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UBC Theses and Dissertations

Numerical studies of some aspects with pressuremeter tests and laterally loaded piles Yan, Li 1986

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NUMERICAL STUDIES OF SOME ASPECTS WITH PRESSUREMETER TESTS AND LATERALLY LOADED PILES by LI YAN B . S c , D a l i a n I n s t i t u t e of Technology, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r , 1986 © L i Yan, 1986 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 o f 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 o r 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 o r p u b l i c a t i o n o f 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 C I V I L E N C V I N E E R I N C X 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 Oci. 10 , i 966 ABSTRACT A n a l y s e s o f l a t e r a l l y l o a d e d p i l e s u s i n g a n o n l i n e a r s u b g r a d e r e a c t i o n method r e l y on t h e r a t i o n a l i t y o f d e s c r i b i n g t h e l a t e r a l s o i l r e a c t i o n i n t e r m s o f f o r c e - d e f l e c t i o n o r P-Y c u r v e s . Most o f t h e e x i s t i n g methods f o r o b t a i n i n g P-Y c u r v e s a r e e i t h e r e m p i r i c a l o r s u f f e r f r o m d i f f i c u l t i e s i n o b t a i n i n g r e l i a b l e s o i l p a r a m e t e r s a s i n p u t . In a d d i t i o n , l i t t l e a c c o u n t i s t a k e n o f t h e e f f e c t s o f p i l e i n s t a l l a t i o n . The p r e s s u r e m e t e r , w h i c h m e a s u r e s l a t e r a l s o i l r e s p o n s e s t o a c a v i t y e x p a n s i o n , o f f e r s a p r o m i s i n g i n - s i t u method f r o m w h i c h P-Y c u r v e s f o r t h e l a t e r a l p i l e p r o b l e m c a n be o b t a i n e d . H e r e i n , a s i m p l e and r a t i o n a l method of c o n s t r u c t i n g P-Y c u r v e s f r o m t h e p r e s s u r e m e t e r c u r v e s i s e x a m i n e d u s i n g a f i n i t e e l e m e n t a n a l y s i s . The P-Y c u r v e i s o b t a i n e d by e x a m i n i n g t h e r e s p o n s e o f t h e s o i l d i s k s u r r o u n d i n g t h e p i l e t o l a t e r a l l o a d s . The p r e s s u r e m e t e r e x p a n s i o n c u r v e i s o b t a i n e d f r o m a n a l y s i s o f t h e c y l i n d r i c a l c a v i t y e x p a n s i o n p r o b l e m . By c o m p a r i n g t h e P-Y a n d p r e s s u r e m e t e r c u r v e s so o b t a i n e d f r om a n a l y s i s , a method o f m o d i f y i n g t h e p r e s s u r e m e t e r c u r v e s t o o b t a i n P-Y c u r v e s i s p r e s e n t e d . The i n s t a l l a t i o n e f f e c t on p r e s s u r e m e t e r c u r v e s a n d P-Y c u r v e s i s e v a l u a t e d f r o m a p a r a m e t r i c s t u d y . The i n s t a l l a t i o n e f f e c t i s m o d e l l e d by a zone o f d i s t u r b e d s o i l h a v i n g d i f f e r e n t s t r e n g t h and modu lu s p r o p e r t i e s f r o m i n t a c t s o i l . A l i n e a r s t r a i n t r i a n g u l a r f i n i t e e l e m e n t p r o g r a m , i n w h i c h t h e n o n l i n e a r s o i l i s m o d e l l e d a s an i n c r e m e n t a l i i e l a s t i c m a t e r i a l w i t h h y p e r b o l i c s t r e s s s t r a i n r e l a t i o n , i s employed throughout. A t h i n i n t e r f a c e element and a s i m p l e t e n s i o n c u t - o f f model s i m u l a t i n g the s o i l - p i l e i n t e r f a c e b e h a v i o r has been i n c o r p o r a t e d i n the program. R e s u l t s of the program and the i n t e r f a c e s i m u l a t i o n a r e i n good agreement w i t h c l o s e d form s o l u t i o n s . From comparison of the responses of p r e s s u r e m e t e r and l a t e r a l p i l e c o n d i t i o n s , the p r e s s u r e m e t e r c u r v e s can be a d j u s t e d t o r e p r e s e n t the P-Y c u r v e s . However, due t o f e a t u r e s of the l o a d i n g mechanism, the i n s t a l l a t i o n has a d i f f e r e n t e f f e c t on p r e s s u r e m e t e r c u r v e s than on P-Y c u r v e s . For pressuremeter c u r v e s , the i n i t i a l s t i f f n e s s i s much a f f e c t e d by the s o i l d i s t u r b a n c e s , but the u l t i m a t e p r e s s u r e i s not s i g n i f i c a n t l y a f f e c t e d . The o p p o s i t e e f f e c t s a r e observed f o r P-Y c u r v e s . The p r a c t i c a l s i g n i f i c a n c e of t h e s e e f f e c t s i n d e v e l o p i n g P-Y c u r v e s from p r e s s u r e m e t e r expansion c u r v e s i s d i s c u s s e d . i i i T a b l e of Co n t e n t s ABSTRACT i i LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENTS xv DEDICATION x v i 1. INTRODUCTION 1 1.1 I n t r o d u c t i o n 1 1.2 Scope of T h e s i s 3 1.3 O r g a n i z a t i o n of T h e s i s 4 2. REVIEWS OF PREVIOUS WORK 6 2.1 I n t r o d u c t i o n 6 2.2 Methods of A n a l y s e s 6 2.3 S p e c i f i c a t i o n of P-Y Curves 13 2.3.1 S e m i - e m p i r i c a l Methods 14 2.3.2 I n - s i t u T e s t i n g Methods 28 2.3.3 C e n t r i f u g e T e s t i n g s 30 2.3.4 F i n i t e Element Methods ....32 3. FINITE ELEMENT PROGRAM 37 3.1 I n t r o d u c t i o n 37 3.2 F i n i t e Element F o r m u l a t i o n 37 3.3 U n i f i e d Approach - An E f f e c t i v e S t r e s s Method ..42 3.3.1 U n d r a i n e d A n a l y s i s 42 3.3.2 D r a i n e d A n a l y s i s 47 3.4 S t r u c t u r e of the Program 48 3.5 S t r e s s - R e d i s t r i b u t i o n 49 4. CONSTITUTIVE RELATIONS 52 4.1 I n t r o d u c t i o n 52 i v 4.2 I n c r e m e n t a l N o n l i n e a r E l a s t i c S o i l Model 53 4.3 B i l i n e a r E l a s t i c - p l a s t i c Model ....58 4.4 I n c o r p o r a t i o n of Tension F a i l u r e 60 5. CYLINDRICAL CAVITY EXPANSION THEORY 62 5.1 I n t r o d u c t i o n 62 5.2 E l a s t o - p l a s t i c C l o s e d Form S o l u t i o n s 63 5.2.1 Problems .....63 5.2.2 C l o s e d Form S o l u t i o n s 65 5.3 F i n i t e Element S i m u l a t i o n 70 5.3.1 F i n i t e element mesh domain 70 5.3.2 Outer boundary e f f e c t s 72 5.4 F i n i t e Element P r e d i c t i o n s 74 5.4.1 M a t e r i a l Models and A n a l y s e s 74 5.4.2 M a t e r i a l P r o p e r t i e s 75 5.4.3 R e s u l t s and Comparison 77 6. FINITE ELEMENT STUDIES OF PRESSUREMETER TESTS 101 6.1 I n t r o d u c t i o n 101 6.2 F i n i t e Element Mesh and Boundary C o n d i t i o n s ... 107 6.3 A n a l y s e s and S o i l Parameters 109 6.4 I n f l u e n c e s of P r e s s u r e m e t e r L/D R a t i o s 114 6.4.1 Cohesive S o i l s 115 6.4.2 C o h e s i o n l e s s S o i l s 128 6.4.3 Summary 1 40 6.5 Comparisons of C y l i n d r i c a l C a v i t y E x p a n s i o n A n a l y s e s and F i e l d P r e s s u r e m e t e r Test Data ....142 6.5.1 Cohesive S o i l s 142 6.5.2 C o h e s i o n l e s s S o i l s 161 7. SOIL-PILE INTERFACE ELEMENTS 175 v 7.1 I n t r o d u c t i o n 175 7.2 Deformation Modes a t I n t e r f a c e 178 7.3 Review on I n t e r f a c e Elements 180 7.3.1 J o i n t Elements w i t h Zero T h i c k n e s s 180 7.3.2 Thi n Layer I n t e r f a c e Element 185 7.4 The Proposed Model f o r S o i l - p i l e I n t e r f a c e ....189 7.4.1 F o r m u l a t i o n of S t i f f n e s s M a t r i x -C o n s t i t u t i v e Laws 190 7.4.2 Deformation and S t r e n g t h C h a r a c t e r i s t i c s 192 7.4.3 I n c o r p o r a t i o n of D e f o r m a t i o n Modes 194 7.4.4 I n t e r f a c e Element - Mesh Layout and I t s T h i c k n e s s 199 7.4.5 P r e l i m i n a r y Assessments - D i r e c t Shear C o n d i t i o n 202 FINITE ELEMENT STUDIES ON LATERALLY LOADED PILES ..216 8.1 I n t r o d u c t i o n 216 8.2 P l a n e S t r a i n . Model 220 8.3 C l o s e d Form S o l u t i o n 222 8.4 F i n i t e Element S i m u l a t i o n 230 8.4.1 F i n i t e Element Mesh Layout 230 8.4.2 Outer Boundary C o n s i d e r a t i o n 233 8.4.3 I n t e r f a c e Element 236 8.5 Cohesive S o i l s 242 8.5.1 S o i l P r o p e r t i e s 243 8.5.2 R e s u l t s and D i s c u s s i o n s 246 8.6 C o h e s i o n l e s s S o i l s 260 INSTALLATION EFFECTS ON PRESSUREMETER CURVES AND P-Y CURVES FOR LATERALLY LOADED PILES IN COHESIVE SOILS 268 9.1 I n t r o d u c t i o n 268 v i 9.2 E x p e r i m e n t a l and A n a l y t i c a l E v i d e n c e s on Ex t e n t of S o i l D i s t u r b a n c e A f t e r P i l e I n s t a l l a t i o n 270 9.3 F i n i t e Element A n a l y s i s 274 9.3.1 D i s t u r b a n c e S i m u l a t i o n 276 9.3.2 F i n i t e Element Mesh 281 9.4 R e s u l t s and D i s c u s s i o n 284 9.4.1 Pre s s u r e m e t e r Curves 284 9.4.2 P-Y Curves 289 9.5 Summary 298 10. SUMMARY AND CONCLUSIONS 300 REFERENCES 306 APPENDIX A 313 APPENDIX B 315 v i i LIST OF TABLES Tab l e 5.1 S o i l Parameters of C o h e s i v e S o i l s f o r C y l i n d r i c a l C a v i t y Expansion S i m u l a t i o n 76 T a b l e 5.2 S o i l Parameters of C o h e s i o n l e s s S o i l s f o r C y l i n d r i c a l C a v i t y Expansion S i m u l a t i o n 76 T a b l e 5.3 R e l a t i o n between Shear Modulus R e d u c t i o n F a c t o r and F a i l u r e P o i s s o n ' s R a t i o 93 T a b l e 6.1 S o i l Parameters f o r A x i s y m m e t r i c A n a l y s e s .... 113 Table 6.2 S o i l Parameters f o r C a v i t y Expansion A n a l y s e s . . 1 1 3 Table 6.3 L/D R a t i o E f f e c t s on I n i t i a l S l o p e s i n C o h e s i v e S o i l s 119 T a b l e 6.4 L/D R a t i o E f f e c t s on D e r i v e d Undrained Shear S t r e n g t h 125 T a b l e 6.5 L/D R a t i o E f f e c t s on D e r i v e d Undrained Shear S t r e n g t h ( B o r s e t t o et a l , 1980) 127 T a b l e 6.6 L/D R a t i o E f f e c t s on I n i t i a l S l o p e s i n C o h e s i o n l e s s S o i l s 132 Table 6.7 L/D R a t i o E f f e c t s on the D e r i v e d F r i c t i o n A n g l e s i n C o h e s i o n l e s s S o i l s 137 Table 6.8 P r e s s u r e m e t e r Test R e s u l t s a t H a m i l t o n Test S i t e , San F r a n c i s c o 153 T a b l e 6.9 S o i l Parameters f o r San F r a n c i s c o Bay Mud .... 155 T a b l e 6.10 S o i l Parameters f o r McDonald Farm S i t e 165 T a b l e 6.11 S o i l Parameters from Byrne and Cheung 166 T a b l e 7.1 Proposed C o e f f i c i e n t s of S k i n F r i c t i o n between S o i l s and C o n s t r u c t i o n M a t e r i a l s 195 Table 7.2 S o i l P r o p e r t i e s f o r I n t e r f a c e Elements 205 Table 7.3 Shear S t r e s s D i s t r i b u t i o n w i t h i n I n t e r f a c e Element at I n t e g r a t i o n P o i n t s ( f o r C o h e s i v e S o i l s ) 206 Table 7.4 A p p l i e d Shear S t r e s s vs R e l a t i v e Displacement of I n t e r f a c e Element under V a r i o u s Aspect R a t i o s 208 T a b l e 7.5 Shear S t r e s s D i s t r i b u t i o n w i t h i n I n t e r f a c e Element a t I n t e g r a t i o n P o i n t s ( f o r C o h e s i o n l e s s S o i l s ) 212 T a b l e 8.1 P l a s t i c i t y S o l u t i o n of U l t i m a t e S o i l R e s i s t a n c e on L a t e r a l C i r c u l a r P i l e i n C o h e s i v e S o i l s 238 v i i i Table 8.2 Parameters of E l a s t o - p l a s t i c S o i l , P i l e and I n t e r f a c e Elements f o r FE A n a l y s i s 245 Table 8.3 I n i t i a l S l o p e s of P r e d i c t e d P-Y Curves 248 Table 8.4 U l t i m a t e S o i l R e s i s t a n c e of P r e d i c t e d P-Y Curves 251 T a b l e 8.5 N o n l i n e a r S o i l Parameters f o r FE A n a l y s e s of P-Y Curves 258 i x LIST OF FIGURES F i g 2.1 W i n k l e r S p r i n g Approach ...9 F i g 2.2 A T y p i c a l Shape of P-Y Curve 11 F i g 2.3 C o n s t r u c t i o n of P-Y Curves f o r S o f t C l a y 17 F i g 2.4 C o e f f i c i e n t A f o r S t i f f anf F i s s u r e d C l a y 19 F i g 2.5 C o n s t r u c t i o n of P-Y Curve f o r S t i f f C l a y 19 F i g 2.6 C o n s t r u c t i o n of P-Y Curve f o r Sand 21 F i g 2 . 7 ( 1 ) , ( I I ) Assumed S o i l F a i l u r e Mechanisms around P i l e 23 F i g 2.8 Nondimensional C o e f f i c i e n t A and B ....25 F i g 2.9 B i l i n e a r P-Y Curve f o r Sand proposed by S c o t t ....26 F i g 2.10 FE Domain employed by Yegian and Wright 33 F i g 3.1 Element Types employed by CONOIL 40 F i g 3.2 S t r e s s S t a t e a s s o c i a t e d w i t h Load Shedding 50 F i g 4.1 H y p e r b o l i c R e p r e s e n t a t i o n of A S t r e s s - s t r a i n Curve : 55 F i g 4.2 S t r e s s S t r a i n Curves f o r D r a i n e d T r i a x i a l T e s t s on Sand 57 F i g 4.3 B i l i n e a r E l a s t i c P l a s t i c Model 59 F i g 5.1 Problem of C y l i n d r i c a l C a v i t y E x p a n s i o n i n S o i l Masses 64 F i g 5.2(a) S o i l Domain used f o r FE A n a l y s i s 71 F i g 5.2(b) FE Mesh f o r C a v i t y E x p a n s i o n S i m u l a t i o n 71 F i g 5.3 I n f l u e n c e s of Outer Boundary Radius i n C a v i t y E xpansion S i m u l a t i o n 73 F i g 5.4 Comparison of O r i g i n a l Program and C l o s e d Form S o l u t i o n 78 F i g 5.5 Displacement D i s t r i b u t i o n under U n d r a i n e d Pl a n e S t r a i n C o n d i t i o n 80 F i g 5.6 Comparison of M o d i f i e d Program and C l o s e d Form S o l u t i o n 82 F i g 5.7 S t r e s s D i s t r i b u t i o n i n Comparison w i t h C l o s e d Form x S o l u t i o n 84 F i g 5.8 P r e s s u r e Expansion Curve f o r C o h e s i o n l e s s S o i l s from O r i g i n a l Program 86 F i g 5.9 S o i l S t r e s s Path i n C a v i t y E x p a n s i o n Problem 88 F i g 5.10 P r e s s u r e Expansion Curve a f t e r D i s c a r d i n g S t r e s s Memory 90 F i g 5.11 P r e s s u r e Expansion Curve w i t h S m a l l e r R e d u c t i o n F a c t o r of Shear Modulus 94 F i g 5.12 Load Shedding E f f e c t on P r e s s u r e Expansion Curve 95 F i g 5.13 P l a s t i c Volume C o r r e c t i o n f o r FE A n a l y s i s 97 F i g 5.14 P r e s s u r e Expansion Curves a f t e r ' P l a s t i c ' Volume C o r r e c t i o n 99 F i g 6.1 L/D R a t i o E f f e c t on D i s p l a c e m e n t P a t t e r n 103 F i g 6.2 L/D R a t i o E f f e c t on E l a s t i c Modulus 105 F i g 6.3 FE Mesh f o r Axisymmetric A n a l y s i s 108 F i g 6.4 FE Mesh f o r C a v i t y Expansion A n a l y s i s 110 F i g 6.5 L/D R a t i o E f f e c t on P r e s s u r e m e t e r Curves i n Cohesive S o i l s 116 F i g 6.6(a) Gibson and Anderson Method f o r d e r i v i n g U n d r a i n e d Shear S t r e n g t h 122 F i g 6.6(b) Pressuremeter Curves i n S e m i - l o g S c a l e 123 F i g 6.7 N o n l i n e a r S o i l Response under D i f f e r e n t L/D R a t i o 129 F i g 6.8 L/D R a t i o E f f e c t on P r e s s u r e m e t e r Curves i n C o h e s i o n l e s s S o i l s 130 F i g 6.9(a) Pressuremeter Curves i n C o h e s i o n l e s s S o i l s . . . 136 F i g 6.9(b) Pressuremeter Curves i n Log-Log S c a l e ( C o h e s i o n l e s s S o i l s ) 138 F i g 6.10 Pressuremeter Curve Measured a t H a m i l t o n S i t e ..144 F i g 6.11 Log of Borehole a t H a m i l t o n S i t e 145 F i g 6.12 E n g i n e e r i n g S o i l P r o p e r t i e s a t H a m i l t o n S i t e . . . . 1 4 7 F i g 6.13 Undrained S t r e n g t h w i t h Depth 148 x i F i g 6 . 1 4 C o e f f i c i e n t of E a r t h P r e s s u r e 151 F i g 6 . 1 5 Comparison w i t h F i e l d Measurements ( 1 ) 1 5 6 F i g 6 . 1 6 Comparison w i t h F i e l d Measurements ( 2 ) 1 5 7 F i g 6 . 1 7 S o i l F a i l u r e Mode a s s o c i a t e d w i t h P r e s s u r e m e t e r T e s t s i n Cohesive S o i l s 1 6 0 F i g 6 . 1 8 S o i l P r o f i l e a t McDonald Farm 1 6 2 F i g 6 . 1 9 Pressuremeter Curve measured a t McDonald Farm...1 6 4 F i g 6 . 2 0 Comparison of FE P r e d i c t i o n w i t h F i e l d Data (SBPMT & CPT data) 1 6 9 F i g 6 . 2 1 Comparison of FE P r e d i c t i o n w i t h F i e l d Data (Byrne & Cheung data) 1 7 0 F i g 6 . 2 2 E f f e c t of the Assumed P o i s s o n ' s R a t i o V a l u e on the FE R e s u l t s 1 7 2 F i g 7 . 1 S o i l Movement a t Sh a l l o w Depth 1 7 6 F i g 7 . 2 S o i l Flows around P i l e a t Depth 1 7 7 F i g 7 . 3 Schematic of De f o r m a t i o n Modes a t I n t e r f a c e 1 7 9 F i g 7 . 4 J o i n t Element w i t h Zero T h i c k n e s s . . 181 F i g 7 . 5 C y l i n d r i c a l I n t e r f a c e Element 1 8 4 F i g 7 . 6 T h i n Layer I n t e r f a c e Element 1 8 6 -F i g 7 . 7 S t r e s s C o n d i t i o n s w i t h V a r i o u s I n t e r f a c e D e f o r m a t i o n Modes 1 9 6 F i g 7 . 8 Mesh Layout f o r T r i a n g u l a r I n t e r f a c e Element 2 0 1 F i g 7 . 9 S i m u l a t i o n of D i r e c t Shear T e s t i n g . . . . . 2 0 3 F i g 7 . 1 0 R e l a t i v e D i s p l a c e m e n t vs I n t e r f a c e R e s i s t a n c e under V a r i o u s L / t R a t i o f o r C o h e s i v e S o i l s . . . . . 2 0 9 F i g 7 . 1 1 R e l a t i v e D i s p l a c e m e n t vs I n t e r f a c e R e s i s t a n c e under V a r i o u s L / t R a t i o f o r C o h e s i o n l e s s S o i l s 2 1 3 F i g 8 . 1 Concept of.P-Y Curves 2 1 8 F i g 8 . 2 'Disk' A n a l y s i s f o r P-Y Curves 2 2 1 F i g 8 . 3 Outer Boundary and P o i s s o n ' s R a t i o E f f e c t s on S t i f f n e s s R a t i o n 2 2 6 F i g 8 . 4 Boundary E f f e c t on the Extreme V a l u e s of M 2 2 7 x i i F i g 8.5 FE Mesh f o r L a t e r a l l y Loaded P i l e 232 F i g 8.6(a) E f f e c t s of I n t e r f a c e Element L / t R a t i o 239 F i g 8.6(b) E f f e c t s of I n t e r f a c e Element L / t R a t i o 241 F i g 8.7 FE P r e d i c t i o n of P-Y Curves u s i n g Model (1) 247 F i g 8.8 I n f l u e n c e of I n t e r f a c e B e h a v i o r 250 F i g 8.9 P-Y Curves from FE P r e d i c t i o n ( I s o t r o p i c Model vs Tension C u t - o f f Model) 253 F i g 8.10 S o i l S t r e s s D i s t r i b u t i o n 256 F i g 8.11 P-Y Curves f o r E l a s t i c P l a s t i c S o i l and N o n l i n e a r S o i l 259 F i g 8.12 H y p e r b o l i c Curve F i t t i n g f o r P-Y Curves i n Cohesive S o i l s 261 F i g 8.13 Outer Boundary E f f e c t on the P-Y Curve i n C o h e s i o n l e s s S o i l 263 F i g 8.14 I n t e r f a c e E f f e c t on The P-Y Curve i n C o h e s i o n l e s s S o i l 265 F i g 8.15 Curve F i t t i n g f o r P-Y Curves i n C o h e s i o n l e s s S o i l s 267 F i g 9.1 F i e l d Data of E x c e s s Pore P r e s s u r e from P i l e D r i v i n g 272 F i g 9.2 T y p i c a l S t r e s s D i s t r i b u t i o n a f t e r P i l e D r i v i n g ..273 F i g 9.3 T y p i c a l V a r i a t i o n of C u a t end of C o n s o l i d a t i o n .275 F i g 9.4 S i m u l a t i o n of I n s t a l l a t i o n f o r P i l e 277. F i g 9.5 Assumed Modulus and S t r e n g t h V a r i a t i o n w i t h i n D i s t u r b e d Zone 278 F i g 9.6 S i m u l a t i o n of I n s t a l l a t i o n f o r Pressuremeter ....280 F i g 9.7 FE Mesh f o r S t u d y i n g I n s t a l l a t i o n E f f e c t on Pressuremeter Curves 282 F i g 9.8 FE mesh f o r S t u d y i n g I n s t a l l a t i o n E f f e c t on P-Y Curves 283 F i g 9.9 Pressuremeter Curves under V a r i o u s S o i l D i s t u r b a n c e s 285 F i g 9.10 R e l a t i v e I n i t i a l Slope of P r e s s u r e m e t e r Curve vs S i z e of D i s t u r b e d Zone 287 x i i i F i g 9.11 R e l a t i v e I n i t i a l S lope of P r e s s u r e m e t e r Curve vs E x t e n t of D i s t u r b a n c e 288 F i g 9.12 P-Y Curves Under V a r i o u s S o i l D i s t u r b a n c e s ( s t i f f c a l y ) 290 F i g 9.13 P-Y Curves Under V a r i o u s S o i l D i s t u r b a n c e s ( s o f t c a l y ) 291 F i g 9.14 R e l a t i v e I n i t i a l Slope of P-Y Curve vs S i z e of D i s t u r b e d Zone 293 F i g 9.15 R e l a t i v e I n i t i a l Slope of P-Y Curve vs E x t e n t of D i s t u r b a n c e 294 F i g 9.16 R e l a t i v e U l t i m a t e R e s i s t a n c e of P-Y Curve vs S i z e of D i s t u r b a n c e 296 F i g 9.17 R e l a t i v e U l t i m a t e R e s i s t a n c e of P-Y Curve vs E x t e n t of D i s t u r b a n c e 297 x i v ACKNOWLEDGEMENTS I w i s h t o ex p r e s s my g r e a t g r a t i t u d e t o my a d v i s o r , P r o f e s s o r P e t e r M. Byrne f o r h i s s u p p o r t , p a t i e n t and i n v a l u a b l e guidance d u r i n g a l l the s t a g e s of t h i s t h e s i s . I am a l s o deeply i n d e b t e d t o P r o f e s s o r W.D.Liam F i n n f o r h i s c a r e f u l r eviews and c o n s t r u c t i v e s u g g e s t i o n s . I a l s o want t o thank P r o f e s s o r s R.G. Campanella and Y.P. V a i d , and Dr. P.K. Robertson f o r t h e i r h e l p s and u s e f u l d i s c u s s i o n s d u r i n g my s t a y a t UBC. The pr e s s u r e m e t e r t e s t r e s u l t s a t McDonald Farm were g e n e r o u s l y p r o v i d e d by Dr. P.K. R o b e r t s o n . The r e s e a r c h a s s i s t a n t s h i p awarded by the Dept. of C i v i l E n g i n e e r i n g d u r i n g the p e r i o d of 1985-1986 i s g r a t e f u l l y acknowledged. Thanks are a l s o extended t o my c o l l e a g u e s , Upul A t u k o r a l a , F r a n c i s c o S a l g a d o , John A l a n Howei, and B l a i r Gohl who share common i n t e r e s t s i n S o i l M echanics. S p e c i a l thanks go t o my w i f e , Chunyan f o r her s u p p o r t , t y p i n g and e d i t t i n g the f i r s t d r a f t . F i n a l l y , but not l e a s t , I w i s h t o thank P r o f e s s o r Shou-I T s i e n , I n s t i t u t e of Mec h a n i c s , C h i n e s e Academy of S c i e n c e s , f o r i n t r o d u c i n g me t o t h i s f a s c i n a t i n g f i e l d , and the M i n i s t r y of E d u c a t i o n of P e o p l e ' s R e p u b l i c of China f o r awarding me a f e l l o w s h i p f o r s t u d y i n g i n Canada. xv DEDICATION TO MY PARENTS x v i 1. INTRODUCTION 1.1 INTRODUCTION P i l e s have been used as p a r t of s t r u c t u r a l f o u n d a t i o n s f o r many c e n t u r i e s . They might have been known as one of the o l d e s t t y p e of f o u n d a t i o n system t o mankind. W i t h the developements i n human c i v i l i z a t i o n , the need t o c o n s t r u c t v a r i o u s t y p e s of massive s t r u c t u r e s , such as h i g h r i s e complex b u i l d i n g , p o r t and h a r b o u r , o f f s h o r e p l a t f o r m , a t more and more u n f a v o r e d s i t e s has i n c r e a s e d the usages of p i l e d f o u n d a t i o n . In o r d e r t o d e s i g n a s a f e and c o s t - e f f e c t i v e p i l e d f o u n d a t i o n system, a l a r g e amount of r e s e a r c h e f f o r t has been p a i d i n t h i s a s p e c t t o improve our knowledge of p i l e b e h a v i o r under d i f f e r e n t g o v e r n i n g f a c t o r s . G e n e r a l f o r c e s a c t i n g on a p i l e d f o u n d a t i o n u s u a l l y c o n s i s t of a x i a l l o a d s , l a t e r a l l o a d s , and b e n d i n g moments. For many y e a r s , v e r t i c a l p i l e s a r e c o n s i d e r e d o n l y a b l e t o r e s i s t the a x i a l l o a d s t h a t were a p p l i e d i n l i n e w i t h p i l e a x i s . T h e r e f o r e , p i l e s t h a t a r e r e q u i r e d t o s u p p o r t l a t e r a l l o a d s , were i n s t a l l e d a t a b a t t e r . However, i t i s now r e a l i z e d t h a t the l a t e r a l r e s i s t a n c e of a v e r t i c a l p i l e i s c o n s i d e r a b l e . A l t h o u g h i n r e c e n t y e a r s l a r g e numbers of a n a l y s i s methods have been proposed, p e o p l e a r e s t i l l not a b l e t o d e s i g n the l a t e r a l l y l o a d e d p i l e w i t h c o n f i d e n c e and e f f e c t i v e n e s s . P e o p l e c o n t i n u e t o seek f o r t h e s i m p l e and 1 2 r a t i o n a l m e t h o d . A t p r e s e n t , t h e s u b g r a d e r e a c t i o n method w i t h n o n l i n e a r P-Y c u r v e i s c o n s i d e r e d t o be t h e most v e r s a t i l e method o f a n a l y s i n g t h e r e s p o n s e s o f l a t e r a l l y l o a d e d p i l e s . The a c c u r a c y o f t h i s m e t h o d , h o w e v e r , d e p e n d s on t h e r a t i o n a l d e v e l o p m e n t o f t h e P-Y c u r v e s . P-Y c u r v e s o f d i f f e r e n t s h a p e s have b e e n p r o p o s e d t o m o d e l t h e b e h a v i o r o f s o i l s a r o u n d a l a t e r a l l y l o a d e d p i l e , s u c h a s t h o s e recommended by A P I ' ( 1 9 7 6 ) . T h e s e c u r v e s a r e g e n e r a l l y o b t a i n e d f r o m b a c k c a l c u l a t i o n o f f i e l d l o a d t e s t s i n s a n d , s t i f f c l a y and s o f t c l a y . T h e i r a d o p t a b i l i t y f o r u n i v e r s a l u s a g e s t h u s i s s t i l l q u e s t i o n a b l e . New p r o c e d u r e s f o r d e v e l o p i n g P-Y c u r v e s h a v e b e e n p r o p o s e d . The d e v e l o p m e n t o f P-Y c u r v e s f r o m 2 - d i m e n s i o n a l f i n i t e e l e m e n t a n a l y s i s i s now g e n e r a l l y c o n s i d e r e d t o be t h e most r i g o r o u s and r a t i o n a l m e t h o d . T h i s m e t h o d c a n e a s i l y i n c o r p o r a t e t h e n o n l i n e a r s o i l b e h a v i o r a n d d e v e l o p t h e c u r v e s f r o m t h e f u n d a m e n t a l s o i l p r o p e r t i e s . H o w e v e r , f o r t h e p r a c t i c a l p u r p o s e s , f a c t o r s s u c h a s p i l e i n s t a l l a t i o n e f f e c t s , s o i l - p i l e i n t e r f a c e s i m u l a t i o n d e s e r v e f u r t h e r c o n s i d e r a t i o n s . D e v e l o p m e n t o f P-Y c u r v e s f r o m t h e p r e s s u r e m e t e r e x p a n s i o n t e s t i s a n o t h e r method t h a t ha s a p p e a r e d r e c e n t l y . T h i s me thod i s s i m p l e and r a t i o n a l , i n t h e l i g h t o f t h e s i m i l a r i t y b e t w e e n t h e l o a d i n g p a t t e r n o f p r e s s u r e m e t e r and l a t e r a l l y l o a d e d p i l e s . I t s p r a c t i c a l a t t r a c t i o n ha s p r o v o k e d t h e e f f o r t o f t h e o r e t i c a l s t u d i e s , w h i c h i n c l u d e 3 the e x a m i n a t i o n of the b a s i c assumption i n i n t e r p r e t i n g t e s t d a t a , and the p o s s i b i l i t y of d e v e l o p i n g P-Y c u r v e s from p r e s s u r e m e t e r c u r v e s . 1.2 SCOPE OF THESIS Main purposes of t h i s t h e s i s a r e t h r e e f o l d : 1. t o examine a n a l y t i c a l l y the v a l i d i t y of a x i s y m m e t r i c p l a n e s t r a i n a ssumption f o r the p r e s s u r e m e t e r t e s t s , and stu d y the p r e s s u r e m e t e r membrane l e n g t h t o d i a m e t e r r a t i o (L/D) e f f e c t on the p r e s s u r e m e t e r t e s t r e s u l t s , 2. t o examine a n a l y t i c a l l y the f a c t o r s a f f e c t i n g the development of P-Y c u r v e s from 2D p l a n e s t r a i n a n a l y s i s , such as the s o i l - p i l e i n t e r f a c e b e h a v i o r . 