during earthquakes. 4.4 PORE PRESSURE PREDICTION FOR SAMPLES WITH PREVIOUS STRAIN HISTORY The primary aim of t h i s section i s to check whether the model can predict pore pressure response for samples with previous s t r a i n h i s t o r i e s . In addition, the underlying mechanism of increase of r e s i s t i v e capacity against l i q u e f a c t i o n due to p r i o r s t r a i n h i s t o r y i s studied. Three seri e s of tests are performed where samples consolidated to o^o = 200 kN/m2 are subjected to c y c l i c shear ( T / C F ^ 0 0.04 to 0.068) u n t i l the porewater pressure r a t i o becomes equal to 0.30, 0.5 and 0.65, r e s p e c t i v e l y . Once the desired porewater pressures are achieved, the experiment i s stopped and the pore pressures allowed to dr a i n . The net change i n volumetric s t r a i n due to drainage i s recorded. Next, the samples which have been subjected to previous s t r a i n h i s t o r i e s are subjected to c y c l i c shear stress u n t i l complete l i q u e f a c t i o n i s reached. In F i g . 4.17 c y c l i c shear stress r a t i o s used for samples with s t r a i n h i s t o r y are plotted against the number of cycles to l i q u e f a c t i o n for various l e v e l s of s t r a i n h i s t o r i e s . Figure 4.16 shows that the resistance to l i q u e f a c t i o n increases with previous s t r a i n h i s t o r y and the increase i s proportional to pore pressure l e v e l developed during shear s t r a i n h i s t o r y . However, i t should be noted that the maximum c y c l i c shear s t r a i n generated during s t r a i n h i s t o r y i n these tests i s always lower than 0.4%. Finn et a l . (1970) suggest a threshhold value of c y c l i c shear s t r a i n of beyond which the resistance to l i q u e f a c t i o n decreases due to 0.20 o o o k. Ui if) 0> w o o o 0.15 0.10 0.05 Ol l i i i 1 r Sand type : Ottawa sand (C-109) cr ' = 200 k N / m 2 , Relative density = 4 5 % • Sample with previous strain history Samples without any strain history U/O" v 0 = Pore pressure developed during the strain history 3 6 10 30 60 100 300 600 1000 Number of cycles to liquefaction , N L FIG. 4.17 Cyclic Stress Ratio vs. Number of Cycles to Liquefaction for Samples with Previous Strain History. strain history. Nonetheless, i t is possible that larger cyclic shear strains (y > 0.05%) developed during shear strain history create a struc-ture in the sand which is more sensitive to liquefaction than the struc-ture created by i n i t i a l consolidation. However, the experimental data confirms the shared conclusion from Finn et a l . (1970) and Seed et a l . (1977) that resistance to liquefaction increases due to previous strain history. Such an increase in resistance is often considered due to change in particle structure, increase in lateral stress and small change in density. Although no attempt is being made in this research to measure the change in particle structure, i t s contribution to increase resistance is evaluated by measuring increase in lateral stress and change in density. In Fig. 4.18 the variation of vertical effective stress against horizontal effective stress is plotted for a test in which an i n i t i a l l y consolidated sample is subjected to 70 cycles of T/O^Q = 0.066 producing a pore pressure ratio of about 0.50. The sample is drained and allowed to reconsolidate to i n i t i a l vertical effective stress. It can be noted from the figure that the difference, k Q, in an i n i t i a l l y consolidated stage and after the application of the strain history is negligible. In addition, the change in relative density due to preshearing is very small (0.30%). Thus, i t can be postulated that since the contribution of increase in rela-tive density and increase in lateral stresses due to cyclic preshearing is very small for the observed increase resistance due to strain history, the primary underlying cause is the change in the particle structure of the sand. To incorporate the influence of strain history in the constitu-tive relation, i t is important to associate a physical variable with the underlying factors involved in increasing the resistance to pore pressure 100 75 50 25 0 1 1 1 Sand type = Ottawa Sand (C-109) I 1 Relative density = 4 5 % 0 - J 0 = 200 k N / m 2 Ko =0.40 (Sample with strain history ) . Cyclic preshearinq I J A U W ^ ^ ^ ^ / Ko = 0.393 (Virgin (T/crv'0 = 0.066, n = 70) . ^ ^ ^ ^ ^ X sample) / j Cyclic shearing Ar (T/cr v o = 0.104 , n = 26.5) 1 i I 0 50 100 150 200 250 Vertical effective stress , o~y. k N / m 2 Horizontal Effective Stress vs. Vertical Effective Stress for Cyclic Stress Controlled Undrained Test on Samples with Previous Shear Strain History. 79 development due to strain history. Such a variable as suggested by Martin et a l . (1975) is a cumulative volumetric strain. The cumulative volumetric strain is used as an index to account for strain hardening effects in the constitutive relationships for stress-strain behaviour of sand by Lee (1975). These relationships are used for dynamic effective stress analysis by Finn et a l . (1977). It i s considered that during the shear strain history no hardening occurs, but once the sample is allowed to drain a small increase in volu-metric strain, e V (j, can take into account a part of increase in stiffness by increasing shear moduli G m n and x m n with hardening constants. In addi-tion, the resulted volumetric strain after the strain hardening decreases the volumetric potential for the sample for further cyclic loading. The evidence of increase in stiffness of sand due to strain history is provided by Toki and Kitago (1974), who observed an increase in the static modulus of a loose dry sand which has undergone several hundred cycles of small-amplitude cyclic stress. The evaluation of the pore pressure predictive capacity of the constitutive relations for the pore pressure model and stress-strain relationship i s performed by subjecting a sample to a loading-drainage-loading sequence. Sample A at Dr = 45% is f i r s t subjected to 70 cycles of a stress ratio T / o y o = 0.066 and then allowed to drain. The rate of development of porewater pressure under this loading i s shown in Fig. 4.19(a) by curve A. After drainage the resulting volumetric strain and change in relative density are recorded (see Table 4.2, Sample 2). The recovered potential plastic strains are used in conjunction with hardening constants to calculate modified G m n and x m n (equation 3.13 and 3.14). When Sample A is next subjected to a cyclic stress ratio Sand type Ottawa sand ( C - 1 0 9 ) o"' = 200 k N / m 2 , Relative density = 4 5 % Number of cycles , N FIG. 4.19(a) Predicted and Measured Porewater Pressures in a Sand with Previous Loading History. TABLE 4-2 INCREASE IN SHEAR MODULUS AND SHEAR STRENGTH DUE TO STRAIN HISTORY SAMPLE NO. Pore Pressure Ratio due to Strain History VOID RATIOS Chang e i n Relative Density Change i n Volumetric S t r a i n G . mn G :mo T mn T mo Before .Strain History After S t r a i n History 1 0.30 0.676 0.674 0.62% 0.108% 1.211 1.0488 2 0.50 0.678 0.675 0.94% 0.295% 1.277 1.1333 3 0.65 0.674 0.664 2.80% 0.4669% 1.445 1.2110 G = G [1 + „ ! V D ] , T = x [1 + „ ! v d ] mn mo H,+H„e , mn mo H„+H.e , 1 2 vd 3 4 vd where H, = 0.947, H„ = 0.394, H_ = 2.212, H, = 0.0001 for D = 45%, Ottawa sand, 1 ' 2 ' 3 ' 4 r 82 T'/a^ = 0.104 porewater pressure develops, as shown by curve C in Fig. 4.19(a). In the same figure, curve B shows the rate O f pore pressure development for a sample which has not been subjected to any shear strain history. It can be observed that the rate of development of pore pressure in Sample C is considerably less than that generated in Sample B. To calculate the pore pressure response for Sample C, the modified values of G m n and x m n for strain hardening effects are used as G m o and t m o for the second application of cyclic loading and volumetric strain constants were adjusted to account for the recovered plastic poten-t i a l strains which occurred during strain hardening. The softening effect of the increasing porewater pressure is also included. The predicted porewater pressure is given by curve D in Fig. 4.19(a). In Fig. 4.19(b), similar results are shown for a sample which was subjected to stress ratio, T/ cvo = 0.068 for 16 cycles and then allowed to drain. The increase in Gmn and x m n due to resulting volumetric strain are shown in Table 4.2 (Sample 1). The predicted and experimental curve for this sample are shown in Fig. 4.19(b) by curves C and D. The comparison between predicted and measured porewater pres-sures, shown in Fig. 4.19(a) and (b), is good and may reasonably be viewed as indicating that .strain hardening due to plastic volumetric strains in sands occurs only after drainage. 4.5 -DISCUSSION The porewater pressure model developed by Martin, Finn and Seed (1975) has been tested under a variety of drained and undrained loading conditions. The basic assumptions of the model appear to be well-founded. There is a strong verification of a unique relationship Sand type : Ottawa sand (C-109) 3 6 10 30 60 100 200 Number of cycles , N FIG. 4.19(b) Predicted and Measured Porewater Pressures in a Sand with Previous Loading History. 84 between volumetric strains in drained tests and porewater changes in undrained tests for samples of given sand with similar strain histories. An important point to emerge from the study is that the rebound modulus used to convert volumetric strains to porewater pressures must be measured under cyclic loading conditions. Moduli measured under static rebound conditions are too s t i f f and for a given volumetric strain generate too much porewater pressure. The model predicts successfully the porewater pressure res-ponse under drained conditions for uniform loading and for irregular cyclic loading histories representative of earthquake loading. When combines with the constitutive stress-strain relations used by Finn et a l . (1977) in their dynamic effective stress analysis, i t predicts successfully the effect of previous cyclic loading history on porewater pressure response. It appears from the test data that in undrained tests strain-hardening or strain history effects do not occur. However, at the conclusion of such tests i f drainage is allowed to take place plastic strains are recovered and the sands strain-harden. If proper values for the rebound modulus are used and the effects of strain-hardening are included whenever drainage occurs, i t appears that the constitutive relationship used for nonlinear effective stress analysis by Finn et a l . (1977), can make good predictions of the development of porewater pressure under f a i r l y general loading patterns and drainage conditions in simple shear. CHAPTER 5 VERIFICATION OF CONSTITUTIVE RELATIONSHIPS FOR  OVERCONSOLIDATED SAND The original porewater pressure model developed by Martin et a l . (1975) is based entirely on tests with normally consolidated sands. In this chapter, performance of the model for overconsolidated sands w i l l be evaluated. It is well known that overconsolidated sands have a greater resistance to liquefaction, the resistance increasing with overconsolidation ratio, OCR (Seed and Peacock, 1971). The process of overconsolidation increases the coefficient of lateral earth pressure, k Q, of sand hence mean effective stress. It is generally considered (Seed and Peacock, 1971; Ishibashi and Sherif, 1974) that the increase in liquefaction potential is caused by higher mean effective stress. The higher mean effective stress results in increased shear modulus and lower shear strainthe tests are generally performed in stress controlled conditions. Lower shear strain procedures smaller potential volumetric strain, hence lower porewater pressure. Ishibashi and Sherif (1974) observed that there is a unique relationship between T / C J ^ Q 1 and the number of cycles to i n i t i a l liquefac-tion, irrespective of the i n i t i a l k Q values and confining pressures (Fig. 5.1). If i t is considered that the process of overconsolidation yields only increased k Q then there should be a unique liquefaction strength curve, plotted in terms of T / a m o and number of cycles to liquefaction, for normally and overconsolidated sand samples. T / ° m o is the ratio of cyclic shear stress to i n i t i a l mean effective stress. 85 86 040 030 Sand type Ottawa sand (C-109) O"J0= 1406 kN/m 2 , Relative density =40.6% (Measure in torsion shear apporatus ) 020 010 Ko = 1.0 Ko = 0.75 Ko = 0.65 10 30 60 100 300 1000 Number of cycles to initial liquefaction , N L 3000 FIG. 5.1 Cyclic Shear/initial Mean Effective Stress vs. Number of Cycles to I n i t i a l Liquefaction for Various k Q values (After Ishibashi and Sherif, 1974). rO 6 t-i - cn O <u a> - C > U o <u u a> O c E 040 030 020 0 OCR Ko o 1 0.40 • 2 075 * 3 1.00 Uniform medium Relative density : _L FIG. 5.2 I 3 10 30 100 300 1000 Number of cycles "to initial liquefaction, N L Cyclic Shear S t r e s s / i n i t i a l Mean Effective Stress vs. Number of Cycles to I n i t i a l Liquefaction for Various OCR (After Seed and Peacock, 1971). 87 However, when the experimental data from Seed and Peacock (1971) are plotted in terms of T / a m Q vs. number of cycles for liquefaction for various overconsolidation ratio, as shown in Fig. 5.2, i t shows that the increase in resistance due to overconsolidation can not be assigned to an increased k Q value. In the comprehensive study by Ishihara and Takatsu (1979) the effect of increase in k Q value and OCR value on liquefaction potential are discussed. The authors observe that for a constant k Q value, the resistance to liquefaction increases with increasing OCR and present an empirical formula to relate the cyclic stress ratio required for liquefaction for both normally and overconsolidated samples. For this study a torsional shear apparatus was used. For their study overconsolidated samples in torsional shear apparatus were prepared where, during overconsolidation, the ratio of horizontal to vertical effective stresses (k Q) were kept constant. However, the overconsolidated samples should be prepared in laterally confined conditions where k Q increases during the overconsolidation process. The contribution of increases in k Q value to the increased resistance to liquefaction could not be obtained from simple shear data as has been done by Seed and Peacock (1971) and Finn et a l . (1978) as the k D value could not be measured after overconsolidation. Since i t is now possible to monitor lateral stress in the cyclic simple shear apparatus, results of a study of the contribution of k Q on increased resistance due to OCR can be presented. 5.1 COMPARISON BETWEEN NORMALLY AND OVERCONSOLIDATED SAND BEHAVIOUR Constant volume tests on normally and overconsolidated samples are performed in the stress controlled conditions. The samples were consolidated to vertical effective stress of 400, 600 and 800 kN/m2 88 9 and, by reducing vertical effective stress to 200 kN/m , samples of OCR = 2,3 and 4 were obtained. During the process of i n i t i a l consolidation and unloading, the lateral stresses were monitored. Figure 5.3 shows the variation of effective lateral stress with respect to vertical effective stress for specimens with OCR = 1,2,3 and 4. It is clear from the figure that, inspite of approximately the same i n i t i a l k Q value for a l l tests the overconsolidation process results in significantly higher k Q which increases with increasing OCR value. The variation of lateral and vertical effective stresses during cyclic shearing for OCR = 1,2,3 and 4 are shown by the dotted curves in Fig. 5.3. The cyclic loading curves are quite similar for normally and overconsolidated samples. In Fig. 5.4 cyclic shear stress ratio, T / O v o , is plotted against the number of cycles for i n i t i a l liquefaction for various OCR values. This figure reveals a significant increase in liquefaction resistance due to overcon-solidation. Average values of k Q from several overconsolidated samples are given in Table 5.1 corresponding to different OCR. In Table 5.2 an empirical relationship between k Q of overconsolidated sample and normally consolidated sample, given by Ishihara and Takatsu (1979), is given where the constant m obtained from experimental data l i e s within the observed range by several investigators. Hence, i t can be assumed that measured ko values are f a i r l y reliable. Using these average values of k Q (Table 5.1), experimental data from Fig. 5.4 could be plotted in terms of T/O^Q versus the number of cycles to liquefaction where am is l+2k o' = a' / (—^-) (5.1) mo vo 3 Figure 5.5 clearly shows that though the increase in resistance to lique-faction due to OCR is partially due to an increase in k Q value, the remain-ing increase in strength can only be attributed to changed particle OJ CO CO <D l _ CO c o Q> > o CD UJ 600 500 400 o 300 200 100 Sand type : Ottawa sand |_ cr v ' 0 = 200 k N / m 2 , T / c r ' 0 = 0.104 Relative density = 4 5 - 4 7 % -OCR = I .0 -OCR = 2.0 OCR = 3.0 XICR =4.0 Ko =0.389 oo VO consolidation 200 300 400 500 600 700 800 Vertical effective stress ,c r v ' , k N / m 2 900 FIG. 5.3 Variation of Effective Horizontal and Vertical Stresses During I n i t i a l Consolidation, Static Unloading and Cyclic Loading for Samples with OCR = 1,2,3 and 4. 90 TABLE 5.1 EXPERIMENTAL k Q VALUES FOR VARIOUS OCR Overconsolidation I?* Ratio fco 1 0 .39 2 0 .67 3 0 .83 4 0 .93 NOTE: k* - these values are the average k Q value obtained from several test data. TABLE 5.2 RELATIONSHIP BETWEEN k^ FOR OVERCONSOLIDATED SAMPLES TO NORMALLY CONSOLIDATED SAMPLES k Q = k 0 (OCR)m (Ishihara and Takatsu, 1979) Experimental Data m - 0.68 Ishihara and Takatsu „ -,, „ ( 1 9 7 9) m - 0.71 to 0.84 Sherif et a l . (1979) m = 0.68 91 c 3 - 1 0 30 100 300 1000 Number of cycles for initial liquefaction , N L 4000 FIG. 5.4 Cyclic Shear S t r e s s / i n i t i a l Vertical Confining Stress vs. Number of Cycles to I n i t i a l Liquefaction for Various Values of OCR. § 040 030 o o 2 E 020 0.10 0 Sand type = Ottawa sand (C-109) O"v'0= 200 kN/m 2 , Relative density =45-47% OCR =4 3 1 0 3 0 'OO 300 1000 3000 Number of cycles for initial liquefaction , N L FIG. 5.5 Cyclic Shear S t r e s s / i n i t i a l Mean Normal Stress vs Number of Cycles to I n i t i a l Liquefaction fc Various Values of OCR. :or 92 structure during the process of overconsolidation. Hence, i t can be stated that i t is not possible to relate the behaviour of normally and overconsolidated samples on the basis of mean normal effective stress. It seems that during the process of overconsolidation specimens acquire a particle arrangement which is more resistant to cyclic loading. However, in the following section, when predictive capacity of Martin et a l . (1975) w i l l be checked for overconsolidated samples, a detailed study of volumetric strain in cyclic loading and dynamic rebound modulus w i l l further distinguish the behaviour of normally and overcon-solidated samples. 5.2 VERIFICATION OF THE CONSTITUTIVE RELATIOUSHIPS The pore pressure model proposed by Martin et a l . (1975) is not restricted to normally consolidated sand samples but can also be used for overconsolidated sand specimens as long as volume change characteristics, rebound modulus characteristics, shear modulus and shear strength used in the model are for the overconsolidation state. In this section an attempt is made to predict porewater pressure for sand specimens with OCR of 2,3 and 4, under stress controlled undrained conditions. The calculated pore-water pressure is then compared with measured porewater pressure. In order to use the model for predicting porewater pressures the volumetric strain characteristics under drained, strain controlled conditions and the dynamic rebound modulus are required for various levels of OCR. In this study the maximum overconsolidation ratio used is 4, thus conclusions derived here are limited to this maximum value. In addition to volume change and dynamic rebound modulus, G Q and x 0 are also required for the constitutive stress-strain law used for the pore pressure model. 93 5.2.1 Volume Change Characteristics To evaluate the volume change characteristics of overconsoli-dated sand specimens, a special series of tests has been performed. Loose specimens of Ottawa sand (C-109) were prepared by the method described in Appendix I and consolidated to vertical effective stresses of 400, 600 and 800 kN/m . During consolidation, lateral stress a n and volumetric strain were monitored. To achieve a specimen with OCR = 2,3 and 4, respectively, vertical effective stress was decreased in increments to a v = 200 kN/m2 and, with monitored a v and a n, i t was possible to plot loading and unloading curves in c ry _ 0' n space and k* value achieved after consolidation was calculated. There were two main observations which can be made from the data obtained during cyclic shear in undrained conditions: 1'. As shown in Fig. 5.6, during the process of cyclic shearing, lateral stresses in the sand specimens gradually decreased in contrast to the increase in lateral stress in normally consolidated samples. For specimens with OCR = 3 or 4 and higher i n i t i a l k D (before cyclic shearing), there was a larger decrease in lateral stress compared to specimens with OCR = 2.0. The drop in lateral stress was generally observed within the f i r s t 15 cycles with the most pronounced drop observed in the f i r s t 5 cycles. With specimens subjected to a higher shear strain amplitude, lateral stresses decreased and stabilized much faster compared to specimens with the same OCR but smaller cyclic shear strain amplitude (cf. Fig. 5.6). 2. For specimens with an overconsolidation ratio of 4, k* = (^T) £h = 0 < 'v is the ratio of horizontal stress to vertical effective stress under the conditions of complete ^late r a l confinement in the horizontal direction. .94 1.00 SS / ss ) <*- V o w str o o > a> > cr o «. Q> o <*— a> b «•£ \ — " i f 2 o D e o o > X 0.75 0.50 0.25 i 1 1 1 r vKo =0.96 to 0 9 3 Overconsolidated sample OCR =4, y = 0.10% OCR = 4 , Y =0.30% o o~ 0 Ko =0.39 Normally consolidated sample / =0 .30% Sand type Ottawa sand ( C - 1 0 9 ) Relative density = 4 5 % , 0 ^ = 2 0 0 k N / m 2 j_ 0 4 6 8 10 Number of cycles , N 12 FIG. 5 . 6 Decrease in the Ratio of Horizontal to Vertical Effective Stress vs. Number of Cycles for Various Cyclic Shear Strain Amplitude for Overconsolidated Sample. 95 the application of a cyclic shear strain amplitude y = 0.30%, caused dilation for the f i r s t half cycle, although the net effect of the f i r s t cycle was a decrease in volume (Fig. 5.7). This typical behaviour was only observed for speciments with OCR = 4. It was observed that for specimens with OCR = 4, the volu-metric strain increment in the f i r s t cycle was always less than in the second cycle of shear strain. It is anticipated that the presence of very high lateral stresses on sand grains restricts the relative move-movement of particles. Since volumetric strains are generated due to interparticle s l i p , presence of higher k D value may reduce i t . This influence is more pronounced for the f i r s t 10 cycles of cyclic shearing during which the lateral stress reduces and stabilizes to a particular value. In Fig. 5.8, where the total volumetric strains are plotted against cycles of shear strain amplitude, y - 0.10% for sand specimens with various overconsolidated ratios, i t can be seen that for the f i r s t 10 cycles of shearing the shape of the curves for OCR = 2,3.and 4 are distinctly different from that for OCR = 1. The most important observation which can be made from Fig. 5.8 is that volumetric strain behaviour con-siderably reduces with increasing overconsolidation ratio inspite of the fact that a l l sand specimens were at the same Dr of 45% to 47%. Pyke (1973) also reports that volumetric strains occurring in cyclic tests on overconsolidated samples are less than those occurring in similar normally consolidated samples. Figure 5.9 illustrates this remarkable decrease in volume change behaviour where incremental volumetric strain for the f i r s t cycle of cyclic shear is plotted against shear strain amplitude for normally and overconsolidated samples. For example, at a cyclic shear strain amplitude, y, of 0.10% the incremental volumetric strain for a 96 0 40 Sond type : Ottawa sand (C-109) Relative density = 4 7 % ,0CR = 4 Ko - 0 960 Shear strain , y % FIG. 5.7 Volumetric Strain Behaviour for First Two Cycles of Shearing for an Overconsolidated Sample. 075 0.50 025 15 20 25 30 Number of cycles , N FIG. 5.8 Volumetric Strain vs. Cycles of Constant Shear Strain Amplitude y - 0.10% for Various OCR. 97 Shear strain amplitude , y % FIG. 5.9 Incremental Volumetric Strain in First Cycle for Ottawa Sand for Various OCR Values. C30 E 0 20 0.10 0 Sand type Ottawa sand (C-109) crv'0 = 200 kN/m 2 , Relative density =45% OCR =2.0 , Ko = 0686 C, =0.459 , C 2 =0.1425 , C 3 =00004 , C 4 = 0.0001 ( / • C u e v d ) 0.10 020 Shear strain amplitude, y% 5? > 0 0.2 04 c o 08 w o a> 1.2 E o > 030 FIG. 5.10(a) Incremental Volumetric Strain vs. Shear Strain Amplitude for Various Levels of Volumetric Strains. 98 sample with OCR = 2 is 28% of that for a normally consolidated sample. Measurement of incremental volumetric strain per cycle vs. shear strain amplitude are plotted in Figs. 5.10(a),(b) and (c) for various OCR. From these data, volume change constants needed for the prediction of porewater pressure from the model are calculated. Figure 5.10(c) shows plainly that the curve corresponding to e v cj = 0.0 or the f i r s t cycle is much lower than curves for e v cj = 0.10, 0.20%, etc. In order to f i t analytical relations given in equation (3.4) to experimental curves for various volumetric strains, data corresponding to e y (j = 0.0 is ignored. Equation (3.4) was also f i t t e d to the data shown in Fig. 5.10(a) ' arid (b) and respective volume change constants were obtained. Values of these constants for each OCR are given in their respective figures (Fig. 5.10(a),(b) and (c)) . 5.2.2 Dynamic Rebound Modulus As described in Chapter 4, to measure the dynamic rebound modulus i t is necessary to perform strain coutrolled undrained and drained tests, hence, for overconsolidated sand specimens in addition to strain controlled drained tests used to calculate volume change constants, strain controlled constant volume undrained tests for specimens with OCR = 2,3 and 4 were performed. When porewater pressure ratios were plotted against recoverable volumetric strains for various levels of cyclic shear strain amplitudes for specimens with OCR = 2, as shown in Fig. 5.11(a), the following points become evident: 1. Apart from experimental discrepancy, there is a unique relationship between porewater pressure developed in the undrained condition and volumetric strains in the drained condition for a l l levels of shear strain amplitude. 99 Sand type Ottawa sand (C-109) tTv'0 = 200 kN/m 2 , Relative density = 45 % OCR = 3 0 , Ko = 0 8 3 C, =0.2768 , C 2 =0 1839, C 3 =0 0019 C 4 = 00001 0 A«7 v d«C, ly-C 2e v d)^C 3€ v d 2/ly.Cue v d) 0 02 0 4 010 020 030 Shear strain amplitude , y% FIG. 5.10(b) Sand type Ottawa sand (C-109) Relative density =47% , OCR = 4, Ko = 096 Shear strain amplitude , / % 5.10(c) Incremental Volumetric Strain vs. Shear Strain Amplitudes for Various Values of Volumetric Strains. Ui Ui O O > CD — C o o o -»— E 1.00 ( 0 CU c 0.75 0.50 0.25 0 0 1 1 1 1 1 1 — Sand type : Ottawa sand (C -109) t r ' =200 k N / m 2 , Relative density =45% OCR = 2 , Ko =0.686 Normally-consolidated sample Over-consolidated sample Legend Shear strain amplitudes Drained Undrained 0.101 % • 0 . 1 0 6 % x 0.191 % 0.211 % A 0.305 % 0.322 % 0.2 0.4 0.6 0 8 1.0 1.2 1.6 1.8 Volumetric strain in percent , e v c j % FIG. 5.11(a) Relationship Between Volumetric Strains and Porewater Pressures in Constant Strain Cyclic Simple Shear Tests. 