3. t o examine a n a l y t i c a l l y t h e p o s s i b l e d i f f e r e n t i n s t a l l a t i o n e f f e c t s on t h e p r e s s u r e m e t e r e x p a n s i o n c u r v e s and t h e l a t e r a l l o a d - p i l e d e f l e c t i o n c u r v e s ( i . e . P-Y c u r v e s ) . Thus, the t h e s i s m a i n l y c o n s i s t s of t h r e e p a r t s : c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y , p r e s s u r e m e t e r r e s p o n s e s , and l a t e r a l l y l o a d e d p i l e . The f i n i t e element program CONOIL ( V a z i r i , 1986) was used t h r o u g h o u t the t h e s i s . The program was f i r s t v e r i f i e d by e x a m i n i n g the c y l i n d r i c a l c a v i t y e x p a n s i o n problem. R e s u l t s f o r both e l a s t o - p e r f e c t l y p l a s t i c c l a y and e l a s t o - p l a s t i c f r i c t i o n a l sand were compared w i t h c l o s e d form s o l u t i o n s . Thus, the r e l i a b i l i t y of the program was e n s u r e d . 4 In the second p a r t , the v a l i d i t y of a x i s y m m e t r i c - p l a n e s t r a i n a s s u m p t i o n f o r p r e s s u r e m e t e r t e s t s was examined t h e o r e t i c a l l y by m o d e l l i n g p r e s s u r e m e t e r response i n a 3D a x i s y m m e t r i c a l domain under d i f f e r e n t L/D r a t i o and a 2D a x i s y m m e t r i - p l a n e s t r a i n domain r e s p e c t i v e l y . P r e s s u r e m e t e r t e s t s a t two s i t e s ( c l a y s and sands) were a l s o a n a l y s e d u s i n g p l a n e s t r a i n f i n i t e element s i m u l a t i o n of c a v i t y e x p a n s i o n . The r e s u l t s were compared w i t h f i e l d d a t a . In the f i n a l p a r t of the t h e s i s , development of P-Y c u r v e s from f i n i t e element p l a n e s t r a i n f o r m u l a t i o n was per f o r m e d and v a l i d a t e d w i t h a v a i l a b l e c l o s e d form s o l u t i o n . The i m p o r t a n c e of p r o p e r l y s i m u l a t i n g s o i l - p i l e i n t e r f a c e p r o p e r t i e s i s i l l u s t r a t e d from the f i n i t e element r e s u l t s . F u r t h e r m o r e , the i n s t a l l a t i o n e f f e c t s of p r e s s u r e m e t e r and p i l e on t h e i r p r e s s u r e - e x p a n s i o n and P-Y c u r v e s were e v a l u a t e d by p e r f o r m i n g a p a r a m e t r i c s t u d y , and the i n s t a l l a t i o n e f f e c t s were d i s c u s s e d f o r the development of P-Y c u r v e s from p r e s s u r e - e x p a n s i o n c u r v e s . 1.3 ORGANIZATION OF THESIS The t h e s i s c o n s i s t s of t e n c h a p t e r s p r e s e n t i n g t h r e e main t o p i c s : c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y , p r e s s u r e m e t e r t e s t , and l a t e r a l l y l o a d e d s i n g l e p i l e . C h a p t e r 2 b r i e f l y r e v i e w s the p r e v i o u s work on the a n a l y s i s of l a t e r a l l y l o a d e d p i l e s . Emphasis i s g i v e n on the e x a m i n a t i o n of t h e i r advantages and s h o r t c o m i n g s . A b r i e f i n t r o d u c t i o n of the program CONOIL i s g i v e n i n Chapter 3. 5 Chapter 4 p r e s e n t s the s t r e s s - s t r a i n r e l a t i o n of s o i l used i n t he t h e s i s . Chapter 5 examines CONOIL i n c a v i t y e x p a n s i o n c o n d i t i o n and some n u m e r i c a l f e a t u r e s which were m o d i f i e d and added i n the program. Comparison of the r e s u l t s w i t h c l o s e d form s o l u t i o n s i l l u s t r a t e s t h e c a p a b i l i t y of the m o d i f i e d program i n s o l v i n g t h e s e p r o b l e m s . Chapter 6 p r e s e n t s the s t u d i e s of p r e s s u r e m e t e r L/D r a t i o e f f e c t on the t e s t r e s u l t s , and the f i n i t e element p r e d i c t i o n s of f i e l d t e s t d a t a . Chapter 7 d i s c u s s e s the i n t e r f a c e element used i n m o d e l l i n g s o i l - p i l e i n t e r f a c e b e h a v i o r . Chapter 8 s t u d i e s the development of P-Y c u r v e s from p l a n e s t r a i n model. Emphasis i s on the p r o p e r s i m u l a t i o n of s o i l - p i l e i n t e r f a c e b e h a v i o r . Chapter 9 examines the i n f l u e n c e of p i l e and/or p r e s s u r e m e t e r i n s t a l l a t i o n s on the P-Y c u r v e s and p r e s s u r e - e x p a n s i o n c u r v e s r e s p e c t i v e l y , and i t s s i g n i f i c a n c e i n c o n v e r t i n g the p r e s s u r e m e t e r c u r v e s t o the P-Y c u r v e s . A g e n e r a l summary of t h i s r e s e a r c h and major c o n c l u s i o n s a r e p r e s e n t e d i n Ch a p t e r 10. P o s s i b l e f u t u r e r e s e a r c h e s a r e a l s o s u g g e s t e d i n t h i s c h a p t e r . 2. REVIEWS OF PREVIOUS WORK 2.1 INTRODUCTION A complete a n a l y s i s of l a t e r a l l y l o a d e d p i l e s i s v e r y d i f f i c u l t and c o m p l i c a t e d , r e q u i r i n g a 3D a n a l y s i s w i t h the n o n l i n e a r s t r e s s - s t r a i n r e l a t i o n of the s o i l . For the p r a c t i c a l a p p l i c a t i o n , s i m p l i f i c a t i o n s of s o i l b e h a v i o u r a r e a l w a y s made i n an a n a l y s i s . In t h i s C h a p t e r , e f f o r t s a r e made t o g i v e a g e n e r a l r e v i e w of the p r e v i o u s work i n the a n a l y s e s of l a t e r a l l y l o a d e d p i l e s . C o n c e n t r a t i o n i s on t h e e x a m i n a t i o n of advantages and l i m i t a t i o n s w i t h each approach. Reviews on t h e d i f f e r e n t methods of a n a l y s i n g the problem a r e g i v e n i n S e c t i o n 2.2 w i t h emphasis on the n o n l i n e a r subgrade r e a c t i o n a p p r o a c h , and the c u r r e n t methods of s p e c i f y i n g the n o n l i n e a r s o i l r e a c t i o n ( o r P-Y c u r v e s ) c u r v e s a r e r e v i e w e d i n S e c t i o n 2.3. 2.2 METHODS OF ANALYSES A l a r g e number of methods f o r the c a l c u l a t i o n of the l a t e r a l d e f l e c t i o n s and of the d i s t r i b u t i o n of bending moments a t wo r k i n g l o a d s and of t h e u l t i m a t e l a t e r a l r e s i s t a n c e of s i n g l e p i l e s and p i l e groups have been p r o p o s e d . To date t h e s e methods can be b r o a d l y c l a s s i f i e d i n t o f o u r groups ( B a n e r j e e and D r i s c o l l , 1976, A t u k o r a l a and B y r n e , 1984) depending on t h e s i m p l i c i t y of models adopted f o r the s o i l r e s p o n s e s , i . e . the modulus of subgrade 6 7 r e a c t i o n m e t h o d , t h e e l a s t i c c o n t i n u u m a p p r o a c h , t h e f i n i t e e l e m e n t f o r m u l a t i o n , and t h e b o u n d a r y e l e m e n t a p p r o a c h . The e l a s t i c c o n t i n u u m a p p r o a c h d e v e l o p e d by P o u l o s (1971) e v a l u a t e s t h e i n t e r a c t i o n b e h a v i o r be tween p i l e s a n d s o i l s i n a f o r m o f t h e i n t e r a c t i o n f a c t o r w h i c h i s c a l c u l a t e d b a s e d on t h e M i n d l i n ' s s o l u t i o n f o r a p o i n t l o a d i n t h e i n t e r i o r o f s e m i - i n f i n i t e , e l a s t i c c o n t i n u u m ( M i n d l i n , 1 9 3 6 ) . The m e t h o d a s s u m e s t h a t t h e s o i l b e h a v e s a s l i n e a r e l a s t i c i s o t r o p i c and homogeneous m a t e r i a l and t h e law o f s u p e r p o s i t i o n h o l d s . C o n s e q u e n t l y t h e method c a n o n l y be u s e d i n t h e c a s e o f s m a l l g r o u p o f p i l e s , a n d i s n o t a p p l i c a b l e t o a n o n l i n e a r a n a l y s i s o f p i l e f o u n d a t i o n . P a s t e x p e r i e n c e s have shown t h a t s i n c e i t i g n o r e s t h e n o n l i n e a r i t y o f s o i l - p i l e s i n t e r a c t i o n , t h e method t e n d s t o o v e r e s t i m a t e t h e i n t e r a c t i o n b e t w e e n s o i l and p i l e s (Novak , 1 9 7 9 ) . Many a t t e m p t s h a v e b e e n made t o u se t h e f i n i t e e l e m e n t m e t h o d i n t h e a n a l y s i s o f l a t e r a l r e s i s t a n c e o f a p i l e f o u n d a t i o n ( B a g u e l i n e t a l , 1 9 7 7 , 1980, D e s a i and A b e l , 1976, K u h l e m e y e r , 1979, Y e g i a n a n d W r i g h t , 1973, A t u k o r a l a a n d B y r n e , 1 9 8 4 ) . In t h e f i n i t e e l e m e n t f o r m u l a t i o n , t h e s u r r o u n d i n g s o i l i s d i s c r e t e d i n t o f i n i t e e l e m e n t s i n w h i c h t h e c o m p l e x s t r e s s - s t r a i n - s t r e n g t h c h a r a c t e r i s t i c s o f s o i l s a n d t h e c o m p l i c a t e d g e o m e t r y o f t h e p r o b l e m c a n be p r o p e r l y s i m u l a t e d . A l t h o u g h r e c e n t d e v e l o p m e n t o f t h e a b i l i t y o f l a r g e c o m p u t e r s and t h e i r s u c c e s s f u l a p p l i c a t i o n i n g e o t e c h n i c a l e n g i n e e r i n g e n a b l e t o p e r f o r m a n o n - l i n e a r 8 a n a l y s i s o f t h r e e d i m e n s i o n a l p r o b l e m s , w h i c h seems t o be t h e o n l y way t o r i g o r o u s l y s i m u l a t e t h e i n t e r a c t i o n o f t h e w h o l e s o i l - p i l e s y s t e m , c o s t and work r e q u i r e d f o r t h i s t y p e o f a n a l y s e s on a r e a l p r o b l e m i s s t i l l e n o r m o u s . The b o u n d a r y e l e m e n t m e t h o d i s a n o t h e r n u m e r i c a l m e t h o d . I t ha s been s u c c e s s f u l l y a p p l i e d t o some s o i l - p i l e i n t e r a c t i o n p r o b l e m s . The m e t h o d d i s c r e t i z e s t h e b o u n d a r y s u r f a c e o f s o i l - p i l e i n t e r f a c e a n d i n t e g r a t e s an a p p r o p r i a t e e l e m e n t a r y p o i n t f o r c e s o l u t i o n f o r t h e s o i l medium ( e . g . M i n d l i n ' s s o l u t i o n f o r homogeneous s o i l s ) o v e r t h e d i s c r e t i z e d s u r f a c e e l e m e n t . The method ha s b e e n u s e d n o t o n l y i n homogeneous s o i l s ( B a n e r j e e e t a l , 1976) b u t a l s o i n nonhomogeneous s o i l s ( B a n e r j e e e t a l , 1978) o f t h e l a t e r a l l y l o a d e d p i l e s . In most c a s e s i t was shown t h a t t h e b o u n d a r y e l e m e n t a n a l y s i s i s l e s s e x p e n s i v e t h a n t h e f i n i t e e l e m e n t a n a l y s i s . However t h e c o s t r e m a i n s v a s t c o m p a r e d t o t h a t r e q u i r e d by r o u t i n e d e s i g n w o r k , a n d t h e d i f f i c u l t i e s r e m a i n i n c h o o s i n g an a p p r o x i m a t e p o i n t f o r c e s o l u t i o n f o r t h e nonhomogeneous s o i l and i n h a n d l i n g t h e n o n l i n e a r s t r e s s - s t r a i n - s t r e n g t h r e l a t i o n o f t h e s o i l , so t h a t t h e m e t h o d i s g e n e r a l l y r e g a r d e d a s a n o t h e r r e s e a r c h t o o l . The m o d u l u s o f s u b g r a d e r e a c t i o n m e t h o d , w h i c h may be one o f t h e o l d e s t m e t h o d s u s e d t o a n a l y s e t h e s o i l - s t r u c t u r e i n t e r a c t i o n p r o b l e m , t r e a t s t h e p i l e a s a l i n e a r l y e l a s t i c beam and d i s c r e t e s t h e s u r r o u n d i n g s o i l s i n t o a bed o f u n c o u p l e d W i n k l e r s p r i n g s . T h o s e u n c o u p l e d W i n k l e r s p r i n g s a s shown i n F i g . 2.1 r e p r e s e n t t h e s o i l l o a d - d e f l e c t i o n M pile segments H -OHM-independent springs F i g . 2.1 W i n k l e r S p r i n g Approach 10 p r o p e r t i e s u n d e r t h e l a t e r a l l o a d i n g s . The g o v e r n i n g e q u a t i o n f o r t h i s t y p e o f s o i l - p i l e s y s t e m i s d e r i v e d b a s e d on t h e c l a s s i c a l H e t e n y i ' s s o l u t i o n f o r a b e a m - c o l u m n on an e l a s t i c f o u n d a t i o n , where s o i l r e a c t i o n i s t a k e n a s a l i n e l o a d ( H e t e n y i , 1 9 4 6 ) . The g o v e r n i n g e q u a t i o n i s i n t h e f o r m : d ' y d 2 y EI T - t + Px — f - p = 0 ; ( 2 . 2 . 1 ) d x q d x 2 w h e r e : Px = a x i a l l o a d on p i l e s ; y = l a t e r a l d e f l e c t i o n o f t h e p i l e a t p o i n t x a l o n g t h e p i l e l e n g t h ; p = s o i l r e a c t i o n p e r u n i t l e n g t h ; a n d EI i s t h e f l e x u r a l r i g i d i t y o f t h e p i l e . In t h e a b o v e a n a l y t i c a l m o d e l , t h e W i n k l e r ' s s p r i n g s r e p r e s e n t i n g t h e s o i l r e a c t i o n , p , c a n be e i t h e r l i n e a r o r more r e a s o n a b l y n o n l i n e a r s p r i n g s u s u a l l y s p e c i f i e d a s P -Y c u r v e s a t p o i n t s a l o n g t h e p i l e l e n g t h ( S c o t t , 1 9 8 1 ) . A t y p i c a l s h a p e o f P-Y c u r v e i s shown i n F i g . 2 . 2 . The P-Y c u r v e , i n w h i c h P i s t h e s o i l r e a c t i o n i n f o r c e p e r u n i t l e n g t h a n d Y i s t h e p i l e l a t e r a l d e f l e c t i o n , e x p r e s s e s s o i l r e a c t i o n a s a f u n c t i o n o f p i l e d e f l e c t i o n and d e p t h v i a t h e s e c a n t s u b g r a d e m o d u l u s o f t h e s o i l , E s , ( R e e s e , 1 9 6 2 ) , i . e . E s = - P / y ( 2 . 2 . 2 ) a s i n d i c a t e d i n F i g . 2 . 2 . The s e c a n t s u b g r a d e m o d u l u s o f s o i l s , E s , w h i c h c a n v a r y i n an a r b i t r a r y manner w i t h d e p t h a n d d e f l e c t i o n , i s n o t a u n i q u e p a r a m e t e r o f t h e s o i l b u t a PILE DEFLECTION, y . 2.2 A T y p i c a l Shape of P-Y c u r v e 1 2 computation d e v i c e to f a c i l i t a t e the s o l u t i o n t h a t i n c o r p o r a t e s the n o n l i n e a r i t y and nonhomogeneity of the s o i l . Then a complete s o l u t i o n i n c l u d i n g bending mement d i s t r i b u t i o n , d e f l e c t i o n d i s t r i b u t i o n and shear s t r e s s e s along the p i l e l e n g t h can be r e a d i l y o b t a i n e d by s o l v i n g a set of d i f f e r e n t i a l equat ions of Eq. (2.2.1) with the s p e c i f i e d P-Y curves at p o i n t s along the p i l e l e n g t h and the ap p r o p r i a t e boundary c o n d i t i o n s . T h i s procedure has been i n c o r p o r a t e d i n computer program so that the v a r i a t i o n of s o i l p r o p e r t i e s along the depth can be e a s i l y acounted f o r in an a n a l y s i s . In such an approach, the s t r e s s t r a n s f e r between the adjacent s p r i n g s i s ignored, which i m p l i e s t h a t s o i l element at a p a r t i c u l a r p o i n t i s not a f f e c t e d by the s o i l elements e i t h e r above i t or below i t . S t r i c t l y speaking, t h i s assumption i s not c o r r e c t s i n c e i t ignores the c o n t i n u i t y of the surrounding s o i l s . T h e r e f o r e , the method may not be r e a d i l y a p p l i c a b l e to the a n a l y s i s of p i l e group a c t i o n . In order to e v a l u a t e the s i g n i f i c a n c e of i g n o r i n g the s t r e s s t r a n s f e r i n the subgrade r e a c t i o n method, Poulos (1971) compared the s o l u t i o n s f o r the s i n g l e p i l e s from the e l a s t i c continuum theory with those from the subgrade r e a c t i o n theory. The comparison showed that the method of subgrade r e a c t i o n does not gi v e s a t i s f a c t o r y r e s u l t s with regards to the magnitudes of displacement, r o t a t i o n s and bending moments, because i t overestimates these q u a n t i t i e s as compared with the e l a s t i c continuum theory, e s p e c i a l l y 1 3 f o r r e l a t i v e f l e x i b l e p i l e s . A n o t h e r l i m i t a t i o n w i t h t h i s method i s t h a t a s t h e method i g n o r e s t h e c o n t i n u i t y of t h e s o i l , t h e s o i l p a r a m e t e r s u s e d i n t h e a n a l y s i s a r e n o t t h e f u n d a m e n t a l s o i l p r o p e r t i e s w i t h t h e h a l f - s p a c e s o i l c o n t i n u u m ( H o r v a t h , 1983). S p e c i a l p a r a m e t e r s have t o be d e v i s e d f o r t h i s a p p r o a c h . However, i n s p i t e o f above l i m i t a t i o n s , n o n l i n e a r W i n k l e r s p r i n g a p p r o a c h i s t h e method most o f t e n u s e d i n t h e d e s i g n o f l a t e r a l l y l o a d e d p i l e s , m a i n l y due t o i t s s i m p l i c i t y i n t h e m a t h e m a t i c a l f o r m u l a t i o n a n d i t s v e r s a t i l i t y i n t h e i n c o r p o r a t i o n o f n o n l i n e a r b e h a v i o u r o f t h e s o i l by a d j u s t i n g t h e s e c a n t s u b g r a d e modulus o f s o i l s , E s , o r P-Y c u r v e s . T h u s , t h e most i m p o r t a n t f a c t o r i n W i n k l e r s p r i n g a p p r o a c h i s t h e a c c u r a t e d e s c r i p t i o n o f s o i l p a r a m e t e r s i n t e r m s of P-Y c u r v e s . 2.3 SPECIFICATION OF P~Y CURVES The c o n c e p t o f P-Y c u r v e s was f i r s t p r o p o s e d by M c C l e l l a n d a n d F o c h t ( 1 9 5 8 ) . S i n c e t h e n t h e r a t i o n a l d e v e l o p m e n t o f P-Y c u r v e s has become an i n t e n s i v e r e s e a r c h a r e a owing t o t h e s u c c e s s f u l a p p l i c a t i o n o f t h e s u b g r a d e r e a c t i o n method i n p r a c t i c e . To d a t e , t h e d i f f e r e n t methods t o o b t a i n t h e P-Y c u r v e s may be g e n e r a l l y g r o u p e d a s f o l l o w s ( A t u k o r a l a a n d B y r n e , 1984): a) s e m i - e m p i r i c a l method; 1 4 b) i n - s i t u t e s t i n g method; c) c e n t r i f u g e t e s t ; and d) f i n i t e element method. 2.3.1 SEMI-EMPIRICAL METHODS In the group of s e m i - e m p i r i c a l methods, the procedures proposed by Matlock (1970) and Reese et a l (1974) to c o n s t r u c t the P-Y curves are most widely used and have been adopted by the American Petroleum I n s t i t u t e (1976). T h e i r methods are based upon back c a l c u l a t i o n of f i e l d l a t e r a l l y loaded p i l e t e s t s and the s t r e s s - s t r a i n c u r ves o b t a i n e d from l a b o r a t o r y compression t e s t , the P-Y c u r v e s are p r o v i d e d a c c o r d i n g to a g e n e r a l d e s c r i p t i o n of s o i l c o n d i t i o n s , i . e . s o f t c l a y , s t i f f c l a y , and sand. For s o f t c l a y , the method proposed by Matlock was developed based on a n a l y s i s of data from l a t e r a l l o a d t e s t s from two p a r t i c u l a r s o f t c l a y s i t e s and from some model t e s t s (Matlock, 1970). The method employs an e x p r e s s i o n to d e s c r i b e the v a r i a t i o n of the u l t i m a t e s o i l r e s i s t a n c e per u n i t l e n g t h , Pu, with depth: Pu = Np Cu D (2.3.1) where: Pu = u l t i m a t e s o i l r e s i s t a n c e per u n i t l e n g t h , Np = a nondimensional u l t i m a t e l a t e r a l s o i l r e s i s t a n c e c o e f f i c i e n t , Cu = undrained s o i l shear s t r e n g t h , and D = p i l e diameter. 15 T h e v a l u e o f t h e u l t i m a t e s o i l r e s i s t a n c e c o e f f i c i e n t , Np , i s a f u n c t i o n o f d e p t h be l ow t h e g r o u n d s u r f a c e . I t r e a c h e s a l i m i t i n g v a l u e o f 9 a t l a r g e d e p t h , w h i c h c o r r e s p o n d s t o t h e a s s u m p t i o n o f h o r i z o n t a l p l a s t i c f l o w f a i l u r e o f s o i l s a r o u n d t h e p i l e . W h i l e n e a r t h e p i l e h e a d , t h e s o i l i n f r o n t o f t h e p i l e i s a s sumed t o be f a i l e d by s h e a r i n g f o r w a r d and upward i n f o r m o f wedge , t h e c o r r e s p o n d i n g v a l u e o f Np r e d u c e s t o t h e r a n g e o f 2 t o 4 d e p e n d i n g upon w h e t h e r t h e p i l e segment i s c o n s i d e r e d a s a p l a t e w i t h o n l y f r o n t a l s o i l r e s i s t a n c e o r w h e t h e r i t i s a s q u a r e c r o s s s e c t i o n w i t h s o i l s h e a r i n g a l o n g t h e s i d e s ( R e e s e , 1 9 5 8 ) . A v a l u e o f 3 was recommended by M a t l o c k (1970) f o r a c y l i n d r i c a l p i l e . T h u s , s u c h a v a r i a t i o n p r e s c r i b e d f o r Np i s : f o r X < X c r Np = 3 + a / C u + J x /D < 9 ( 2 . 3 . 2 ) f o r X > X c r Np = 9 ( 2 . 3 . 3 ) w h e r e : a = e f f e c t i v e o v e r b u r d e n s t r e s s a t d e p t h X , J = an e m p i r i c a l c o n s t a n t w i t h an a p p r o x i m a t e v a l u e o f 0 .5 f o r s o f t o f f s h o r e c l a y s a n d 0 .25 f o r s t i f f e r c l a y s , X = d e p t h be l ow g r o u n d s u r f a c e , a n d X c r = c r i t i c a l d e p t h a t w h i c h s u r f a c e f a i l u r e t r a n s f o r m s t o c o n f i n e d p l a s t i c p l a n e s t r a i n f l o w f a i l u r e . 1 6 M a t l o c k f o u n d t h a t he was b e s t a b l e t o m a t c h t h e e x p e r i m e n t a l P-Y c u r v e s by u s i n g a c u b i c p a r a b o l i c f u n c t i o n , a s shown i n F i g . 2.3 : p = 0 .5 Pu ( Y / Y c ) ° ' 3 3 ( 2 . 3 . 4 ) w h e r e : Pu = u l t i m a t e s o i l r e s i s t a n c e , y = p i l e d e f e c t i o n , a n d y c = r e f e r e n c e d e f l e c t i o n o f t h e p i l e . And t h e r e f e r e n c e d e f l e c t i o n o f t h e p i l e , y , i s d e t e r m i n e d f r o m l a b o r a t o r y t e s t s v i a : y „ = 2.5 e D ( 2 . 3 . 5 ) c c w h e r e : e c = m a j o r p r i n c i p a l s t r a i n a t one h a l f t h e maximum d e v i a t o r s t r e s s i n a UU t r i a x i a l c o m p r e s s i o n t e s t . The b a s i s f o r t h e a b o v e e m p l o y m e n t o f « c i n t h e p a r a b o l i c f u n c t i o n i s t h e a s s u m p t i o n t h a t t h e P-Y c u r v e s a r e s i m i l a r i n s h a p e t o t h e l a b o r a t o r y s t r e s s - s t r a i n c u r v e s o f t h e s o i l s . T h e r e f o r e , t h e v a l u e o f e c p l a y s an i m p o r t a n t p a r t i n t h e c o n s t r u c t i o n o f P-Y c u r v e s f o r c l a y . B e c a u s e e i s d e t e r m i n e d f r o m UU t r i a x i a l t e s t s , t h i s c v a l u e o f s t r a i n may n o t t r u e l y r e p r e s e n t c o n d i t i o n s i n g r o u n d . In t h e a b s e n c e o f t r i a x i a l t e s t d a t a , M a t l o c k (1970) s u g g e s t e d t h a t v a l u e s o f e c p r o p o s e d by Skempton (1951) c o u l d be u s e d . H o w e v e r , J i a m i o l k o w s k i a n d G a r a s s i n o (1977) commented t h a t more p r e c i s e t e s t s a r e r e q u i r e d t o o b t a i n e . T h e y F i g . 2.3 Construction of P-Y curves for Soft Clay ( a f t e r M a t l o c k , 1970) 18 s u g g e s t e d t o u se C K 0 U 1 t r i a x i a l c o m p r e s s i o n t e s t s w i t h t h e most a p p r o p r i a t e s t r e s s p a t h t o be f o l l o w e d d u r i n g c o n s o l i d a t i o n a n d s h e a r i n g . F o r s t i f f a n d f i s s u r e d c l a y , t h e c o n s t r u c t i o n m e t h o d s u g g e s t e d by R e e s e e t a l ( 1 9 7 5 ) , R e e s e a n d W e l c h (1975) i s b a s e d on a s e r i e s o f l a t e r a l l o a d t e s t s a t a s i t e c o n s i s t i n g o f h e a v i l y o v e r c o n s o l i d a t e d , j o i n t e d c l a y . The m e t h o d f o l l o w s t h e same g e n e r a l l o g i c a s i n t h e M a t l o c k ' s m e t h o d f o r s o f t c l a y b u t w i t h s e v e r a l n o t a b l e d i f f e r e n c e s i n t h e f o r m u l a u s e d f o r t h e maximum s o i l r e s i s t a n c e a n d t h e m a t h e m a t i c a l f u n c t i o n f o r t h e s h a p e o f P-Y c u r v e s . The f o r m u l a e m p l o y e d f o r t h e c a l c u l a t i o n o f maximum s o i l r e s i s t a n c e by R e e s e e t a l ( 1975) i s : Pu = [2 Cu D + a + 2 .83 Cu x ] < 11 Cu D , ( 2 . 3 . 6 ) A g r e e m e n t o f t h e a b o v e e q u a t i o n w i t h t h e f i e l d m e a s u r e m e n t s was v e r y p o o r . An e m p i r i c a l c o e f f i c e n t , A i s t h e n i n t r o d u c e d i n t o E q . ( 2 . 3 . 6 ) . V a r i a t i o n o f A w i t h d e p t h i s shown i n F i g . 2 . 4 . The m a t h e m a t i c a l f u n c t i o n u s e d by R e e s e a n d W e l c h (1975) t o d e f i n e t h e P-Y c u r v e f o r s t i f f c l a y i s a l m o s t t h e same a s i n t h e M a t l o c k ' s m e t h o d f o r s o f t c l a y b u t a p a r a b o l a o f f o u r t h o r d e r . In p r a c t i c e , t h e a b o v e p a r a b o l i c P-Y c u r v e s may r a i s e n u m e r i c a l p r o b l e m s a s t h e i n i t i a l m o d u l u s o f t h e p a r a b o l i c 1 C K 0 U : K 0 c o n s o l i d a t e d , u n d r a i n e d t r i a x i a l c o m p r e s s i o n t e s t s . F i g . 2.4 C o e f f i c i e n t A f o r S t i f f and F i s s u r e d C l a y ( a f t e r Reese e t a l , 1975) 0 Ay c y e 6 Ay c !8Ay c DEFLECTION, > . in. F i g . 2.5 C o n s t r u c t i o n o f P-Y c u r v e f o r S t i f f C l a y ( a f t e r R e e s e e t a l , 1975) 20 c u r v e t e n d s t o i n f i n i t y . Due t o t h i s r e a s o n , R e e s e e t a l (1975) c o n s t r u c t t h e P-Y c u r v e f o r s t i f f a n d f i s s u r e d c l a y u s i n g a c o m p o s i t e c u r v e c o m p o s e d o f s t r a i g h t l i n e s and p a r a b o l i c s e g m e n t s . The i n i t i a l p o r t i o n o f t h e c u r v e i s s t r a i g h t l i n e d e f i n e d b y : E . = k -x ( 2 . 3 . 7 ) s i s w h e r e : E = i n i t i a l s u b g r a d e m o d u l u s , x = d e p t h , a n d k g = c o e f f i c i e n t o f s u b g r a d e r e a c t i o n d e s c r i b i n g t h e g r a d i e n t o f t h e i n i t i a l s u b g r a d e m o d u l u s w i t h d e p t h . The v a l u e s o f k g f o r c l a y a r e u s u a l l y r e l a t e d t o t h e u n d r a i n e d s h e a r s t r e n g t h ( R e e s e e t a l , 1 9 7 5 ) . The s h a p e s o f t h e P-Y c u r v e s f o r t h e s t i f f c l a y a r e shown i n F i g . 2.5 f o r s t a t i c l o a d i n g s . As i n d i c a t e d i n t h e f i g u r e , t h e e x t r e m e d e g r e d a t i o n o c c u r s i n t h e c u r v e s , w h i c h may r e p r e s e n t t h e s t r a i n s o f t e n i n g phenomenon i n t h e s t i f f , f i s s u r e d c l a y s . A l t h o u g h t h i s a p p r o a c h w i t h an i n i t i a l l i n e a r P-Y r e l a t i o n c a n o v e r c o m e t h e p r o b l e m m e n t i o n e d a b o v e , t h e p r o b l e m t o o b t a i n a r e l i a b l e e s t i m a t e o f v a l u e , e„ s t i l l e x i s t s , c The s e m i - e m p i r i c a l p r o c e d u r e s f o r c o n s t r u c t i o n o f P-Y c u r v e s i n c o h e s i o n l e s s s o i l s were d e v e l o p e d by R e e s e e t a l (1974) t o f i t d a t a f r o m a p a r t i c u l a r l a t e r a l l o a d t e s t a t M u s t a n g I s l a n d , T e x a s (Cox e t a l , 1 9 7 4 ) . P-Y c u r v e s a r e c o n s t r u c t e d f o r e a c h d e s i r e d d e p t h . E a c h c u r v e c o n s i s t s o f t h r e e s t r a i g h t l i n e s and a p a r a b o l a , a s shown i n F i g . 2 . 6 . F i g . 2.6 C o n s t r u c t i o n of P-Y curve f o r Sand ( a f t e r Reese et a l , 1974) 22 The i n i t i a l p o r t i o n i s a s t r a i g h t l i n e r e p r e s e n t i n g t h e " e l a s t i c " b e h a v i o u r o f t h e s a n d , a n d t h e f i n a l s t r a i g h t p o r t i o n i s h o r i z o n t a l , r e p r e s e n t s t h e " p l a s t i c " b e h a v i o u r o f t h e s a n d . T h e s e two s t r a i g h t l i n e s a r e c o n n e c t e d w i t h a p a r a b o l a a n d a s l o p i n g s t r a i g h t l i n e . The p a r a b o l a a n d t h e i n t e r m e d i a t e s t r a i g h t l i n e were s e l e c t e d e m p i r i c a l l y t o f i t t h e s h a p e o f e x p e r i m e n t a l P-Y c u r v e s w h i c h were d e t e r m i n e d f r o m t h e b e n d i n g moment m e a s u r e m e n t s i n f i e l d . ' The s l o p e o f t h e i n i t i a l s t r a i g h t l i n e i s d e f i n e d by t h e i n i t i a l s o i l m o d u l u s , E . , a s i n t h e same f o r m a s i n s i E q . ( 2 . 