101 2. Curves shown in Fig. 5.11(a) for normally and over-consolidated sand samples l i e very close to each other. The dynamic rebound modulus which is the slope of these curves,..normalized with respect to i n i t i a l confining pressure, is the same for normally and overconsolidated samples. This observation is further strengthened by the data shown in Fig. 5.11(b) for OCR = 3. The above information again confirms the basic assumption made in the model of Martin et a l . (1975) about the relationship between porewater pressure and volume change. This assumption has now been verified for normally consolidated samples with different relative densities and for overconsolidated samples with various overconsolidation ratios. However, before i t can be stated that there is a unique value for dynamic rebound modulus for normally and overconsolidated state, i t was considered appropriate to check the validity of OCR = 4. The experi-mental data obtained for a sample with overconsolidation ratio 4 are shown in Fig. 5.11(c). The data l i e on the curve for normally consoli-dated sand u n t i l the pore pressure ratio has reached a value of 0.50. Beyond that the experimental data for OCR = 4 l i e on a different curve. Aside from the possibility of experimental error, the reason for this divergence may be that the dynamic rebound moduli for normally consolidated and heavily overconsolidated sands are different. It can be concluded that the dynamic rebound modulus value for normally consolidated samples at various relative densities and over-consolidated samples (OCR =2,3,4) is the same for Ottawa sand when samples are being dynamically unloaded from a' = 200 kN/m^ . ' _ o i f 3 3 CO CO CD CD O CD i _ O 0_ co co CD CO c c o o 1.00 075 £ 0 5 0 w 0.25 h 1 1 1 1 : — r — Sand type : Ottawa sand (C-109) cr vo OCR = 3 200 k N / m 2 , Relative density =45% Ko = 0 8 3 Cr— Over - consolidated sample Normally consolidated sample Legend Shear strain amplitudes Drained Undrained x A 0.128 % 0 . 2 0 4 % 0 . 3 1 0 % 0.122 % 0.208 % 0.290 % 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Volumetric strain in percent , e v c j % 1.8 o FIG. 5.11(b) Relationship Between Volumetric Strains and Porewater Pressures in Constant Strain Cyclic Simple Shear Tests. 1.00 Sand type • Ottawa sand (C-109) o~v'0 = 200 k N / m 2 , Relative density OCR = 4 , Ko = 0.96 = 4 7 % Normally - consolidated sample Over-consolidated sample Legend Shear strain amplitudes Drained Undrained 0 . 1 0 3 % 0 . 2 0 9 % 0.305 % 0.103 % 0.192 % 0.299 % 0 0.2 0 4 0.6 0.8 1.0 1.2 1.4 1.6 Volumetric strain in percent , € y d % 1.8 .11(c) Relationship Between Volumetric Strains and Porewater Pressures in Constant Strain Cyclic Simple Shear Tests. 104 5.2.3 I n i t i a l Shear Modulus and Shear Strength Apart from the porewater pressure model, constitutive stress-strain relations are also required to calculate pore pressure analytically and for those G M O and x m o are needed. As mentioned in the previous chapter, i n i t i a l shear modulus can be calculated by the equation suggested by Hardin and Drnevich (1972) . They showed that the overconsolidation ratio is a relatively unimportant parameter for the shear modulus of clean sand. The effect of overconsolidation is to increase G^Q depending on the plasticity index, PI, and for s o i l with no plasticity there are almost no effects from overconsolidation. In addit ion, A f i f i and Richart (1973) show that the increase in shear modulus of clean sand for y = 2.5 x 10"^ due to overconsolidation history is quite small. In Fig. 5.12 results obtained by A f i f i and Richart are shown for sand with a varying percentage of fines. For sands similar to Ottawa sand, the increase in shear modulus due to OCR 1.32 shown as 0.08% which is an insignificant amount. However, i t should be noted that these results,were obtained by A f i f i and Richart in a resonant column apparatus in which overconsolidation was achieved by increasing and decreasing the a l l round pressure of the sand specimen. It is anticipated that by this method the overconsolidation did not result in any increase in k Q. The influence of overconsolidation of i n i t i a l shear modulus, where overconsolidation is achieved under laterally confined conditions, has.never been." studied. The Roscoe-type simple shear apparatus available at the University of B r i t i s h Columbia is not suitable for measurement of shear modulus at low strain (y<0.05%), the influence of overconsolidation on i n i t i a l shear modulus cannot be checked. However, from the experimental data obtained for shear strain 105 10.0 7.50 o o 2 5.00 o o 2.50 0 1 1 1 Air-dry Ottawa sand : 1 (30-50) 1 : i e = (1.01- 0.47) , OCR = (1.32) 40 tt 30 " 20 lb° o OCR = 133 - l / ' oCR = 2 T i m e in s t a g e i -0.08% i i i i 1 . i 30 60 1.0 D50 , mm Average Value of Shear Modulus vs. Mean Particle Size for Soils with OCR = 1.33 to 2 (After A f i f i and Richart, 1973). cn r--o d m O > o O 10.0 7.5 5.0 2.5 0 Sand type : Ottawa sand (C-109) O"v'0= 200 kN/m 2 .Relative density = 4 5 - 4 7 % 010 030 .06 .10 .30 Shear strain amplitude , y% .60 FIG. 5.13 Av. Shear Modulus vs. Shear Strain for First Cycle of Shearing at Various OCR. 106 amplitudes y > 0.05%, i t was observed that for a given shear strain amplitude, the corresponding shear stress in the f i r s t quarter of cycle is much higher for overconsolidated samples than for normally consoli-dated samples. In Fig. 5.13, an average shear modulus is plotted against shear strain amplitude at various OCR values where the shear modulus is calculated for the f i r s t quarter of cycle of shear loading. This figure illustrates the increase in shear modulus with increasing OCR. However, since this data is only restricted to y > 0.06%, behaviour of these curves at very small shear strain (y~.001%) is not known. Hence, the increase in G m o due to increasing OCR can not be evaluated from this data. In the present analysis, the shear modulus and shear strength for sand specimens with OCR were calculated by equations (3.10) and (3.11). For calculating G m Q and x m o the appropriate value of k Q corresponding to OCR was used in equations (3.8) and (3.9). 5.2.4 Pore Pressure Prediction The a b i l i t y of the model to predict porewater pressures in overconsolidated sand was tested by predicting the liquefaction strength curves for various OCR and comparing the results with experimental curves. The strength curves, as shown in Fig. 5.14, are plots of the cyclic stress ratio, T / a V Q versus the number of cycles to i n i t i a l liquefaction, Nj^ ; x being the applied cyclic shear stress. Liquefaction strength curves for OCR = 1,2,3 and 4 are computed using measured volumetric strain charac-te r i s t i c s , dynamic rebound characteristics and appropriate G m o and T m o . As a note of cl a r i f i c a t i o n , i t should be pointed out that no strain harden-ing effect has been included. The points in Fig. 5.14 are experimental data from undrained constant volume cyclic simple shear tests. The i n i t i a l effective vertical pressure, cr v o, in a l l tests after the OCR was established 0.40 o - > b 0.30 o o £ 0.20 <u CO o 0.10 o Sand type • Ottawa sand ( C - 1 0 9 ) cry'0 = 200 k N / m 2 , Relative density = 4 5 - 4 7 % Analytical curve O , » , A , A Experimental data 3 6 10 30 60 100 Number of cycles for initial l iquefact ion, N L o FIG. 5.14 Cyclic Stress Ratio vs. Number of Cycles for I n i t i a l Liquefaction for Various OCR Ratios. 108 as 200 kN/m2. The comparison between the computed and measured liquefaction strengths i s good, although the performance of the model for overconsoli-dated samples is not as good as for normally consolidated samples. The difference between analytical and experimental results for higher over-consolidation ratios (OCR = 4) may be due to two reasons - f i r s t l y , the inability to account for the correct behaviour of OCR = 4 for the f i r s t few cycles; secondly, the unknown increase in shear modulus due to OCR = 4. 5.3 DISCUSSION It can be concluded that the set of constitutive relationships, both for the pore pressure model and for stress-strain behaviour, developed for normally consolidated sand samples are applicable to overconsolidated sand samples where overconsolidation ratio ranges between 2 to 4. However, i t is observed that more research is required to study the effects of over-consolidation on i n i t i a l shear modulus and shear strength. CHAPTER 6 POREWATER PRESSURE MODEL BASED ON ENDOCHRONIC THEORY In Chapters IV and V i t has been shown that the constitutive relationships used in effective stress analysis proposed by Finn et a l . (1977) can predict r e a l i s t i c pore pressure activity in stress controlled undrained tests. However, the pore pressure model requires the incre-mental volumetric strain obtained under strain controlled drained condi-tions given by equation (3.1) and the dynamic rebound modulus as discussed in section 4.2. It is further shown that the rebound modulus cannot be measured with a conventional oedometer as suggested by Martin et a l . (1975), because the rebound response of sand under undrained cyclic loading is different from that under static unloading. Thus, measurement of the dynamic rebound modulus requires both undrained and drained cyclic loading tests, a more complicated process than conven-tional methods of porewater prediction. If a different link could be found between porewater pressure and the dynamic response parameter of the sand-water.system, this would obviate the need for measuring the rebound modulus of the sand skeleton under cyclic loading. Such a relationship is found in the endochronic theory proposed by Valanis (1971) . Zienkiewicz et a l . (1978) point out that endochronic theory has the capability of relating volumetric strains using a single variable which w i l l account for the dynamic response parameter. In endochronic theory the nonlinearity of s o i l is represented by a variable which describes the complete sequence of events of loading through successive states of the material. Although for sand endochronic variables are 109 110 independent of time, they incorporate aspects of the strain history of the sand and are thus termed endochronic 1. Endochronic variables are mathematical transformations of real physical variables, though they themselves have no direct physical interpretation. For liquefaction i t w i l l be seen that these variables are a transformation of deformation increments. In this chapter pore pressure data obtained from conventional undrained strain controlled and stress controlled tests are related with endochronic variables. The endochronic formulation is a function of a single variable which is uniquely calculated from strain or stress his-tories. The proposed formulation is verified against irregular strain history data and used to predict the pore pressure response in stress controlled undrained tests. Endochronic formulation of porewater pressure obtained for various relative densities, overconsolidation ratios and types of sands is presented. In addition, the endochronic formulation of volumetric strains obtained in strain controlled drained tests is presen-ted and the predictive capacity of the formulation is checked. Such a formulation can be used to calculate settlement due to irregular loading during earthquakes. 6.1 ENDOCHRONIC THEORY Valanis (1971) proposes that nonlinearity in a material can be characterized by an independent scalar variable which is a function of deformation and time. The potential of this theory for modelling the liquefaction of sand was f i r s t recognised by Bazant and Krizek (1976), who use endochronic variables to represent the densification or volumetric Endochronic: Endo - within; Chronic - related with time. strains caused by cyclic shearing (cf. Section 2.13). The independent variable, called the rearrangement measure, is related with shear strain in the following relationship (Valanis, 1971): d? = 4 de.. de..)h ( 6 - 0 2 I J I J where d£ is the incremental length of rearrangement measure and de^j is the deviatoric shear strain. To relate volumetric strain or densification with rearrangement measure (Q, another variable called the damage para-meter is proposed. The relationship between the damage parameter and the rearrangement measure takes account of strain hardening 2 and strain softening 3 effects. Note that Bazant and Krizek (1976) do not present a unique relationship between volumetric strains and the damage parameter. As discussed in section 2.1.3, Zienkiewicz et a l . (1978) relate volumetric strain caused by cyclic loading with an endochronic variable and rearrangement measure with damage parameter using the transformation shown in equation 2.8. They observe that volumetric strain can be uniquely related with damage parameter for a l l levels of shear stress ratio. In Chapters IV and V above i t has been shown emphatically that a relationship exists between volumetric strains in the drained condition and porewater pressure in the undrained condition, when both are subjected to the same strain history. Hence, there is a good possibi-l i t y that an endochronic formulation which uniquely relates volumetric strains with an endochronic variable can be found for porewater pressure. This endochronic formulation for pore pressure data is derived in the 2For constant cyclic shear strain the decrease in volumetric strain increment with increasing number of cycles is called strain hardening. 3The increase in volumetric strain with increasing amplitude of shear strain is called strain softening. 112 following section. 6.2 ENDOCHRONIC FORMULATION:OF PORE PRESSURE DATA Data on porewater pressures, u, developed in Ottawa Sand at a relative density Dr = 45% during undrained constant strain cyclic loading tests in simple shear are shown in Fig. 6.1 for four different shear strain amplitudes, y, ranging from 0.056 to 0.314%. Figure 6.1 shows that the non-dimensional porewater pressure ratio, u/a v o, increases with increasing strain amplitude, and incremental change in u/a^ 0 per cycle decreases with increasing number of cycles of constant cyclic shear strain amplitude. This behaviour is analogous to the volumetric strain obtained during strain controlled drained tests as discussed in section 3.2.1. Thus, i t i s clear that the porewater pressure ratio is a function of shear strain amplitude, y, and the number of cycles, N. u/o^ = f(y,N) (6.2) An alternative to N in describing the strain history applied to the sample is the length, E,, of the strain path corresponding to N cycles of y. As the main source of pore pressure increase is shear strain, i t is a lik e l y index for porewater pressure. The variable, E,^. is a monotonically increasing and continuous variable. Various definitions of the length of strain path are possible. A definition which equates an increment in the length of strain path with an increment in deviatoric strain, given in equation (6.1) has been chosen. For the simple shear condition, y = 2e-^ 2> a n d equation (6.1) degenerates to: Note that the same variable is. called rearrangement measure by Bazant and Krizek (197 6). — I — ; 1 : r H — i r 1 r Sand type : Ottawa sand (C-109) Number of Cycles , N FIG. 6.1 Porewater Pressure Ratio, vs. Strain Cycles. 