3 . 7 ) . Bu t h e r e t h e v a l u e s o f E . a r e r e l a t e d t o t h e s i r e l a t i v e d e n s i t y o f t h e s a n d . The u l t i m a t e s o i l r e s i s t a n c e Pu f o r s t a t i c l o a d i n g was c a l c u l a t e d t h e o r e t i c a l l y f o r two c a s e s . ( R e e s e , 1962, a n d R e e s e e t a l , 1 9 7 4 ) . A t t h e s u r f a c e , t h e s o i l i s a s sumed t o f a i l upward s i n a wedge, w h i l e a t d e p t h t h e s o i l i s a s sumed t o f a i l by f l o w i n g a r o u n d t h e p i l e i n a h o r i z o n t a l p l a n e . The a s sumed m e c h a n i s m s o f s o i l f a i l u r e by t h e r e s e a r c h e r s i s shown i n F i g . 2 . 7 ( 1 ) , ( I I ) . The f o r m u l a e d e r i v e d by R e e s e e t a l a r e a s f o l l o w s : 1 ) . Above t h e c r i t i c a l d e p t h (X < X c r ) , Pu = a {D (Kp - Ka) + x tan/3 [Kp t a n a + Kz ( t a n $ ) - t a n a ] } ( 2 . 3 . 8 ) Direction ol Pile Movement Pile of Ft <c) F i g . 2 .7(1) Assumed P a s s i v e Wedge-Type F a i l u r e Near S u r f a c e (a) G e n e r a l Shape of Wedge (b) F o r c e s on Wedge (c) F o r c e s on P i l e - f -Mudhne-7 H Load (b) . 2 .7 (11 ) Assumed Mode of S o i l F a i l u r e by L a t e r a l F low (a) S e c t i o n t h r o u g h t h e P i l e (b) E l e v a t i o n of the P i l e ( a f t e r Reese et a l , 1974) 24 2 ) . Be low t h e c r i t i c a l d e p t h (X > X c r ) , Pu = 7 D z ( K p 3 + 2 Kz K p 2 t a n ^ + 2 Kz tanc? - Ka) ( 2 . 3 . 9 ) w h e r e : Pu = u l t i m a t e s o i l r e s i s t a n c e p e r u n i t l e n g t h , a = e f f e c t i v e o v e r b u r d e n s t r e s s a t d e p t h x , 7 = a v e r a g e e f f e c t i v e u n i t w e i g h t o f s o i l s up t o d e p t h x , D = p i l e d i a m e t e r , Kp = c o e f f i c i e n t o f p a s s i v e e a r t h p r e s s u r e = t a n 2 (45 + ^>), Ka = c o e f f i c i e n t o f a c t i v e e a r t h p r e s s u r e = t a n 2 ( 4 5 - ^ ) , Kz = c o e f f i c i e n t o f l a t e r a l e a r t h p r e s s u r e r e g a r d i n g h o r i z o n t a l e f f e c t i v e s t r e s s e s , g e n e r a l l y t a k e n a s t h e c o e f f i c i e n t o f e a r t h p r e s s u r e a t - r e s t , K 0 , vV3 < a ^ ? ; 0 = 45 + ~$/2 , a n d X c r = t h e c r i t i c a l d e p t h . The c o m p a r i s o n o f t h e a b o v e c o m p u t e d u l t i m a t e s o i l r e s i s t a n c e s w i t h f i e l d d a t a was p o o r . R e e s e e t a l p r o p o s e d t h e e m p i r i c a l c o e f f i c i e n t s , A a n d B t o o b t a i n t h e P a n d P r m u v a l u e s i n t h e P-Y c u r v e s . The v a l u e s o f A and B a r e shown i n F i g . 2 . 8 . A n o t h e r d i f f e r e n t s e m i - e m p i r i c a l a p p r o a c h t o d e v e l o p P-Y c u r v e s f o r c o h e s i o n l e s s s o i l i s t h e m e t h o d p r o p o s e d by S c o t t (1980) b a s e d on t h e c e n t r i f u g e t e s t r e s u l t s on m o d e l p i l e s . The m e t h o d h a s two s i g n i f i c a n t f e a t u r e s d i f f e r e n t f r o m o t h e r s . One o f t h e s e f e a t u r e s i s t h a t t h e P-Y c u r v e s f o r c o h e s i o n l e s s s a n d s i s s i m p l y r e p r e s e n t e d by b i l i n e a r c u r v e s a s shown i n F i g . 2 . 9 . By c o m p a r e d w i t h t h e R e e s e e t a l p r o c e d u r e s f o r c o n s t r u c t i n g P-Y c u r v e s f o r F i g . 2.8 Nondimensional C o e f f i c i e n t A and B ( a f t e r Reese et a l , 1974) p F i g . 2.9 B i l i n e a r P-Y curve for Sand proposed by Scott (a f t e r Murchinson and O ' N e i l l , 1984) to 27 s a n d s , S c o t t c o n c l u d e d t h a t s u c h a s i m p l e b i l i n e a r P-Y c u r v e c a n s e r v e j u s t a s w e l l t o g i v e s a t i s f a c t o r y r e s u l t s . In h i s m e t h o d t h e s l o p e o f t h e i n i t i a l segment i s d e f i n e d c l o s e t o t h e Y o u n g ' s m o d u l u s , E o f s o i l , i . e . E s ^ E . The s e c o n d segment o f t h e P-Y c u r v e i s e m p i r i c a l l y d e f i n e d by a s l o p e o f k x / 4 . As t h e s e c o n d segment h a s a c o n s t a n t - n o n z e r o 5 s l o p e , t h e m e t h o d i m p l i e s t h a t s o i l r e s i s t a n c e i n c r e a s e s l i n e a r l y w i t h t h e l a t e r a l d i s p l a c e m e n t w i t h no u l t i m a t e r e s i s t a n c e v a l u e . The u l t i m a t e s o i l r e s i s t a n c e c o n c e p t i s t h e r e f o r e n o t a p p l i e d i n t h e m e t h o d . As shown i n t h e a b o v e , a l l t h e s e m i - e m p i r i c a l P-Y c u r v e s a r e b a s e d on d a t a f r o m a s e t o f p a r t i c u l a r l a t e r a l l o a d t e s t s a t s p e c i f i c s i t e s o r p a r t i c u l a r m o d e l t e s t s on s i m i l a r s o i l c o n d i t i o n s . The good a g r e e m e n t be tween t h e p r e d i c t e d and m e a s u r e d r e s p o n s e s a s r e p o r t e d by R e e s e a n d M a t l o c k i s t o be e x p e c t e d , a s t h e c u r v e s were d e v e l o p e d f r o m t h e t e s t s a t t h e s e s i t e s . The m e t h o d s d e v e l o p e d a r e t o some e x t e n t s i t e - o r i e n t e d , t h e i r u n i v e r s a l v a l i d i t y i s q u e s t i o n a b l e . B o t h f o r c o h e s i v e a n d c o h e s i o n l e s s s o i l s , l i m i t e d a s s e s s m e n t s on t h e v a l i d i t y o f a p p l y i n g a b o v e methods t o o t h e r l o a d t e s t c a s e s h a v e shown t h a t t h e c o n f i d e n c e i n p r e d i c t i n g d e f l e c t i o n s a n d moments i s u n f o r t u n a t e l y r a t h e r p o o r ( M u r c h i n s o n and O ' N e i l , 1984; G a z i o g l u and O ' N e i l , 1 9 8 4 ) . T h o s e s t u d i e s s u g g e s t e d t h a t f u r t h e r r e s e a r c h i n t o f u n d a m e n t a l m e c h a n i s m s o f l a t e r a l p i l e - s o i l i n t e r a c t i o n i s w a r r a n t e d . 28 2 . 3 . 2 I N - S I T U TEST ING METHODS As s o i l e l e m e n t s i n f r o n t o f p i l e u n d e r g o s i m i l a r l o a d i n g p a t t e r n a s i n p r e s s u r e m e t e r t e s t s , p r e s s u r e m e t e r t e s t s may be a b l e t o p r o v i d e i n - s i t u m e a s u r e m e n t s o f s o i l d e f o r m a t i o n c h a r a c t e r i s t i c s i n t h e d i r e c t i o n r e l a t e d t o t h e l a t e r a l l y l o a d e d p i l e s . G e n e r a l l y , t h e r e a r e two ways t o make u se o f t h e p r e s s u r e m e t e r t e s t r e s u l t s f o r t h e d e s i g n o f l a t e r a l l y l o a d e d p i l e s ( B r i a u d e t a l , 1 9 8 2 ) : 1. t o u s e t h e p r e s s u r e m e t e r t e s t r e s u l t s t o o b t a i n a m o d u l u s o f h o r i z o n t a l s u b g r a d e s o i l r e a c t i o n . 2. t o make u s e o f t h e w h o l e p r e s s u r e - e x p a n s i o n c u r v e t o p r e d i c t t h e P-Y c u r v e s . In t h e f i r s t c h o i c e , b a s e d on t h e r e s u l t s f r o m M e n a r d p r e s s u r e m e t e r t e s t s , M e n a r d a n d Gamb in p r o p o s e d a s e t o f e m p i r i c a l f o r m u l a e t o o b t a i n t h e m o d u l u s o f h o r i z o n t a l s u b g r a d e r e a c t i o n , E s , f r o m p r e s s u r e m e t e r m o d u l u s , Em, o b t a i n e d f r o m M e n a r d p r e s s u r e m e t e r c u r v e s ( G a m b i n , 1 9 7 9 ) . In t h e s e c o n d c h o i c e , t h e P-Y c u r v e s d e s c r i b i n g t h e l a t e r a l s o i l r e a c t i o n a s a f u n c t i o n o f p i l e l a t e r a l d e f l e c t i o n a r e d i r e c t l y c o n s t r u c t e d f r o m t h e s h a p e s o f p r e s s u r e - e x p a n s i o n c u r v e s . A t p r e s e n t , t h e r e a r e two g e n e r a l w a y s : ( a ) . The P-Y c u r v e s a r e c o n s t r u c t e d by s c a l i n g t h e s h a p e s o f t h e e n t i r e p r e s s u r e - e x p a n s i o n w i t h c e r t a i n f a c t o r s ( R o b e r t s o n e t a l , 1983, A t u k o r a l a and B y r n e , 1 9 8 4 ) , 29 ( b ) . A l t e r n a t i v e l y , t h e m e c h a n i s m o f s o i l r e s i s t a n c e t o t h e l a t e r a l movement o f p i l e s i s s e p a r a t e d i n t o two c o m p o n e n t s : t h e f r o n t a l r e a c t i o n ( c a l l e d q - y c u r v e s ) a n d t h e f r i c t i o n a l r e a c t i o n ( f - y c u r v e s ) on t h e s i d e s ( B r i a u d e t a l , 1982, 1 9 8 3 ) . The q - y c u r v e s a r e o b t a i n e d f r o m t h e p r e s s u r e m e t e r c u r v e s , b u t t h e o r y must be u s e d t o o b t a i n t h e f - y c u r v e s . T h e n , t h e e n t i r e P-Y c u r v e c a n be c o n s t r u c t e d f r o m t h e f - y a n d q - y c u r v e s . So f a r , i t h a s b e e n shown t h a t b o t h a p p r o a c h e s a r e p r o m i s i n g and o f p r a c t i c a l i n t e r e s t . However none o f t h e a b o v e methods h a v e p r o p e r l y t a k e n a c c o u n t t h e f a c t o r o f d i f f e r e n t i n s t a l l a t i o n e f f e c t s b e t w e e n t h e p r e s s u r e m e t e r and p i l e s . As d i f f e r e n t l o a d i n g m e c h a n i s m s a r e a s s o c i a t e d w i t h t h e p r e s s u r e m e t e r a n d t h e l a t e r a l l y l o a d e d p i l e , s o i l d i s t u r b a n c e s due t o t h e i n s t a l l a t i o n may a f f e c t t h e p r e s s u r e m e t e r c u r v e s a n d t h e P-Y c u r v e s t o a d i f f e r e n t e x t e n t . T h i s d i f f e r e n t e f f e c t may be i m p o r t a n t when t h e p r e s s u r e m e t e r c u r v e s a r e s i m p l y c o n v e r t e d t o o b t a i n t h e P-Y c u r v e s f o r t h e l a t e r a l l y l o a d e d p i l e by some m u t i p l y i n g f a c t o r s . Bu t t h e methods m e n t i o n e d a b o v e i m p l i c i t l y n e g l e c t s u c h c o n s i d e r a t i o n , a s s u m i n g s o i l d i s t u r b a n c e s i n d u c e d by i n s t a l l a t i o n o f t h e p r e s s u r e m e t e r a n d / o r t h e p i l e have t h e same c o n s e q u e n t i a l e f f e c t s on t h e p r e s s u r e m e t e r c u r v e s and t h e P-Y c u r v e s f o r p i l e s . No work h a s b e e n done on t h i s a s p e c t a s y e t t o e v a l u a t e s u c h an a s s u m p t i o n a n d t o i n c o r p o r a t e t h e d i f f e r e n t e f f e c t i n t h e c o n v e r t i n g f a c t o r s , 30 i f n e c e s s a r y . 2.3.3 CENTRIFUGE TESTINGS Due t o the complex n a t u r e of the problem, the u n d e r s t a n d i n g of d e f o r m a t i o n mechanisms a s s o c i a t e d w i t h the l a t e r a l l y l o a d e d p i l e s u s u a l l y r e q u i r e s h i g h - q u a l i t y e x p e r i m e n t a l o b s e r v a t i o n . The s e m i - e m p i r i c a l development of P-Y c u r v e s so f a r , however, a r e m a i n l y based on the l i m i t e d f i e l d l a t e r a l l o a d t e s t d a t a as mentioned i n S e c t i o n 2.3.1. Such a s i t u a t i o n i s p a r t i c u l a r f o r o f f s h o r e p i l i n g , where the d i f f i c u l t i e s and expenses al m o s t p r o h i b i t the p o s s i b i l i t y of c o n d u c t i n g e x p e r i m e n t a l p a r a m e t r i c s t u d i e s on f u l l s i z e p i l e s . T h i s r e q u e s t has prompted the e v o l u t i o n of c e n t r i f u g a l m o d e l l i n g t e c h n i q u e s r e c e n t l y . The c e n t r i f u g a l t e s t i n g has been shown t o be a c o n v e n i e n t and v i a b l e way of making p a r a m e t r i c s t u d i e s of the response of p i l e s t o l a t e r a l l o a d i n g t o e n r i c h our c u r r e n t l i m i t e d d a t a base. However, the g e o t e c h n i c a l c e n t r i f u g e s a r e not p o p u l a r a t p r e s e n t because of the h i g h l y s o p h i s t i c a t e d and e x p e n s i v e equipments. As i n the f i e l d l o a d t e s t s , the development of the P-Y c u r v e s from the c e n t r i f u g a l t e s t s c o n s i s t s of measuring the bending moments, M, a l o n g the model p i l e . Then, the P-Y c u r v e s a t v a r i o u s d e p t h s of s o i l s a r e g e n e r a t e d by p l o t t i n g v a l u e s of P and Y a t each d e p t h f o r i n c r e a s i n g l e v e l s of l a t e r a l l o a d i n g . The v a l u e s of P and Y a r e d e r i v e d by: 31 Y = / / ( M/EI ) dx d 2 M ( 2 . 3 . 1 0 ) ( 2 . 3 . 1 1 ) As i n d e r i v i n g P-Y c u r v e s f r o m t h e f i e l d l o a d t e s t s on an i n s t r u m e n t e d p i l e , t h i s p r o c e d u r e a l s o s u f f e r s f r o m d i f f i c u l t i e s i n t h e d e t e r m i n a t i o n o f s o i l r e a c t i o n , P, f r o m E q . ( 2 . 3 . 1 1 ) i n w h i c h t h e d e r i v a t i v e o f t h e c u r v e u s e d f o r f i t t i n g t h e m e a s u r e d b e n d i n g moment d a t a i s v e r y s e n s i t i v e t o t h e s h a p e o f t h e c u r v e , a n d t h e e r r o r s i n t e s t d a t a w o u l d be m a g n i f i e d d u r i n g t h e d o u b l e d i f f e r e n t i a t i o n . C o n s e q u e n t l y , t h e a c c u r a c i e s o f t h e P-Y c u r v e s d e v e l o p e d f r o m c e n t r i f u g a l t e s t s a r e s o m e t i m e s i n q u e s t i o n . B a r t o n a n d F i n n e t a l ( 1983 ) p e r f o r m e d a s e r i e s o f c e n t r i f u g a l t e s t s on m o d e l p i l e s i n s a n d s s i m i l a r t o t h o s e a t M u s t a n g i s l a n d s i t e (Cox e t a l , 1 9 7 4 ) . The e x p e r i m e n t a l P-Y c u r v e s were c o m p a r e d w i t h t h o s e recommended by R e e s e e t a l ( 1 9 7 4 ) , M a t l o c k e t a l ( 1 9 8 0 ) , and t h o s e d i r e c t l y by t h e f i n i t e e l e m e n t a n a l y s i s u s i n g r e p r e s e n t a t i v e c o n s t i t u t i v e law f o r s a n d . C o m p a r i s o n s w i t h R e e s e e t a l r e c o m m e n d a t i o n showed t h a t t e s t d a t a seems t o q u a l i t a t i v e l y s u p p o r t t h e g e n e r a l s h a p e o f t h e P-Y c u r v e s s u g g e s t e d by R e e s e e t a l , b u t i n d i c a t e d t h a t R e e s e e t a l m e t h o d u n d e r e s t i m a t e s t h e u l t i m a t e s o i l r e s i s t a n c e n e a r t h e p i l e h e a d , and o v e r e s t i m a t e s i t a t d e p t h . I t a l s o o v e r e s t i m a t e s t h e i n i t i a l s t i f f n e s s a n d i t s d i s t r i b u t i o n w i t h d e p t h . C o m p a r i s o n s w i t h M a t l o c k e t a l c r i t e r i a showed t h a t t h e r e i s l i t t l e r e s e m b l e n c e b e t w e e n t h e P-Y c u r v e s f r o m t h e two 32 m e t h o d s , t h e M a t l o c k ' s c u r v e s o v e r e s t i m a t i n g t h e i n i t i a l s t i f f n e s s , u n d e r e s t i m a t i n g t h e u l t i m a t e s o i l r e s i s t a n c e n e a r t h e p i l e h e a d a n d o v e r e s t i m a t i n g i t a t d e p t h s g r e a t e r t h a n a b o u t 4 d i a m e t e r s . C o m p a r i s o n s w i t h t h e f i n i t e e l e m e n t p r e d i c t i o n , h o w e v e r , d i d c o n f i r m t h e f e a s i b i l i t y o f c o m p u t i n g P-Y c u r v e s d i r e c t l y f r o m b a s i c s o i l p a r a m e t e r s . 2 . 3 . 4 F I N I T E ELEMENT METHODS ' In r e c e n t y e a r s , a p p l i c a t i o n o f t h e f i n i t e e l e m e n t method t o t h e n u m e r i c a l d e v e l o p m e n t o f P-Y c u r v e s f o r l a t e r a l l y l o a d e d p i l e s ha s r e c e i v e d more a n d more a t t e n t i o n due t o t h e c o n t i n u o u s n e e d t o d e v e l o p t h e P-Y c u r v e s more r i g o r o u s l y . The f i n i t e e l e m e n t m e t h o d h a s b e e n p r o v e d t o be a p o w e r f u l t o o l f o r a n a l y s i n g t h e s o i l - s t r u c t u r e i n t e r a c t i o n p r o b l e m s i n a more d e t a i l a n d f u n d a m e n t a l l e v e l , b u t h i g h c o s t and t e d i o u s i n p u t d a t a s t i l l p r o h i b i t 3D a n a l y s i s b e i n g r o u t i n e d e s i g n means . H o w e v e r , i t s r e c e n t a p p l i c a t i o n i n d e v e l o p m e n t o f t h e P-Y c u r v e s u s i n g 2D f o r m u l a t i o n h a s i n d i c a t e d t h a t t h e f i n i t e e l e m e n t method may become a d i r e c t d e s i g n t o o l f o r l a t e r a l l y l o a d e d p i l e s . The f i r s t a t t e m p t i n t h i s r e g a r d was a t t r i b u t e d t o Y e g i a n a n d W r i g h t ( 1 9 7 3 ) , where t h e y a n a l y s e d t h e r e s p o n s e o f a s i n g l e p i l e u n d e r t h e s h o r t t e r m s t a t i c l o a d s i n s o f t s a t u r a t e d c l a y . In t h e i r a n a l y s e s , a q u a r t e r o f 2D d o m a i n t h a t c o n t a i n e d a s y m m e t r i c a l b o u n d a r y and an a n t i s y m m e t r i c a l b o u n d a r y c o n d i t i o n s was e m p l o y e d t o s i m u l a t e t h e s o i l - p i l e i n t e r a c t i o n , a s shown i n F i g . 2 . 1 0 , i n w h i c h p l a n e s t r a i n 33 Direction of Pil« Ditploctmtnt F i g . 2.10 F i n i t e E lement Domain employed by Y e g i a n and Wr i gh t (1973) 34 c o n d i t i o n was a s sumed f o r p l a c e s a t d e p t h a n d p l a n e s t r e s s c o n d i t i o n f o r p l a c e s n e a r t h e s u r f a c e . The o u t e r b o u n d a r y o f t h e f i n i t e e l e m e n t d o m a i n u s e d was c o n s i d e r e d t o be f i x e d c y l i n d r i c a l l y u n d e r t h e a s s u m p t i o n t h a t t h e r e i s l i t t l e s o i l d e f o r m a t i o n o c c u r r i n g b e y o n d t h i s b o u n d a r y . By v a r y i n g t h e r a d i u s o f o u t e r b o u n d a r y away f r o m t h e c e n t e r o f t h e p i l e s e c t i o n , t h e y c o n c l u d e d t h a t t h e z o n e o f p i l e i n f l u e n c e i s a b o u t 8 t i m e s t h e p i l e d i a m e t e r s . However t h e i r c o n c l u s i o n was b a s e d on a c o m p a r i s o n o f s o i l r e s i s t a n c e s f r o m t h e f i n i t e e l e m e n t p l a n e s t r e s s a n a l y s e s a n d t h e M a t l o c k ' s m e t h o d . T h e y d i d n o t s t u d y t h e s i z e e f f e c t o f t h e f i x e d o u t e r b o u n d a r y on t h e f i n i t e e l e m e n t p l a n e s t r a i n s o l u t i o n , w h i c h ha s been f o u n d t h e o r e t i c a l l y t o be s i g n i f i c a n t f o r t h e p i l e l a t e r a l s t i f f n e s s ( B a r d e t , 1 9 7 9 ) . In a d d i t i o n , t h e i r f i n i t e e l e m e n t a n a l y s e s i n b o t h p l a n e s t r e s s and p l a n e s t r a i n f o r m u l a t i o n s d i d n o t a l l o w f o r t h e t h e s e p a r a t i o n a t t h e s o i l - p i l e i n t e r f a c e w h i c h , i n r e a l i t y , w o u l d p r o b a b l y o c c u r when t h e t e n s i l e c r a c k i n g o c c u r s i n s o i l s a t t h e b a c k o f p i l e s . In t h e i r a n a l y s i s , a h y p e r b o l i c r e l a t i o n s h i p (Duncan and C h a n g , 1970) was e m p l o y e d t o d e s c r i b e t h e n o n l i n e a r s t r e s s s t r a i n b e h a v i o r f o r t h e s o i l e l e m e n t s a n d t h e c y l i n d r i c a l s l i p e l e m e n t s . A l t h o u g h t h e n o n l i n e a r s t r e s s s t r a i n b e h a v i o u r o f s o i l s was t a k e n i n t o a c c o u n t , t h e i r r e s u l t s were m a i n l y a p p r o p r i a t e f o r t h e p i l e r e s p o n s e s a t l i n e a r e l a s t i c s t a g e s , a s t h e y e m p l o y e d t h e a n t i s y m m e t r i c a l b o u n d a r y c o n d i t i o n t h a t w i l l n o t e x i s t f o r t h e n o n l i n e a r 35 s o i l r e s p o n s e s . F rom t h e c o m p a r i s o n w i t h t h e P-Y c u r v e s recommended by M a t l o c k ( 1 9 7 0 ) , i t was shown t h a t a t s h a l l o w d e p t h s , a r e a s o n a b l y a c c u r a t e P-Y c u r v e c a n be p r e d i c t e d f r o m t h e f i n i t e e l e m e n t m e t h o d u s i n g p l a n e s t r e s s c o n d i t i o n s , a l t h o u g h t h e u l t i m a t e r e s i s t a n c e by M a t l o c k ' s c r i t e r i a was o v e r e s t i m a t e d by t h e f i n i t e e l e m e n t m e t h o d . T h i s d i f f e r e n c e i n t h e u l t i m a t e r e s i s t a n c e s , h o w e v e r , may be m a i n l y a t t r i b u t e d t o t h e l i m i t a t i o n t h a t t h e y d i d n o t c o n s i d e r t h e s o i l - p i l e s e p a r a t i o n . F o r t h e a r e a s a t d e p t h , i t was f o u n d t h a t t h e c r i t e r i a p r o p o s e d by M a t l o c k y i e l d an u l t i m a t e p i l e r e s i s t a n c e a p p r o x i m a t e l y midway b e t w e e n t h e u l t i m a t e r e s i s t a n c e s c o r r e s p o n d i n g t o p l a n e s t r a i n and p l a n e s t r e s s p r e d i c t i o n s . The d i f f e r e n c e s a r e v e r y s i g n i f i c a n t s i n c e , a s c o m p a r e d w i t h M a t l o c k ' s r e s u l t s , p l a n e s t r a i n c o n d i t i o n s o v e r p r e d i c t a b o u t 31% w h i l e p l a n e s t r e s s c o n d i t i o n s u n d e r p r e d i c t a b o u t 29% o f t h e u l t i m a t e r e s i s t a n c e . F rom a p a r a m e t r i c s t u d y i n w h i c h i n t e r f a c e p r o p e r t i e s were v a r i e d , Y e g i a n a n d W r i g h t (1973) f o u n d o u t t h a t s o i l l a t e r a l r e s i s t a n c e s a r e much d e p e n d e n t on t h e s o i l - p i l e i n t e r f a c e c h a r a c t e r i s t i c s a s sumed f o r t h e i n t e r f a c e e l e m e n t s , t h e v a r i a t i o n o f i n t e r f a c e p r o p e r t i e s i t s e l f may y i e l d a g r e a t c h a n g e i n t h e p r e d i c t e d u l t i m a t e s o i l r e s i s t a n c e a s much a s 37 p e r c e n t w h i c h i s b i g g e r t h a n t h e d i f f e r e n c e b e t w e e n t h e f i n i t e e l e m e n t a n a l y s e s a n d M a t l o c k ' s c r i t e r i a . T h i s r e s u l t t h e r e f o r e s u g g e s t s t h a t t h e a c c u r a c y o f P -Y c u r v e s p r e d i c t e d f r o m f i n i t e e l e m e n t m e t h o d d e p e n d s 36 on t h e a c c u r a t e d e s c r i p t i o n o f s o i l b e h a v i o u r a n d s o i l - p i l e i n t e r f a c e p r o p e r t i e s . B a r t o n and F i n n (1983) a l s o p r e d i c t e d t h e P-Y c u r v e s f o r l a t e r a l l y l o a d e d p i l e s i n d e n s e s a n d u s i n g f i n i t e e l e m e n t m e t h o d . The f i n i t e e l e m e n t a n a l y s i s was c o n d u c t e d u n d e r t h e p l a n e s t r a i n c o n d i t i o n s . The s o i l was m o d e l l e d a s an e l a s t i c - p l a s t i c m a t e r i a l i n c a p a b l e o f t e n s i o n . The c o m p u t e d P-Y c u r v e s were t h e n c o m p a r e d w i t h t h e i r e x p e r i m e n t a l c u r v e s f r o m c e n t r i f u g e t e s t s . In t h e a r e a s c l o s e t o t h e p i l e h e a d , t h e c o m p u t e d P-Y c u r v e s were i n a c l o s e a g r e e m e n t w i t h t h e e x p e r i m e n t a l c u r v e s , h o w e v e r , a t d e p t h s g r e a t e r t h a n a b o u t 5 d i a m e t e r s , t h e c o m p u t e d P-Y c u r v e s were l e s s s t i f f e r t h a n t h e e x p e r i m e n t a l c u r v e s , and t h e d i f f e r e n c e became more s e v e r e a s d e p t h s i n c r e a s e d t o 10 d i a m e t e r s . In g e n e r a l , a l t h o u g h t h e f i n i t e e l e m e n t m e t h o d o f f e r s a g r e a t p o t e n t i a l t o p r e d i c t t h e P-Y c u r v e s f o r l a t e r a l l y l o a d e d p i l e s f r o m t h e f u n d a m e n t a l s o i l p r o p e r t i e s , s e v e r a l f a c t o r s d e s e r v e f u r t h e r s t u d i e s , s u c h a s t h e d e s c r i p t i o n o f s o i l b e h a v i o r , s o i l t e n s i l e f a i l u r e , a n d t h e s o i l - p i l e i n t e r f a c e s i m u l a t i o n . T h e s e f a c t o r s w i l l be f u r t h e r c o m p l i c a t e d i n t h e a n a l y s i s i f t h e p i l e i n s t a l l a t i o n e f f e c t i s t o be c o n s i d e r e d . T h e r e f o r e f u r t h e r r e s e a r c h work i s n e c e s s a r y w i t h r e g a r d t o t h e a p p l i c a t i o n o f f i n i t e e l e m e n t m e t h o d s i n t h e d e v e l o p m e n t o f P-Y c u r v e s f o r t h e l a t e r a l l y l o a d e d p i l e s . 3. F I N I T E ELEMENT PROGRAM 3.1 INTRODUCTION The f o l l o w i n g f i n i t e e l e m e n t a n a l y s e s o f p r e s s u r e m e t e r e x p a n s i o n t e s t s a n d t h e l a t e r a l l y l o a d e d p i l e s a r e b a s e d on t h e new h i g h o r d e r e l e m e n t p r o g r a m CONOIL d e v e l o p e d by V a z i r i ( 1986) a t UBC. T h i s p r o g r a m c a n p e r f o r m d r a i n e d , u n d r a i n e d , a n d c o n s o l i d a t i o n a n a l y s e s f o r l a r g e v a r i e t i e s o f g e o t e c h n i c a l p r o b l e m s . A c o m p l e t e d o c u m e n t a t i o n o f t h e p r o g r a m i s c o n t a i n e d i n V a z i r i ' s D o c t o r a l D i s s e r t a t i o n . In o r d e r t o u s e CONOIL f o r p r e s e n t s t u d i e s , v e r i f i c a t i o n s o f t h i s p r o g r a m were made, some m o d i f i c a t i o n s were made t o i m p r o v e t h e a c c u r a c y and e f f i c i e n c i e s i n s o l v i n g c u r r e n t i n t e r e s t e d p r o b l e m s . The v e r i f i c a t i o n and m o d i f i c a t i o n o f t h e p r o g r a m w i l l be g i v e n i n C h a p t e r 5 i n c o m p a r i s o n w i t h c l o s e d f o r m s o l u t i o n . H e r e i n , a b r i e f p r e s e n t a t i o n o f t h e f i n i t e e l e m e n t f o r m u l a t i o n , t h e methods o f a n a l y s e s , and t h e n u m e r i c a l p r o c e d u r e s i s g i v e n . In t h e a p p l i c a t i o n o f f i n i t e e l e m e n t p r o g r a m t o a n y p r a c t i c a l p r o b l e m , i t i s e s s e n t i a l t o u n d e r s t a n d t h e a d o p t e d p r o g r a m , a n d e n s u r e i t s a c c u r a c y . 3.2 F I N I T E ELEMENT FORMULATION In t h e f i n i t e e l e m e n t a n a l y s e s , t h e s o i l c o n t i n u u m i s r e p l a c e d w i t h an e q u i v a l e n t f i n i t e a s s e m b l a g e o f d i s c r e t e s m a l l e r c o n t i n u a c a l l e d f i n i t e e l e m e n t s . T h o s e e l e m e n t s a r e c o n n e c t e d a t a f i n i t e number o f n o d e s . T h e r e f o r e s p a c i a l 37 38 v a r i a t i o n of s o i l p r o p e r t i e s can be a p p r o x i m a t e l y r e p r e s e n t e d by the average (or w e i g h t e d a verage) p r o p e r t i e s i n each element. The f i n i t e element method assumes a d i s t r i b u t i o n f i e l d f o r the unknown q u a n t i t y over the domain of each element. The unknown q u a n t i t y used can be t h e d i s p l a c e m e n t , s t r e s s or b o t h . When the unknown i s d i s p l a c e m e n t , s t r e s s or b o t h , the f o r m u l a t i o n i s u s u a l l y r e f e r r e d t o as d i s p l a c e m e n t method, s t r e s s method or h y b r i d method r e s p e c t i v e l y . These d i s t r i b u t i o n f i e l d s a r e u s u a l l y s p e c i f i e d u s i n g p o l y n o m i a l s . G e n e r a l l y , h i g h o r d e r p o l y n o m i a l g i v e s o l u t i o n s t h a t a r e c l o s e r t o the r e a l answer. Elements w i t h h i g h e r o r d e r p o l y n o m i a l s a r e c a l l e d h i g h e r o r d e r e l e m e n t s . P a s t e x p e r i e n c e s have shown t h a t the lower o r d e r elements sometimes would g i v e u n r e a l i s t i c answers i n the p r e d i c t i o n of l i m i t p r e s s u r e , e s p e c i a l l y i n a x i s y m m e t r i c a l u n d r a i n e d c o n d i t i o n ( N a g t e g a a l e t a l , 1974, S l o a n and Randolph, 1982, and De B o r s t and Vermeer, 1984). I t i s f o r t h i s reason t h a t h i g h e r o r d e r elements a r e employed i n CONOIL. In a d d i t i o n t o the r e q u i r e m e n t of o r d e r of the p o l y n o m i a l , the a p p r o x i m a t i n g f i e l d must be c o n t i n u o u s and a l s o must have c o n t i n u o u s d e r i v a t i v e s a t l e a s t w i t h i n e l e m e n t s . Such a c o n d i t i o n i s g e n e r a l l y r e g a r d e d as the a d m i s s i b i l i t y of the a p p r o x i m a t i n g f i e l d . For the d i s p l a c e m e n t f o r m u l a t i o n , the d i s p l a c e m e n t must be c o n t i n u o u s a c r o s s the edges of e l e m e n t s . Such elements a r e c o n f o r m i n g . The d i s p l a c e m e n t f i e l d s h o u l d a l s o be a b l e t o 39 r e p r e s e n t b o t h r i g i d body motion and the c o n s t a n t s t r a i n s t a t e of the element. These two c o n d i t i o n s a r e u s u a l l y r e f e r r e d t o as completeness of the e l e m e n t s . The above c o n d i t i o n s a r e r e q u i r e d f o r the convergence of the f i n i t e element s o l u t i o n t o the c o r r e c t answer as the mesh i s r e f i n e d . Based on the f i n i t e element t h e o r y , d i s p l a c e m e n t f o r m u l a t i o n w i t h c o n f o r m i n g elements would g i v e upperbound s o l u t i o n . The elements t h a t a r e complete but nonconforming ar e a l s o w i d e l y used. In some c a s e s , they may g i v e b e t t e r r e s u l t s than the c o n f o r m i n g elements (such as the nonconforming p l a t e e l e m e n t ) . However, t h o s e nonconforming elements can not g i v e the bounded s o l u t i o n , i t i s not c l e a r whether the r e s u l t i s an upperbound or lower bound s o l u t i o n . For CONOIL, the f o r m u l a t i o n i s based on the d i s p l a c e m e n t method. The element can be e i t h e r l i n e a r s t r a i n or c u b i c s t r a i n t r i a n g u l a r , as shown i n F i g . 3.1. The assumed d i s p l a c e m e n t f i e l d i s a d m i s s i b l e , complete and the elements a r e c o n f o r m i n g . T h e r e f o r e i n t h e o r y , the r e s u l t s from the program s h o u l d be the upperbound of the r e a l answer. Based on the assumed d i s p l a c e m e n t f i e l d , the e q u a t i o n s r e l a t i n g the element n o d a l f o r c e s and d i s p l a c e m e n t s can be d e r i v e d u s i n g v a r i a t i o n a l p r i n c i p l e , or the p r i n c i p l e of v i r t u a l work. The e q u a t i o n can be w r i t t e n i n the form : {Q} = [K] {q} (3.2.1) a) L i n e a r s t r a i n (b) Cubic s t r a i n F i g . 3.1 Element Types 41 where: [K] i s the element s t i f f n e s s m a t r i x {Q} i s the n o d a l f o r c e v e c t o r {q} i s the unknown d i s p l a c e m e n t v e c t o r Then the e q u a t i o n f o r each element i s combined i n the s t a n d a r d manner t o o b t a i n the g l o b a l e q u a t i o n f o r e n t i r e f i n i t e element mesh. The method adopted f o r the above e q u a t i o n s o l u t i o n i s a major f a c t o r i n f l u e n c i n g the e f f i c i e n c y and computer c o s t of the e n t i r e f i n i t e element a n a l y s i s . As the h i g h o r d e r elements a r e employed i n the program, a d r a s t i c i n c r e a s e i n the bandwidth of s t i f f n e s s m a t r i x e x i s t s . T h e r e f o r e a method o t h e r than t r a d i t i o n a l bandwidth s o l v e r s i s adopted. In CONOIL, the f r o n t a l s o l u t i o n t e c h n i q u e ( I r o n s , 1970) i s employed f o r s o l v i n g the above s i m u l t a n e o u s e q u a t i o n s . In t h i s p r o c e d u r e , assembly and e l i m i n a t i o n of the e q u a t i o n a r e combined, then the c o r e s t o r a g e r e q u i r e m e n t and the number of a r i t h m e t i c a l o p e r a t i o n s a r e s i g n i f i c a n t l y reduced. As a r e s u l t , the g l o b a l s t i f f n e s s m a t r i x i s never e x p l i c i t l y formed. B e s i d e s , w i t h the f r o n t a l s o l u t i o n the numbering of the element nodes becomes i m m a t e r i a l . U s e r s can o r d e r the nodes i n the way they w i s h , the r e f i n e m e n t of the mesh does not r e q u i r e renumbering the e n t i r e mesh. Under g i v e n d i s p l a c e m e n t f i e l d , the f o r m u l a t i o n of s t i f f n e s s m a t r i x , [ K ] , of each element (see Eq. (3.2.1)) depends upon the s t r e s s - s t r a i n laws of the m a t e r i a l . In s o i l s , the s t r e s s - s t r a i n r e l a t i o n s a r e o f t e n governed by 42 e f f e c t i v e s t r e s s e s r a t h e r than t o t a l s t r e s s e s . T h e r e f o r e , e f f e c t i v e s t r e s s approach appears more r a t i o n a l i n the f i n i t e element f o r m u l a t i o n w h i c h would widen the scope of a n a l y s i s . 3.3 UNIFIED APPROACH - AN EFFECTIVE STRESS METHOD The p r i n c i p l e of e f f e c t i v e s t r e s s i s an i m p o r t a n t concept i n g e o t e c h n i c a l a n a l y s i s . A n a l y t i c a l methods based on t h i s c o ncept have been p r o p o s e d by many r e s e a r c h e r s . An e l e g a n t approach proposed by N a y l o r (1973) w i l l be p r e s e n t e d h e r e i n . As shown l a t e r , t h i s a p p r o a c h p r o v i d e s a u n i f i e d p r o c e d u r e w i t h which the u n d r a i n e d a n a l y s i s i n terms of e i t h e r t o t a l s t r e s s or e f f e c t i v e s t r e s s , and the d r a i n e d a n a l y s i s can be performed based on the same m a t h e m a t i c a l f o r m u l a t i o n . 3.3.1 UNDRAINED ANALYSIS A c c o r d i n g t o T e r z a g h i ' s t h e o r y , the p r i n c i p l e of e f f e c t i v e s t r e s s can be e x p r e s s e d by : {ACT} {ACT'} + {m} Au (3.3.1) where {ACT} v e c t o r of t o t a l s t r e s s changes {ACT'} v e c t o r of e f f e c t i v e s t r e s s changes Au - pore p r e s s u r e changes {m} = e i t h e r { 1 , 1 , 0 } T or {1,1,1,0,0,0} T, depending upon whether i t i s p l a n e s t r a i n or 3D a n a l y s i s 43 Other r e l a t i o n s have a l s o been proposed f o r the e f f e c t i v e s t r e s s t h a t t a k e a c c o u n t of the i n t e r g r a n u l a r c o n t a c t a r e a (Skempton, 1960), but f o r most e n g i n e e r i n g problems a t low p r e s s u r e s , T e r z a g h i ' s d i f i n i t i o n i s s u f f i c i e n t l y a c c u r a t e . The t o t a l s t r e s s changes can be r e l a t e d t o the s t r a i n changes, {Ae}, by a c o n s t i t u t i v e m a t r i x [D] f o r u n d r a i n e d c o n d i t i o n s , i . e . {Aa} = [D] {Ae} (3.3.2) Based on t h e p r i n c i p l e of e f f e c t i v e s t r e s s , the b e h a v i o r of s o i l s k e l e t o n i s governed by c o n s t i t u t i v e m a t r i x [D'] i n terms of e f f e c t i v e s t r e s s p a r a m e t e r s . Then the e f f e c t i v e s t r e s s changes a r e r e l a t e d t o s t r a i n changes by : {Aa'} = [D'] {Ae} (3.3.3) Suppose the pore f l u i d element undergoes a v o l u m e t r i c s t r a i n change Ae^ as the pore p r e s s u r e changes by Au i n u n d r a i n e d c o n d i t i o n s , then Au = B f Ae^ (3.3.4) where B^ i s the apparent pore f l u i d b u l k modulus. B^ i s r e l a t e d t o the b u l k modulus of the pore f l u i d , B w, and the s o l i d p a r t i c l e s t i f f n e s s , B , by the p o r o s i t y , n, by 44 ( N a y l o r e t a l , 1981) 1 /B f = n / B w + ( 1 - n ) / B s ( 3 . 3 . 5 ) I t i s a s sumed t h a t f o r t h e f u l l y u n d r a i n e d c o n d i t i o n t h e r e w i l l be no movement o f p o r e f l u i d r e l a t i v e t o t h e s k e l e t o n , a n d c o n s e q u e n t l y t h e s k e l e t o n and t h e p o r e f l u i d u n d e r g o e s t h e same d e f o r m a t i o n , i . e . t h e c o m p a t i b i l i t y o f t h e u n d r a i n e d c o n d i t i o n . So t h a t s u b s t i t u t i o n o f E q . ( 3 . 3 . 2 ) , ( 3 . 3 . 3 ) a n d ( 3 . 3 . 7 ) i n t o t h e p r i n c i p l e o f e f f e c t i v e s t r e s s e q u a t i o n ( 3 . 3 . 1 ) , t h e n t h e c o n s t i t u t i v e m a t r i x [D] i n t e r m s o f t o t a l s t r e s s p a r a m e t e r s i s r e l a t e d t o t h e e f f e c t i v e c o n s t i t u t i v e m o d e l and t h e a p p a r e n t p o r e f l u i d b u l k m o d u l u s by : A e L = Ae / n = {m} 1 A e / n v v ( 3 . 3 . 6 ) a n d t h e r e f o r e E q . ( 3 . 3 . 4 ) b e c o m e s : Au = B, {m} T A e / n ( 3 . 3 . 7 ) [D] = [ D ' ] + {m} {m} T B f / n ( 3 . 3 . 8 ) I t s h o u l d be n o t e d t h a t i n E q . ( 3 . 3 . 8 ) , t h e r e i s no r e s t r i c t i o n i m p o s e d on t h e f o r m s o f [ D ] , [ D ' ] and B^. T h e y c a n r e p r e s e n t l i n e a r e l a s t i c p r o p e r t i e s , o r any f o r m o f 45 n o n l i n e a r , e l a s t i c p l a s t i c r e l a t i o n . The same f o r m u l a t i o n can be a p p l i e d t o any c o n s t i t u t i v e laws of m a t e r i a l s . For the e l a s t i c i s o t r o p i c c o n s t i t u t i v e law ( e i t h e r l i n e a r or i n c r e m e n t a l l i n e a r ) , [D] and [D'] i n v o v l e o n l y two e l a s t i c p a r a m e t e r s . For the p a i r of b u l k modulus B (o r B ' ) , and shear modulus, G (or G'), then Eq. (3.3.8) would g i v e the f o l l o w i n g r e l a t i o n s f o r p l a n e s t r a i n c o n d i t i o n ( N a y l o r e t a l , 1981) : G = G' (3.3.9) B = B' + B f/n (3.3.10) The r e l a t i o n s of Eq. (3.3.9) and (3.3.10) p r o v i d e a c o n v e n i e n t way of c o n v e r t i n g between t o t a l and e f f e c t i v e s t r e s s p arameters f o r e l a s t i c i s o t r o p i c u n d r a i n e d a n a l y s i s . They a l l o w f o r e i t h e r t o t a l or e f f e c t i v e s t r e s s a n a l y s i s t o be p erformed f o r the same u n d r a i n e d c o n d i t i o n . V a z i r i adopted N a y l o r ' s approach and e x t e n d th e above model t o the u n d r a i n e d a n a l y s i s of u n s a t u r a t e d o i l sand w i t h the c oncept of 'Homogenized C o m p r e s s i b l e Phase'. D e t a i l s of t h i s c o ncept were c o v e r e d i n V a z i r i (1986), and w i l l not p r e s e n t e d h e r e i n as s a t u r a t e d c o h e s i v e s o i l s a r e assumed i n the p r e s e n t s t u d i e s . E f f e c t i v e S t r e s s and T o t a l Stress A n a l y s i s As shown b e f o r e , the above e f f e c t i v e s t r e s s method p r o v i d e s the f l e x i b i l i t y and ease f o r t o t a l s t r e s s and 46 e f f e c t i v e s t r e s s a n a l y s i s o f u n d r a i n e d l o a d i n g c o n d i t i o n b a s e d on t h e same f o r m u l a t i o n . In e f f e c t i v e s t r e s s a n a l y s i s , t h e m e t h o d a l l o w s f o r e x p l i c i t d e t e r m i n a t i o n o f t h e p o r e p r e s s u r e a n d e f f e c t i v e s t r e s s c o m p o n e n t s i n t h e u n d r a i n e d c o n d i t i o n . The p o r e p r e s s u r e c h a n g e i s o b t a i n e d t h r o u g h t h e u n d r a i n e d c o m p a t i b i l i t y o f v o l u m e t r i c s t r a i n . The a n a l y s i s c a n be a c c o m p l i s h e d by t h e e f f e c t i v e s t r e s s p a r a m e t e r s w h i c h f o r m t h e e f f e c t i v e c o n s t i t u t i v e m a t r i x [ D ' ] , a n d a n o n - z e r o a p p a r e n t p o r e f l u i d b u l k m o d u l u s B^. C h o i c e o f B^, i n f a c t , i s n o t c r i t i c a l a s l o n g a s B^ i s much l a r g e r t h a n t h e e f f e c t i v e b u l k m o d u l u s B ' . I t c a n be shown t h a t a v a l u e o f B^ i n t h e r a n g e o f 100 t o 500 B' i s e q u i v a l e n t t o u s i n g an u n d r a i n e d P o i s s o n ' s r a t i o , v i n t h e r a n g e o f 0 .495 t o 0 . 4 9 9 i n t h e t o t a l s t r e s s a n a l y s i s . A n o t h e r way o f e f f e c t i v e s t r e s s a n a l y s i s f o r u n d r a i n e d c o n d i t i o n i s b a s e d on t h e f i n i t e e l e m e n t f o r m u l a t i o n p r o p o s e d by C h r i s t i a n ( 1 9 6 8 ) , i n w h i c h t h e p o r e p r e s s u r e i s e v a l u a t e d f r o m t h e t o t a l s t r e s s c h a n g e s b a s e d on t h e Skempton p o r e p r e s s u r e p a r a m e t e r s . T h i s a p p r o a c h h a s b e e n f r e q u e n t l y u s e d by many r e s e a r c h e r s ( e . g . B y r n e a n d J a n z e n , 1 9 8 4 ) . H o w e v e r , t h i s a p p r o a c h i s r e l a t i v e i n e f f i c i e n t , d e t e r m i n a t i o n o f p o r e p r e s s u r e f r o m t h e t o t a l s t r e s s c h a n g e g e n e r a l l y r e q u i r e s i t e r a t i o n p r o c e d u r e . In f a c t , t h e a p p r o a c h o f o b t a i n i n g p o r e p r e s s u r e f r o m t o t a l s t r e s s c h a n g e i s l i n k e d t o t h e N a y l o r ' s m e t h o d . T h i s i s b e c a u s e t h e r e i s a r e l a t i o n b e t w e e n t h e a p p a r e n t p o r e 47 f l u i d b u l k m o d u l u s and t h e S k e m p t o n ' s p a r a m e t e r , B ( B y r n e , 1985) ( s e e A p p e n d i x A f o r t h e d e r i v a t i o n ) : skem a s B f / n = B ' B skem' / ( 1 - B skem ( 3 . 3 . 1 1 ) In most c a s e s , h o w e v e r , t h e a b o v e e f f e c t i v e s t r e s s a p p r o a c h i s u n n e c e s s a r i l y c o m p l i c a t e d f o r u n d r a i n e d l o a d i n g i n c o h e s i v e s o i l s . The l o a d i n g s o f t e n t a k e p l a c e i n s u c h a manner t h a t l i t t l e o r no d r a i n a g e t a k e s p l a c e , a n d t h e r e i s no i n t e r e s t i n t h e s e p a r a t i o n o f e f f e c t i v e s t r e s s a n d p o r e p r e s s u r e . F o r t h o s e c o n d i t i o n s , B y r n e (1983) h a s i n d i c a t e d t h a t i t i s o f t e n d e s i r a b l e t o work w i t h t o t a l r a t h e r t h a n e f f e c t i v e s t r e s s e s , i n w h i c h c a s e t h e p r o c e d u r e o f d r a i n e d a n a l y s i s i s a d o p t e d w i t h z e r o p o r e p r e s s u r e , a n d t h e a p p r o p r i a t e s o i l p a r a m e t e r s a r e o b t a i n e d f r o m t e s t s e v a l u a t e d i n t e r m s o f t o t a l r a t h e r t h a n e f f e c t i v e s t r e s s e s , s u c h a s u n d r a i n e d s h e a r s t r e n g t h C^. And t h e n t h e u n d r a i n e d c o n d i t i o n i s s i m u l a t e d by u s i n g a h i g h b u l k m o d u l u s w h i c h i s e q u i v a l e n t t o P o i s s o n ' s r a t i o c l o s e t o 0 . 5 . 3 . 3 . 2 DRAINED ANALYS IS As shown i n E q . ( 3 . 3 . 8 ) , i n t h e e f f e c t i v e s t r e s s m e t h o d , t h e t o t a l s t r e s s m a t r i x [D] a n d t h e e f f e c t i v e s t r e s s m a t r i x [ D ' ] a r e r e l a t e d by t h e e q u i v a l e n t p o r e f l u i d b u l k m o d u l u s , B j . T h e r e f o r e f o r s i m u l a t i o n o f t h e d r a i n e d l o a d i n g c o n d i t i o n where t h e r e i s no c h a n g e i n p o r e p r e s s u r e , a d r a i n e d a n a l y s i s c a n be i m p l e m e n t e d s i m p l y by s e t t i n g 48 f = 0. In t h i s c a s e , as f a r as the s t r e s s , changes a r e c o n c e r n e d , t h e r e becomes no d i s t i n c t i o n between e f f e c t i v e and t o t a l s t r e s s a n a l y s i s , and the a p p r o p r i a t e s o i l parameter from d r a i n e d t e s t s a r e used (such as 0 ' , C , E', and u') i n t h e a n a l y s i s . 3.4 STRUCTURE OF THE PROGRAM CONOIL c o n s i s t s of two s e p a r a t e subprograms : Geometry program and Main program. Geometry program i n t e r p r e t s the f i n i t e element mesh d a t a , and p r o v i d e s the geometry i n f o r m a t i o n f o r t h e Main program. The i n f o r m a t i o n i s co n n e c t e d t h r o u g h a l i n k f i l e . T h i s p r o c e d u r e can reduce the p i t f a l l s u s u a l l y i n v o l v e d i n the s e t - u p of f i n i t e element mesh. CONOIL i s an i n c r e m e n t a l l i n e a r program u s i n g tangent s t i f f n e s s method. T h i s method d i v i d e s the a p p l i e d l o a d i n t o s e v e r a l s m a l l i n c r e m e n t s , and assumes t h a t s o i l s behave l i n e a r - e l a s t i c i n each l o a d i n c r e m e n t . In the program, each l o a d increment i s a n a l y s e d t w i c e , the f i r s t time u s i n g modulus v a l u e s f o r the s o i l element based on the s t r e s s e s a t the b e g i n n i n g of the in c r e m e n t , and the second time u s i n g modulus v a l u e s based on the average s t r e s s e s d u r i n g t h e in c r e m e n t . The changes i n s t r e s s and s t r a i n i n s o i l elements and the changes i n n o d a l p o i n t d i s p l a c e m e n t d u r i n g each increment a r e added t o the v a l u e s a t the b e g i n n i n g of the in c r e m e n t . F o r the a n a l y s i s w i t h o u t shear volume c o u p l i n g or s t r e s s r e d i s t r i b u t i o n u s u a l l y o n l y two i t e r a t i o n s a r e 4 9 p e r f o r m e d . F o r a l o a d i n c r e m e n t , s o i l p r o p e r t i e s a r e p r i n t e d o u t f o r t h e a v e r a g e s t r e s s l e v e l o f t h e i n c r e m e n t , w h i l e d i s p l a c e m e n t s , s t r a i n s and s t r e s s e s a r e p r i n t e d o u t f o r t h e e n d o f t h e i n c r e m e n t . CONOIL p o s s e s s e s s e v e r a l o t h e r a b i l i t i e s o f p e r f o r m i n g t e m p e r a t u r e e f f e c t s , s h e a r v o l u m e c o u p l i n g , a n d s t r e s s r e d i s t r i b u t i o n a n a l y s e s . Among t h e m , o n l y s t r e s s r e d i s t r i b u t i o n i s f r e q u e n t l y u s e d i n t h e t h e s i s . 3 .5 STRESS -RED ISTR IBUT ION F o r an i n c r e m e n t a l l i n e a r p r o g r a m , p r o b l e m s w o u l d a r i s e when t h e s t r e s s p a t h r e a c h t h e f a i l u r e e n v e l o p e ( i . e . M o h r - c o u l u m b c r i t e r i o n ) o f t h e s o i l . F o r s u c h a s t r e s s s t a t e , t h e s h e a r m o d u l u s i s s e t t o a s m a l l v a l u e c l o s e t o z e r o , w h i c h means t h a t t h e s h e a r s t r e s s i n t h a t e l e m e n t w i l l n o t c h a n g e f o r a f u r t h e r i n c r e m e n t o f l o a d i n g . The o v e r s t r e s s o f f e n d i n g t h e s t r e n g t h e n v e l o p e i n t h i s i n c r e m e n t w o u l d , h o w e v e r , s t a y i n t h a t e l e m e n t . M o r e o v e r , i f t h e i n c r e m e n t o f l o a d i n g i s s u c h t h a t t h e n o r m a l s t r e s s d e c r e a s e s , s u c h a s i n t h e s o i l s b e h i n d t h e l a t e r a l l y l o a d e d p i l e , t h e n u n l e s s t h e s t r e n g t h e n v e l o p e i s h o r i z o n t a l , t h e p r e d i c t e d s t r e s s s t a t e w i l l v i o l a t e t h e f a i l u r e c r i t e r i o n , a s shown i n F i g . 3 . 2 . A m e t h o d o f s h e d d i n g t h e o v e r s t r e s s t o a d j a c e n t e l e m e n t s s h o u l d be u s e d so t h a t t h e o v e r p r e d i c t e d s t r e s s s t a t e w o u l d be b r o u g h t back t o t h e f a i l u r e e n v e l o p e . S e v e r a l me thods e x i s t i n t h e c o r r e c t i o n o f o v e r s t r e s s e s . The method b a s e d on c o n s t a n t mean n o r m a l 50 (a) S t r e s s Path around L a t e r a l . P i l e (b) S t r e s s C o r r e c t i o n based on Constant Mean Normal S t r e s s F i g . 3.2 S t r e s s S t a t e a s s o c i a t e d w i t h Load Shedding 51 s t r e s s i s employed i n the t h e s i s . T h i s method seems more c o n s i s t e n t w i t h the common assum p t i o n t h a t the s o i l element undergoes f a i l u r e w i t h no volume but d i s t o r t i o n change. However, t h i s approach may meet problems i n boundary elements when major or minor p r i n c i p a l s t r e s s i n tho s e elements i s the s p e c i f i e d s t r e s s c o n d i t i o n a t b o u n d a r i e s . Based on F i g . 3 . 2 ( b ) , the c o r r e c t i o n of the o v e r s t r e s s w i l l be: A a = A a X y A a = a „ - a ' X X X A a = a , - a ' y y y A r = T - T 1 xy xy xy a = t a n " 1 ( 2 A T / ( a - a )) xy x y A a y = [ ( a , - a 3 ) / 2 - ( ( a , +a 3 ) sin0+c*cos<£) ]cos0 AT = [ ( a 1 - a 3 ) / 2 - (( a , + a 3 ) sin0+c*cos0) ] sin<£ xy The removal of t h e s e o v e r s t r e s s e s can be a c h i e v e d by a p p l y i n g a s e t of n o d a l f o r c e s w h i c h i s o b t a i n e d by the p r i n c i p l e of v i r t u a l work. Due t o the g e n e r a l n a t u r e of s t r e s s l e v e l dependency of the f a i l u r e s t r e n g t h e n v e l o p e , the computed s t r e s s s t a t e may v i o l a t e the f a i l u r e e n v e l o p e a g a i n a f t e r t h e a p p l i c a t i o n of t h e n o d a l f o r c e s . T h e r e f o r e i t e r a t i o n s may be r e q u i r e d t o b r i n g the s t r e s s s t a t e t o the a s s i g n e d t o l e r a n c e . W i t h the i n c o r p o r a t i o n of s t r e s s r e d i s t r i b u t i o n (or l o a d shedding) t e c h n i q u e , the s o i l can be m o d e l l e d a p p r o x i m a t e l y as n o n l i n e a r e l a s t i c p e r f e c t l y p l a s t i c mater i a l . 4. CONST ITUT IVE RELATIONS 4.1 INTRODUCTION In p r e d i c t i n g d e f o r m a t i o n a n d s t r e s s d i s t r i b u t i o n i n l o a d e d s o i l m a s s e s , t h e f i n i t e e l e m e n t method p r o v i d e s a v e r y p o w e r f u l t e c h n i q u e . The method r e q u i r e s s e v e r a l s t e p s o f i d e a l i z i n g t h e s o i l mass and m o d e l l i n g i t s s t r e s s - s t r a i n b e h a v i o r . T h e r e f o r e , t h e a c c u r a t e p r e d i c t i o n s a r e d e p e n d e n t n o t O n l y on t h e a c c u r a c y o f t h e d i s c r e t i z a t i o n a n d n u m e r i c a l p r o c e d u r e s , b u t a l s o on t h e a b i l i t y t o i d e a l i z e t h e s u b s o i l c o n d i t i o n s and f o r m u l a t e m a t h e m a t i c a l m o d e l s s i m u l a t i n g t h e s t r e s s - s t r a i n b e h a v i o r o f r e a l s o i l . The s t r e s s - s t r a i n c h a r a c t e r i s t i c s o f r e a l s o i l a r e v e r y c o m p l e x , b e i n g n o n l i n e a r , i n e l a s t i c a n d s t r e s s l e v e l d e p e n d e n t . V a r i o u s s t r e s s - s t r a i n m o d e l s h a v e b e e n p r o p o s e d t o r e p r e s e n t t h e b e h a v i o r o f s o i l . T h e s e r a n g e f r o m s i m p l e l i n e a r e l a s t i c m o d e l s t o h i g h l y s o p h i s t i c a t e d e l a s t i c p l a s t i c m o d e l s . H o w e v e r , i t g e n e r a l l y a p p e a r s t h a t no r e a l s o i l c a n be a c c u r a t e l y r e p r e s e n t e d by a u n i q u e s t r e s s - s t r a i n m o d e l , and e a c h o f t h e p r o p o s e d m o d e l s c a n o n l y a t b e s t r e p r e s e n t some p a r t i c u l a r c o n s t i t u t i v e f e a t u r e s o f r e a l s o i l s . In many g e o t e c h n i c a l e n g i n e e r i n g a p p l i c a t i o n s , t h e n o n l i n e a r i t y , s t r e s s l e v e l d e p e n d e n c y , a n d i n e l a s t i c i t y a r e t h r e e i m p o r t a n t c h a r a c t e r i s t i c s o f t h e s t r e s s - s t r a i n b e h a v i o r o f s o i l s . M o d e l l i n g t h e s e a s p e c t s becomes a p r i m a r y r e q u i r e m e n t i n t h e s o i l s t r e s s - s t r a i n r e l a t i o n s h i p . Among 52 53 s e v e r a l methods of m o d e l l i n g n o n l i n e a r s o i l b e h a v i o r , c u r v e f i t t i n g methods i n v o l v i n g h y p e r b o l i c f u n c t i o n s a r e s i m p l e , and have been w i d e l y used w i t h some s u c c e s s e s . 4.2 INCREMENTAL NONLINEAR ELASTIC SOIL MODEL CONOIL employs the i n c r e m e n t a l n o n - l i n e a r e l a s t i c , i s o t r o p i c s t r e s s - s t r a i n model proposed by Duncan and Chang (1970). In t h i s model, t h e two e l a s t i c parameters a r e r e q u i r e d t o c h a r a c t e r i z e t h e n o n l i n e a r s t r e s s - s t r a i n and volume change b e h a v i o r . The independent e l a s t i c p a rameters commonly used a r e the Young's modulus, E, and P o i s s o n ' s r a t i o , u. The b u l k modulus, B, and the shear modulus, G, a r e perhaps more fundamental p a r a m e t e r s as they r e p r e s e n t the volume and d i s t o r t i o n components of s o i l r e s p o n s e s , and would be the most d e s i r a b l e ones t o use. However, d e t e r m i n a t i o n of the shear modulus from l a b o r a t o r y t e s t i n g s i s d i f f i c u l t , r e q u i r i n g s p e c i a l equipments. U n l i k e shear modulus, G, both b u l k modulus, B, and Young's modulus, E, can be d e t e r m i n e d from t h e c o n v e n t i o n a l t r i a x i a l t e s t . The Young's modulus i s v e r y s i m i l a r i n c h a r a c t e r t o the shear modulus as bo t h a r e a measure of d i s t o r t i o n a l r e s p o n s e . For t h i s r e a s o n , Modulus, E, and B a r e used i n t h i s t h e s i s . In t he i n c r e m e n t a l n o n l i n e a r s o i l r e s p o n s e , the a p p r o p r i a t e v a l u e s of E and B depend upon the l e v e l of s t r e s s , and they a r e u s u a l l y d e t e r m i n e d from l a b o r a t o r y t e s t s . I n the i n t e r p r e t a t i o n of t h e s e t e s t s , i t i s common t o e x p r e s s t h e d i s t o r t i o n a l r e s p o n s e i n terms of m o d i f i e d 54 h y p e r b o l a s and the v o l u m e t r i c response i n e x p o n e n t i a l form. Based on Konder's (1963) f i n d i n g s t h a t the s t r e s s - s t r a i n c u r v e s f o r a number of s o i l s can be app r o x i m a t e d r e a s o n a b l y a c c u r a t e l y by a h y p e r b o l a , Duncan and Chang (1970) proposed a m o d i f i e d h y p e r b o l i c s t r e s s - s t r a i n c u r v e as : ( 0 , - 0 - 3 ) = — j - p j (4.1.1) — + E. ( a , - a 3 ) f A t y p i c a l shape of the m o d i f i e d h y p e r b o l a i s shown i n F i g . 