114 dg = ( d e 1 2 d e 1 2 + d e 2 1 d e 2 1 ) 1 5 / /2 = \ |d Y| (6.3) When sinusoidal c y c l i c shear s t r a i n s are applied and y = y Q s i n wt, then 5 at the end of the N*-*1 cycle w i l l be: E, = 2y N (6.4) o where Yo ^ s t n e c y c l i c shear s t r a i n amplitude. From t h i s equation the t o t a l length of the s t r a i n path i s calculated for each c y c l i c shear s t r a i n amplitude as shown i n F i g . 6.1. The pore pressure r a t i o i n a constant s t r a i n test may now be obtained by u/o; Q = g(Y,C) (6.5) The data i n F i g . 6.1 i s shown i n F i g . 6.2 with u/a^ Q plotted against the s t r a i n length E,. A natural logarithmic p l o t i s used to expand the plotted path length at small values of I t can be observed from F i g . 6.2 that the pore pressure r a t i o has a l i n e a r r e l a t i o n s h i p with the logarithm of £ for the range of shear s t r a i n amplitude used i n t e s t i n g . In equation (6.5) the number of cycles, N, of equation (6.2) has been replaced by the continuous v a r i a b l e , £. However, before constant shear s t r a i n data can be generalised to i r r e g u l a r s t r a i n patterns, e x p l i c i t dependence on the shear s t r a i n amplitude, Y» must be removed. For t h i s purpose endochronic v a r i a b l e s are used. Our main purpose i s to express u/°vo a s a f u n c t i ° n °f a s i n g l e monotonically increasing function of a v a r i a b l e , K, which can be defined f o r both constant-strain and i r r e g u l a r s t r a i n h i s t o r i e s . Hence, the v a r i a b l e K must represent a l l parameters defining the s t r a i n h i s t o r y , including varying s t r a i n ampli-tudes and number of c y c l e s . The parameter i s c a l l e d the damage parameter because the e f f e c t of shear s t r a i n h i s t o r y i s to induce pore pressure and weaken the resistance of sand to deformation as discussed by Finn and .001 .003 .005 .01 03 .05 Length of strain path , £ FIG. 6.2 Porewater Pressure Ratio vs. Natural Logarithm of Length of Strain Path. 116 and Bhatia (1980). The porewater pressure r a t i o , u/a^ Q, can be expressed as a function of the damage parameter K, as given i n equation (6.6), where the damage parameter i s a transformation of E, i f a transformation T exi s t s so that f o r K = T£ u/a' = G(K) (6.6) vo The transformation T and the function G are foundiusing the data given i n Fi g . 6.2. If the pore pressure data i s to be e x p l i c i t l y independent of shear s t r a i n amplitude then the transformation should be such that a l l the curves shown i n F i g . 6.2 collapse into a sing l e curve giving u/0^ O as a function of K, i . e . , u/o\^.0 = G ( K ) . For a p a r t i c u l a r porewater pressure r a t i o , u / o v o, the length of s t r a i n path required to cause t h i s porewater r a t i o i s d i f f e r e n t f o r d i f f e r e n t s t r a i n amplitudes. Hence, i f the trans-formation can collapse curves corresponding to d i f f e r e n t s t r a i n amplitudes, the transformation T should be a function of shear s t r a i n amplitude, y. Referring to F i g . 6.2, consider a p a r t i c u l a r value of u/a^ D which occurs at E,^ for a shear s t r a i n amplitude Yl > a n a a t 5 2 f ° r a shear s t r a i n amplitude Y2* The question a r i s e s , can t h i s value of u / a v o be associated with the value tc-^ of a new v a r i a b l e K such that KX = T ^ = T5 2 (6.7) for a l l (Yi>?i) a n a (Y2>52)? s o ' t-h e n a H the curves can be collapsed into one curve giving u/o^ Q as a function of K. Consider T = e A Y (6.8) X Y I A Y 2 KI = h e = h e 1 1 7 *(Y1-Y2> _ / C = 52/51 or A = £n (fc^/fc^) I ( Y 1 ~ Y 2 ) ( 6 - 9 ) in which A is called the transformation factor and y is expressed in percen-tage. The existence of a unique porewater pressure function G(K) requires a unique value of A for a given sand at a given relative density. However, when A is applied to many different data pairs ( Y ^ J C ^ ; Y 2 > ? 2 ^ ^ N 6 . 2 , i t is observed that a range of values of A results. In Fig. 6 . 3 , A values calculated at various levels of porewater pressure ratio and shear strain amplitudes are shown. About f i f t y values have been calculated, ranging from 3 . 0 to 7 . 0 . A weighted average of 4 . 9 9 is obtained from these values with values of A obtained from curves corresponding to shear strains of 0 . 3 1 4 % and 0 . 0 5 6 % having been weighted most heavily. The calculation of A values has been restricted to experimental data below a pore pressure ratio, u/a^ Q of 0 . 9 0 . The experimental data in Fig. 6 . 2 shows that the pore pressure curves bend at theO.90 pore pressure ratio and tend to merge into one curve, hence this part of the curves cannot be used for analysis. Using the average value of A , each data point (u/a^0,y,5) is converted to a data point (u/a^ D,K) using the following transformation K = S e 4 , 9 V ( 6 . 1 0 ) The new data are shown plotted in Fig. 6 . 4 against the natural logarithm of K. Because a unique value of A does not exist, the plotted points define a narrow band rather a single curve. However, despite this range of A, the use of the mean value of A for the given sand has consistently yielded data points f a l l i n g within a narrow band such as shown in Fig. 6 . 4 . A nonlinear least square curve f i t t i n g method has been used to determine the curve shown in Fig. 6.4 describing the relationship between 1.00 0.75 0.50 0 2 5 0 Sand type : Ottawa sand (C -109 ) cr v ' o = 200 k N / m 2 .Re la t i ve density =45% M o / ' Av. X = 4.99 001 005 01 Length of strain path , £ 05 10 FIG. 6.3 Various Values of Transformation Factor, A . r 1 1 — : 1 1 r Sand type : Ottawa sand ( C -109 ) cr' =200 k N / m 2 , Relative density =45% .001 .003 .006 .01 .03 .06 1.0 3.0 Damage parameter , K FIG. 6.4 Pore Pressure Ratio vs. Natural Logarithm of Damage Parameter. 120 u/a v Q and K. The equation of this curve is u/o' = G(K) = (A/B) Ln(1+BK) (6.11) vo where A and B, called endochronic constants are 111.5 • and 452.5, respectively. The same data are plotted in Fig. 6.5 on a natural scale. The data shown in Fig. 6.5 have been fitt e d by the method of nonlinear least squares. The equation of the curve is u/o' = K(DK+C) / (AK+B) (6.12) vo with A=79.42, B=0.93, C=93.58, and 5=71.86. As is usually the case with least squares f i t t i n g procedures, other values of the constants A,B,C and D can be obtained depending on the i n i t i a l values assumed, however, the relative values of the constants w i l l always be such as to yield the best least square approximation to the data. Therefore, i t is possible to relate porewater ratios with a mono-tonically increasing function of a single variable, K. The question can be asked, can this formulation be applicable to a l l levels of i n i t i a l confining pressure? In Fig. 6.6 the porewater pressure ratio is plotted against constant strain cycles of 0.10 and 0.20 for i n i t i a l confining stresses, with a^Q at 1.0, 2.0 and 3.0 kN/m . Points corresponding to various i n i t i a l confining stresses l i e on one curve within the range of experimental error. With this supporting evidence in hand i t can be assumed that the unique relationship (equation 6.11) between pore pressure ratio and damage parameter is applicable for shear strains in the range of 0.3 to .05% and an i n i t i a l vertical confining stress, o\ Q^, of 1.0 to 3.0 kN/m^ . The range of shear strains and vertical confining pressures are the most useful ranges for liquefaction analysis of saturated sands during earthquakes. 1 1 1 1 1 Sand type = Ottawa sand ( C - 1 0 9 ) i i i c r j 0 = 2 0 0 k N / m 2 , Relative density = 4 5 % U / c r J 0 = K ( D K + O / A / C + B , A = 7 9 . 4 2 , B = 0 . 9 3 , C = 9 3 . 5 8 , D= 7 1 . 8 6 A • i i •——~~a x ^ i — Legend : • y = 0 . 0 5 6 % -x / = 0 . I 0 0 % A / = 0 . 2 0 0 % — • 7 = 0 . 3 1 4 % -f i i i i i i' i 0 .01 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 7 . 0 8 Damage parameter , K FIG. 6.5 Porewater Pressure Ratio vs. Damage Parameter. FIG. 6.6 Pore Pressure Ratio vs. Number of Strain Cycles at Various Confining Stresses. 123 6.2.1 Inverse Transformation The accuracy with which the basic test data in Fig. 6.1 and 6.2 is represented by equations (6.11) and (6.12) needs to be tested by using the inverse transformation of T to transfer points from curves defined by equations (6.11) and (6.12) back to those of Figs. 6.1 and 6.2 and comparing computed results with the original test data. This process of inverse transformation checks the accuracy of assuming a unique rela-tionship between u/o^0 and K. Analytical and experimental porewater pressure curves are shown in Fig. 6.7 plotted against E, and in Fig. 6.8 plotted against N. In Fig. 6.8 the dotted curves which represent the analytical inverse transform curve are f a i r l y close to the experimental curves. For a shear strain.amplitude of y = 0.056%, the analytical curve overestimates the pore pressure ratio whereas for a shear strain of y = 0.10%, i t under-estimates the pore pressure ratio by 4 to 5%. It should be noted that the analytical pore pressure curve by inverse transformation was only obtained for values of pore pressure ratio less than 0.90. It is assumed that the analytical function given in equation (6.11) is also applicable for pore pressure ratios greater than 0.90. This assumption may create errors in calculating pore pressure ratios beyond 0.90, however, from a practical point of view pore pressure response beyond 0.90 of i n i t i a l confining stress is not very important. Sometimes, the process of converting pore pressure data from 5-space to K-space requires two t r i a l s . Suppose that with one set of endochronic constants and average value of transformation factor, the representation of pore pressure is not considered satisfactory in a parti-cular strain range, then additional values of A should be computed in this .001 003 006 01 03 06 10 30 Length of strain path,£ FIG. 6.7 Comparison of Computed and Experimental Porewater Pressure in t--plot. 1.00 0.75 0.50 0.25 1 1 : — i 1 n • — r -Sand type = Ottawa sand (C -109) cr v' 0= 200 k N / m 2 , Relative density =45% U/c r v ; 0 =A /B L n (l + B/c) , A = 111.70 , B = 452.46 , X =4.99 Inverse transformation from K- Space . / = 0 . 0 5 6 % / = 0 I 0 % ° Experimental curve . / = 0 . 2 0 % Inverse analytical curve ^ = 0.314% 0 10 15 20 25 Number of cycles , N 30 35 40 FIG. 6.8 Comparison of Computed and Measured Porewater Pressures in N-plot. 126 region. These additional values w i l l weight a new mean value of A towards this strain range and improve the accuracy of data representation in the range. Experience today indicates that, provided a reasonable number of data pairs are used i n i t i a l l y in determining the average A, most of the time no further adjustments are necessary. It should be emphasized that G(K) represents not just the four curves shown in Fig. 6.1 but any test curves that might be determined within the given range. The function G(K) blankets the entire strain amplitude range for which the experimental data used for analysis was obtained. Application of the formulation to strain amplitudes beyond this range, however, should be done only after a preliminary check of the curve for the particular value of shear strain amplitude. 6.2.2 Endochronic Representation of Porewater Pressure Data for Various  Relative Densities, Overconsolidation Ratios and Types of Sand In this section, pore pressure data obtained under strain controlled undrained conditions for various shear strain amplitudes w i l l be presented as a monotonically increasing function of the damage para-meter. These data have been obtained for several different conditions, each of which w i l l be discussed separately in the following sections. 6.2.2.1 Various relative densities A series of tests have been performed on Ottawa Sand at rela-tive densities of 54%, 60% and 68% for shear strain amplitudes y = 0.10 to y = 0.40%. A typical plot of porewater ratio, u/a^ Q, against the number of cycles of a shear strain amplitude of y = 0.20% is shown in Fig. 6.9 for various relative densities. Fig. 6.9 shows that the rate of pore pressure generation decreases with increasing relative density in strain controlled undrained tests. 127 o - > b 00 3 CO w a. in t £ 0.75 * 1 0.50 a> c >r O E 025 Sand type : Ottawa sand (C-109) cr v o = 200 kN/m 2 , Shear strain = 0 2 0 % 9 0 % Pore pressure line 8 12 16 Number of cycles , N 20 FIG. 6.9 Porewater Pressure Ratio vs. Strain Cycles of 0.20% at Various Relative Densities. . o Damage parameter , K FIG. 6.10 Porewater Pressure Ratio vs. Natural Logarithm of Damage Parameter at Var ious Relative Densities. 