4.1. D i f f e r e n t i a t i n g t he above h y p e r b o l i c s t r e s s - s t r a i n c u r v e , the tangent Young's modulus i s e x p r e s s e d a s : , Rf ( a , - a 3 ) ., E. = E. (1 - — 3—f-1- ) 2 (4.1.2) t 1 (0^-03)^ — The shear s t r e n g t h ( a , - a 3 ) f i s governed by Mohr-Coulumb c r i t e r i o n : / \ 2 C cos</> + 2 a 3 s i n 0 ( a i _ a 3 > f = ( i - s i n t f ) (4.1.3) and 0 = 0 , - A</>log(a3/P ) (4.1.4) The v a r i a t i o n of the i n i t i a l Young's modulus E^ w i t h c o n f i n i n g s t r e s s , a 3 i s e x p r e s s e d u s i n g Janbu's (1961) f o r m u l a : ( 0r° 3W " 1 . ^ — « v ° 3 > f -/ ^ iference A E . / l\ 3. / Stress Dii R f ( 0 r ° 3 > u i t A x i a l S t r a i n , c F i g . 4.1 Hype r b o l i c R e p r e s e n t a t i o n of A S t r e s s - s t r a i n Curves cn cn 56 E. l n ( 4 . 1 . 5 ) Pa F o r t h e n o n l i n e a r v o l u m e t r i c b e h a v i o r , t h e f o l l o w i n g e x p o n e n t i a l e x p r e s s i o n i s commonly u s e d : A t y p i c a l e x a m p l e o f n o n l i n e a r b e h a v i o r f o r s a n d i s shown i n F i g . 4 . 2 . T h e r e f o r e t h e c o m p l e t e s t r e s s - s t r a i n b e h a v i o r o f t h e s o i l i s d e f i n e d by two e l a s t i c s o i l p a r a m e t e r s E f c a n d B^. T h e s e p a r a m e t e r s i n t u r n d e p e n d upon t h e t y p e o f s o i l a n d t h e l e v e l o f s t r e s s w i t h i n t h e s o i l , and a r e s p e c i f i e d i n t e r m s o f s e v e n s o i l c o n s t a n t s : K E i s t h e Y o u n g ' s m o d u l u s number n i s t h e Y o u n g ' s m o d u l u s e x p o n e n t K D i s t h e b u l k m o d u l u s number a m i s t h e b u l k m o d u l u s e x p o n e n t Rf i s t h e f a i l u r e r a t i o 0, i s t h e peak f r i c t i o n a n g l e o f s a n d a t a c o n f i n i n g s t r e s s o f 1 a tm L\<P i s t h e d e c r e a s e i n f a i l u r e a n g l e o f s a n d f o r a t e n f o l d i n c r e a s e i n c o n f i n i n g s t r e s s C i s t h e c o h e s i o n i n t e r c e p t o f t h e s t r e n g t h e n v e l o p e 0 i s t h e s l o p e o f t h e s t r e n g t h e n v e l o p e The p r o c e d u r e s f o r e v a l u a t i n g t h e s e p a r a m e t e r s f r o m l a b o r a t o r y t e s t s a r e d e s c r i b e d i n d e t a i l by Duncan e t a l ( 1 9 8 0 ) . B y r n e e t a l (1983) a l s o p r e s e n t p a r t i c u l a r v a l u e s m ( 4 . 1 . 6 ) 57 16 _ S t r t t t D i f f t r inc t v« A i i o l Stroin O Oj • 4) kg /cm* & 0*e • 3kg/cm* * crc • I kg/cm* 2 3 4 5 6 Axiol Stroin, % (a) (b) O 10 2 0 Volumetric Stroin, c ¥ - % F i g . 4.2 S t r e s s s t r a i n Curves f o r d r a i n e d t r i a x i a l t e s t s on sand ( a f t e r Byrne & Cheung, 1984) 58 f o r c o h e s i v e s o i l w h i l e B y r n e a n d E l d r i d g e (1983) f o r s a n d s . U s i n g t h e a b o v e h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n w i t h t h e i n c o r p o r a t i o n o f l o a d s h e d d i n g i t e r a t i o n t e c h n i q u e , n o n l i n e a r e l a s t i c - p l a s t i c b e h a v i o r o f s o i l s c a n be s i m u l a t e d . 4 .3 B IL INEAR E L A S T I C - P L A S T I C MODEL In some c a s e s , s o i l s a p p e a r b i l i n e a r e l a s t i c - p l a s t i c s t r e s s s t r a i n c u r v e , s u c h a s o v e r c o n s o l i d a t e d c l a y s u n d e r t h e u n d r a i n e d l o a d i n g c o n d i t i o n s . S u c h a b i l i n e a r s t r e s s - s t r a i n c u r v e ( a s shown i n F i g . 4 . 3 ) cart a l s o be s i m u l a t e d u s i n g t h e a b o v e g e n e r a l h y p e r b o l i c s t r e s s - s t r a i n m o d e l w i t h s l i g h t m o d i f i c a t i o n . As shown i n F i g . 4 . 3 , t h e i n i t i a l l i n e a r e l a s t i c c u r v e ( i . e . AB segment ) c a n be o b t a i n e d by p u t t i n g Rf c l o s e t o z e r o i n E q . ( 4 . 1 . 1 ) . In t h i s c a s e , t h e e l a s t i c p a r a m e t e r s a r e i n d e p e n d e n t o f t h e s t r e s s l e v e l , t h u s , t h e m o d u l u s e x p o n e n t s m and n a r e b o t h s e t t o z e r o . The p l a s t i c f a i l u r e o c c u r s when t h e d e v i a t o r i c s t r e s s r e a c h e s t h e u n d r a i n e d s h e a r s t r e n g t h C^. T h i s i s e q u i v a l e n t t o t h e T r e s c a ' s f a i l u r e c r i t e r i o n ( i . e . a^-o2 = 2 C u ) i n t h e c l a s s i c p l a s t i c i t y . When s o i l e l e m e n t s a r e i n p l a s t i c f a i l u r e , t h e l a r g e p l a s t i c d e f o r m a t i o n ( i . e . BC l i n e i n F i g . 4 . 3 ) i s s i m u l a t e d by u s i n g s m a l l s h e a r m o d u l u s a n d t h e l o a d s h e d d i n g i t e r a t i o n p r o c e d u r e . T h e r e f o r e a s i m p l e b i l i n e a r e l a s t i c - p l a s t i c m o d e l c a n be o b t a i n e d f o r c o h e s i v e s o i l s . T h i s m o d e l i s a l s o u s e f u l i n t h e e x a m i n a t i o n o f t h e f i n i t e e l e m e n t p r o g r a m . F i g . 4.3 B i l i n e a r E l a s t i c P l a s t i c model 60 4.4 INCORPORATION OF TENSION FAILURE In g e n e r a l , s o i l s a r e v e r y weak i n s u s t a i n i n g t e n s i o n s t r e s s . I t i s , t h e r e f o r e , o f t e n t o n e g l e c t the t e n s i l e s t r e n g t h of s o i l s i n the e a r t h s t r u c t u r e a n a l y s i s . In o r d e r t o s i m u l a t e the t e n s i l e c r a c k i n g or c a v i t y i n s o i l s b e h i n d the l a t e r a l p i l e s , a t e n s i o n f a i l u r e c r i t e r i o n s h o u l d be i n c o r p o r a t e d i n the above s o i l model. H e r e i n , a s i m p l e t e n s i o n c u t - o f f model i s p r e s e n t e d . When a s o i l element i s i n c o m p r e s s i o n , the element i s assumed t o f o l l o w the above n o n l i n e a r h y p e r b o l i c / o r b i l i n e a r e l a s t i c p l a s t i c s t r e s s - s t r a i n c u r v e s whereas, when a s o i l element i s s u b j e c t t o a t e n s i l e s t r e s s , i . e . minor p r i n c i p a l s t r e s s becomes n e g a t i v e , or l e s s than t h e s o i l t e n s i l e s t r e n g t h ( f o r c o h e s i v e s o i l , i . e . s o i l c o h e s i o n ) , the element i s assumed t o l o s e e n t i r e c a p a b i l i t y of s u s t a i n i n g any subsequent s t r e s s . At t h i s t i m e , shear and b u l k moduli of the element a r e both d e f a u l t e d t o s m a l l v a l u e s , so t h a t the element undergoes l a r g e shear d i s t o r t i o n and v o l u m e t r i c change i n the f o l l o w i n g l o a d i n g p r o c e s s . Load shedding i t e r a t i o n p r o c e d u r e i s employed t o r e d i s t r i b u t e e x c e s s s t r e s s e s t o the a d j a c e n t e l e m e n t s . The f o r e g o i n g s i m p l e t e n s i o n c u t - o f f model based on the minor p r i n c i p a l s t r e s s s t a t e , s t r i c t l y s p e a k i n g , may not be adequate f o r the res p o n s e s of r e a l s o i l s . I n r e a l i t y , s o i l i s a n i s o t r o p i c m a t e r i a l , i t may l o s e i t s s t r e n g t h i n one d i r e c t i o n due t o the t e n s i l e c r a c k i n g i n t h a t d i r e c t i o n , but s t i l l p o s s e s s c e r t a i n s t r e n g t h t o r e s i s t t he s t r e s s i n o t h e r 61 d i r e c t i o n s . To cope w i t h t h i s k i n d of a n i s o t r o p i c response of n a t u r a l s o i l s , an a n i s o t r o p i c s t r e s s - s t r a i n model such as c r o s s - a n i s o t r o p i c model s h o u l d be used. However, the above s i m p l e t e n s i o n c u t - o f f model i s c o n s i s t e n t w i t h the framework of i n c r e m e n t a l i s o t r o p i c e l a s t i c i t y a p p r o a c h , and would be the f i r s t s t e p t o s i m u l a t e the t e n s i o n f a i l u r e u s i n g h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n s . 5. CYLINDRICAL CAVITY EXPANSION THEORY 5.1 INTRODUCTION In g e o t e c h n i c a l e n g i n e e r i n g , t h e r e a r e v a r i o u s p r a c t i c a l problems which c o n c e r n an a n a l y t i c a l problem of the e x p a n s i o n of a c a v i t y i n a s o i l mass, such as the i n t e r p r e t a t i o n of the p r e s s u r e m e t e r t e s t (Gibson and Anderson, 1961, Hughes e t a l , 1977, B a g u e l i n et a l , 1978), the e f f e c t of p i l e i n s t a l l a t i o n ( R a n d o l p h , C a r t e r and Wroth, 1979) and the b e a r i n g c a p a c i t y of deep f o u n d a t i o n ( V e s i c , 1972). In g e n e r a l t h e r e a r e two t y p e s of problems i n the e x p a n s i o n of a c a v i t y i n a s o i l mass, i . e . c y l i n d r i c a l c a v i t y e x p a n s i o n , and s p h e r i c a l c a v i t y e x p a n s i o n . In the former case the c a v i t y i s assumed t o be expanded c y l i n d r i c a l l y under c o n d i t i o n s of axisymmetry and p l a n e s t r a i n w h i l e i n the l a t t e r case t h e c a v i t y i s assumed t o be expanded under c o n d i t i o n of s p h e r i c a l symmetry. However, i n e i t h e r case the problem i s o n l y o n e - d i m e n s i o n a l because the d i s p l a c e m e n t s i n the medium a r e everywhere r a d i a l . T h e r e f o r e the problem i s s i m p l i f i e d , and i n some c a s e s , under c e r t a i n i d e a l i z a t i o n s of the s o i l b e h a v i o r , the problem can be s o l v e d i n the c l o s e d form. Due t o i t s a v a i l a b i l i t y i n c l o s e d form s o l u t i o n s , the c a v i t y e x p a n s i o n t h e o r y i s a l s o o f t e n used t o e v a l u a t e the a c c u r a c y of f i n i t e element program i n t h e a n a l y s e s of s i m i l a r but more complex problems. For p r e s e n t i n t e r e s t s , 62 63 o n l y c y l i n d r i c a l c a v i t y e x p a n s i o n problem w i l l be d i s c u s s e d e x c l u s i v e l y i n t h i s c h a p t e r . The purpose of t h i s c h a p t e r i s t o examine the a c c u r a c y of CONOIL. Based on the e x a m i n a t i o n , some f e a t u r e s of the o r i g i n a l program were m o d i f i e d so t h a t t h e program can a c c u r a t e l y and e f f i c i e n t l y s i m u l a t e t h o s e p r o b l e m s . 5.2 ELASTO-PLASTIC CLOSED FORM SOLUTIONS 5.2.1 PROBLEMS For the c y l i n d r i c a l c a v i t y e x p a n s i o n problem, the s o i l mass s u b j e c t e d t o a c y l i n d r i c a l p r e s s u r e , P, a t the w a l l of c a v i t y w i l l move r a d i a l l y outwards under the axisymmetry and p l a n e s t r a i n c o n d i t i o n s . T h e r e f o r e , an i n f i n i t e d i s k of u n i t t h i c k n e s s w i t h an i n i t i a l c a v i t y , r0, can be taken a c r o s s t h e s o i l mass f o r the a n a l y s i s . Such an a n a l y t i c a l model i s shown i n F i g . 5.1. I n i t i a l l y , the c a v i t y has a r a d i u s r0, and the e n t i r e s o i l mass i s s u b j e c t e d t o an i n - s i t u i s o t r o p i c s t r e s s s t a t e , P 0 . As the p r e s s u r e , P, i n c r e a s e s , the c a v i t y i s expanded r a d i a l l y outwards. At the b e g i n n i n g , the e n t i r e s o i l mass i s d i s p l a c e d e l a s t i c a l l y . ' L a t e r , a t some s t a g e s the p l a s t i c d e f o r m a t i o n i s i n i t i a t e d a t the w a l l of c a v i t y , and the p l a s t i c r e g i o n s p e c i f i e d by the e l a s t o - p l a s t i c i n t e r b o u n d a r y , i n F i g . 5.1 w i l l e n l a r g e as the a p p l i e d p r e s s u r e c o n t i n u e s t o i n c r e a s e . The p r e s s u r e vs the r a d i a l d i s p l a c e m e n t or the c i r c u m f e r e n t i a l s t r a i n forms the so 64 Elastic F i g . 5.1 Problem of C y l i n d r i c a l C a v i t y E xpansion i n S o i l Mass 65 c a l l e d p r e s s u r e e x p a n s i o n c u r v e . 5.2.2 CLOSED FORM SOLUTIONS a) C o h e s i v e S o i l s Many c l o s e d form s o l u t i o n s have been proposed f o r c o h e s i v e s o i l s ( G i bson and Anderson, 1961 and B a g u e l i n e t a l , 1978). For p r e s e n t i n t e r e s t s , o n l y the e l a s t o - p l a s t i c s o l u t i o n by B a g u e l i n e t a l (1978) i s p r e s e n t e d h e r e i n : I t i s assumed t h a t the c o h e s i v e s o i l f a i l s i n accordance w i t h the T r e s c a ' s c r i t e r i o n , i . e . o r - ar = 2 C u (5.2.1) In the e l a s t i c r e g i o n , the s t r e s s changes i n s o i l s f o l l o w the e l a s t i c i t y t h e o r y (Timoshenko and G o o d i e r , 1951): A a r = - Ao-0 (5.2.2) and s o i l s s t a r t f a i l u r e when re a c h e s a p a r t i c u l a r v a l u e P f , i . e . P f = P 0 + C u (5.2.3) which e x i s t s i n t h e e l a s t o - p l a s t i c i n t e r f a c e , r ^ . T h i s c o n d i t i o n i s f i r s t r e a ched a t the w a l l of c a v i t y and then p r o p a g a t e s i n t o the s o i l medium as the a p p l i e d p r e s s u r e P 66 c o n t i n u e s t o i n c r e a s e . S t r e s s F i e l d s When s o i l i s i n t h e e l a s t o - p l a s t i c range, s t r e s s e s i n the e l a s t i c r e g i o n f o l l o w t h e e q u a t i o n s : ar = P ° + ( P f " p o H r f / r ) 2 (5.2.4) o9 = P0 - ( P f - P 0 ) ( r f / r ) 2 (5.2.5) s t a r t i n g from the boundary of e l a s t o - p l a s t i c r e g i o n , r ^ . S t r e s s e s i n the a n n u l a r p l a s t i c r e g i o n , however, would f o l l o w the f o l l o w i n g e q u a t i o n s f o r the s m a l l s t r a i n t h e o r y : a r = P f + 2 C u l n ( r f / r ) (5.2.6) CTg = P f - 2 C u l n ( r f / r ) (5.2.7) As the s t r a i n of p r e s s u r e m e t e r e x p a n s i o n t e s t s i s seldom beyond 10% i n p r a c t i c e , the e r r o r of s m a l l s t r a i n t h e o r y i s of s m a l l s i g n i f i c a n c e . From the Eq. ( 5 . 2 . 6 ) , P = ar when r = r 0 . T h e r e f o r e , the a p p l i e d p r e s s u r e AP i s AP = C u + 2 C u l n ( r f / r 0 ) (5.2.8) 67 S i n c e i n u n d r a i n e d (no volume change) c o n d i t i o n s : ( r f / r 0 ) 2 = 2 G e 0 / C u (5.2.9) T h e r e f o r e the p r e s s u r e e x p a n s i o n c u r v e i s i n form of AP = 2 G U 0 / r 0 (5.2.10) f o r e l a s t i c r e s p o n s e , and AP = C u + 2 C u l n ( 2 G e 0 / C u ) (5.2.11) f o r e l a s t o - p l a s t i c r e s p o n s e . T h e r e f o r e the i n i t i a l s l o p e of the p r e s s u r e e x p a n s i o n c u r v e i s 2G, and the l i m i t p r e s s u r e P L would be : P T = P 0 + C„[1 + ln(G/C„)] (5.2.12) Li U U b) C o h e s i o n l e s s S o i l s Many r e s e a r c h e r s have d e v e l o p e d c l o s e d form s o l u t i o n s f o r c o h e s i o n l e s s s o i l s ( G i b s o n and Anderson, 1961, V e s i c , 1972, Hughes et a l , 1977). H e r e i n Hughes e t a l s o l u t i o n i s s e l e c t e d f o r the c o m p a r i s o n , which i s p r e s e n t e d b r i e f l y as f o l l o w s : The s o i l continuum i s assumed as an i s o t r o p i c e l a s t i c , f r i c t i o n a l p l a s t i c and shear-volume c o u p l i n g m a t e r i a l i n Hughes e t a l f o r m a u l a t i o n . The s o i l behaves e l a s t i c a l l y and 68 obeys Hooke's law u n t i l the onset of p l a s t i c y i e l d i n g , which i s governed by Mohr-Coulumb c r i t e r i o n , i . e . a r = N oe (5.2.13) where 1 + S i n 0 N = ;—* , and 1 - S i n 0 <f> i s the f r i c t i o n a l a n g l e , For s i m p l i c i t y , t h e s m a l l s t r a i n t h e o r y i s a d o p t e d , and the e q u i l i b r i u m e q u a t i o n r e q u i r e d i s then : da a - aft V + — - = 0 (5.2.14) dr S t r e s s f i e l d In the e l a s t i c r e g i o n , the s t r e s s e s f o l l o w the same e l a s t i c e q u a t i o n s as f o r c o h e s i v e s o i l s ( e . g . Eq. (5.2.6) and ( 5 . 2 . 7 ) ) . In the p l a s t i c r e g i o n , however, c o m b i n i n g the Eq. (5.2.12) and (5.2.14) w i t h the o u t e r boundary c o n d i t i o n s of the p l a s t i c zone, i . e . a t r = r ^ , = o^, the s t r e s s s o l u t i o n i s : I n ( a r / a f ) = ( 1 - N ) I n ( r f / r ) (5.2.15) 69 F u r t h e r m o r e , t h e d i l a t a n c y e f f e c t o f g r a n u l a r m a t e r i a l s o c c u r r i n g i n t h e f a i l u r e s t a g e i s i g n o r e d h e r e i n f o r s i m p l i c i t y . T h e r e f o r e , u s i n g t h e a s s u m p t i o n o f no v o l u m e c h a n g e i n t h e p l a s t i c r e g i o n , a n d t h e b o u n d a r y c o n d i t i o n s : r = r f , e = e f r = r 0 , e = e 0 ( 5 . 2 . 1 6 ) t h e s o l u t i o n f o r s t r a i n s w i t h i n t h e a n n u l a r p l a s t i c z o n e i s : , V 2 / ( 1 - N ) e 0 38 « f U r / a f ) . ( 5 . 2 . 1 7 ) where e 0 i s t h e s t r a i n m e a s u r e d a t t h e w a l l o f c a v i t y , a n d a f = P 0 ( 1 + S i n 0 ) ( 5 . 2 . 1 8 ) e f = P 0 S i n 0 / 2 G ( 5 . 2 . 1 9 ) T h u s t h e p r e s s u r e e x p a n s i o n c u r v e f o r i n c o m p r e s s i b l e e l a s t o - f r i c t i o n a l p l a s t i c c o h e s i o n l e s s s o i l s i s : AP + P 0 = ( e 0 / e f ) ( 1 N ) / 2 ( 5 . 2 . 2 0 ) where AP i s t h e a p p l i e d p r e s s u r e a t t h e w a l l o f c a v i t y , and P 0 i s t h e i n s i t u i s o t r o p i c p r e s s u r e . 70 5.3 FINITE ELEMENT SIMULATION 5.3.1 FINITE ELEMENT MESH DOMAIN As d i s c u s s e d b e f o r e , the c y l i n d r i c a l c a v i t y e x p a n s i o n i s an a x i s y m m e t r i c a l and p l a n e s t r a i n problem, the s t r e s s and s t r a i n f i e l d s a r e o n l y dependent upon the r a d i a l d i s p l a c e m e n t . An i n f i n i t e d i s k of u n i t t h i c k n e s s of the s o i l c r o s s s e c t i o n i s then taken f o r t h e a n a l y s i s . However, f o r the f i n i t e element s i m u l a t i o n of the problem, a l a r g e but f i n i t e d i s k of t h e s o i l c r o s s s e c t i o n i s t a k e n f o r the a n a l y s e s , w h i c h i s shown i n F i g . 5 . 2 ( a ) . S i n c e the problem i s a x i s y m m e t r i c about the c e n t r a l a x i s , an a x i s y m m e t r i c f i n i t e elememt mesh was employed i n the a n a l y s e s , as shown i n F i g . 5 . 2 ( b ) . The p l a n e s t r a i n c o n d i t i o n i s o b t a i n e d by imposing the d i s p l a c e m e n t c o n s t r a i n t s i n the v e r t i c a l d i r e c t i o n . F o r t h i s purpose, a s e r i e s of r o l l e r s were p l a c e d a t nodes of b o t h upper and bottom b o u n d a r i e s of the mesh domain. However, f o r the i n n e r and o u t e r b o u n d a r i e s s t r e s s boundary c o n d i t i o n s were assumed. At t h e i n n e r boundary, the p r e s s u r e P i s a p p l i e d on the c a v i t y , w h i l e a t the o u t e r boundary the s t r e s s e s a c t i n g on a r e e q u a l t o t h e i n s i t u s t r e s s s t a t e , assuming no s t r e s s changes i n t h e s o i l s beyond t h i s boundary. Hence, b o t h b o u n d a r i e s a r e f r e e t o move r a d i a l l y outwards under the p r e s s u r e P. The p r e s s u r e e x p a n s i o n c u r v e i s o b t a i n e d from the a p p l i e d p r e s s u r e and the r e s u l t i n g d i s p l a c e m e n t , or the c i r c u m f e r e n t i a l s t r a i n a t the w a l l of c a v i t y . The F i g . 5.2(a) S o i l Domain used f o r F i n i t e Element A n a l y s i s F i g . 5.2(b) F i n i t e Element Mesh f o r C a v i t y E x p a n s i o n S i m u l a t i o n 72 c i r c u m f e r e n t i a l s t r a i n i s c a l c u l a t e d as A U 0 / r 0 based on the s m a l l s t r a i n t h e o r y . In a l l the f o l l o w i n g a n a l y s e s , an i n i t i a l c a v i t y r a d i u s , r 0 = 50 mm. i s assumed. 5.3.2 OUTER BOUNDARY EFFECTS The i n f l u e n c e s of the o u t e r boundary R of the mesh domain on the f i n i t e element p r e d i c t i o n was examined f o r c o h e s i v e s o i l s under u n d r a i n e d c o n d i t i o n . Two f i n i t e element a n a l y s e s w i t h o u t e r r a d i a of 5 0 r o and l 0 0 r o were performed u s i n g t h e d e s c r i b e d mesh domain and boundary c o n d i t i o n s . The r e s u l t s a r e shown i n F i g . 5.3. As e x p e c t e d , the f i n i t e element p r e d i c t i o n w i t h the l a r g e r o u t e r r a d i u s , 1 0 0 r o , g i v e s a s o f t e r p r e s s u r e e x p a n s i o n c u r v e than t h a t w i t h the s m a l l e r o u t e r r a d i u s , 5 0 r o . T h i s i s because when the o u t e r r a d i u s i s l a r g e r , more s o i l r e g i o n i s s t r e s s e d , and c o n s e q u e n t l y more s o i l d e f o r m a t i o n i s c a l c u l a t e d a t the w a l l of c a v i t y , l e a d i n g t o a s o f t e r r e s p o n s e . As shown i n the f i g u r e , t he two mesh models g i v e almost i d e n t i c a l response i n the v e r y s m a l l s t r a i n l e v e l where the s o i l medium i s c o m p l e t e l y i n the e l a s t i c s t a g e . When p a r t s of s o i l medium b e g i n t o y i e l d , the d i f f e r e n c e between the two models becomes l a r g e r , but g e n e r a l l y l e s s than 7%. T h e r e f o r e , a l t h o u g h the l a r g e r the o u t e r r a d i u s R of the mesh domain i s , the f i n i t e element r e s u l t s a r e c l o s e r t o the r e a l answer, the e r r o r due t o the f i n i t e mesh domain may not be s i g n i f i c a n t i f the o u t e r r a d i u s i s s u f f i c i e n t l y APPLIED PRESSURES P - KPfl 0.0 40.0 80.0 120.0 160,0 200 CD 74 l a r g e . The d i f f e r e n c e w o u l d come i n t o t h e p i c t u r e o n l y when t h e s o i l medium i s i n t h e e l a s t o - p l a s t i c s t a g e . F o r p r e s e n t i n t e r e s t s , an o u t e r r a d i u s o f l 0 0 r o was s e l e c t e d f o r t h e f o l l o w i n g a n a l y s i s . As s e e n i n l a t e r c o m p a r i s o n s , t h e f i n i t e e l e m e n t a n a l y s e s u s i n g s u c h a mesh d o m a i n c a n p r o v i d e r e s u l t s i n good a g r e e m e n t w i t h t h e c l o s e d f o r m s o l u t i o n . 5.4 F I N I T E ELEMENT PREDICTIONS 5.4.1 MATERIAL MODELS AND ANALYSES F i n i t e e l e m e n t a n a l y s e s u s i n g t h e f o r e g o i n g e l e m e n t mesh were p e r f o r m e d on b o t h c o h e s i v e s o i l s a n d c o h e s i o n l e s s s o i l s . In o r d e r t o c o m p a r e t h e e l a s t o - p l a s t i c c l o s e d f o r m s o l u t i o n p r e s e n t e d i n S e c . 5 . 2 , t h e e l a s t o - p e r f e c t l y p l a s t i c m a t e r i a l m o d e l a n d e l a s t o - f r i c t i o n a l p l a s t i c one were e m p l o y e d f o r c o h e s i v e s o i l s a n d c o h e s i o n l e s s s o i l s r e s p e c t i v e l y . The s t r a i n h a r d e n i n g , s t r a i n s o f t e n i n g , a n d t h e s h e a r vo l ume c o u p l i n g e f f e c t o f t h e r e a l s o i l b e h a v i o r , h o w e v e r , were no t c o n s i d e r e d h e r e i n f o r s i m p l i c i t y . F o r c o h e s i v e s o i l s , t h e u n d r a i n e d l o a d i n g c o n d i t i o n was a s s u m e d i n t h e a n a l y s i s , w h i c h was o b t a i n e d by u s i n g an a r b i t r a r i l y h i g h b u l k m o d u l u s e q u i v a l e n t t o a P o i s s o n r a t i o o f 0 . 4 9 9 . T o t a l s t r e s s a n a l y s i s was a d o p t e d , i . e . t h e p o r e p r e s s u r e was n o t e v a l u a t e d d u r i n g t h e l o a d i n g p r o c e s s . The p l a s t i c y i e l d i n g o f s o i l s i s s p e c i f i e d by t h e u n d r a i n e d s h e a r s t r e n g t h C^. F o r t h e s h e a r s t r e s s b e l o w t h e u n d r a i n e d s h e a r s t r e n g t h C , t h e s o i l w i l l b e h a v e e l a s t i c a l l y w h i l e 75 f o r t h e s h e a r s t r e s s a b o v e C , t h e s o i l b e h a v e s a s a p e r f e c t l y p l a s t i c m a t e r i a l . F o r c o h e s i o n l e s s s o i l s , t h e d r a i n e d l o a d i n g c o n d i t i o n was e m p l o y e d i n t h e a n a l y s i s . The P o i s s o n ' s r a t i o a s sumed i s 0 . 2 , w h i c h i s no t u n u s u a l f o r t h e p r a c t i c a l s a n d s . T h e s t r e n g t h c h a r a c t e r i s t i c s a r e g o v e r n e d by t h e M o h r - C o u l o m b c r i t e r i o n . H e n c e , t h e c o h e s i o n l e s s s o i l b a s i c a l l y b e h a v e s a s e l a s t o - f r i c t i o n a l p l a s t i c m a t e r i a l . 5 . 4 . 2 MATERIAL PROPERTIES The s o i l p a r a m e t e r s r e q u i r e d i n t h e h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n s h i p a r e p r e s e n t e d i n T a b l e 5.1 and 5.2 f o r c o h e s i v e s o i l s a n d c o h e s i o n l e s s s o i l s r e s p e c t i v e l y . The i n i t i a l s t r e s s c o n d i t i o n a s s u m e d i n t h e s o i l medium was i s o t r o p i c , w h i c h i s e q u i v a l e n t t o t h e c l o s e d f o r m s o l u t i o n . The s m a l l v a l u e o f Rf i s a f l a g i n t h e p r o g r a m t o i n d i c a t e t h e e l a s t o - p l a s t i c s o i l b e h a v i o r . U n l e s s f o r n o n l i n e a r s o i l b e h a v i o r , t h e e l a s t i c m o d u l i o f t h e s o i l a r e n o t d e p e n d e n t on t h e s t r e s s l e v e l i n t h e e l a s t o - p l a s t i c m o d e l , t h e r e f o r e t h e m o d u l u s numbers i n t h e e l a s t o - p l a s t i c m o d e l were a s s i g n e d t o z e r o . F o r c o h e s i o n l e s s s o i l , t h e v a r i a t i o n o f t h e f r i c t i o n a n g l e <t> w i t h t h e s t r e s s l e v e l i s n o t c o n s i d e r e d i n t h e c l o s e d f o r m s o l u t i o n , n o r t h e d i l a t a n c y e f f e c t o f g r a n u l a r m a t e r i a l s a t l a r g e s t r a i n l e v e l , t h e d i l a t i o n i t e r a t i o n f u n c t i o n o f t h e p r o g r a m was , t h e r e f o r e , n o t u s e d i n t h e a n a l y s i s , a n d t h e was a s s i g n e d t o z e r o i n t h e T a b l e 5.1 S o i l P arameters of C o h e s i v e S o i l s f o r C y l i n d r i c a l C a v i t y E x p a n s i o n S i m u l a t i o n S o i l E l a s t o - P l a s t i c N o n l i n e a r Model Parameters Model C u (Kpa) 7.5 7.5 K E 59.21 59.21 KB 9869 9869 V 0.499 0.499 n 0.0 0.0 m 0.0 0.0 Rf 0.0 0.9 K0 1 .0 1 .0 cr0 (Kpa) 80.0 80.0 T a b l e 5.2 S o i l Parameters of C o h e s i o n l e s s S o i l s f o r C y l i n d r i c a l C a v i t y E x p a n s i o n S i m u l a t i o n S o i l E l a s t o - P l a s t i c N o n l i n e a r Model Parameters Model ( % ) 50 50 0 36° 36° KE 600 600 KB 360 360 0.2 0.2 n 0.0 0.5 m 0.0 0.25 Rf 0.0 0.9 K0 1 .0 1 .0 (KN/m 2) 50.0 50.0 1. vQ i s the i n i t i a l p o i s s o n ' s r a t i o of sands e l a s t o - p l a s t i c model. 77 5.4.3 RESULTS AND COMPARISON In the f o l l o w i n g s t u d i e s , r e s u l t s from the o r i g i n a l program were f i r s t compared with the c l o s e d form s o l u t i o n . Based on t h i s comparison, the program was then m o d i f i e d , and good agreement with the c l o s e d form s o l u t i o n was o b t a i n e d . The m o d i f i c a t i o n s are pre s e n t e d i n d e t a i l h e r e i n . A.COHESIVE SOILS O r i g i n a l P r o g r a m The p r e d i c t e d p r e s s u r e expansion curve from the o r i g i n a l program using the e l a s t o - p e r f e c t l y p l a s t i c s o i l behavior i s shown i n F i g . 5.4 i n comparison with the c l o s e d form s o l u t i o n , I t i s c l e a r l y shown i n the f i g u r e t hat i n the e l a s t i c , e a r l y small s t r a i n l e v e l the program g i v e s the pr e s s u r e expansion curve c l o s e l y agreeable to the c l o s e d form. However, at l a r g e s t r a i n l e v e l the program p r e d i c t s much l e s s displacement. The d e v i a t i o n of the p r e d i c t e d curve s t a r t s to grow at the displacement of 0.8mm, i . e . 1.8% s t r a i n , where the p l a s t i c s o i l deformation medium has oc c u r r e d near the c a v i t y . The d i f f e r e n c e between the curves seems to accumulate as more s o i l r e g i o n becomes p l a s t i c . The l a r g e s t e r r o r i n displacement p r e d i c t i o n appeared i n the f i g u r e i s more than 35%, which occurs at the end of a n a l y s i s , and the c o r r e s p o n d i n g e r r o r i n the pr e s s u r e p r e d i c t i o n i s about 10%. I t i s found out that i n the o r i g i n a l program the e l a s t i c shear and bulk moduli are d e f a u l t e d by a f a c t o r of F i g . 5.4 C o m p a r i s o n o f O r i g i n a l P r o g r a m a n d C l o s e d F o r m S o l u t i o n CD 79 1000 a n d 10 r e s p e c t i v e l y when t h e s o i l e l e m e n t r e a c h e s f a i l u r e . The s m a l l d e f a u l t e d s h e a r modu lu s i s t o s i m u l a t e t h e l a r g e p l a s t i c d e f o r m a t i o n i n t h e s u b s e q u e n t l o a d i n c r e m e n t s . H o w e v e r , d e f a u l t i n g t h e b u l k m o d u l u s f o r t h e s h e a r - f a i l e d e l e m e n t h a s l a c k o f p h y s i c a l b a s i s , a n d i s c o n s i d e r e d t o be i n a p p r o p r i a t e a s i t w i l l r e l a x t h e s i m u l a t i o n o f u n d r a i n e d c o n d i t i o n s f o r c o h e s i v e s o i l s . In o u r a n a l y s i s , t h e u n d r a i n e d c o n d i t i o n i s s i m u l a t e d u s i n g a h i g h b u l k m o d u l u s e q u i v a l e n t t o t h e P o i s s o n ' s r a t i o o f 0 . 4 9 9 . T h e r e f o r e , t h e r e d u c t i o n o f b u l k m o d u l u s f o r f a i l e d e l e m e n t s w i l l p r o d u c e some vo lume c h a n g e s , l e a d i n g t o a s o f t e r p r e s s u r e e x p a n s i o n c u r v e . In f a c t t h e q u a l i t y o f t h e u n d r a i n e d c o n d i t i o n s i m u l a t i o n c a n be c h e c k e d f r o m t h e c a l c u l a t e d d i s p l a c e m e n t s a t t h e i n n e r a n d o u t e r b o u n d a r i e s . As shown i n F i g . 5 . 5 , t h e u n d r a i n e d ( o r no v o l u m e c h a n g e ) c o n d i t i o n w i l l r e s u l t i n a r e l a t i o n s h i p b e t w e e n i n n e r and o u t e r b o u n d a r y d i s p l a c e m e n t s : U 0 = R - 6 / r 0 ( 5 . 4 . 1 ) I t was f o u n d t h a t f o r t h e f i n i t e e l e m e n t r e s u l t s shown i n F i g . 5 . 4 , E q . ( 5 . 4 . 1 ) was n o t f u l l y s a t i s f i e d . T h e r e f o r e , t h e r e a l p r e s s u r e e x p a n s i o n c u r v e p r e d i c t e d f r o m t h e p r o g r a m w o u l d h a v e b e e n e v e n s t i f f e r t h a n what i s shown i n F i g . 5.4 i f t h e b u l k m o d u l u s was n o t d e f a u l t e d by a f a c t o r o f 10. The s t i f f e r r e s p o n s e p r e d i c t e d by t h e p r o g r a m i s c o n s i d e r e d t o r e s u l t f r o m t h e h i g h s t r e s s r a t i o a t t r a c t e d i n F i g . 5.5 Displacement D i s t r i b u t i o n under Undrained Plane S t r a i n C o n d i t i o n 81 f a i l u r e e l e m e n t s . Due t o t h e n a t u r e o f i n c r e m e n t a l a n a l y s i s , e x t r a s t r e s s e s o f f e n d i n g t h e f a i l u r e e n v e l o p e i n some f a i l u r e e l e m e n t s a r e a l m o s t i n e v i t a b l e , t h e y w i l l r e m a i n i n t h e f a i l u r e e l e m e n t s , g i v i n g a s t i f f e r r e s p o n s e i n t h e p r e s s u r e e x p a n s i o n c u r v e . An e f f e c t i v e method t o s p r e a d t h e e x c e s s s t r e s s f r o m t h e f a i l u r e e l e m e n t s t o t h e a d j a c e n t e l e m e n t s i s t h e l o a d s h e d d i n g i t e r a t i o n t e c h n i q u e . However, s u c h a p r o c e d u r e was n o t i n c l u d e d i n t h e o r g i n a l p r o g r a m f o r c o h e s i v e s o i l s . M o d i f i c a t i o n s of the Program W i t h r e s p e c t t o t h e above l i m i t a t i o n s i n t h e o r i g i n a l p r o g r a m , m o d i f i c a t i o n s were made t o improve t h e a b i l i t y and a c c u r a c y o f t h e p r o g r a m i n t h e a n a l y s i s f o r c o h e s i v e s o i l s . The f e a t u r e o f d e f a u l t i n g t h e b u l k modulus f o r f a i l u r e e l e m e n t s was d i s c a r d e d , and t h e l o a d s h e d d i n g i t e r a t i o n t e c h n i q u e was i n c o r p o r a t e d so t h a t a g e n e r a l c, <t> m a t e r i a l c a n be h a n d l e d by t h e p r o g r a m . The m o d i f i e d p r o g r a m was t h e n employed t o a n a l y s e t h e same p r o b l e m u s i n g t h e same d a t a a s b e f o r e . The r e s u l t s a r e shown i n F i g . 5.6 i n c o m p a r i s o n w i t h t h o s e f r o m t h e c l o s e d f o r m s o l u t i o n a n d t h e o r i g i n a l p r o g r a m . I t i s s e e n t h a t t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e f r o m t h e m o d i f i e d p r o g r a m i s i n e x c e l l e n t a g r e e m e n t w i t h t h e c l o s e d f o r m s o l u t i o n h a v i n g an i n i t i a l s l o p e a b o u t 1.0226(2G^). The e r r o r i s n e a r l y i n v i s i b l e . In t h i s c a s e , t h e u n d r a i n e d l o a d i n g c o n d i t i o n i s a c c u r a t e l y s i m u l a t e d a s t h e E q . (5.4.1) i s : — CLOSED FORM SOLUTION c r o H h ORIGINAL PROGRAM ^ 0 MODIFIED PROGRAM n _ x x NONLINEAR SOILS o ' -^ I I I I I I I I I I I I I 1 I I I I I 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40 CIRCUMFERENTIAL STRAIN [%) F i g . 5.6 Comparison of M o d i f i e d Program and C l o s e d Form S o l u t i o n 83 s a t i s f i e d w i t h an e r r o r l e s s than 1%. In a d d i t i o n , the n o n l i n e a r , s t r e s s dependent s o i l b e h a v i o r was a l s o employed i n the a n a l y s i s u s i n g the m o d i f i e d program, the r e s u l t i s a l s o shown i n F i g . 5.6. As e x p e c t e d , i t g i v e s a s o f t e r p r e s s u r e e x p a n s i o n c u r v e as compared w i t h the c l o s e d form s o l u t i o n . The c u r v e does not e x h i b i t an i n i t i a l l i n e a r e l a s t i c p o r t i o n as i n e l a s t i c p l a s t i c s o l u t i o n . T h e r e f o r e , the i n i t i a l s l o p e c a l c u l a t e d a t the end of t h e f i r s t l o a d increment i s l e s s than 2G. i n the I n o n l i n e a r , s t r e s s l e v e l dependent a n a l y s i s . In the meantime, the s t r e s s d i s t r i b u t i o n p r e d i c t e d from the m o d i f i e d program i n the e l a s t o - p l a s t i c s o i l mass i s compared w i t h the c l o s e d form s o l u t i o n p r e s e n t e d i n Sec. 5.2.2. The comparisons a r e shown i n F i g . 5 . 7 ( a ) , ( b ) . As shown i n the f i g u r e s , the p a t t e r n of the s t r e s s d i s t r i b u t i o n and the p r e d i c t e d s i z e of p l a s t i c zone a r e a l l i n good agreement w i t h the c l o s e d form s o l u t i o n . B. COHESIONLESS SOILS I t s h o u l d be n o t e d f i r s t t h a t the e l a s t o - p l a s t i c s o l u t i o n of t h e p r e s s u r e e x p a n s i o n c u r v e from the f i n i t e element a n a l y s i s f o r the c y l i n d r i c a l c a v i t y e x p a n s i o n i n c o h e n s i o n l e s s s o i l s has not been examined so f a r by the p r e v i o u s r e s e a r c h e r s (She, 1986) due t o i t s i n h e r e n t d i f f i c u l t i e s . An attempt was made i n t h i s t h e s i s t o u n d e r s t a n d the d i f f e r e n t f a c t o r s which a f f e c t the f i n i t e element a n a l y s i s by means of comparing the f i n i t e element 84 CC" - CLOSED 1 FORM S O L U T I O N + F I N I T E E L E M E N T (a) c/v_:-or -c r CD " d o i — 32.0 0.0 4.0 8.0 12.0 16.0 20.0 24.0 R A D I A L D I S T A N C E (R/RO) 28.0 36.0 40.0 CC c n . , _ U 5 cn CCo or or C L O S E D FORM S O L U T I O N (b) aQ + + F I N I T E E L E M E N T 1-0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0 R A O I H L D I S T A N C E (R/RO) F i g . 5.7 Stress D i s t r i b u t i o n i n Comparison with Closed Form S o l u t i o n 85 p r e d i c t i o n w i t h i t s c o u n t e r p a r t , c l o s e d form s o l u t i o n . The o r i g i n a l program was then m o d i f i e d t o o b t a i n an agreement between t h e s e two s o l u t i o n s . O r i g i n a l Program The p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s from the o r i g i n a l program u s i n g e l a s t o - p l a s t i c f r i c t i o n a l m a t e r i a l model i s p r e s e n t e d i n F i g . 5.8. For the sake of c o m p a r i s o n s , the e l a s t o - p l a s t i c c l o s e d form s o l u t i o n and the n o n l i n e a r f i n i t e element p r e d i c t i o n a r e a l s o shown i n the F i g . 5.8. The c l o s e d form s o l u t i o n i s of the form proposed by Hughes et al (1977), which has been p r e s e n t e d i n Sec. 5.2. The i n i t i a l P o i s s o n r a t i o used i n a l l the a n a l y s e s was 0.2, which remains c o n s t a n t f o r the e l a s t o - p l a s t i c a n a l y s e s but v a r i e s f o r the n o n l i n e a r a n a l y s i s . As shown i n F i g . 5.8, the e l a s t o - p l a s t i c f i n i t e element a n a l y s i s g i v e s r e s u l t s a g r e e a b l e w i t h c l o s e d form s o l u t i o n o n l y i n the s m a l l s t r a i n l e v e l , i . e . up t o 0.1%, where the e l a s t i c s o i l medium i s i n t h e e l a s t i c d e f o r m a t i o n s t a g e . Beyond t h a t l e v e l , the p l a s t i c d e f o r m a t i o n of s o i l s o c c u r s , the f i n i t e element a n a l y s i s then p r e d i c t s much s o f t e r p r e s s u r e e x p a n s i o n c u r v e . The d e v i a t i o n of t h e s e two c u r v e s seems t o accumulate as more s o i l r e g i o n i s i n the p l a s t i c d e f o r m a t i o n s t a g e . Comparing the two f i n i t e element p r e d i c t i o n s , the e l a s t o - p l a s t i c s o l u t i o n i s s t i f f e r than the n o n l i n e a r one u n t i l t he s t r a i n l e v e l r eaches 2.4%, which i s u s u a l l y cr Q_ o (NI Closed Form Solution • + — r- FEU: Elast ic P last ic o e> FEtl: Nonlinear Soi ls i i i i i i i i r 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 CIRCUMFERENTIAL STRAIN IX) F i g . 5.8 P r e s s u r e E x p a n s i o n C u r v e f o r C o h e s i o n l e s s S o i l s f r o m O r i g i n a l P r o g r a m 3.2 3.6 4.0 oo cn 87 e x p e c t e d . H o w e v e r , a s shown i n t h e f i g u r e , t h e e l a s t o - p l a s t i c s o l u t i o n becomes e v e n s o f t e r t h a n t h e n o n l i n e a r f o r t h e s t r a i n l e v e l b e y o n d 2 .4%, S u c h an i n c o r r e c t t r e n d o f F E p r e d i c t i o n s q u e s t i o n s t h e a c c u r a c y o f t h e o r i g i n a l p r o g r a m i n t h e e l a s t i c - p l a s t i c a n a l y s i s . Modifications of the Program 1) Permanent failure stress record ' As s t a t e d e a r l i e r , t h e e l a s t o - p l a s t i c p r e s s u r e e x p a n s i o n c u r v e f r o m f i n i t e e l e m e n t a n a l y s i s i s t o o s o f t . T h i s was f o u n d t o be due t o t h e i m p r o p e r n u m e r i c a l t r e a t m e n t o f f a i l e d s o i l e l e m e n t s i n t h e o r i g i n a l p r o g r a m . In t h e o r i g i n a l p r o g r a m , t h e r e i s a s t r e s s s t a t e memory w h i c h r e c o r d s t h e s h e a r s t r e s s r a t i o o f f a i l e d e l e m e n t s . Once a s o i l e l e m e n t i s f a i l e d , f a i l u r e s t r e s s s t a t e o f t h e e l e m e n t w i l l be p e r m a n t l y m e m o r i z e d , a n d t h e r e f o r e t h e e l e m e n t i s r e g a r d e d t o be f a i l e d f o r e v e r . T h i s i m p l i e s t h a t t h e s t r e s s s t a t e o f t h e f a i l u r e s o i l e l e m e n t w i l l s t a y on t h e f a i l u r e e n v e l o p e a t t h e same p o i n t f o r a l l t h e f o l l o w i n g l o a d i n g p r o c e s s , e . g . P o i n t A i n F i g . 5 . 9 . In r e a l i t y , s u c h a t r e a t m e n t i s o n l y a p p r o p r i a t e f o r t h e u n d r a i n e d c o h e s i v e s o i l s where t h e u n d r a i n e d s h e a r s t r e n g t h i s i n d e p e n d e n t o f t h e c u r r e n t n o r m a l s t r e s s s t a t e . F o r c o h e s i o n l e s s f r i c t i o n a l s o i l s , h o w e v e r , t h e s t r e n g t h i s d e p e n d e n t upon t h e c u r r e n t s t a t e o f s t r e s s e s , a s shown i n t h e m o d i f i e d M o h r - C o u l o m b d i a g r a m i n F i g . 5 . 9 . 00 00 89 As shown i n t h e f i g u r e , i n i t i a l l y t h e s o i l medium i s i n e l a s t i c r e g i o n i n w h i c h t h e mean n o r m a l s t r e s s r e m a i n s c o n s t a n t , t h e s t r e s s p a t h i n c r e m e n t i s t h e r e f o r e s t r a i g h t l i n e v e r t i c a l l y up t o P o i n t A on t h e f a i l u r e e n v e l o p e . When t h e s t r e s s s t a t e o f an e l e m e n t r e a c h t h e f a i l u r e e n v e l o p e a t p o i n t A , a l a r g e amount o f p l a s t i c d e f o r m a t i o n o c c u r s . H o w e v e r , t h e s t r e n g t h o f t h e e l e m e n t w i l l i n c r e a s e a s t h e mean n o r m a l s t r e s s i n c r e a s e i n t h e f o l l o w i n g l o a d i n c r e m e n t s . The s t r e s s p a t h i n c r e m e n t o f t h e e l e m e n t w i l l move a l o n g t h e i n c l i n e d f a i l u r e e n v e l o p e i n s t e a d o f s t a y i n g a t p o i n t A f o r e v e r a s a s s u m e d i n t h e p r o g r a m . T h e r e f o r e , i t i s i m p r o p e r t o h a v e a p e r m a n e n t f a i l u r e s t r e s s s t a t e r e c o r d f o r c o h e s i o n l e s s f r i c t i o n a l s o i l s . A f t e r c o r r e c t i n g t h e e r r o r i n t h e p r o g r a m , t h e r e s u l t i s shown i n F i g . 5 . 1 0 . As e x p e c t e d , t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e becomes s t i f f e r t h a n t h e p r e v i o u s o n e , m o v i n g t o w a r d s t h e c l o s e d f o r m s o l u t i o n . H o w e v e r , h i g h o s c i l l a t i o n a p p e a r s i n t h e c u r v e , w h i c h i n d i c a t e s t h a t n u m e r i c a l i n s t a b i l i t y o c c u r r e d i n t h e a n a l y s i s . T h i s p r o b l e m i s f r e q u e n t l y r e p o r t e d i n t h e p r e v i o u s work ( A t u k o r a l a a n d B y r n e , 1984, S h e , 1 9 8 6 ) . However t h e r e a s o n and t r e a t m e n t s have n o t b e e n f u l l y e x p l o r e d . 2) R e d u c t i o n f a c t o r f o r shear modulus F l u c t u a t i o n o f t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e i s c o n s i d e r e d t o be due t o t h e t o o much r e d u c t i o n o f s h e a r m o d u l u s f o r t h e f a i l u r e e l e m e n t s . In t h e o r i g i n a l p r o g r a m , cr o_ ^ -1 (\J co — C l o s e d Form S o l u t i o n -+ FEM: w i t h s t r e s s memory -O FEM: wit h o u t s t r e s s memory |— | | | | | | | | 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 CIRCUMFERENTIAL STRAIN it) 3.2 3.6 4.0 F i g . 5.10 P r e s s u r e Expansion Curve a f t e r D i s c a r d i n g S t r e s s Memory o 91 t h e s h e a r m o d u l u s i s d e f a u l t e d t o a s m a l l v a l u e by a f a c t o r o f 1000, w h i c h i s t h e same a s f o r c o h e s i v e s o i l s . T h e r e f o r e , f o r c o h e s i o n l e s s s o i l s where t h e s t r e n g t h i n c r e a s e s w i t h t h e mean n o r m a l s t r e s s , t h e f a i l u r e e l e m e n t s u n d e r g o an a b r u p t b a c k - a n d - f o r t h c h a n g e i n s h e a r m o d u l u s f r o m a l a r g e number t o a s m a l l n u m b e r . T h i s amount s t o s a y i n g t h a t t h e s t r e s s p a t h f o l l o w e d by t h e f a i l e d e l e m e n t s c h a n g e s s u d d e n l y a n d i n c r e m e n t a l l y f r o m a v e r t i c a l l i n e t o a n e a r l y h o r i z o n t a l l i n e , z i g z a g g i n g a l o n g t h e i n c l i n e d f a i l u r e e n v e l o p e . As a r e s u l t , n u m e r i c a l i n s t a b i l i t y w i l l be i n c u r r e d , e s p e c i a l l y f o r e l a s t o - p l a s t i c m a t e r i a l . By i n s p e c t i o n , t h e s t a b i l i t y c o n d i t i o n w o u l d , t h e r e f o r e , be i m p r o v e d i f a s m a l l e r r e d u c t i o n f a c t o r i s a d o p t e d f o r t h e c o h e s i o n l e s s s o i l s . In t h i s c a s e , t h e s t r e s s p a t h i n c r e m e n t s f o r e l a s t o - p l a s t i c m a t e r i a l w o u l d c h a n g e more g r a d u a l l y f r o m a v e r t i c a l l i n e t o an i n c l i n e d s l o p e r a t h e r t h a n a n e a r l y h o r i z o n t a l l i n e . In t h e o r y , i t seems t h a t an o p t i m a l r e d u c t i o n f a c t o r e x i s t s , w h i c h d e p e n d s upon t h e f r i c t i o n a l a n g l e 0 o f t h e f a i l u r e e n v e l o p e , a n d t h e l o a d i n g p a t h a s w e l l . F o r c o h e s i v e s o i l s , t h e f a i l u r e e n v e l o p e o f t h e u n d r a i n e d s h e a r s t r e n g t h i s a h o r i z o n t a l l i n e w i t h 0 = 0 , t h e n i n t h e a n a l y s i s a l a r g e v a l u e o f t h e r e d u c t i o n f a c t o r i s u s e d t o a l l o w t h e s t r e s s p a t h i n c r e m e n t t o move n e a r l y h o r i z o n t a l l y a l o n g t h e f a i l u r e e n v e l o p e . F o r c o h e s i o n l e s s s o i l s , i n s t e a d , a s m a l l e r r e d u c t i o n v a l u e s h o u l d be u s e d i n v i e w o f t h e i n c l i n e d f a i l u r e e n v e l o p e , so t h a t t h e s t r e s s p a t h i n c r e m e n t c a n 92 approach the f a i l u r e e nvelope more c l o s e l y and g r a d u a l l y . Such a concept i s c o n s i s t e n t w i t h p a s t e x p e r i e n c e s i n the a n a l y s i s f o r sands. In the dense sand, i t was found t h a t the n u m e r i c a l i n s t a b i l i t y problem i s more l i k e l y t o o c c u r than i n the l o o s e sand i f the same r e d u c t i o n as f o r c l a y i s used. T h i s i s because i n the dense sand, the f r i c t i o n a n g l e <t> i s h i g h , a s m a l l e r r e d u c t i o n f a c t o r i s t h e r e f o r e e x p e c t e d than i n the l o o s e sand. For the n o n l i n e a r a n a l y s i s , as the s t r e s s path' increment i s c u r v e d , g r a d u a l l y a p p r o a c h i n g the f a i l u r e e n v e l o p e , t h e r e f o r e t h e n u m e r i c a l problem may not be en c o u n t e r e d even a l a r g e v a l u e of r e d u c t i o n f a c t o r i s used. However, such an o p t i m a l r e d u c t i o n f a c t o r i s d i f f i c u l t t o dete r m i n e a p r i o r i f o r an a n a l y s i s . S i n c e i n t h e h y p e r b o l i c s t r e s s s t r a i n r e l a t i o n , the e l a s t i c m o duli a r e r e l a t e d t o each o t h e r v i a P o i s s o n ' s r a t i o , t h e r e i s a r e l a t i o n s h i p between t h e r e d u c t i o n f a c t o r and the P o i s s o n ' s r a t i o i f the b u l k modulus remains unchanged. Such a r e l a t i o n s h i p i s shown as : 1 .5B - G f u f = zrz — (5.4.2) f 3B + G f where G^ = G/Rd, and Rd i s the r e d u c t i o n f a c t o r f o r the f a i l u r e shear modulus. T a b l e 5.3 i l l u s t r a t e s the r e l a t i o n between the r e d u c t i o n f a c t o r and the P o i s s o n ' s r a t i o f o r the f a i l u r e e l e m e n t s . As shown i n the t a b l e , a r e d u c t i o n f a c t o r of 1000 i s c o r r e s p o n d e n t t o a P o i s s o n ' s r a t i o of 0.49965. 93 T a b l e 5.3 R e l a t i o n s h i p Between Shear Modulus R e d u c t i o n F a c t o r and F a i l u r e P o i s s o n R a t i o «d 1000 350 70 35 25 20 0.49965 0.499 0.495 0.490 0.485 0.480 In view of t h i s , a f a i l u r e P o i s s o n ' s r a t i o of 0.48 c o r r e s p o n d i n g t o a r e d u c t i o n f a c t o r of about 20 i n Ta b l e 5.3, was used i n the a n a l y s i s . The r e s u l t s a r e shown i n F i g . 5.11. As e x p e c t e d , the p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e i s c l e a r l y smoothed up w i t h the s m a l l e r v a l u e of r e d u c t i o n f a c t o r , which p r o v e s the f o r e g o i n g a n a l y s i s . 3) Load s h e d d i n g e f f e c t Due t o the i n c r e m e n t a l n a t u r e of the f i n i t e element a n a l y s i s , the s t r e s s p a t h increment z i g z a g s a l o n g w i t h the f a i l u r e e n v e l o p e when the s o i l element i s f a i l e d . A c c u r a c y of t h e a n a l y s i s t h e r e f o r e i n p a r t depends on the s i z e of each l o a d i n c r e m e n t . S i n c e the o p t i m a l s i z e of l o a d increment i s d i f f i c u l t t o d e t e r m i n e a p r i o r i , l o a d shedding i t e r a t i o n t e c h n i q u e i s c o n s i d e r e d as n e c e s s i t y i n the a n a l y s i s t o r e d i s t r i b u t e the e x c e s s shear s t r e s s a t t r a c t e d i n f a i l u r e elements t o the a d j a c e n t u n f a i l e d e l e m e n t s . The i n f l u e n c e of the s t r e s s r e d i s t r i b u t i o n on t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e i s shown i n F i g . 5.12. A f t e r s t r e s s cr CNJ ro 4-- Closed Form So lu t ion -+ FEN: large reduction •o FEM: small reduction i 1 1 1 1 1 i i r 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 CIRCUMFERENTIAL STRAIN {%) F i g . 5.11 P r e s s u r e E x p a n s i o n C u r v e w i t h S m a l l e r R e d u c t i o n F a c t o r f o r S h e a r M o d u l u s 3.2 3.6 4.0 96 r e d i s t r i b u t i o n , the h i g h shear s t r e s s r a t i o s a t t r a c t e d i n f a i l u r e elements a r e load-shedded t o t h e a d j a c e n t e l e m e n t s , t h e r e f o r e p r o d u c i n g a s o f t e r r e s p o n s e . 4) P l a s t i c - v o l u m e c o r r e c t i o n As shown i n F i g . 5.12, the p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e w i t h s m a l l e r r e d u c t i o n f a c t o r and l o a d s h e d d i n g i t e r a t i o n i s much s o f t e r than the c l o s e d form. I t i s aware t h a t f o r m u l a t i o n of the c l o s e d form s o l u t i o n shown i n Sec. 5.2 employs the no volume change a s s u m p t i o n t h r o u g h o u t the e n t i r e e l a s t o - p l a s t i c s o i l r e g i o n . For t h e f i n i t e element a n a l y s i s , a c c o r d i n g t o the e l a s t i c i t y t h e o r y , the no volume change w i l l be i n h e r e n t l y t r u e i n e l a s t i c r e g i o n where the mean normal s t r e s s remains c o n s t a n t . However, i n the a n n u l a r p l a s t i c r e g i o n the program w i l l a u t o m a t i c a l l y c a l c u l a t e the " p l a s t i c " volume change i n c r e m e n t a l l y t h r o u g h the e l a s t i c b u l k modulus as i n t h a t r e g i o n the mean normal s t r e s s w i l l i n c r e a s e w i t h the l o a d i n c r e m e n t s . T h e r e f o r e , i f the f i n i t e element p r e d i c t i o n i s t o be r e a l l y compared w i t h the c l o s e d form s o l u t i o n , the " p l a s t i c " volume change s h o u l d be c o r r e c t e d from the f i n i t e element r e s u l t s . As shown i n F i g . 5.13, the e x t r a d i s p l a c e m e n t a t the c a v i t y due t o the volume change i n t h e a n n u l a r p l a s t i c r e g i o n of f i n i t e element a n a l y s i s can be c a l c u l a t e d as : 1 n AU f = - E (e . r . Ar.) (5.4.3) f r o i = 1 V 1 1 1 97 A P 2ir r 0 AU f = I (2w e . r- Ar. ) f " i . i v i 1 i F i g . 5.13 P l a s t i c Volume C o r r e c t i o n f o r F i n i t e Element A n a l y s i s 9 8 where: AU^ i s t h e r a d i a l d i s p l a c e m e n t a t the w a l l of c a v i t y due t o the volume change i n a n n u l a r p l a s t i c zone, e v ^ i s the v o l u m e t r i c s t r a i n of elements i n the a n n u l a r p l a s t i c zone, Ar. i s the w i d t h of the element, r . i s the r a d i u s of 1 ' 1 elements t o t h e c e n t r a l a x i s , r 0 i s the i n i t i a l r a d i u s of the c a v i t y . Such an e x t r a d i s p l a c e m e n t i s then s u b t r a c t e d from the p r e d i c t e d d i s p l a c e m e n t a t the w a l l of c a v i t y . The p r e s s u r e e x p a n s i o n c u r v e a f t e r the p l a s t i c volume c o r r e c t i o n i s shown i n F i g . 5.14 i n comparison w i t h the c l o s e d form s o l u t i o n . As shown i n the f i g u r e , the response i s s t i f f e r than the p r e v i o u s one, and i t s agreement w i t h the c l o s e d form s o l u t i o n i s r e m a r k a b l e . The i n i t i a l s l o p e i s a p p r o x i m a t e l y e q u a l t o 2G^. The n o n l i n e a r e l a s t i c p r e s s u r e e x p a n s i o n c u r v e from m o d i f i e d program i s a l s o i n c l u d e d i n F i g . 5.14. As e x p e c t e d , i t i s s o f t e r than the e l a s t i c p l a s t i c c u r v e , i t s i n i t i a l s l o p e i s l e s s than 2G^. T h e r e f o r e the a c c u r a c y of the m o d i f i e d program i s a s s u r e d . In summary, the m o d i f i e d program i s c a p a b l e of p r o v i d i n g the r e s u l t s t h a t a r e i n remarkable agreement w i t h the c l o s e d form s o l u t i o n s f o r b o t h c o h e s i v e and c o h e s i o n l e s s s o i l s . D i f f e r e n t f a i l u r e shear modulus r e d u c t i o n f a c t o r s may be r e q u i r e d f o r c o h e s i v e s o i l s and c o h e n s i o n l e s s s o i l s , e s p e c i a l l y f o r dense sand i n which a s m a l l e r v a l u e may be n e c e s s a r y . I t i s a l s o shown i n t h e f o r e g o i n g a n a l y s i s t h a t the l o a d shedding i t e r a t i o n t e c h n i q u e i s a p o w e r f u l t o o l t o f a c i l i t a t e the program t o s i m u l a t e the s o i l f a i l u r e o — ~ ] CE C l o s e d Form S o l u t i o n w " H h FEM: b e f o r e volume c o r r e c t i o n l~ 1 -ro — O O FEM: a f t e r volume c o r r e c t i o n I _ X X FEM: n o n l i n e a r s o l u t i o n 4.0 Volume C o r r e c t i o n c o n d i t i o n s u s i n g the s i m p l e h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n . 