128 The pore pressure data for various shear strain levels for each relative density are used to calculate the average transformation factor and endochronic constants. For each relative density the f i n a l formulation of pore pressure ratio against damage parameter is shown in Fig. 6.10 wherein for each relative density experimental data corresponding to various shear strain amplitudes l i e in very narrow bands. The set of endochronic constants and average transformation factor for each relative density is given in Table 6.1. It is apparent that though these sets of constants for the endochronic formulation give good agreement with experi-mental data, the constants themselves cannot be said to have a uniform tendency with increasing relative density. No concrete conclusions can be drawn about the average transformation factor with various relative densities because in an indirect way, A also depends on the shape of the porewater pressure curve in £-space since pore pressure data when plotted in 5-space for various relative densities does not always vary linearly with Ln(5). In general, the tabulated values of average A for various relative densities indicate that this constant does not differ much for loose to dense Ottawa Sand. Furthermore, Table 6.1 does not indicate any trend in endochronic constants A and B considered alone. However, the ratio A/B decreases with increasing relative density. Since denser sand requires a larger number of cycles of constant shear strain amplitude, compared to loose sand (Fig. 6.9), the length of strain path required to cause .a certain level of u/a v o w i l l increase with increasing relative den-sity. Even i f A a v e as shown in Table 6.1 does not change much with increas-ing relative density, the damage parameter K required to create u^/a^.Q would increase with increasing relative density. This general trend can be seen in Fig. 6.10. TABLE 6.1 ENDOCHRONIC CONSTANTS FOR VARIOUS RELATIVE DENSITIES Relative Density A A B A/B D% r 45 4.99 111.70 452.46 0.246 54 3.72 100.09 498.40 0.201 60 3.83 92.43 559.11 0.165 68 2.964 148.79 1397.08 0.1065 130 6.2.2.2 Overconsolidation ratios As discussed in Chapter V, any increase in overconsolidation ratio causes an increase in the resistance to liquefaction. In Fig. 6.11 pore pressure ratio is plotted against natural logarithm of length of strain path 5 for various overconsolidation ratios and the experimental data shows a considerable reduction in porewater pressure ratio for samples with different overconsolidation ratio at a particular value of 5. Curves plotted in Fig. 6.11 also indicate that the curvature of the curves becomes more and more pronounced with increasing overconsolidation ratio. Endochronic transformation has been performed on the porewater pressure data for various shear strain amplitudes and overconsolidation ratios. Results of two such analyses are shown in Fig. 6.12(a) and (b) for overconsolidation ratios of 2 and 4. For both overconsolidation ratios, the data f a l l in very narrow bands and, again, confirm that pore pressure ratios can be related with damage parameter. The analytical function given by equation (6.11) when fitte d to experimental data yields endochronic constants A and B for each overconsolidation ratio. These are presented in Fig. 6.12. 6.2.2.3 Endochronic representation for other sands In addition to Ottawa Sand, two other types of sands, Crystal S i l i c a Sand and Toyoura Sand were tested. It should be noted that a l l types of sand are clean sands with quartz particles, though Ottawa Sand has subrounded to rounded particles whereas the other two have subangular to angular particles. In order to make some comparison between the behaviour of a l l three types of sands, tests on Crystal S i l i c a and Toyoura sands are made at a relative density of 45%. The procedure discussed in section 6.3.1 o Length of strain path , £ FIG. 6.11 Pore Pressure Ratio vs. Ln (Length of Strain Path) for Various OCR. 132 o - > b ^ CO co co co £ o» Q. C E OJ -o c CL O o .00 0.75 0.50 0.25 - 0 1 I I 1 Ottawa sand (C-109) i Ant, Dr = 4 7 % , o - V ' 0 = 200 k N / m 2 A =14.9 , B = 35.46, X = 3.62 J $ OCR = 4 v # Legend : • / = 0.I0% x 7 = 0 2 0 % A 7= 0 . 3 0 % i i i 1 (D .001 0 0 5 .01 .05 .10 Damage parameter , K .50 1.0 FIG. 6.12 Porewater Pressure Ratio vs. Ln (Damage Parameter) for Overconsolidated Sand. 133 was again used to obtain a relationship between u/a v o and damage para-meter, and results of this procedure are shown in Fig. 6.13. The main observation which can be made from this figure is that pore pressure data corresponding to various shear strain amplitudes can be represented by a monotonically increasing function of damage parameter for each type of sand at a particular relative density. In Fig. 6.14, the pore pressure ratio is plotted against the number of cycles of cyclic shear strain amplitude, y = 0.20%, for a l l three types of sand at a relative density of 45%. The rate of pore pressure generation is significantly different for each type of sand. Pore pres-sures generated for Toyoura sand for a given number of cycles are 50% of those for Ottawa sand as shown in Fig. 6.14. Endochronic constants obtained from the analysis of a l l three types of sands are given in Table 6.2. In Fig. 6.13 curves for Crystal S i l i c a and Toyoura sands shift to the right, side or towards the higher values of damage parameter compared to Ottawa sand. This behaviour i s ju s t i f i e d in two ways, f i r s t l y , to generate the same porewater pressure ratio, higher number of cycles of constant strain or a longer strain path is required for Crystal S i l i c a and Toyoura sands compared to Ottawa sand; secondly, the A.ave is greater for Toyoura and Crystal S i l i c a sands. Higher value of the length of strain path and A a v e yields greater values of damage parameter, hence, the pore pressure curve in Fig. 6.13 shifts towards the higher K-values. It can be concluded that for each relative density (45,54,60 and 68%) and overconsolidation ratio (OCR = 2,3,4) and different types of sands pore pressure data for each case of various shear strain amplitudes can be represented as a continuously increasing function of damage para-meter. Pore pressure curves (u/c?vo = G(K)) for various relative densities and types of sands when plotted shows their relative behaviour as shown in 6.13 Porewater Pressure Ratio vs. Ln (Damage Parameter) for Various Types of^Sajads 135 FIG. 6.14 Porewater Pressure Ratio vs. Strain Cycles of y " 0.20% for Various Types of Sands. 136 TABLE 6.2 ENDOCHRONIC CONSTANTS FOR VARIOUS TYPES OF SANDS SAND TYPE Dr% A ^ Ottawa Sand 45 4.99 111.50 452.46 Toyoura Sand 45 9.36 15.17 82.95 Crystal S i l i c a Sand 45 5.36 99.70 543.60 137 Figs. 6.10 and 6.13. 6.2.3 Verification of Endochronic Pore Pressure Formulation Pore pressure generated by a series of symmetrical and unsymmetrical shear strain cycles can readily be computed from the endo-chronic formulation. However, since the formulation given by equation (6.11) is based on pore pressure data obtained under strain conditions with regular strain cycles, i t s validity must be checked for irregular strain histories. For this purpose, the shear strain histories shown in Fig. 6.15(a) and (b) have been used to predict pore pressure ratios. To calcu-late the pore pressure ratio for each cycle, the incremental length of strain path applied in that cycle is calculated and then converted to the incremental damage parameter. Subsequently, by using the cummulative damage parameter with appropriate values for endochronic constants, pore pressure ratios are calculated using equation (6.11). A typical set of calculations for the strain histories of Fig. 6.15(b) is given in Table 6.3. Calculated and experimental curves for both strain histories are given in Fig. 6.15 showing good agreement. A more severe check of the endochronic formulation is the c a l -culation of porewater pressure for stress controlled undrained tests. This calculation can use either the shear strain, history as recorded experimen-tal l y or the constitutive relationships for stress-strain as discussed in Chapter III. Since special care was taken in the undrained stress controlled tests to record the shear strain amplitude, which can f a l l as low as .03%, the measured shear strain history has been used to calculate the porewater pressure ratio. Pore pressure/ Initial confining stress, U/O"v'0 M o t—« H i c or § 3 cr M CD l-i l-i -\ H H -fD cn o 00 o -t> C 3 1—1 o PJ W >"< H fD <-> r t CD to «! cn r t fl> i-i m to 3 H -3 n tfl M H - n CO 3 r t o SB H r t ro • to 3 &• w X fD l-i an: fD 3 r t CO I-1 Z >d c Ul 3 fD cr t CD r t fD o H T) o t i «< fD o cn CD cn CA C i-i fD 8? iti o cn Pore pressure/ Initial confining stress ,U/O~ v 0 O ro cn 1 (Ji O O cn O O t/% , Shear strain amplitude in percent .1 .1 | . . OJ ro — — ro OJ O O O O O O O Shear strain amplitude in percent — ro OJ o o o 8£T TABLE 6.3 PORE PRESSURE CALCULATION FOR IRREGULAR STRAIN HISTORY USING THE ENDOCHRONIC FORMULATION, OTTAWA SAND, Dr=45% u / a v o = A/B Ln(l+BK) A = 111.50 B = 452.4 X = 4.99 N i Y% I AK± u/°vo Analytical u/a v o Experimental 1 0.018 .00036 1.093 .000394 .000394 .0404 .0160 2 0.042 .00084 1.233 .00103 .00143 .1220 .058 3 0.06 .00120 1.349 .00106 .00304 .2130 .170 4 0.084 .00168 1.520 .00255 .00559 .3108 .297 5 .108 .00216 1.714 .00370 .00929 .4064 .416 6 .108 .00216 1 .714 .00370 .01290 .4754 .523 7 .120 .0024 1.819 .00438 .01728 .5364 .599 8 .144 .00288 2.051 .00590 .0231 .6016 .676 9 .168 .00336 2.312 .00770 .0308 .666 .747 10 .168 .00336 2.312 .0077 .0385 .718 .805 11 .180 .0036 2.455 .0088 .0475 .766 .835 12 .180 .0036 2.455 .0088 .0565 .807 .877 13 .204 .00408 2.768 .01129 .0675 .850 .890 14 .216 .00432 2.938 .01269 .0801 .891 .899 15 .216 .00432 2.938 .01269 .0927 .9266 .917 16 .240 .00480 3.312 .01589 .1087 .9645 .923 17 .240 .00480 3.312 .01589 .1247 .9980 .928 NOTE: Yi is i n percentage. 140 The calculated and experimental curves for a relative density of 45% are shown in Fig. 6.16, and those for 60% in Fig. 6.17. Both show very good agreement between the experimental and analytical results. A typical calculation is shown in Table 6.4 for a stress ratio of 0.089. It i s important to note that the analytical porewater pressure ratio w i l l exceed the value of 1.0 once the cyclic shear strain amplitude reaches 0.8% to 1.0%. However, the porewater pressure ratios corresponding to these shear strains are 0.75 or higher, meaning that the sample requires only one or two more cycles for complete liquefaction, hence this discre-pancy is not serious. Similar analyses have been performed where shear strains calculated using equations (3.10) and (3.14) without hardening produced analytical and experimental curves lying close to each other, though the agreement i s not as good as that shown in Fig. 6.16. Pore pressure response has also been calculated for overconsolidated Ottawa sand samples and good agreement between analytical and experimental curves observed. In conclusion, the proposed endochronic formulation for pore-water pressures obtained from experiments performed with constant cyclic shear strain amplitude is capable of predicting porewater pressures for irregular strain histories. 6.2.4 Endochronic Representation of Porewater Pressure from Stress  Controlled Undrained Tests In preceding sections the porewater pressure data used for endochronic formulation has been obtained under constant strain conditions and the verification of the formulation is performed by predicting pore-water pressure in stress controlled undrained tests. In this section pore pressure data obtained under stress controlled conditions are used o > b 3 Sand type : Ottawa sand (C-109) a v ' 0 = 200 k N / m 2 , Relative density =45% o Experimental curve — Analytical curve ^ ^ ^ ^ (Predictions based on measured strains") 6 10 Number of cycles , N 30 o 60 100 200 FIG. 6.16 Predicted and Measured Porewater Pressure in Constant Stress Cyclic Simple Shear Tests, Dr = 45%. FIG. 6.17 Predicted and. Measured Porewater Pressure in Constant Stress Cyclic Simple Shear Tests, Dr = 60%. TABLE 6.4 CALCULATION OF PORE PRESSURE RATIO FOR STRESS CONTROLLED UNDRAINED TESTS ON OTTAWA SAND AT Dr=45% at T/oyO=0.089 u/a' = A/B Ln(1+BK) A = 111.50 B = 452.4 A = 4.99 N y% A? ± e 1 AK± K+AK± u/a v o u/a v o Calculated Experimental 1 .0409 .000818 1.226 .00103 .00103 .0935 .101 2 .0409 .00818 1.226 .00103 .00206 .163 .166 3 .0409 .000818 1.226 .00103 .00303 .212 .210 4 .0409 .000818 1.226 .00103 .00406 .256 .248 5 .0415 .00083 1.23 .00102 .00508 .294 .281 6 .042 .00084 1.233 .001036 .00611 .326 .312 7 .0431 .000864 1.239 .001069 .00717 .356 .338 8 .0437 .00087 1.243 .00108 .00825 .383 .362 9 .0447 .00089 1.249 .00111 .00936 .407 .386 10 .0464 .00092 1.260 .00117 .0105 .431 .405 11 .0484 .00097 1.273 .00123 .0117 .453 .429 12 .0486 .00103 1.295 .00134 .0130 .476 .450 13 .0519 .00107 1.305 .00139 .0143 .496 .472 14 .0535 .00120 1.349 .00162 .0159 .518 .493 15 .06011 .00125 1.369 .00718 .0176 .540 .514 16 .062 .00144 1.432 .00212 .01970 .565 .544 17 .0801 .00160 1.491 .00271 .0224 .594 .570 18 .1147 .00229 1.772 .00452 .0269 .635 .613 19 0.5247 .01040 13.710 .0324 .0593 .819 .673 20 1.967 .0394 18312.0 128.02 128.07 >1.00 .857 NOTE: Y- is in percentage. 144 for the endochronic formulation. The analytical formulations obtained in this section are obtained by two methods: 1. By relating porewater pressure with the length of stress path, and 2. By relating porewater pressure with the length of strain path in which generated shear strain in stress controlled conditions are required to c a l -culate the length of strain path. 6.2.4.1 Pore pressure as a function of stress path Porewater pressure generated under uniform stress controlled undrained conditions are related with the length of stress path. The definition of length of stress path is chosen as dn = /do..do.. (6.13) in which a y is deviatoric stress and, for the simple shear condition, i s j = 1,2. For the special case where cyclic shear strain is applied sinusoidally and x = T q sin wt, n at the end of the N1-*1 cycle w i l l be n = 4x N (6.14) o Figure 6.18 shows that porewater pressure ratio is plotted against the natural logarithm of the number of cycles of constant shear stress for various cyclic shear stress ratios, j/a^Q. This data was obtained in constant volume simple shear tests performed on Ottawa sand at a relative density of 45%. The pore pressure ratio in constant stress undrained tests may now be defined by u /°vo " MT/°vo>^  ( 6 ' 1 5 ) The general definition of the length of stress path given in equation (6.13) for the simple shear condition w i l l contain the units of shear FIG. 6.18 Porewater Pressure Ratio vs. Number of Cycles for Various Cyclic Shear Stress Ratios. 