6. FINITE ELEMENT STUDIES OF PRESSUREMETER TESTS 6.1 INTRODUCTION In the f i e l d of s o i l mechanics, t h e r e has l o n g been an emphasis on l a b o r a t o r y t e s t i n g f o r d e f i n i t i o n of the d e s i g n parameters f o r e a r t h s t r u c t u r e s . However, because of the r e c o g n i t i o n of b a s i c problems c r e a t e d by d i s t u r b a n c e s and s t r e s s - r e l i e f i n the s a m p l i n g and sample p r e p a r a t i o n p r o c e s s , which a r e more common i n the case of c o h e s i o n l e s s sands, i n - s i t u t e s t i n g p r o c e d u r e s have g a i n e d i n c r e a s i n g a t t e n t i o n . One of the common and a t t r a c t i v e d e v i c e s f o r i n - s i t u t e s t i n g of s o i l s i s the p r e s s u r e m e t e r . In t h e o r y , the r e s u l t s from a p r e s s u r e m e t e r t e s t can be used t o de t e r m i n e the i n - s i t u l a t e r a l s t r e s s i n t h e s o i l , and t h e s t r e s s - s t r a i n response and shear s t r e n g t h of the s o i l . The b a s i c i d e a of t h e p r e s s u r e m e t e r t e s t i s r e l a t i v e l y s i m p l e , i t i n v o l v e s e x p a n d i n g a c y l i n d r i c a l , f l e x i b l e , f i n i t e membrane a g a i n s t the s i d e s of a h o l e w i t h i n the i n f i n i t e s o i l medium. However, from p a s t e x p e r i e n c e s , i t has been r e c o g n i z e d t h a t t h e t e s t c o u l d be c o m p l i c a t e d by i s s u e s such as the amount of s o i l d i s t u r b a n c e b e f o r e and d u r i n g t h e probe i n s e r t i o n p r i o r t o t e s t i n g , the k i n d of s o i l d e f o r m a t i o n p a t t e r n around the f i n i t e p r e s s u r e m e t e r membrane under g i v e n p r e s s u r e m e t e r L/D r a t i o , and the method t o de t e r m i n e the b a s i c e n g i n e e r i n g parameters from t h e t e s t d a t a . 101 102 A t p r e s e n t , t h e common methods o f i n t e r p r e t i n g p r e s s u r e m e t e r t e s t d a t a t o d e t e r m i n e t h e e n g i n e e r i n g p r o p e r t i e s o f t h e s o i l i n v o l v e t h e c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y , i n w h i c h a c y l i n d r i c a l c a v i t y i s e x p a n d e d u n d e r t h e c o n d i t i o n s o f a x i s y m m e t r y a n d p l a n e s t r a i n ( G i b s o n a n d A n d e r s o n , 1961 , B a g u e l i n , e t a l , 1972, L a d a n y i , 1963 and P a l m e r , 1 9 7 2 ) . H o w e v e r , s u c h a t h e o r y c a n o n l y be e s t a b l i s h e d f o r p r e s s u r e m e t e r t e s t w i t h a s u f f i c i e n t membrane l e n g t h t o d i a m e t e r r a t i o ( L /D ) a s t h e p r e s s u r e m e t e r t e s t i n v o l v e s e x p a n d i n g a f i n i t e membrane w i t h i n t h e i n f i n i t e s o i l med ium, a n d t h e end e f f e c t o f t h e membrane s h o u l d be c o n s i d e r e d . The i n f l u e n c e o f t h e l e n g t h t o d i a m e t e r r a t i o L /D on t h e d e t e r m i n a t i o n o f s o i l p a r a m e t e r s f r o m t h e p r e s s u r e m e t e r t e s t d a t a was e x a m i n e d by many r e s e a r c h e r s . L i v n e h e t a l (1971) c o n d u c t e d a m a t h e m a t i c a l a n a l y s i s , b a s e d on t h e c o n v e n t i o n a l l i n e a r e l a s t i c t h e o r y , o f t h e p r e s s u r e p a t t e r n s g e n e r a t e d i n an i d e a l e l a s t i c m a t e r i a l by a f i n i t e p r e s s u r e m e t e r membrane i n an i n f i n i t e b o r e h o l e . U s i n g T r a n t e r ' s (1946) e q u a t i o n f o r r a d i a l d i s p l a c e m e n t i n a b o r e h o l e , he c o n c l u d e d t h a t p r o b e l e n g t h h a d l i t t l e e f f e c t on p r e s s u r e m e t e r t e s t r e s u l t s . H a r t m a n a n d S chmer tmann ( 1 9 7 5 ) , a l s o b a s e d on T r a n t e r ' s w o r k , c o n f i r m e d L i v n e h ' s c o n c l u s i o n s u s i n g f i n i t e e l e m e n t a n a l y s e s on c o h e s i o n l e s s s a n d s . F i g . 6.1 shows t h e r e s u l t s f r o m H a r t m a n ' s a n a l y s e s u s i n g a b r o a d o f r a n g e o f L/D and P o i s s o n ' s r a t i o v a l u e s . H a r t m a n ' s r e s u l t s shown i n F i g . 6.1 s u g g e s t e d t h a t u n d e r F i g . 6.1 I n f l u e n c e of L/D R a t i o on Displacement P a t t e n ( a f t e r L a i e r et a l , 1975) 1 04 n o r m a l t e s t i n g c o n d i t i o n s i n t h e e l a s t i c r a n g e , d e f l e c t i o n o v e r t h e c e n t r a l t w o - t h i r d s o f t h e p r e s s u r e m e t e r membrane r e m a i n e d e s s e n t i a l l y u n a l t e r e d by t h e end e f f e c t s . The p r e d i c t e d d i s p l a c e m e n t c u r v e s showed t h a t f o r L /D r a t i o g r e a t e r t h a n 4, t h e a v e r a g e e l a s t i c r a d i a l d i s p l a c e m e n t c o m p u t e d o v e r t h e c e n t e r 1/3 o f t h e f i n i t e membrane i s g r e a t e r t h a n 94% o f t h e r a d i a l d i s p l a c e m e n t c o m p u t e d f o r an i n f i n i t e l y l o n g p r o b e . T h i s s u g g e s t e d t h a t t h e d e f o r m a t i o n m o d u l u s , E o r G c a n be e v a l u a t e d f r o m t h e i n i t i a l e l a s t i c r e s p o n s e o f t h e p r e s s u r e m e t e r b a s e d on i n f i n i t e c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y w i t h o n l y m i n o r e r r o r i n v a l u e s . B a s e d on t h e i r t r i a x i a l chamber t e s t r e s u l t s f o r s a n d s , w h i c h i s shown i n F i g . 6 . 2 , L a i e r e t a l ( 1975) c o n c l u d e d t h a t f o r t h e M e n a r d p r e s s u r e m e t e r , t h e L/D r a t i o had no c o n s i s t e n t o r s i g n i f i c a n t i n f l u e n c e on t h e p r e s s u r e m e t e r m o d u l u s v a l u e s . T h i s c o n f i r m e d t h e a f o r e - m e n t i o n e d t h e o r e t i c a l p r e d i c t i o n s by L i v n e h e t a l ( 1 9 7 1 ) , H a r t m a n and Schmer tmann ( 1 9 7 5 ) . H o w e v e r , u n d e r c o n t r o l l e d d e n s i t y o f t e s t e d s a n d s , L a i e r e t a l (1975) were a b l e t o f i n d t h a t p r e s s u r e m e t e r membrane l e n g t h d o e s h a v e a m a r k e d e f f e c t on t h e m e a s u r e d l i m i t p r e s s u r e s . The i n c r e a s e o f t h e p r e s s u r e m e t e r membrane l e n g t h r e s u l t e d i n t h e r e d u c t i o n o f t h e l i m i t p r e s s u r e . T h e r e f o r e , L a i e r ' s e x p e r i m e n t a l r e s u l t s seem t o i n d i c a t e t h a t t h e p r e s s u r e m e t e r L /D r a t i o w o u l d s i g n i f i c a n t l y i n f l u e n c e t h e t e s t r e s u l t s a f t e r t h e s o i l medium i s i n t h e e l a s t o - p l a s t i c r a n g e , a n d c o n s e q u e n t l y t h e a x i s y m m e t r i c a l 105 F i g . 6.2 I n f l u e n c e of L/D R a t i o on E l a s t i c Modulus V a l u e s ( a f t e r L a i e r et a l , 1975) 106 p l a n e s t r a i n c a v i t y e x p a n s i o n t h e o r y s h o u l d be u s e d w i t h s u f f i c i e n t L/D t o d e t e r m i n e t h e s h e a r s t r e n g t h c h a r a c t e r i s t i c s f r o m t h e p r e s s u r e m e t e r t e s t d a t a . In a d d i t i o n , i n t h e a p p l i c a t i o n o f t h e p r e s s u r e m e t e r e x p a n s i o n c u r v e s f o r t h e d e v e l o p m e n t o f P -Y c u r v e s f o r t h e l a t e r a l l y l o a d e d p i l e s , i t i s u s u a l l y a s s u m e d t h a t a t d e p t h , t h e p r e s s u r e m e t e r e x p a n s i o n and t h e p i l e l a t e r a l l o a d i n g a r e a l l i n p l a n e s t r a i n , a n d t h e r e f o r e t h e s i m i l a r i t y i n t h e l o a d i n g p a t t e r n e x i s t s b e t w e e n t h e t w o . U n d e r t h i s a s s u m p t i o n , t h e p r e s s u r e m e t e r e x p a n s i o n c u r v e s c a n be s i m p l y c o n v e r t e d t o t h e P-Y c u r v e s by c e r t a i n s c a l i n g f a c t o r s ( A t u k o r a l a a n d B y r n e , 1 9 8 4 ) . T h e r e f o r e , t h e a x i s y m m e t r i c a l p l a n e s t r a i n d i s p l a c e m e n t p a t t e r n i s t h e key a s s u m p t i o n f o r t h e i n t e r p r e t a t i o n o f p r e s s u r e m e t e r d a t a a n d t h e d e v e l o p m e n t o f t h e P-Y c u r v e s b a s e d on t h e p r e s s u r e e x p a n s i o n c u r v e s . More work i s w a r r a n t e d t o e x a m i n e t h e i n f l u e n c e o f p r e s s u r e m e t e r L /D r a t i o i n a t h e o r e t i c a l v i e w p o i n t . Such a s t u d y i s s u i t a b l e f o r f i n i t e e l e m e n t p a r a m e t r i c s t u d i e s . In t h i s c h a p t e r , an a x i s y m m e t r i c a l f i n i t e e l e m e n t p a r a m e t r i c s t u d y i s f i r s t p e r f o r m e d w i t h a b r o a d r a n g e o f L /D r a t i o , and t h e r e s u l t s a r e c o m p a r e d w i t h t h e f i n i t e e l e m e n t a x i s y m m e t r i c a l p l a n e s t r a i n c a v i t y e x p a n s i o n s i m u l a t i o n . T h e n a t t e m p t s a r e made t o r e p r e d i c t some f i e l d t e s t d a t a u s i n g t h e c y l i n d r i c a l c a v i t y e x p a n s i o n m o d e l . The s i g n i f i c a n c e o f t h e a n a l y s e s a r e d i s c u s s e d i n d e t a i l s . The p r e s s u r e m e t e r t e s t h e r e i n i s a s s u m e d t o be c a r r i e d o u t w i t h no s o i l d i s t u r b a n c e , w h i c h i s e q u i v a l e n t t o t h e 107 t e s t w i t h a " i d e a l " s e l f - b o r i n g p r e s s u r e m e t e r d e v i c e . In r e a l i t y , some s o i l d i s t u r b a n c e s e v e n f o r a s e l f - b o r i n g p r e s s u r e m e t e r may be i n e v i t a b l l y i n t r o d u c e d a n d c o n s e q u e n t l y h a v e an i n f l u e n c e on t h e t e s t r e s u l t s . S u c h an e f f e c t w i l l be e x a m i n e d e x c l u s i v e l y l a t e r i n C h a p t e r 9. 6.2 F I N I T E ELEMENT MESH AND BOUNDARY CONDITIONS The f i n i t e e l e m e n t mesh f o r t h e a x i s y m m e t r i c a n a l y s e s i s Shown i n F i g . 6 . 3 . The d e p t h o f t h e p r e s s u r e m e t e r t e s t i s a s s u m e d t o be 5.0 m, a n d t h e d i a m e t e r , D, o f t h e p r e s s u r e m e t e r i s a s sumed t o be 0.1 m. T h e r e f o r e c o m p a r i n g w i t h t h e p r e s s u r e m e t e r d i a m e t e r , t h e t e s t d e p t h i s c o n s i d e r e d t o be r e l a t i v e l y d e e p . As shown i n t h e f i g u r e , s u f f i c i e n t l y f i n e e l e m e n t s a r e u s e d i n t h e s o i l a d j a c e n t t o t h e p r e s s u r e m e t e r membrane, a n d s p a r s e mesh i n t h e f a r f i e l d . The p r e s s u r e i s a p p l i e d on t h e l e f t b o u n d a r y a t t h e b o t t o m e n d . In t h e a n a l y s e s , t h e d i a m e t e r o f t h e p r e s s u r e m e t e r i s k e p t c o n s t a n t , w h i l e t h e p r e s s u r e d l e n g t h w h i c h i s h a l f o f t h e p r e s s u r e m e t e r membrane l e n g t h i s v a r i e d , a n d t h e r e f o r e p r o d u c i n g d i f f e r e n t v a l u e s o f L/D r a t i o f o r t h e a n a l y s e s . F o r t h e u p p e r a n d o u t e r b o u n d a r i e s , z e r o s t r e s s a n d z e r o s t r e s s c h a n g e c o n d i t i o n s a r e e m p l o y e d r e s p e c t i v e l y , w h i l e f o r t h e b o t t o m b o u n d a r y , no v e r t i c a l d i s p l a c e m e n t i s a s s u m e d . T h i s i s r e a s o n a b l e f o r t h e p r e s s u r e m e t e r t e s t a t d e p t h where t h e h i g h o v e r b u r d e n p r e s s u r e o f s o i l s p r e v e n t s s o i l s f r o m b e i n g d i s p l a c e d v e r t i c a l l y . The b o t t o m b o u n d a r y 109 i s s e l e c t e d a t t h e m i d p o i n t o f t h e p r e s s u r e m e t e r membrane, a n d t h e o u t e r b o u n d a r y i s 100 rQ away f r o m t h e c e n t r a l a x i s . E x c e p t f o r t h e p r e s s u r e d a r e a , t h e z e r o s t r e s s c h a n g e c o n d i t i o n i s a l s o e m p l o y e d f o r t h e l e f t hand b o u n d a r y . As i n c h a p t e r 5, an a x i s y m m e t r i c a l p l a n e s t r a i n f i n i t e e l e m e n t mesh i s u s e d f o r t h e s i m u l a t i o n o f c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n , w h i c h i s shown i n F i g . 6 . 4 . S i m i l a r l y , a s u f f i c i e n t l y f i n e mesh i s e m p l o y e d i n t h e s o i l n e x t t o t h e p r e s s u r e m e t e r t o r e p r e s e n t t h e h i g h s t r e s s d i s t r i b u t i o n a r o u n d t h e p r o b e , w h e r e a s a s p a r s e mesh i s u s e d i n t h e f a r f i e l d . The o u t e r b o u n d a r y h a s t h e same o u t e r r a d i u s , a n d t h e same b o u n d a r y c o n d i t i o n a s f o r t h e a x i s y m m e t r i c c a s e . H o w e v e r , i n t h i s c a s e a s e r i e s o f r o l l e r s a r e p l a c e d on b o t h u p p e r a n d b o t t o m b o u n d a r i e s t o e n s u r e t h e p l a n e s t r a i n c o n d i t i o n . The p r e s s u r e i s a p p l i e d i n c r e m e n t a l l y on t h e i n n e r b o u n d a r y . In t h e a b o v e f i n i t e e l e m e n t mesh t h e e l e m e n t s a r e a l l o f t h e l i n e a r s t r a i n t r i a n g u l a r t y p e w i t h 7 i n t e r g r a t i o n p o i n t s . 6.3 ANALYSES AND SOIL PARAMETERS A n o n l i n e a r , s t r e s s d e p e n d e n t h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n s h i p o f s o i l s i s a d o p t e d f o r t h e a n a l y s e s . F o r c o h e s i v e s o i l s , u n d r a i n e d p r e s s u r e m e t e r t e s t c o n d i t i o n i s a s s u m e d . The a n a l y s i s i s b a s e d on t o t a l s t r e s s e s , and t h e u n d r a i n e d c o n d i t i o n i s s i m u l a t e d by a l a r g e v a l u e o f b u l k m o d u l u s e q u i v a l e n t t o a P o i s s o n ' s r a t i o o f 0 . 4 9 9 . F o r F i g . 6.4 F i n i t e Element Mesh f o r C a v i t y Expansion A n a l y s i s 111. c o h e s i o n l e s s s o i l s , the p r e s s u r e m e t e r t e s t i s assumed s u f f i c i e n t l y slow t h a t the d r a i n e d l o a d i n g c o n d i t i o n i s p r e v a i l i n g . In the a x i s y m m e t r i c a n a l y s e s , the nonhomogeneity of t h e s o i l parameters w i t h d e p t h i s c o n s i d e r e d , and the i n - s i t u s t r e s s i s assumed i s o t r o p i c and i s p r o p o r t i o n a l l y v a r i e d from z e r o a t the s u r f a c e t o c e r t a i n v a l u e s a t 5 m d e p t h , w h i l e i n the a x i s y m m e t r i c a l p l a n e s t r a i n a n a l y s e s the homogeneous, i s o t r o p i c s o i l medium i s assumed. In the l a t t e r c a s e , the s o i l p arameters and the i n i t i a l s t r e s s s t a t e c o r r e s p o n d t o the de p t h of 5 m i n the a x i s y m m e t r i c a l c a s e . C o h e s i v e S o i l s In the case of a x i s y m m e t r i c a n a l y s e s f o r c o h e s i v e s o i l s , the N o r m a l l y C o n s o l i d a t e d c l a y i s assumed. The s o i l p arameters such as u n d r a i n e d shear s t r e n g t h , C y and i n i t i a l modulus, E a r e p r o p o r t i o n a l t o the e f f e c t i v e overburden p r e s s u r e a t t h e de p t h of i n t e r e s t , i . e . C = 0.25 a' u v E = M^C U (6.3.1) (6.3.2) where M i s the modulus m u l t i p l i e r . I t u s u a l l y v a r i e s i n range of 200-1200, depending on the methods chosen t o det e r m i n e i t . L a b o r a t o r y t e s t s where some d i s t u r b a n c e s i n sa m p l i n g and p r e p a r i n g samples a r e i n v o l v e d , u s u a l l y g i v e s m a l l e r v a l u e s w h i l e i n - s i t u t e s t s and computer back 1 1 2 a n a l y s e s u s u a l l y show h i g h e r v a l u e s (Clough and Denby, 1977). H e r e i n a r e p r e s e n t a t i v e v a l u e of M = 800 i s s e l e c t e d . For the s i m p l i c i t y i n c a l c u l a t i o n , the 5 m N.C c l a y d e p o s i t i s d i v i d e d i n t o 4 l a y e r s i n the c a l c u l a t i o n of s o i l p a r a m e t e r s . In the l a y e r s e x c e p t the bottom one, the s o i l p a r a m e t e r s a r e c a l c u l a t e d a t r e p r e s e n t a t i v e depths which a r e a t the m i d p o i n t of t h e l a y e r s . For the bottom l a y e r , however, the s o i l p a r a m e t e r s a r e c a l c u l a t e d a t the d e p t h of 5 m,' i . e . a t t h e bottom boundary. The s o i l p a rameters f o r t h i s l a y e r a r e a l s o employed f o r the a x i s y m m e t r i c a l p l a n e s t r a i n a n a l y s e s . For c o h e s i v e s o i l s , b o t h e l a s t o - p l a s t i c and n o n l i n e a r s o i l b e h a v i o r a r e c o n s i d e r e d . The e l a s t o - p l a s t i c s o i l b e h a v i o r i s o b t a i n e d u s i n g h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n w i t h v e r y s m a l l v a l u e of Rf (see Chapter 3 ) . The s o i l p a rameters used i n t h e a n a l y s e s f o r c o h e s i v e s o i l s a r e shown i n T a b l e 6.1 and 6.2. C o h e s i o n l e s s S o i l s In the c a s e of a x i s y m m e t r i c a l a n a l y s e s f o r c o h e s i o n l e s s s o i l s , the nonhomogeneity of s o i l parameters w i t h d epth i s i n i t i a l s t r e s s dependent, and i t would be a u t o m a t i c a l l y a c c o u n t e d f o r i f the i n i t i a l s t r e s s s t a t e assumed f o r the s o i l d e p o s i t i s v a r i e d w i t h d e p t h . In t h e a n a l y s e s , the i n i t i a l i s o t r o p i c s t r e s s s t a t e i s assumed t o be p r o p o r t i o n a l l y v a r i e d from z e r o a t s u r f a c e t o 50 Kpa a t d e p t h of 5 m, and a dense sand of r e l a t i v e d e n s i t y D = 75% 1 13 T a b l e 6.1 S o i l Parameters f o r A x i s y m m e t r i c A n a l y s e s S o i l P arameters C o h e s i v e E l a s t i o - P l a s t i c S o i l s N o n l i n e a r C o h e s i o n l e s s S o i l N o n l i n e a r 7 (KN/m 3) 16 16 20 C u (Kpa) 1 .5 H 1 .5 H <t> (°) • 42° 4>cv (o) 33° L\<$> 4° K E 11.84 H 11.84 H 1500 KB 1973.79 H 1973.79 H 900 n 0.0 0.0 0.5 m 0.0 0.0 0.25 Rf 0.0 0.9 0.7 K0 1 .0 1 .0 1 .0 a v (Kpa) 6 H 6 H 10 H G.W.T (m) 0.0 0.0 0.0 T a b l e 6.2 S o i l Parameters f o r C a v i t y E x p a n s i o n A n a l y s e s S o i l C o h e s i v e S o i l s C o h e s i o n l e s s Parameters E l a s t i o - P l a s t i c N o n l i n e a r S o i l N o n l i n e a r 7 (KN/m 3) 16 16 20 C u (Kpa) 7.5 7.5 <j> (°) 42° <t>cv (o) 33° L\<f> 4° K E 59.21 59.21 1 500 KB 9868.95 9868.95 900 n 0.0 0.0 0.5 m 0.0 0.0 0.25 Rf 0.0 0.9 0.7 K0 a v (Kpa) 1 .0 1 .0 1 .0 30 30 50 G.W.T (m) 0.0 0.0 0.0 Depth (m) 5 5 5 1. H - Depth of i n t e r e s t (m) , 2. G.W.T - Ground water t a b l e . 1 1 4 i s s i m u l a t e d . The s o i l p a r a m e t e r s r e q u i r e d by h y p e r b o l i c s t r e s s s t r a i n r e l a t i o n a r e a l s o i n c l u d e d i n T a b l e 6.1 and 6 . 2 . T h e s e p a r a m e t e r s a r e r e f e r r e d t o i n v a l u e s f r o m B y r n e a n d Cheung (1984) . F o r t h e a x i s y m m e t r i c a l p l a n e s t r a i n a n a l y s e s , t h e same s o i l p a r a m e t e r s a r e u s e d . The d i f f e r e n c e l i e s i n t h e i n i t i a l s t r e s s s t a t e a s s i g n e d f o r t h e e l e m e n t s . As i n t h e c a s e o f c o h e s i v e s o i l s , t h e i n i t i a l s t r e s s s t a t e i s a s s u m e d i s o t r o p i c a n d homogeneous t h r o u g h o u t t h e mesh d o m a i n , and i s c o r r e s p o n d e n t t o t h e d e p t h o f 5 m i n a x i s y m m e t r i c a l a n a l y s e s . 6.4 INFLUENCES OF PRESSUREMETER L/D RATIOS B a s e d on t h e a x i s y m m e t r i c a l f i n i t e e l e m e n t a n a l y s e s w i t h d i f f e r e n t v a l u e s o f L /D r a t i o and t h e a x i s y m m e t r i c a l p l a n e s t r a i n a n a l y s i s u s i n g t h e same s o i l p a r a m e t e r s , t h e i n f l u e n c e o f t h e p r e s s u r e m e t e r L/D r a t i o on t h e p r e s s u r e m e t e r t e s t r e s u l t s c a n be e v a l u a t e d , and c o n s e q u e n t l y t h e a x i s y m m e t r i c a l p l a n e s t r a i n a s s u m p t i o n f o r t h e p r e s s u r e m e t e r t e s t c a n be v a l i d a t e d t h e o r e t i c a l l y . F o r t h e f o l l o w i n g d i s c u s s i o n , 3 v a l u e s o f L /D r a t i o , i . e . L/D = 1 , 4 , 12 were u s e d i n t h e a n a l y s e s . T h e s e v a l u e s a r e b e l i e v e d t o c o v e r t h e common r a n g e o f t h e L /D r a t i o i n t h e d e s i g n o f p r a c t i c a l p r e s s u r e m e t e r membrane. 1 15 6.4.1 COHESIVE SOILS 1) P r e s s u r e e x p a n s i o n c u r v e s For the sake of c o m p a r i s o n s , the p r e s s u r e e x p a n s i o n c u r v e s from the a x i s y m m e t r i c a l p l a n e s t r a i n a n a l y s i s and the a x i s y m m e t r i c a n a l y s i s w i t h L/D r a t i o of 1, 4, 12 a r e a l l shown i n F i g 6.5. The p r e s s u r e e x p a n s i o n c u r v e s a r e p l o t t e d i n terms of the a p p l i e d p r e s s u r e and the r e s u l t e d r a d i a l d i s p l a c e m e n t a t the f i r s t node. For the i d e a l c o n d i t i o n , i . e . t h e a x i s y m m e t r i c a l p l a n e s t r a i n c o n d i t i o n , the p r e s s u r e e x p a n s i o n c u r v e i s curve 1. I t i s shown i n the f i g u r e t h a t i n g e n e r a l , s m a l l e r L/D r a t i o s r e s u l t i n s t i f f e r p r e s s u r e e x p a n s i o n c u r v e s , e s p e c i a l l y i n the case of L/D = 1; the p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e i s much h i g h e r than t h e o t h e r c u r v e s . However, the c u r v e s w i t h L/D r a t i o of 4 and 12 a r e p r a c t i c a l l y c l o s e t o the i d e a l c u r v e ( i . e . Curve 7 ) . The above r e s u l t s a r e r e a s o n a b l e . When t h e p r e s s u r e m e t e r membrane i s s m a l l , the a p p l i e d p r e s s u r e w i l l b a s i c a l l y s p r e a d out as i n the case of s p h e r i c a l c a v i t y e x p a n s i o n w i t h i n s o i l medium, then h i g h e r p r e s s u r e s a re r e q u i r e d t o overcome the s u r r o u n d i n g s o i l r e s i s t a n c e i n t h r e e d i m e n s i o n , so c a l l e d 'end e f f e c t s ' ( B a g u e l i n e t a l , 1978). In t h i s c a s e , the p r e s s u r e m e t e r t e s t s a r e c l o s e r t o the s p h e r i c a l c a v i t y e x p a n s i o n c o n d i t i o n r a t h e r than the e x p e c t e d c y l i n d r i c a l c o n d i t i o n . As the p r e s s u r e m e t e r membrane l e n g t h i s i n c r e a s e d , the a d d i t i o n a l t e s t energy (or a p p l i e d p r e s s u r e ) t o overcome the 'end e f f e c t s ' becomes a 4-X--+ F E M : C V . E X P . E - P -o FEMrPNT L/D=12 E~P FEM:PMT L/D=4 E - P -a FEM:PMT L/D=1 E - P Pi = 6.6 C u E l a s t i c P l a s t i c S o i l s 1 I I I I I r~ 0.0 1.0 2.0 3.0 4.0 RADIAL DISPLACEMENT ~i 1 r 5.0 (MM) D=0 6.0 7.0 1CM T E S T AT 8.0 9.0 5M DEPTH 10 F i g . 6.5 The i n f l u e n c e o f L/D r a t i o on P r e s s u r e E x p a n s i o n C u r v e s i n C o h e s i v e S o i l s 1 1 7 p r o g r e s s i v e l y s m a l l e r f r a c t i o n o f t h e t o t a l , a n d more p r e s s u r e s s p r e a d o u t u n d e r t h e a x i s y m m e t r i c a l p l a n e s t r a i n c o n d i t i o n s . T h e r e f o r e , i t i s c l e a r t h a t t h e v a l i d i t y o f t h e c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y f o r p r e s s u r e m e t e r t e s t s i s d e p e n d e n t upon t h e v a l u e s o f L /D r a t i o . I d e a l l y , t h e t h e o r y i s o n l y v a l i d f o r t h e p r e s s u r e m e t e r w h i c h h a s t h e i n f i n i t i v e L/D r a t i o . H o w e v e r , b a s e d on t h i s s t u d y , t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s w i t h L /D r a t i o e q u a l t o o r g r e a t e r t h a n 4 w o u l d be a p p r o x i m a t e t o t h e t h e o r e t i c a l c u r v e , a n d t h e i n c r e a s e o f L /D r a t i o f r o m 4 t o 12 w i l l o n l y r e s u l t i n a m a r g i n a l i m p r o v e m e n t i n t h e a c c u r a c y . In p r a c t i c e , t h e common s e l f - b o r i n g p r e s s u r e m e t e r s a r e d e s i g n e d w i t h L /D r a t i o g r e a t e r t h a n 4, u s u a l l y a r o u n d 8. T h e r e f o r e , a s f a r a s t h e p r e s s u r e e x p a n s i o n c u r v e s f o r t h e s o f t c o h e s i v e s o i l s i s c o n c e r n e d , s u c h a s f o r t h e d e v e l o p m e n t o f P - Y c u r v e , t h e commonly u s e d L/D r a t i o a r e s u f f i c i e n t l y l a r g e , and t h e e x p e r i m e n t a l p r e s s u r e e x p a n s i o n c u r v e w o u l d c l o s e t o t h a t u n d e r t h e t h e o r e t i c a l p l a n e s t r a i n c o n d i t i o n . 2) I n i t i a l s l o p e s of t h e p r e s s u r e m e t e r c u r v e s U n d e r t h e i d e a l c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n , t h e i n i t i a l s l o p e s o f t h e p r e s s u r e e x p a n s i o n c u r v e s a r e e q u a l t o 2G^ ( s e e C h a p t e r 5 ) . T h e r e f o r e , t h e i n i t i a l s l o p e s o f t h e e x p e r i m e n t a l p r e s s u r e e x p a n s i o n c u r v e s a r e u s u a l l y u s e d t o d e t e r m i n e t h e i n i t i a l e l a s t i c m o d u l u s f o r t h e t e s t e d 118 s o i l m e d i u m . A n o t h e r m e t h o d t o d e t e r m i n e t h e i n i t i a l e l a s t i c m o d u l u s i s t o u t i l i s e t h e s l o p e o f u n l o a d - r e l o a d l o o p . F rom e l a s t o - p l a s t i c t h e o r y , t h e s e two m e t h o d s a r e t h e same. The i n i t i a l s l o p e s o f t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s shown i n F i g . 6 .5 a r e t a b u l a t e d i n T a b l e 6 . 3 . The i n i t i a l s l o p e o f e a c h c u r v e i s c a l c u l a t e d a t t h e e n d o f t h e f i r s t l o a d i n c r e m e n t , where t h e s t r a i n i s g e n e r a l l y i n a l e v e l o f 0 . 0 8 % . In s u c h a s m a l l s t r a i n l e v e l , t h e e n t i r e s o i l medium i s i n t h e e l a s t i c r a n g e . As shown i n T a b l e 6 . 3 , e x c e p t f o r t h e L/D r a t i o e q u a l t o u n i t y , t h e o t h e r p r e d i c t e d i n i t i a l s l o p e s a r e c l o s e t o t h e t h e o r e t i c a l v a l u e o f 2G^ w i t h e r r o r s l e s s t h a n 7%, where i s t h e i n i t i a l s h e a r m o d u l u s . The r e s u l t s show t h a t t h e f i n i t e p r e s s u r e m e t e r membrane l e n g t h w i t h L /D r a t i o e q u a l t o o r g r e a t e r t h a n 4 d o e s n o t h a v e g r e a t i n f l u e n c e s on t h e e l a s t i c m o d u l u s d e r i v e d f r o m t h e p r e s s u r e m e t e r t e s t d a t a i n t h e e l a s t i c s t a g e . The i n c r e a s e o f t h e L/D r a t i o f r o m 4 t o 12 may n o t be j u s t i f i e d t o i m p r o v e t h e a c c u r a c y , a s f r o m t h i s s t u d y , t h e e r r o r i s o n l y m a r g i n a l l y r e d u c e d f r o m 6.2% t o 1.5%. F o r t h e L/D r a t i o l e s s t h a n 4, h o w e v e r , t h e d e r i v e d e l a s t i c m o d u l u s i s g r e a t l y a f f e c t e d by t h e v a l u e o f L /D r a t i o . As shown i n T a b l e 6 . 3 , t h e i n i t i a l s l o p e s o b t a i n e d w i t h t h e membrane l e n g t h o f L /D r a t i o = 1 i s a b o u t 30% l a r g e r t h a n t h a t w i t h t h e L/D r a t i o = 4. 3) L i m i t p r e s s u r e s 119 T a b l e 6.3 I n f l u e n c e s of L/D R a t i o on I n i t i a l S l o p e s of P r e s s u r e m e t e r Curves i n C o h e s i v e S o i l s L/D r a t i o S l o p e s 4 1 2 1 .289(2Gi) 1 .062(2Gi) 1 .0!5(2Gi) - I n i t i a l Shear Modulus E x t r a p o l a t i n g the p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s shown i n F i g . 