146 stress. It is considered suitable to define n as given in equation (6.14), where the term, length of stress path, is actually the length of the shear stress ratio n = 4 ^- N (6.16) v The data plotted in Fig. 6.18 is converted in n-space, hence the number of cycles, N, is replaced by the continuous variable n. The transformation T, which is required to convert from stress path space to damage parameter space, is chosen as T = e V ° (6.17) in which T / O ^ Q is the i n i t i a l cyclic shear stress ratio and A is the influence factor. In order to evaluate values of A, experimental data plotted in terms of porewater pressure ratio and length of stress path for various T / O ^ 0 have been used. The range of A value obtained have the mean value of 49.72. Using this average value of A, the data from n-space has been transformed to K-space as shown in Fig. 6.19. The experimental points corresponding to various T/O^ 0, as shown in Fig. 6.19, f a l l in a band that becomes wider after pore pressures have reached 60% of i n i t i a l confining stress. With the assumed form of transformation given in equation (6.17), the assumption of a unique relationship between u/o^0 and K is not very good after the pore pressure ratio has exceeded a value of 0.60. However, for practical purposes this error is not significant. A nonlinear least squares curve f i t t i n g method has been used to determine the curve shown in Fig. 6.19, describing the relationship between u/a^.Q and K. This equation is of the same form as given in equation (6.11). The values of the endochronic constants A and B are given in Fig. 6.19. Hence, i t can be concluded that pore pressure measured H O ON I—• o 1 r t (D H >-a H fl> to co c H ID 8? r t < a v r t c H Co O ere co H o Hi o CO g 00 ro CO ro r t ro H Pore pressure / In i t ia l confining stress, U / c r ' vo o Q 3 Q CD Q Q 3 CD CD 8 CD o s o CD o o OJ o o o p ro o CP o o C7i o o • • > X • r-<n p o o o O H • i O » — 6 6 b 2 o o CO 00 -vl CD 9 a. CO CM OJ < -Ol cn o q < -o CD II II > O CO 6 o u> r-OJ *• + >" CD II . -P> CO > CO II O 6 o ro cn a> o II ro O O CO Q OL »< CD Z O 3 ro 33 CD < . CD a. CD c/> II $2 o o CA O o I o cfi 148 in stress controlled undrained tests for various stress ratios can be represented by a continuous increasing function of the damage parameter. Since, in a general sense, stress controlled tests are a conventional way to study the liquefaction behaviour of sand in cyclic loading, any formulation based on data from these tests is always important. Such formulation obviates the need of performing special tests. 6.2.4.2 Pore pressure as a function of strain path Porewater pressure ratio measured in stress controlled undrained tests can also be expressed as a function of the length of strain path and the transformation required to convert from strain path to damage para-meter can have the form given in equation (6.17). The shear strains generated during cyclic simple shear tests shown in Fig. 6.18 are plotted in Fig. 6.20(a). The experimental data in this figure show that cyclic shear strains remain almost constant un t i l u/o^0 is less than 0.30. Beyond this point, shear strain gradually increases with increasing porewater pressure u n t i l u/a^Q. reaches a value of 0.65. After this point the sample requires one or two additional cycles to generate shear strains of 5 to 10%. Cyclic shear strains shown in Fig. 6.20(a) are used to calculate the length of strain path where the length of strain path is again given by equation (6.1). Porewater pressure data given in Fig. 6.18 are used in conjunction with the calculated length of strain path and a plot of u/a^ 0 and £ obtained. The transformation of porewater pressure data from £-space to K-space was obtained by assuming the transformation given in equation (6.17). The average value of A was calculated and f i n a l results of this analysis are given in Fig. 6.21, where scatter in data corresponding to various shear stress ratios is significant to conclude the existence of a unique relationship between. 1.00 \Z 0 7 5 0.50 0 2 5 10 Sand type : Ottawa sand (C -109) tTv'0 = 200 k N / m 2 , Relative density =45% U/cr' = A / B L n ( l + B ) , A = 0 0 0 2 6 7 8 , B = 0 .0093 , X = 49719 Legend; T / C T V 0 • 0 0 6 3 5 X 0 0 7 6 5 A 0 0 8 3 4 • 0 0 8 9 0. 1034 30 60 100 300 600 1000 Damage parameter , 3 0 0 0 FIG. 6.19 Porewater Pressure Ratio vs. Natural Logarithm of Damage Paramet 150 u/a^ 0 and K. A similar analysis is performed by Zienkiewicz et a l . (1978) as discussed in section. 2 .1.2. In Appendix III, i t has been shown that volumetric strains obtained under strain controlled drained conditions can also be presented as a monotonically increasing function of the damage parameter. Such a formulation is capable of predicting volumetric strains for irregular strain histories. 6.3 DISCUSSION A new method for processing porewater pressure data of a saturated sand at a particular relative density from conventional cyclic simple shear under strain or stress controlled conditions has been presen-ted. Usually, the description of such data involves a curve of the pore-water pressure ratio, u/a^ Q vs. the number of uniform load cycles, N, for each shear stress or shear strain amplitude. In the approach presented here a l l the curves can be replaced by a single curve, u/o^0 = G(K) in which K is a variable that encompasses the effects of both shear or strain ampli-tude and cyclic loading. The procedure to obtain this variable K, is quite simple and uses test data from conventional tests. In addition, the function G(K) is a super-efficient represen-tation of pore pressure and volumetric strain data, requiring only one curve to describe a l l the test data at one relative density for the range of cyclic stresses or strains of interest. The variable K which is easily defined for irregular stress or strain conditions allows the direct use of data from uniform cyclic loading tests to predict the porewater pressure caused by irregular stress or strain histories generated by earthquake loading. 151 Moreover, the new porewater pressure function, u/o^Q = G(K) simplifies greatly the estimation of porewater pressures in the f i e l d . The endochronic formulation of volumetric strains can be used to calculate settlement due to irregular strain or stress cycles. CHAPTER 7. . SUMMARY AND CONCLUSIONS 7.1 SUMMARY The main purpose of this research is twofold. First, to verify the most crucial assumption made for the constitutive relation-ships behind the dynamic effective stress analysis by Finn, Lee and Martin (1977). Second, to propose a simple method of porewater pressure prediction based on results from conventional tests. The dynamic effective stress (Finn, Lee and Martin, 1977) is based on a pore pressure model proposed by Martin et a l . (1975) after the incorporation of the r e a l i s t i c constitutive relationships proposed by Lee (1975) for the nonlinear, hysteretic stress-strain behaviour of sand during cyclic loading. In this thesis the performance of these constitu-tive relationships, both for the pore pressure model and for stress-strain behaviour, is evaluated for normally consolidated and overconsolidated sand s. A simple formulation is proposed that expresses porewater pressure as a monotonically increasing function of a single .variable. Data for this formulation are available from conventional constant strain or stress undrained tests. The formulation can be coupled with dynamic response analysis to predict porewater pressures in irregular stress or strain histories such as those that result from earthquakes. 7.2 CONCLUSIONS After testing under a variety of drained and undrained loading conditions, the porewater pressure model of Martin et a l . (1975) appears 152 153 to be based on sound assumptions. The most crucial questions raised about the model in Chapter III can now be anwered as follows: 1. The experimental data indicates that the potential volumetric strain in undrained conditions is the same as that in drained, when both drained and undrained samples are subj ected to the same shear strain history. It has been strongly v e r i -fied that there is unique relationship between volumetric strain in the drained condition and porewater change in the undrained condition for a given sand and relative density when samples have been subjected to similar strain histories. This observation based on experimental data obtained for Ottawa sand (C-109) at various relative densities and overconsolidation ratios. 2. An important point to emerge from this study is that the rebound modulus used to convert volumetric strains to porewater pressure (equation 3.2) must be measured under cyclic loading conditions. The static rebound modulus used in conjunction with volumetric strain data overestimates porewater in stress controlled undrained conditions. However, i t has been found that static rebound modulus can be adjusted towards the dynamic modulus by a factor 3 to 5 and this adjustment can be made in constant k/?. 3. For the application of the model under undrained conditions, the strain hardening effect should not be included for the analysis. However, at the conclusion of such tests, i f drainage is allowed to take place, plastic strains are recovered and sands strain-harden. Provided proper values of the rebound modulus are used and effects of strain-hardening are included whenever drainage occurs, the analysis clearly shows that the set of constitutive relationships can 154 make good predictions of the development of porewater under f a i r l y general loading patterns and drainage conditions in simple shear. 4. The increase in resistance due to overconsolidation cannot be completely attributed to increase in mean effective stress. It may be possible that during the process of overconsolidation, in addition to increased horizontal stress, sands attain more stable structures which would contribute to increased resistance. However, the effective stress pore pressure model can make good prediction of the development of porewater pressure for overconsolidated sand, provided appropriate volumetric strain constants and rebound modulus constants are used. It i s fe l t that the measurement of volume change constants and dynamic rebound constants require tests which are not considered conven-tional. In particular, the evaluation of dynamic rebound characteristics is quite time consuming. For this reason a simple and efficient method is proposed by which porewater pressures measured in routine laboratory stress or strain controlled tests can be utilized in dynamic effective stress analysis by a single curve, u/a^Q = f ( K ) . The proposed formulation, based on endochronic theory has been developed from extensive data and it s performance thoroughly evaluated. The main conclusions which can be drawn from this study are (cf. Chapter VI): 1. The proposed method for processing porewater pressure data on saturated sand at a particular relative density from conventional cyclic simple shear tests, involves a single curve u/o\ 0^ = G(K) in which K is a transformed variable that encompasses the effects of both shear stress or strain amplitude and cyclic loading. This observation is based on experimental data for various relative densities, overconsolidation ratios and types of sand. 155 2. The procedure to obtain the variable K is simple using conventional test results. The function G(K) is a super-efficient representation of porewater pressure data, requiring only one curve to describe a l l the test data for the range of cyclic stresses and strains of interest. 3. The variable K which is easily defined for irregular stress or strain conditions allows the direct use of the data from the uniform cyclic loading tests to predict the porewater pressure caused by irregular strain or stress histories similar to those generated by earthquakes. 4. In addition, i t is possible to represent volumetric strains obtained under strain controlled drained conditions as a monotonically increasing function of the damage parameter. Moreover, such a formulation is capable of accurately predicting volumetric strain under irregular strain history or stress controlled conditions. Finally, for the dynamic effective stress analysis a porewater pressure model based on fundamental properties of the s o i l skeleton and water is no longer required. During any time increment At in the dynamic analysis, the increment in porewater pressure can be determined from the incremental change in K. The s o i l properties may now be modified for this change in porewater pressure and the analysis continued for the next time increment At. The use of the new procedure means that no special tests are required for dynamic effective stress analysis. Moreover, i t is an extremely efficient way of storing large amounts of data. For prediction of seismic settlement, the endochronic formulation of volumetric strain can be used. 7.3 SUGGESTIONS FOR FUTURE RESEARCH WORK The proposed endochronic formulation should be coupled with 156 dynamic response analysis to perform effective stress analysis. In this way, the performance of this formulation for random and irregular loading patterns can be evaluated. Predictive capacity of the constitutive relationship and proposed endochronic formulation for pore pressure should be checked with f i e l d data. LIST OF REFERENCES 1. A f i f i , S.S. and Richart, F.E. (1973), "Stress History Effects on Shear Modulus of Soils", Soils and Foundations, Vol. 13, No. 1, March, pp. 78-95. 2. Bazant, Z.P. and Krizek, R.J. (1976), "Endochronic Constitutive Law for Liquefaction of Sand", Journal of the Engineering Mechanics Division, ASCE, Vol. 102, No. EM2, April, pp. 225-238. 3. Biot, M.A. (1956), "Theory of Propagation of Elastic Waves in Fluid-Saturated Porous Solid-I. Low-Frequency Range", Journal of the Acoustical Society of America, Vol. 28, pp. 168-197. 4. Biot, M.A. (1957), "The Elastic Coefficient of the Theory of Consoli-dation", Journal of Applied Mechanics, Vol. 24, pp. 594-601. 5. Bjerrum, L. and Landva, A. (1966), "Direct Simple Shear Tests on a Norwegian Quick Clay", Geotechnique _1_6 1, pp. 1-20. 6. Casagrande, A. (1936), "Characteristics of Cohesionless Soils Affecting the Stability of Earth F i l l s " , Journal of the Boston Society of C i v i l Engineers, January 1936. Reprinted in "Contribu-tions to Soil Mechanics, 1925-1960", Boston Society of C i v i l Engineers, October 1940. 7. Castro, G. (1969), "Liquefaction of Sands", Harvard Soil Mechanics Series, No. 81, Cambridge, Mass., January. 8. Cuellar, V. (1977), "A Simple Shear Theory for the One-Dimensional Behaviour of Dry Sand Under Cyclic Loading", Proceedings of DMSR 77, Karlsruhe, Vol. 2, Sept. 5-16, pp. 101-111. 9. Dafalias, Y.F. and Popov, E.P. (1976), "Plastic Internal Variables Formalism of Cyclic Plasticity", Journal of Applied Mechanics, 98(4), pp. 645-650. 10. Finn, W.D. Liam (1979), " C r i t i c a l Review of Dynamic Effective Stress Analysis", Proceedings, 2nd U.S. National Conference on Earthquake Engineering, Stanford, Calif., August 22-24, pp. 11. Finn, W.D. Liam (1980), "Dynamic Response Analysis of Saturated Sands", State-of-the-Art volume, International Symposium on Soils Under Cyclic and Transient Loading, Swansea, Wales, January 7-11, John Wiley & Sons Ltd., London, (in press). 12. Finn, W.D. Liam and Bhatia, S.K. (1980), "Verification of Non-linear Effective Stress Model in Simple Shear", Accepted for publication, Proceedings, ASCE Special 2-Session Series, Hollywood-by-the-Sea, Florida, October 27-31, (in press). 13. Finn, W.D. Liam, Bhatia, S.K. and Pickering, D.J. (1980), "The Cyclic Simple Shear Test", State-of-the-Art volume, International Symposium on Soils Under Cyclic and Transient Loading, Swansea, Wales, January 7-11, John Wiley & Sons Ltd., London, (in press). 157 158 14. Finn, W.D. Liam and Bhatia, S.K. (1980), "Prediction of Seismic Pore-water Pressures", Accepted for publication, Proceedings, 10th International Conference on Soil Mechanics & Foundation Engineering (X.ICSMFE), Stockholm, Sweden, June 15-19, (in press). 15. Finn, W.D. Liam, Bransby, P.L. and Pickering, D.J. (1970), "Effect of Strain History on Liquefaction of Sand", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM6, Proc. Paper 7670, November, pp. 1917-1934. 16. Finn, W.D. Liam, Lee, K.W. and Martin, G.R. (1976,1977), "An Effec-tive Stress Model for Liquefaction", ASCE Annual Convention and Exposition, Philadelphia, Pa., Sept. 22-Oct. 1, 1976, Preprint 2752, also in Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT6, Proc. Paper 13008, June 1977, pp. 517-533. 17. Finn, W.D. Liam and Martin G.R. (1980), "Soil as an Anisotropic Kinematic Hardening Solid", Accepted for publication, Proceedings, ASCE Special 2-Session Series, Hollywood-by-the-Sea, Florida, October 27-31, (in press). 18. Finn, W.D. Liam, Pickering, D.J. and Bransby, P.L. (1971), "Sand Liquefaction in Triaxial and Simple Shear Tests", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, SM4, pp. 639-659. 19. Finn, W.D. Liam and Vaid, Y.P. (1977), "Liquefaction Potential from Drained Constant Volume Cyclic Simple Shear Tests", Proceedings, 6th World Conference on Earthquake Engineering, New Delhi, India, Session 6, pp. 7-12. 20. Finn, W.D. Liam, Vaid, Y.P. and Bhatia, S.K. (1978), "Constant Volume Cyclic Simple Shear Testing", 2nd International Conference on Microzonation, San Francisco, Calif., Nov. 26-Dec. 1, Vol. 2, pp. 839-851. 21. Florin, V.A. and Ivanov, P.L. (1961), "Liquefaction of Saturated Sandy Soils", Proceedings, 2nd. International Conference on Soil Mechanics and Foundation Engineering, Vol. I, pp. 102-111. 22. Ghaboussi, Jamshid and Dikmen, Umit S. (1978), "Liquefaction Analysis of Horizontally Layered Sands", Journal of the Geotechnical Engineering Division, ASCE, Vol. 104, No. GT3, March, pp. 341-357. 23. Hardin, B.O. and Drnevich, V.P. (1972), "Shear Modulus and Damping in Soils" Design Equations and Curves", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol 98, No. SM7, Proc. Paper 9006, July, pp. 667-692. 24. Ishibashi, I. and Sherif, M.A. (1974), "Liquefaction of a Loose Saturated Sand by Torsional Simple Shear Device", Soil Engineering Research Report No. 8, Department of C i v i l Engineering, University of Washington, Seattle, Washington. 159 25. Ishibashi, I., Sherif, M.A. and Tsuehiya, C. (1977), "Pore Pressure Rise Mechanism and Soil Liquefaction", Soils and Foundations, Vol. 17, No. 2, June, pp. 17-27. 26. Ishihara, K. and Takatsu, H. (1979), "Effects of Overconsolidation and k D Conditions on the Liquefaction Characteristics of Sands", Soils and Foundations, Vol. 19, No. 4, pp. 59-68. 27. Ishihara, K., Tatsuoka, F. and Yasuda, S. (1975), "Undrained Deforma-tion and Liquefaction of Sand Under Cyclic Stresses", Soils and Foundations, Vol. 15, No. 1, pp. 29-44. 28. Khosla, V.K. and Wu, T.H. (1976), "Stress-Strain Behaviour of Sand", Journal of the Geotechnical Engineering Division, ASCE, Vol. 102, No. GT4, Proc. Paper 12079, April, pp. 303-322. 29. Kishida, H. (1966), "Damage to Reinforced Concrete Buildings in Niigata Cith With Special Reference to Foundation Engineering", Soils and Foundations, Vol. VII, No. 1. 30. Kolbuszewski, J.J. (1948a), "An Experimental Study of the Maximum and Minimum Porosities of Sands", Proceedings, 2nd International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, Vol. 1, pp. 158-165. 31. Kondner, R.L. and Zelasko, J.S. (1963), "A Hyperbolic Stress-Strain Formulation for Sands", Proceedings, 2nd Pan American Conference on Soil Mechanics and Foundations Engineering, pp. 289-324. 32. Lee, K.L. and Chan, K. (1972), "Number of Equivalent Significant Cycles in Strong Motion Earthquake", Proceedings, International Conference on Microzonation, October, Seattle, Wash., Vol. II, pp. 609-627. 33. Lee, K.W. (1975), "Mechanical Model for the Analysis of Liquefaction of Horizontal Soil Deposits", Ph.D. Thesis, University of British Columbia, Vancouver, Canada. 34. Liou, CP., Streeter, V.L. and Richart, F.E., Jr. (1977), "Numerical Model for Liquefaction", Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT6, Proc. Paper 12998, June, pp. 589-606. 35. Lucks, A.S., Christian, J.T., Brandow, G.E. and Hoeg, K. (1972), "Stress Conditions in NGI Simple Shear Test", Proceedings, ASCE ^8, SMI, pp. 155-160. 36. Martin, G.R., Finn, W.D. Liam and Seed, H.B. (1974,1975), "Fundamen-tals of Liquefaction Under Cyclic Loading", Soil Mechanics Series, No. 23, Department of C i v i l Engineering, University of Br i t i s h Columbia, Vancouver, Canada, 1974, also in Journal of the Geotechnical Engineering Division, ASCE, Vol. 101, No. GT5, Proc. Paper 11284, May 1975, pp. 423-438. 160 37. Martin, P.P. and Seed, H.B. (1979), "Simplified Procedure for Effective Stress Analysis of Ground Response", Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT6, Proc. Paper 14659, June, pp. 739-758. 38. Masing, G. (1926), "Eigenspannungen und Verfestigung beim Messing", Proceedings, 2nd International Congress of Applied Mechanics, Zurich, Switzerland. 39. Moussa, A.A. (1975), "Equivalent Drained-Undrained Shearing Resis-tance of Sand to Cyclic Simple Shear Loading", Geotechnique 25, No. 3, pp. 485-494. 40. Mroz, Z., Norris, V.A. and Zienkiewicz, O.C. (1978), "An Anisotropic Hardening Model for Soils and Its Application to Cyclic Loading", International Journal of Numerical and Analytical Methods in Geomechanics, 2:203-221. 41. Newmark, N.M. and Rosenblueth, E. (1971), Fundamentals of Earthquake  Engineering, Prentice-Hall, Inc., Englewood C l i f f s , N.J., pp. 162-163. 42. Ohsaki, Yorihiko (1966), "Niigata Earthquakes, 1964, Building Damage and Soil Conditions", Soils and Foundations, Vol. VI, No. 2, pp. 14-37. 43. Peacock, William H. and Seed, H. Bolton (1968), "Sand Liquefaction Under Cyclic Loading Simple Shear Conditions", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SM3, pp. 689-708. 44. Pickering, D.J. (1969), "A Simple Shear Machine for Soil", Ph.D. Dissertation, University of British Columbia, Vancouver, Canada. 45. Pickering, D.J. (1973), "Drained Liquefaction-Testing in Simple Shear", Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 99, SM12, pp. 1179-1184. 46. Poorooshasb, H.B. (1971), "Deformation of Sand in Triaxial Compression", Proceedings, 4th Asian Regional Conference on Soil Mechanics and Foundation Engineering, Bangkok, Thailand, Vol. 1, pp. 63-66. 47. Pyke, R.M. (1973), "Settlement and Liquefaction of Sands Under Multi-directional Loading", Ph.D. Thesis, University of California, Berkeley. 48. Roscoe, K.H. (1953), "An Apparatus for the Application of Simple Shear to Soil Samples", Proceedings, 3rd International Conference on Soil Mechanics, Zurich, Switzerland, '1_ 2, pp. 186-191. 49. Schnabel, P.B., Lysmer, J. and Seed, H.B. (1972), "SHAKE: A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites", Report No. EERC 72-12, Earthquake Engineering Research Center, University of California, Berkeley, December. 161 50. Schofield, A.N. and Wroth, CP. (1968), C r i t i c a l State Soil  Mechanics, McGraw-Hill, London. 51. Seed, H.B. (1968), "Landslides During Earthquakes Due to Soil Liquefaction", Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. 94, No. SM5, Proc. Paper 6110, September, pp. 1055-1122. 52. Seed, H.B. (1976), "Evaluation of Soil Liquefaction Effects on Level Ground During Earthquake", ASCE National Convention, Specialty Session, Liquefaction Problems in Geotechnical Engineering, Philadelphia, Pa., September, pp. 1-104. 53. Seed, H.B. and Idriss, I.M. (1967), "Analysis of Soil Liquefaction: Niigata Earthquake", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM3, pp. 83-108. 54. Seed, H.B. and Idriss, I.M. (1971), "Simplified Procedure for Evalua-ting Soil Liquefaction Potential", Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. 97, No. SM9, pp. 1249-1273. 55. Seed, H.B-., Idriss, I.M., Makdisi, F. and Banerjee, N. (1975), "Representation of Irregular Stress Time Histories by Equivalent Uniform Stress Series in Liquefaction Analyses", Report No. EERC 75-29, Earthquake Engineering Research Center, University of California, Berkeley, October. 56. Seed, H.B., Mori, K. and Chan, C.K. (1977), "Influence of Seismic History on Liquefaction of Sands", Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT4, Proc. Paper 12841, April, pp. 246-270. 57. Seed, H.B., Pyke, R.M. and Martin, G.R. (1979), "Effect of Multi-directional Shaking on Pore Pressure Development in Sand", Journal of the Geotechnical Engineering Division, ASCE, Vol. 104, No. GT1, Proc. Paper 13485, January, pp. 27-44. 58. Silver, M.L. and Seed, H.B. (1971), "Deformation Characteristics of Sands Under Cyclic Loading", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM8, August, pp. 1081-1098. 59. Silver, M.L. and Seed, H.B. (1971), "Volume Changes in Sands During Cyclic Loading", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM9, September, pp. 1171-1182. 60. Streeter, V.L., Wylie, E.B. and Richart, F.E. (1973), "Soil Motion Computations by Characteristics Method", Meeting Preprint 1952, ASCE National Structural Engineering Meeting, San Francisco, Calif., April. 61. Tatsuoka, F. and Ishihara, K. (1974), "Yielding of Sand in Triaxial Compression", Soils and Foundations, Vol. 14, No. 2, pp. 63-76. 62. Toki, S. and Kitago, S. (1974), "Effect of Repeated Loading on Deformation Behaviour of Dry Sand", Journal of the Japanese Society of Soil Mechanics and Foundation Engineering, Vol. 14, No. 1, pp. 95-103 (in Japanese). 162 63. Vaid, Y.P. and Finn, W.D. Liam (1979), "Effect of Static Shear on Liquefaction Potential", Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, GT10, Proc. Paper 14909, pp. 1233-1246. 64. Valanis, K.C. (1971), "A Theory of Viscoplasticity Without a Yield Surface", Archivum Mechaniki Stosowanej, Vol. 23, No. 4, pp. 517-533. 65. Wood, D.M. and Budhu, M. (1980), "The Behaviour of Leighton Buzzard Sand in Cyclic Simple Shear Test", International Symposium on Soils Under Cyclic and Transient Loading, Swansea, Wales, January 7-11, Vol. 1, pp. 9-21. 66. Wood, D.M., Drescher, A. and Budhu, M. (1979), "0n the Determination of the Stress State in the Simple Shear Apparatus", Submitted for publication. 67. Youd, T.L. (1972), "Compaction of Sands by Repeated Shear Straining", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 98, No. SM7, July, pp. 709-725. 68. Youd, T.L. and Craven, T.N. (1975), "Lateral Stress in Sands During Cyclic Loading", Journal of the Geotechnical Engineering Division, ASCE, Vol. 100, GT2, Technical Note, February, pp. 217-223. 69. Zienkiewicz, O.C., Chang, C.T. and Hinton, E. (1978), "Non-linear Seismic Response and Liquefaction", International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 2, pp. 381-404. APPENDIX I DESCRIPTION OF THE CYCLIC SIMPLE SHEAR APPARATUS  AND SAMPLE PREPARATION Description of the Cyclic Simple Shear Apparatus The University of British Columbia's (UBC) simple shear apparatus is an improved version of the apparatus originally designed by Roscoe (1953). The simple shear apparatus consists of horizontal carriage, vertical carriage and body frame with a sample of dimensions, 5.08 x 5.08 x 2.78 cm. The UBC simple shear apparatus was designed by Pickering (1969) and detailed description of components are given by Finn et a l . (1970). In Plate 1-1, the UBC cyclic simple shear apparatus is shown. The most recent advances in the cyclic simple shear apparatus is the development of the constant volume cyclic simple shear test by Finn and Vaid (1977) . In evaluating the results of cyclic loading tests on saturated sands carried out under undrained conditions, i t is assumed that no volume changes occur during the test. However, compliance in the test system allows volume expansion to occur in the supposedly constant volume saturated sample. This volume change, having the same effect as partial drainage, decreases the tendency for the porewater pressure to rise during cyclic loading. Therefore, undrained tests overestimate the resistance to liquefaction. Finn and Vaid (1977) showed that the error due to system compliance are always on the unsafe side and may range up to 100%. During this study, an alternative procedure for determining 163 P l a t e 1 - 1 C y c l i c s i m p l e s h e a r a p p a r a t u s 165 the undrained behaviour of sand, ie., constant volume tests on dry sands are performed. In these tests, the change in confining pressure to maintain constant volume are equivalent to the changes in porewater pressure in the corresponding test. Finn et a l . (1978) have shown that tests performed on dry and saturated sands gave exactly the same results, hence, in this research a l l tests were performed on dry sands. The modified apparatus is shown in Fig. 1-1 and Plate 1-2. The two components of linear horizontal strain are identically zero in this simple shear apparatus. Thus, a constant volume condition is achieved by clamping the loading head to prevent vertical strain. A horizontal reaction plate is clamped to four vertical posts which are threaded into the body of the simple shear apparatus. Thus, a constant volume condition is achieved by clamping the loading head and carrying on i t s upper side a heavy loading bolt which passes through a central hole in the reaction plate. The desired vertical pressure on the sample is applied by tightening the loading bolt nut on the underside of the reaction plate. Simultaneously, the loading head is clamped in position by tightening the loading bolt nut on the top side of the reaction plate. Another important innovation was the incorporation of two small s t i f f pressure transducers (350 kPa capacity and f u l l scale deflection of 0.0015 cm) on one of the moveable lateral boundaries in order to monitor the lateral stresses during cyclic loading. Maximum gross volume change introduced at the onset of lique-faction in this so called constant volume test is very small and arises as a result of the recovery of elastic deformation in the vertical loading components when the load on the clamped loading head is reduced to zero. 166 FIG. I - l Constant Volume Cyclic Simple Shear Apparatus. 167 Plate I- % Constant Volume Simple 168 The use of a thick reaction plate, heavy vertical posts and loading bolt, and a very s t i f f load transducer reduces the vertical movement of the clamped head to a negligible amount. For liquefaction tests with i n i t i a l a^ Q = 196 kN/m^  this movement amounted to a maximum of 5 x 10-^ cm which was only 5% of the movement of the floating head due to the system compliance in liquefaction tests on saturated undrained samples in the same equipment and is equivalent to a total vertical strain of the order of 0.02%. Recently, Finn, Bhatia and Pickering (1980) presented a complete analysis of the UBC simple shear apparatus including the effect of the boundary conditions on the test results. Method of Sample Preparation The tests conducted for this research were performed on dry sand samples. To prepare a sample with the cyclic simple shear apparatus, the membrane mounted was clamped to the pedestal by the bottom plate (which was inside the membrane) and the rubber membrane was held wide open by metal hooks as shown in Plate 1-3. A weighted amount of dry sand was deposited within the membrane in the apparatus through a funnel top permitting a f r e e f a l l . However, to achieve very loose samples, the height of f a l l of the sand particles was kept at 1 cm and the funnel was gradually raised as the sand was poured within the membrane. The funnel was trans-formed across the plane area of the sample so that the sand surface remained almost level. Kolbuszewski (1948) has shown that the density achieved by pluvial compaction such as this depends on the intensity of the rain of sand particles, higher intensities yielding lower densities. Since the aim was to make identical and loose samples for each specimen the total pouring time was one minute. By'this process, i t was possible 169 Plate 1-4 Placing top plate on sand sample -4-0.18 s -4 -0 .12 s E i i l i l i I I I l I I 1 1 1 L i i r i I I I i i i i — i — L to i I l l I I I I L ci- 0.06 s f = 750 cycles/s I 1 I L 4.0 g 3.0g 2.0g I.Og 0.0 1.0 g 2.0g 3.0g Acceleration (g) , c m / s FIG. 1-2 Vibrations Applied to Sand Sample. 171 to prepare a sample of relative density of 32 to 35% for Ottawa sand. Once the sand had been poured, the excess sand over the fin a l elevation was siphoned off using a small vacuum. The top ribbed plate was then placed on the sand surface (Plate 1-4) and extra sand particles sandwiched between the plate and rubber membrane were sucked in. Then, the rubber membrane was closed over i t and sealed to the loading head. The desired f i n a l relative density was then obtained by hitting the simple shear apparatus bed with a plastic hammer. By this method, the sample was subjected to high frequency vibrations (Fig. 1-2) and these vibrations were applied when the sample was kept under a seating pressure of approximately 0.2 kg/cm^. The top ribbed plate resting on top of the sand sample thus follows the settlement of the sand surface and resumes a proper seating, while the entire sample gets uniformly densified without development of a loose thin layer at the top of the sample. Finn, Vaid and Bhatia (1978) showed that the development of a thin layer on top of the sample under-estimates the liquefaction potential for sand samples. The vertical confining pressure was then increased to the required value of the overburden pressure and during consolidation lateral stresses (pressure transducers from top and bottom) were recorded. Tests Performed For this research the following types of tests were performed: 1. Drained test under cyclic strain controlled and cyclic stress controlled conditions; 2. Constant volume test under cyclic strain controlled and cyclic stress controlled conditions; 172 3. Static unloading tests in simple shear apparatus and consolidation equipment. For cyclic loading tests, both for drained and undrained tests, cyclic stress and strains were applied by an MTS servo-controlled electro-hydraulic piston using a sinusoidal waveform at a frequency of 0.2 Hz. During each test vertical stress, lateral, stress, cyclic shear stress and cyclic shear strain were continuously monitored with electronic transducers and records obtained on chart recorders. In Plate 1-5, the cyclic simple shear appartus is shown with a l l recording equipments. To measure the static unloading curve for the sand sample consolidated to i n i t i a l vertical confining stress, vertical was decreased in small increments and recovered volumetric strains were measured. To completely unload the sample in the simple shear apparatus (since wt. of the loading frame was 0.396 kg/cm ), a double-acting piston was used (Plate 1-6). Most of the tests were on Ottawa sand at relative densities of Dr = 45,54,60 and 68%. Tests were on Crystal S i l i c a and Toyoura sands at D r = 45%. In most of the tests, the samples were consolidated to i n i t i a l v e rtical confining pressure 0 V O = 200 kN/m but a few tests were a l performed at a y o =100 kN/m2 and 300 kN/m2. !' Plate 1-5 Cyclic simple shear apparatus with recording equipments 174 Plate 1-6 Set up with double acting piston to measure static rebound moduLs 175 APPENDIX II PHYSICAL PROPERTIES OF OTTAWA SAND (C-109), CRYSTAL SILICA  SAND (NO. 20) AND TOYOURA SAND Most of the tests were performed on Ottawa Sand (C-109). This is a natural S i l i c a Sand consisting of round to subround particles as shown in Plate I I - l . Grain size distribution curve and physical properties of the sand are given in Fig. I I - l and Table II, respectively. This sand has been tested by Finn et a l . (1971) to relate behaviour of saturated sand in cyclic simple shear and cyclic t r i a x i a l . Finn and Vaid (1977) used i t for constant volume simple shear tests. The maximum void ratio was determined in accordance with ASTM D2049-69 and minimum void ratio was obtained by the Kolbuszewski (1948) method. Crystal S i l i c a Sand (No. 20) is a uniform angular quartz sand. Silver and Seed (1971) used this sand and physical properties listed in Table Il-a were obtained from that reference. Toyoura Sand is the most widely used sand in Japan for research purposes. This sand consists of quartz (80%), chert (30%) and feldspar (17%) and particles are subangular to angular (Plate I I - l ) . 176 • • >• 6 • • » <7 • • • • 4 J I I 2 mm Ottawa sand (C-109) I L I 2 mm Crystal silica sand No. 20 J I I 2 mm Toyoura sand GRAIN SHAPE Plate H-l FIG. I I - l Particle Size Distribution of Soils Used in the Tests. TABLE I I - l PROPERTIES OF SANDS SAND TYPE OTTAWA (ASTM C-109) CRYSTAL SILICA SAND TOYOURA SAND GRAIN SIZE D,_ = 0.40 mm 60 D 1 Q = 0.19 mm D,„ = 0.65 mm 60 D 1 Q = 0.52 mm D,_ = 0.16 mm 60 D 1 Q = 0.11 mm GRAIN SHAPE Subround to Round Subangular to Angular Subangular UNIFORMITY COEFFICIENT, C u 2.10 1.25 1.44 SPECIFIC GRAVITY, G s 2.67 2.65 2.65 MAXIMUM VOID RATIO, e max 0.82* 0.973 0.960 MINIMUM VOID RATIO, e . min 0.50 0.636 0.64 *Ishibashi and Sherif (1974) reported e^^ = 0.76 for the same sand. APPENDIX III ENDOCHRONIC REPRESENTATION OF VOLUMETRIC STRAINS In this "appendix volumetric strain, e V (j, obtained under constant strain cyclic loading in the simple shear condition for loose Ottawa sand w i l l formulate in endochronic form. The purpose of this formulation is two-fold: f i r s t l y , to check whether volumetric strain can also be pre-sented as a continuous function of the drainage parameter; secondly, i f so, to check whether this simple formulation can be used to calculate volume-t r i c strain, hence settlement for the irregular strain history encountered by s o i l during an earthquake. II I - l Endochronic Formulation of Volumetric Strains The volumetric strains generated in cyclic simple shear at various constant strain amplitudes ranging from y = 0.056% to y = 0.314% are shown in Fig. II I - l versus the number of load cycles N. Definition of length of strain path, E,, is again assumed as given in equation (6.1). Thus, for simple shear conditions, dE, is given by equation (6.2). The data shown in Fig.III-2 is plotted against the length of the strain path, E,. We again seek a transformation, exactly of the same form as equation (6.8), where the transformation factor, A, is defined by equation (6.9). When equation (6.9) is applied to the data shown in Fig. III-l, a range of values for A results with a mean value of 5.71. Using this average value of A, the data in Fig. I-LI—2- is transformed to £*-space, where £ is again called the damage parameter, and transformed results are plotted in *For the pore pressure transformation, the symbol used for damage parameter is k . In order to differentiate the numerical values of damage parameter for volumetric strain to that of porewater pressure for the same sand at the same relative density, the symbol £ is referred to as damage parameter in section 6.3. . 180 FIG. I I I - l Volumetric S t r a i n vs. S t r a i n Cycles f o r Ottawa Sand. . 15 C o o 1.0 4> e £ 0.5 0 1 1 Sand type : Ottawa sand (C-109) l i cr v ' 0 = 200 k N / m 2 .Relative density =45% y$y-1 ' 001 .005 01 05 10 Length of strain path , £ FIG. I I I - 2 Volumetric S t r a i n vs. Natural Logarithm of Length of S t r a i n Path f o r Ottawa Sand. 'fj 181 Fig.III-3. Experimental data plotted in Fig.III-4 for various shear strain amplitudes l i e in a very narrow band. The same data are plotted on a natural scale where a nonlinear least square curve f i t t i n g method has been used to define the analytical curve £vd = ? ( D 1 C + C D ) 1 ( A 1 ? + B l ) (III-l) . with Ax = 0.078, B1 = 0.0038, Cx = 0.0716 and Dj_ = 0.138. Although i t is quite evident that volumetric strain obtained for a type of sand and rela-tive density under constant cyclic strain can be represented as a unique function of damage parameter, £, i t seem appropriate that the inverse transformation as discussed in section 6.2.2 should again be applied. Hence, the inverse transformation is performed to transfer points back as shown in Figs. I I I - l and III-2. In Figs.III-5 and III-6 both analytical and experimental volumetric curves are shown in £-space and N-space which show that the unique relationship between volumetric strain and damage parameter £ as given in equation (III-l) represents experimental data correctly. It should be emphasized that the relationship given in equa-tion (III-l) represents not just the four curves shown in Fig. I I I - l but any test curve that might be determined within the same range. Finn (1979) presents an endochronic formulation of volumetric strain obtained for Nakashima sand and some details about the endochronic formulation of volu-metric strain data are discussed by Finn and Bhatia (1980). III-2 Verification of the Endochronic Representation of Volumetric Strain In order to check the validity of volumetric strain given by equation (III-l) , the formulation is used to calculate volumetric strains in drained stress controlled tests on Ottawa sand. In drained stress con-trolled tests with a continuous application of.cyclic shear stress, shear 182 Sond type : Ottawa sand (C-109) o-' v 0 = 200 kN/m 2 , Relative density =45% « v d = A/B L n ( l+Bf) A = 16 994 , B = 34.718 X = 5.71 Legend : • / =0056 % X X =0100 % A X = 0 200 % O X = 0314 % Analytical curve 003 006 01 03 06 .10 Damage parameter , f .30 60 FIG.III - 3 Volumetric S t r a i n vs. Ln (Damage Parameter) for Ottawa Sand. Sand i i r type Ottawa sand (C-109) 2.0 " °" vo = 2 0 kg/cm2 , Relative density = • 45 % -> € vd = f(D,f +C,)/(Air + B1) , A,= 0078 , B,= 00038, Ml X = 5.71 , C,= 0 0716 , D,= 0. 138 c 1.5 • -o w «i o \- IO Legend : 0) E JS^ »—• » 0 X = 0 056 % olui X ^ ° Analytical function X X =0.100 % > 0.5 a X = 0 2 0 0 % a X = 0 300% 0 i i I i 1 1 1 1 I I 0 010 0 20 0 30 0.40 050 Damage parameter, <T FIG. III-4 Volumetric S t r a i n vs. Damage Parameter f o r Ottawa Sand. 183 > e O > 20 1.5 10 05 0 Sand type Ottawa sand (C-109) crv'0 =200 kN/m2 .Relative density =45% « v d = f(Df •C)/(Af + B) , A = 0078 ,B = 0 0038, C =0 0716 , D =0138 , X = 571 Analytical curve Experimental curve .001 003 .01 03 Length of strain path , £ FIG. III-5 Comparison of Computed and Experimental Volumetric Strains in £-plot. CP > c o m u k_ cu E 3 o > Sand type Ottawa sand (C-109) crv'0 = 200 kN/m 2 , Relative density =45% (Inverse transformation from K - space ) = f(Df + C)/(Ar +B) , A = 0.078, B = 0 0038. , D = 0 138 , X =5.71 Analytical curve IB) Experimental curve 10 20 30 40 Number of cycles , N 50 FIG. III -6 Comparison of Computed and Experimental Volumetric Strains in N-plot. 184 strain decreases as shown in Fig. III-7 for two stress ratios, T/O^ 0 = 0.146 and T/O^ 0 = 0.119. This condition is obviously different from the original test condition that supplied the data for the endochronic formulation. For the present analysis, average shear strains are calculated for each cycle as shear strains generated in each half cycle are different. Shear strain for each cycle is converted to incremental length of strain path and the incremental damage parameter generated for the cycle is calcu-lated. The cumulative damage parameter is used in equation (III-l) with the set of constants given for Ottawa sand to calculate the total volume-t r i c strain. The analytical and experimental volumetric strains plotted in Fig. III-8 for two cyclic stress ratios show that f a i r l y good predictions of volumetric strains can be made by the endochronic formulation of volu-metric strain data. Some discrepancy in the experimental and analytical results in Fig. III-8 may be attributed to the fact that the endochronic formulation used for an analysis corresponding to D r = 45% whereas the stress controlled tests have been performed at Dr = 47%. The volumetric strains represented in endochronic form can be used in conjunction with dynamic rebound modulus for the pore pressure prediction. In addition, such a formulation can be used to calculate settlements. 185 3 I i i i i i i i I 0 5 10 15 20 25 30 35 40 N u m b e r of c y c l e s , N FIG. I'IT-7 Av. Shear Stress vs. Number of Cycles for Constant Stress Drained Test. FIG. I I I - 8 Comparison of Calculated and Experimental Volumetric Strains. 

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