6.5, i t i s shown t h a t , u n l i k e the i n i t i a l s l o p e s where s o i l s a r e i n the e l a s t i c s t a g e , the L/D r a t i o has g r e a t e r e f f e c t s on t h e p r e d i c t e d l i m i t p r e s s u r e s . S m a l l e r L/D r a t i o r e s u l t s i n h i g h e r p r e d i c t e d l i m i t p r e s s u r e . As shown i n F i g . 6.5, f o r L/D r a t i o e q u a l t o 1, the l i m i t p r e s s u r e i s p r e d i c t e d much h i g h e r than the o t h e r s , w h i l e f o r the L/D r a t i o s e q u a l t o or g r e a t e r than 4, the p r e d i c t e d l i m i t p r e s s u r e s a r e c l o s e , and seem t o approach the t h e o r e t i c a l l i m i t p r e s s u r e , P L = 6.6 C^ which i s o b t a i n e d from Eq. ( 5 . 2 . 1 2 ) . The e f f e c t s of L/D r a t i o on t h e p r e s s u r e m e t e r c u r v e s and t h e i r l i m i t p r e s s u r e s w i l l i n t u r n a f f e c t the u n d r a i n e d shear s t r e n g t h , shear s t r e s s - s t r a i n c u r v e , and the f a i l u r e s t r a i n o b t a i n e d from the p r e s s u r e m e t e r t e s t . These tend t o i n d i c a t e t h a t the L/D r a t i o becomes i m p o r t a n t when the p l a s t i c d e f o r m a t i o n o c c u r s i n t h e s o i l medium. 1 20 4) U n d r a i n e d shear s t r e n g t h U s i n g the p r e s s u r e m e t e r t e s t d a t a i n c o h e s i v e s o i l s , t h e u n d r a i n e d shear s t r e n g t h and the shear s t r e s s - s t r a i n c u r v e s can be e v a l u a t e d t h r o u g h d i f f e r e n t a v a i l a b l e methods ( G i b s o n and Anderson, 1961, L a d a n y i , 1963, B a g u e l i n e t a l , 1972, Palmer, 1972, P r e v o s t and Hoeg, 1975, and Denby, 1978). In Gibson and Anderson method, the e l a s t o - p l a s t i c s o i l b e h a v i o r i s assumed f o r the a n a l y s e s , w h i l e i n the o t h e r methods, no p r e a s s u m p t i o n on s o i l b e h a v i o r i s r e q u i r e d , but v a r i o u s c u r v e f i t t i n g t e c h n i q u e s a r e employed f o r the a n a l y s e s . At p r e s e n t , the a c c u r a c y i n d e r i v i n g u n d r a i n e d shear s t r e n g t h from the p r e s s u r e m e t e r t e s t s i s s t i l l q u e s t i o n a b l e , and i t depends on s e v e r a l f a c t o r s . Among them t h e r e i n c l u d e : 1. the i n t e r p r e t a t i o n method, 2. the r a t i o of membrane l e n g t h t o d i a m e t e r , as a l l the methods a r e based on p l a n e s t r a i n c o n d i t i o n s . The i n f l u e n c e of the v a r i o u s i n t e r p r e t a t i o n methods on t h e o b t a i n e d u n d r a i n e d shear s t r e n g t h have been examined by Denby (1978). I t was found t h a t t h e d i f f e r e n c e i s u s u a l l y w i t h i n 10%. Thus, the method i t s e l f may not be a l a r g e f a c t o r i n the d e t e r m i n a t i o n . In t h e f o l l o w i n g a n a l y s e s of the i n f l u e n c e of L/D r a t i o on the u n d r a i n e d shear s t r e n g t h , t h e method proposed by G i b s o n and Anderson (1961) i s employed, as i n p r e s e n t s t u d i e s t h e e l a s t o - p l a s t i c c o h e s i v e s o i l i s assumed. 121 In t h e o r i g i n a l a n a l y s i s o f G i b s o n and A n d e r s o n method (1961) f o r t h e i n t e r p r e t a t i o n o f an u n d r a i n e d p r e s s u r e m e t e r t e s t i n c l a y , i t i s a s sumed t h a t t h e c l a y b e h a v e s a s a p e r f e c t l y e l a s t i c , p e r f e c t l y p l a s t i c m a t e r i a l c h a r a c t e r i z e d by a s h e a r m o d u l u s , G , a P o i s s o n ' s r a t i o o f 0 .5 and an u n d r a i n e d s h e a r s t r e n g t h C^. T h e a n a l y s i s l e a d s t o a r e s u l t t h a t , a f t e r f a i l u r e has b e e n i n i t i a t e d i n t h e c l a y , t h e a p p l i e d p r e s s u r e , P, i s l i n e a r l y r e l a t e d t o t h e l o g a r i t h m o f t h e r e s u l t e d v o l u m e t r i c s t r a i n A V / V , where V i s t h e c u r r e n t v o l u m e o f t h e c a v i t y o r p r e s s u r e m e t e r p r o b e , AV i s t h e r e s u l t e d c h a n g e i n t h e v o l u m e o f t h e c a v i t y . M o r e o v e r , i f t h e n a t u r a l l o g a r i t h m s a r e u s e d t h e g r a d i e n t o f t h e l i n e ( o r t h e p r e s s u r e m e t e r c u r v e s ) w i l l be e q u a l t o C u . T h i s p r o c e d u r e i s i l l u s t r a t e d i n F i g . 6 . 6 ( a ) . F o r p r e s e n t i n t e r e s t s , t h e p r e s s u r e e x p a n s i o n c u r v e s i n F i g . 6.5 p r e d i c t e d f r o m t h e f i n i t e e l e m e n t a n a l y s i s w i t h L /D = 1, 4, 12 have b e e n r e p l o t t e d i n F i g . 6 . 6 ( b ) , i n t h e a b o v e manner a s t h e a p p l i e d p r e s s u r e , P, a g a i n s t l n ( A V / V ) . The v o l u m e t r i c s t r a i n i s c a l c u l a t e d a c c o r d i n g t o t h e p r e d i c t e d c i r c u m f e r e n t i a l s t r a i n , e^ , by t h e r e l a t i o n s h i p : A V / V = 1 - ( 1 + e ^ ) " 2 = 2 eQ ( 6 . 4 . 1 ) As e x p e c t e d , t h e p o i n t s l i e r e m a r k a b l y c l o s e t o a s t r a i g h t l i n e e x c e p t f o r t h o s e p r i o r t o t h e i n i t i a t i o n o f f a i l u r e s . E x c e p t f o r t h e l i n e w i t h L /D r a t i o o f 1, t h e o t h e r kN/m* 2000 1900 1800 1700 1600 1500 / ' D - i 1 1 — I — u _ i I l I • • • • 7 8 9 10 11 12 H 16 18 20 A V . , F i g . 6.6 (a) G i b s o n and Ander son P r o c e d u r e f o r D e t e r m i n a t i o n of U n d r a i n e d Shear S t r e n g t h ( a f t e r W r o t h , 1982) i n CD . (0 a 0) u 3 M cn <u u cu x> CD . co o (M a a < CD CD . o <D Closed Form Solution + + Cav.Exp.Solution(FErl) o o L/D = 1 (FEU) x x L/D = 4 (FEM) • • L/D =12 (FEM) o xa o • xm+ o . <!> X C H - O O XG4- O <!> X D f O <!> >0- O O X 3 f O <I> XLD-1- o xa+o XD-KD >• 83 O o o 0 e< 1 I 1 1 7 IO2 10-' " I i i I 7 10° ' I ' M 7 10' Volumetric S t r a i n AV/V (%) in l o g 1 0 _ s c a l e F i g . 6.6 (b) Pressure Expansion Curves in Semi-log Scale ( E l a s t i c P l a s t i c Clays) 124 l i n e s w i t h L /D r a t i o = 4 o r 12 a r e c l o s e t o t h e c a v i t y e x p a n s i o n c o n d i t i o n . The s e l e c t e d g r a d i e n t s f o r t h o s e l i n e s , w h i c h a r e t h e d e r i v e d u n d r a i n e d s h e a r s t r e n g t h s , a r e s u m m a r i z e d i n T a b l e 6 . 4 . F o r t h e s a k e o f c o m p a r i s o n s , t h e a b o v e p r o c e d u r e t o d e t e r m i n e t h e u n d r a i n e d s h e a r s t r e n g t h i s a l s o a p p l i e d t o t h e c a v i t y e x p a n s i o n c l o s e d f o r m s o l u t i o n . In t h i s c a s e , t h e t h e o r e t i c a l c a v i t y e x p a n s i o n c u r v e i s f i r s t c a l c u l a t e d a c c o r d i n g t o E q . ( 5 . 2 . 1 1 ) , a n d t h e n t r a n s f o r m e d t o t h e f o r m Shown i n F i g . 6 . 6 . The r e p r e d i c t e d u n d r a i n e d s h e a r s t r e n g t h f o r t h e c a v i t y e x p a n s i o n c l o s e d f o r m s o l u t i o n i s a l s o i n c l u d e d i n T a b l e 6 . 4 . As shown i n t h e s e c o n d c o l u m n o f t h e t a b l e , a l l t h e d e r i v e d u n d r a i n e d s h e a r s t r e n g t h s a r e h i g h e r t h a n t h e i n i t i a l i n p u t one w h i c h i s 7.5 K p a . The r e s u l t s f r o m f i n i t e e l e m e n t a n a l y s e s w i t h t h e L/D r a t i o o f 1, 4, 12 a r e h i g h e r t h a n t h e i n p u t v a l u e o f C u by 50%, 26% a n d 20% r e s p e c t i v e l y . H o w e v e r , i t s h o u l d be n o t e d t h a t t h e e r r o r s may come f r o m t h e L/D r a t i o a s w e l l a s t h e e v a l u t i o n m e t h o d u s e d . The f r a c t i o n o f t h e e r r o r s due t o t h e m e t h o d i t s e l f may be e s t i m a t e d a p p r o x i m a t e l y f r o m t h e e r r o r i n v o l v e d i n t h e d e r i v e d C y f o r t h e c a v i t y e x p a n s i o n c l o s e d f o r m s o l u t i o n . As shown i n T a b l e 6 . 4 , t h e d e r i v e d C f r o m t h e c l o s e d f o r m u s o l u t i o n i s a b o u t 7.7% h i g h e r , w h i c h may r o u g h l y i n d i c a t e t h e e r r o r i n v o l v e d i n t h e method i t s e l f . T h i s e r r o r l i e s i n t h e r a n g e r e p o r t e d by Denby ( 1 9 7 8 ) . 125 T a b l e 6.4 I n f l u e n c e of L/D R a t i o on Determined U n d r a i n e d Shear S t r e n g t h L/D r a t i o D e r i v e d C u E r r o r E r r o r E r r o r (Kpa) ( 1 ) ( % ) ( 2 ) ( % ) ( 3 ) ( % ) C l o s e d Form Cav.Exp.(FE) 8.078 8.333 11.249 9.476 8.972 7.7 1 1 50 26 20 3 39 1 7 1 1 4 1 2 35 1 4 8 1. The i n p u t u n d r a i n e d shear s t r e n g t h C u = 7.5 Kpa 2. C l o s e d form s o l u t i o n i s o b t a i n e d t h r o u g h Eq. (5.2.10) & Eq. (5.2.11) f o r c y l i n d r i c a l c a v i t y e x p a n s i o n 3. Cav.Exp.(FE) i s the f i n i t e element s o l u t i o n of the c y l i n d r i c a l c a v i t y e x p a n s i o n 4. D r i v e d C u i s o b t a i n e d u s i n g G i b s o n and Anderson method 5. E r r o r (1) i s the comparison of d e r i v e d C u w i t h C u = 7.5 Kpa 6. E r r o r (2) i s the comparison of d e r i v e d C u w i t h C u = 8.078 Kpa 7. E r r o r (3) i s the comparison of d e r i v e d C u w i t h C u = 8.333 Kpa In o r d e r t o e l i m i n a t e the method e r r o r mentioned above, the d e r i v e d u n d r a i n e d shear s t r e n g t h s a r e compared w i t h t h a t from the c l o s e d form s o l u t i o n , and the f i n i t e element c a v i t y e x p a n s i o n s o l u t i o n . The r e s u l t s a r e shown i n t h e l a s t two columns of T a b l e 6.4. I t i s shown t h a t a l l the d e r i v e d u n d r a i n e d shear s t r e n g t h s a r e s t i l l h i g h e r than the i n p u t v a l u e . The e r r o r p e r c e n t a g e depends upon the v a l u e of L/D r a t i o . S m a l l e r v a l u e of L/D r a t i o g i v e s h i g h e r d e r i v e d u n d r a i n e d shear s t r e n g t h . As compared w i t h the d e t e r m i n a t i o n of e l a s t i c modulus, t h e L/D r a t i o g e n e r a l l y has a g r e a t e r i n f l u e n c e on 126 t h e u n d r a i n e d s h e a r s t r e n g t h . F o r t h e p r e s e n t d e s i g n s o f p r e s s u r e m e t e r s where t h e L/D r a t i o i s a b o u t 6, t h e d e r i v e d u n d r a i n e d s h e a r s t r e n g t h u s i n g c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y may be o v e r p r e d i c t e d by a b o u t 10% due t o t h e f i n i t e membrane l e n g t h e f f e c t s . S i m i l a r r e s u l t s were r e p o r t e d by many r e s e a r c h e r s . B o r s e t t o e t a l (1983) a l s o p e r f o r m e d a f i n i t e e l e m e n t a n a l y s i s , and t h e r e s u l t s a r e shown i n T a b l e 6 . 5 . I t i s i n t e r e s t i n g t o n o t e t h a t t h e o r d e r o f t h e i n f l u e n c e r e p o r t e d by B o r s e t t o e t a l i s s i m i l a r t o t h e f o r e g o i n g r e s u l t s i n t h e t h i r d c o l u m n o f T a b l e 6 . 4 . The i n f l u e n c e o f t h e L/D r a t i o was a l s o i n v e s t i g a t e d by J a m i o l k o w s k i e t a l (1980) i n a t e s t p r o g r a m i n t h e P o r t o T o l l e n o r m a l l y c o n s o l i d a t e d s i l t y c l a y . T e s t s c o n d u c t e d w i t h L/D = 2 l e d t o u n d r a i n e d s h e a r s t r e n g t h s 35% h i g h e r t h a n t h o s e c a r r i e d o u t w i t h an L/D = 4 p r o b e . T h e r e f o r e t h e p r e s e n t f i n i t e e l e m e n t a n a l y s e s d i d p r e d i c t t h e i n f l u e n c e o f L/D r a t i o o b s e r v e d f r o m t h e f i e l d t e s t d a t a . 5) N o n l i n e a r s o i l b e h a v i o r In t h e p r e v i o u s d i s c u s s i o n s , t h e e l a s t o - p l a s t i c s o i l b e h a v i o r was a s s u m e d . H o w e v e r , i n r e a l i t y , most c o h e s i v e s o i l s a r e n o n l i n e a r , s t r e s s d e p e n d e n t . In o r d e r t o e v a l u a t e t h e n o n l i n e a r s o i l r e s p o n s e o f t h e p r e s s u r e m e t e r t e s t s u n d e r d i f f e r e n t L /D r a t i o , s i m i l a r a n a l y s e s were p e r f o r m e d w i t h n o n l i n e a r s o i l m o d e l . The n o n l i n e a r s o i l p a r a m e t e r s a r e a l s o i n c l u d e d i n T a b l e 6.1 and 6.2 f o r t h e a x i s y m m e t r i c a l 1 27 Table 6.5 Influence of Length to Diameter Ratio on Derived Undrained Shear Strength (after Borsetto et a l . , 1983) LENGTH TO UNDRAINED SHEAR STRENGTH DIAMETER RATIO OVERPREDICTION (%) 2 36 4 26 6 22 128 a n a l y s e s and the a x i s y m m e t r i c a l p l a n e s t r a i n a n a l y s e s r e s p e c t i v e l y . The n o n l i n e a r a n a l y s i s r e s u l t s a r e shown i n F i g . 6.7. As compared w i t h the r e s u l t s f o r e l a s t o - p l a s t i c s o i l model shown i n F i g . 6.5, I t i s shown t h a t the n o n l i n e a r s o i l model g i v e s the responses s i m i l a r t o those o b t a i n e d from the e l a s t o - p l a s t i c s o i l b e h a v i o r . T h e r e f o r e , t h e p r e v i o u s c o n c l u s i o n s drawn from the e l a s t o - p l a s t i c a n a l y s e s seem t o be v a l i d f o r the n o n l i n e a r s o i l b e h a v i o r as w e l l . In summary, based on the f o r e g o i n g a n a l y s e s , the i n f l u e n c e of the p r e s s u r e m e t e r L/D r a t i o on the d e t e r m i n a t i o n of shear s t r e n g t h c h a r a c t e r i s t i c s i s more pronounced than i n the d e t e r m i n a t i o n of e l a s t i c modulus. As f a r as the p r e s s u r e m e t e r e x p a n s i o n c u r v e s and the e l a s t i c modulus a r e c o n c e r n e d , the L/D r a t i o e q u a l t o 4 would be s u f f i c i e n t t o p r o v i d e the p r e s s u r e m e t e r t e s t d a t a t h a t a r e p r a c t i c a l l y c l o s e t o the c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n s a t d e p t h . However, th e u n d r a i n e d shear s t r e n g t h p r e d i c t i o n w i l l be about 15% h i g h e r than t h a t o b t a i n e d from the c y l i n d r i c a l c a v i t y e x p a n s i o n s o l u t i o n . 6.4.2 COHESIONLESS SOILS 1) P r e s s u r e e x p a n s i o n c u r v e s For t h e c o h e s i o n l e s s s o i l s , the p r e d i c t e d c u r v e s from the axisymmetry p l a n e s t r a i n a n a l y s i s and t h e a x i s y m m e t r i c a n a l y s e s w i t h L/D r a t i o of 1, 4, 12 a r e a l l shown i n F i g . 6.8. The a n a l y s e s f o r the c o h e s i o n l e s s s o i l were x-- FEM -+ FEM -x FEM -Q FEM CAV.EX PAN SI ON 3DPMT L/D=1 3DPMT L/D=4 3DPMT L/D=12 4.0 C I R C U M . 6.0 S T R A I N 8.0 (X) i 1 r 10.0 12.0 =7.5KPA N - L T .14.0 C L A Y T 16.0 D = 0 . I M 18.0 20 F i g . 6.7 N o n l i n e a r S o i l Response under D i f f e r e n t L/D R a t i o o- •« 3DPMT L / D - 1 -+ i ^ i l / 3DPMT L / D = 4 •+ FEM 3DPMT L / D = 1 2 - FEM CYLIN.CAV.EXP. N o n l i n e a r S o i l s 1 .8 CIRCUM 2.4 STRAIN 3.0 [%) N-3.6 4.2 L SAND ( D r = 75.v) 4.8 5.4 F i g . 6.8 I n f l u e n c e of L/D R a t i o on P r e s s u r e E x p a n s i o n C u r v e s i n C o h e s i o n l e s s S o i l s 131 p e r f o r m e d u s i n g a n o n l i n e a r s o i l m o d e l . The p r e s s u r e e x p a n s i o n c u r v e s a r e p l o t t e d i n t e r m s o f t h e a p p l i e d p r e s s u r e a n d t h e c i r c u m f e r e n t i a l s t r a i n ( A U 0 / r 0 ) a t t h e w a l l o f p r o b e . The d i s p l a c e m e n t a t t h e f i r s t node o f b o t t o m b o u n d a r y was u s e d f o r t h e c a l c u l a t i o n . In g e n e r a l , t h e r e s u l t s a r e s i m i l a r t o t h o s e o b t a i n e d f o r c o h e s i v e s o i l s . The p r e d i c t e d c u r v e w i t h L /D r a t i o = 1 i s much s t i f f e r t h a n o t h e r c u r v e s . The c u r v e i s a l m o s t l i n e a r , s h o w i n g much h i g h e r s o i l r e s i s t a n c e s . W i t h h i g h e r L /D r a t i o , t h e p r e d i c t e d c u r v e s become p r o g r e s s i v e l y s o f t e r , a p p r o a c h i n g t h e c u r v e f r o m t h e c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n . U n l i k e t h e c a s e f o r c o h e s i v e s o i l s , t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s f o r t h e L/D r a t i o f r o m 4 t o 12, d e v i a t e s i g n i f i c a n t l y f r o m t h e a x i s y m m e t r i c a l p l a n e s t r a i n c u r v e . The d e v i a t i o n seems t o a c c u m u l a t e w i t h t h e i n c r e a s e o f s t r a i n l e v e l . E v e n f o r t h e L /D r a t i o = 12, t h e b e s t a g r e e m e n t o f t h e p r e d i c t e d c u r v e w i t h t h e c a v i t y e x p a n s i o n one i s o b s e r v e d o n l y up t o s t r a i n a b o u t 3 .0% . A t l a r g e s t r a i n s , t h e i n f l u e n c e o f t h e L/D r a t i o i s much more s i g n i f i c a n t t h a n i t was f o r t h e s o f t c o h e s i v e s o i l s . 2) I n i t i a l s l o p e s of the c u r v e s T h e i n i t i a l s l o p e s o f t h e p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s f o r d i f f e r e n t L/D r a t i o s shown i n F i g . 6.8 a r e t a b u l a t e d i n T a b l e 6 . 6 . The i n i t i a l s l o p e i s c a l c u l a t e d a t t h e e n d o f t h e f i r s t l o a d i n c r e m e n t , w h i c h i s g e n e r a l l y i n a s t r a i n l e v e l l e s s t h a n 0 . 0 5 % . In t h i s s t r a i n l e v e l , t h e 1 32 T a b l e 6.6 L/D R a t i o E f f e c t s on I n i t i a l S l o p e s i n C o h e s i o n l e s s S o i l s L/D r a t i o I n i t i a l S l o p e s 1 4 1.075(2Gi) 12 0.844(2Gi) Cav.Exp.(FE) 0.843(2Gj) 0.844 1. G i - I n i t i a l s h e ar modulus of the s o i l , 2. Cav.Exp.(FE) - The f i n i t e element s o l u t i o n of c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n e n t i r e s o i l medium i s i n the e l a s t i c range. As shown i n the t a b l e , t h e p r e d i c t e d i n i t i a l s l o p e s , e x c e p t f o r the L/D r a t i o = 1, a r e g e n e r a l l y l e s s than the t h e o r e t i c a l v a l u e of 2G^. T h i s i s due t o the n o n l i n e a r e l a s t i c , s t r e s s dependent model of s o i l s assumed i n the a n a l y s i s , and t h e r e f o r e the p r e d i c t d p r e s s u r e e x p a n s i o n c u r v e s l a c k an i n i t i a l e l a s t i c p o r t i o n . A l t h o u g h the i n i t i a l s l o p e s from the n o n l i n e a r e l a s t i c model a r e g e n e r a l l y l e s s than t h e l i n e a r e l a s t i c s o l u t i o n , the i n f l u e n c e of L/D r a t i o can be examined by comparing the i n i t i a l s l o p e from the c a v i t y e x p a n s i o n case w i t h the o t h e r s from d i f f e r e n t L/D r a t i o s . I t can be seen from T a b l e 6.6 t h a t the s l o p e s of the L/D r a t i o l a r g e r than 4 a r e v e r y c l o s e t o t h a t of c a v i t y e x p a n s i o n c a s e , but the i n i t i a l s l o p e of the L/D r a t i o = 1 i s about 27% h i g h e r than the e x p e c t e d . T h i s seems t o suggest t h a t the L/D r a t i o e q u a l t o or l a r g e r than 4 has no 1 33 s i g n i f i c a n t i n f l u e n c e on the i n i t i a l e l a s t i c p o r t i o n of the p r e s s u r e m e t e r c u r v e , and c o n s e q u e n t l y , the i n i t i a l e l a s t i c modulus d e t e r m i n e d w i l l be l e s s a f f e c t e d . T h i s seems t o c o n f i r m the p r e v i o u s a n a l y t i c a l and e x p e r i m e n t a l r e s u l t s o b t a i n e d by L i v n e h e t a l (1971), Hartman and Schmertmann (1975), and L a i e r e t a l (1975) f o r the Menard p r e s s u r e m e t e r . 3) L i m i t p r e s s u r e s As shown i n F i g . 6.5, f o r a l l the L/D r a t i o s t u d i e d h e r e i n , the d e v i a t i o n of the p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s from the c y l i n d r i c a l c a v i t y e x p a n s i o n c u r v e becomes s i g n i f i c a n t i n the l a r g e r s t r a i n l e v e l . E x t r a p o l a t i n g t h e s e c u r v e s , the p r e d i c t e d l i m i t p r e s s u r e s seem not t o approach the c y l i n d r i c a l c a v i t y e x p a n s i o n c a s e , even f o r the L/D r a t i o = 12. The reason f o r t h i s i s t h e f a c t t h a t a p r e s u r e m e t e r w i t h f i n i t e membrane l e n g t h was b e i n g expanded. As t h e s o i l medium g e t s i n t o the p l a s t i c f a i l u r e s , a zone of s o f t p l a s t i c s o i l forms around the p r e s s u r e m e t e r membrane. I n such an e l a s t i c p l a s t i c c o m p o s i t e system, most of the a p p l i e d p r e s s u r e s l a t e r a r e t a k e n by the s t i f f e r e l a s t i c s o i l r e g i o n s u r r o u n d i n g the p l a s t i c s o f t zone. In the t h r e e d i m e n s i o n a l a x i s y m m e t r i c a l c o n d i t i o n , the s t i f f e r e l a s t i c s o i l r e g i o n out of the l o a d i n g p l a n e p a r t i c i p a t e s i n t h i s a c t i o n , as a r e s u l t , more s o i l r e s i s t a n c e comes from the out of p l a n e s o i l r e g i o n , l e a d i n g t o much h i g h e r l i m i t p r e s s u r e than i n the p l a n e s t r a i n c a s e . In o t h e r words, the above 1 34 h i g h e r p r e d i c t i o n s o f l i m i t p r e s s u r e s f o r t h e f i n i t e membrane l e n g t h a r e due t o t h e m e c h a n i s m o f t h r e e d i m e n s i o n a l e l a s t i c p l a s t i c s o i l s t r e s s r e d i s t r i b u t i o n . The l a r g e i n f l u e n c e o f t h e L/D r a t i o on t h e l i m i t p r e s s u r e were a l s o f o u n d i n many e x p e r i m e n t a l r e s u l t s ( L a i e r e t a l , 1975, J e w e l l e t a l , 1 9 8 0 ) . In J e w e l l e t a l r e s u l t s , t h e p r e s s u r e m e t e r t e s t w i t h L/D r a t i o =4 was f o u n d t o g i v e a m e a s u r e d l i m i t p r e s s u r e 18% h i g h e r t h a n t h e t h e o r e t i c a l v a l u e . H o w e v e r , t h e y r e p o r t e d t h a t t h e m e a s u r e d l i m i t p r e s s u r e became c l o s e t o t h e t h e o r e t i c a l v a l u e when t h e p r e s s u r e m e t e r w i t h L /D r a t i o = 6.2 was u s e d . T h e r e f o r e , t h e o p t i m a l L/D r a t i o f o r t h e p r e s s u r e m e t e r t e s t s on c o h e s i o n l e s s s o i l s r e g a r d i n g t h e c o r r e c t l i m i t p r e s s u r e i s s t i l l f a r f r o m c o n c l u s i v e . 4 ) Determination of shear strength c h a r a c t e r i s t i c s The s h e a r s t r e n g t h s o f c o h e s i o n l e s s s o i l s a r e u s u a l l y c h a r a c t e r i z e d by t h e f r i c t i o n a l a n g l e <j>. U s i n g t h e p r e s s u r e m e t e r t e s t d a t a i n c o h e s i o n l e s s s o i l s , i t i s p o s s i b l e t o d e t e r m i n e t h e f r i c t i o n a l a n g l e b a s e d on t h e m e t h o d s u c h a s Hughes e t a l ( 1 9 7 7 ) . In Hughes e t a l m e t h o d f o r t h e i n t e r p r e t a t i o n o f a d r a i n e d p r e s s u r e m e t e r t e s t i n s a n d , i t i s a s sumed t h a t t h e s a n d b e h a v e s e l a s t i c a l l y b e f o r e f a i l u r e , a n d f a i l s a t a c o n s t a n t r a t i o o f e f f e c t i v e s t r e s s e s and a c o n s t a n t r a t e o f d i l a t i o n . The a n a l y s i s l e a d s t o t h e r e s u l t t h a t , a f t e r f a i l u r e h a s b e e n i n i t i a t e d i n t h e s a n d , t h e l o g a r i t h m o f t h e 1 35 e f f e c t i v e r a d i a l s t r e s s i s l i n e a r l y r e l a t e d t o t h e l o g a r i t h m o f t h e s t r a i n e^ , w h i c h i s i l l u s t r a t e d i n F i g . 6 . 9 ( a ) . The s l o p e o f t h e l i n e i s g i v e n by t h e f o l l o w i n g e x p r e s s i o n : s = s i n 0 (1 + sinv) . , B 1 + s i n 0 l b . 4 . 2 ; where <t> i s t h e a n g l e o f e f f e c t i v e f r i c t i o n a l a n g l e , a n d v i s t h e a n g l e o f d i l a t i o n . E m p l o y i n g t h e s t r e s s d i l a t a n c y t h e o r y o f Rowe ( 1 9 6 1 , 1 9 7 1 ) , t h e f r i c t i o n a l a n g l e <t> a n d t h e d i l a t i o n a n g l e v c a n be o b t a i n e d v i a : S i n * " [ ( K - l ) S + 2 ] ( 6 - 4 - 3 ) [ 2 K S - ( K - 1 ) ] . S i n v = - — ^ R + ^ ) ( 6 . 4 . 4 ) where K = (1 + s i n 0 c ^ , ) / ( 1 - s i n < £ c ^ ) , <j>c^ i s t h e f r i c t i o n a l a n g l e o f t h e m a t e r i a l a t c o n s t a n t v o l u m e . In o r d e r t o d e t e r m i n e a n g l e s o f <j> a n d v w i t h Hughes e t a l m e t h o d ( 1 9 7 7 ) , a v a l u e o f must be o b t a i n e d e x p e r i m e n t a l l y o r a s s u m e d . In g e n e r a l , t h e v a l u e s o f d e p e n d s upon m i n e r a l c o m p o s i t i o n o f t h e g r a n u l a r m a t e r i a l . F o r p r e s e n t s t u d i e s , a v a l u e o f 3 3 ° i s a s s u m e d . T h i s v a l u e i s r e f e r r e d t o B y r n e a n d Cheung ( 1 9 8 4 ) , and i s b e l i e v e d t o be r e p r e s e n t a t i v e f o r most p r a c t i c a l s a n d s . The s l o p e s o f t h e p r e s s u r e e x p a n s i o n c u r v e s i n l o g - l o g s c a l e a n d t h e d e r i v e d f r i c t i o n a l a n g l e s i n t h e f o r e g o i n g manner a r e s u m m a r i z e d i n T a b l e 6 . 7 . p kN/m' SOO ISO too 350 300 250 -i 1 i 1 ». t V . S 6 7 8 F i g . 6.9 (a) Pressure Expansion Curves in C ohesionless S o i l s ( a f t e r Wroth, 1982) co CTi 1 37 T a b l e 6 .7 L/D R a t i o E f f e c t s on t h e D e r i v e d F r i c t i o n A n g l e f r o m P r e s s u r e m t e r C u r v e s i n C o h e s i o n l e s s S o i l s L /D R a t i o S l o p e s (S) P r e d i c t e d 0 0 E r r o r E r r o r (1) % (2) % 4 12 C a v . E x p . 0 . 7 0 8 6 0 . 5 7 8 6 0 .5492 0 . 5 3 3 9 5 7 . 4 ° 5 0 . 0 ° 4 6 . 7 ° 4 5 . 7 ° 40 % 22 % 14 % 1 1 % 26 % 10 % 2 % 1. C a v . E x p . - t h e f i n i t e e l e m e n t s o l u t i o n o f c y l i n d r i c a l ' c a v i t y e x p a n s i o n c o n d i t i o n ; 2. t h e i n p u t m a t e r i a l f r i c t i o n a n g l e 0 = 4 1 ° ; 3. E r r o r (1) - t h e c o m p a r i s o n o f p r e d i c t e d 0 w i t h 0 = 4 1 ° w h i c h i s t h e i n p u t v a l u e ; 4. E r r o r (2) - t h e c o m p a r i s o n o f p r e d i c t e d 0 w i t h 0 = 4 5 . 7 ° w h i c h i s o b t a i n e d f r o m c y l i n d r i c a l c a v i t y e x p a n s i o n s o l u t i o n . As shown b e f o r e , t h e s h a p e s o f t h e p r e s s u r e e x p a n s i o n c u r v e s f r o m p r e s s u r e m e t e r t e s t s d e p e n d upon t h e L /D r a t i o . T h e r e f o r e , t h e s l o p e s o f t h e s e c u r v e s i n l o g - l o g s c a l e a n d t h e d e r i v e d f r i c t i o n a l a n g l e a r e a l s o a f f e c t e d by t h e L/D r a t i o . T h e s e a r e i l l u s t r a t e d i n F i g . 6 . 9 ( b ) a n d T a b l e 6 . 7 . I t i s s e e n t h a t t h e s l o p e s o f l i n e s w i t h L /D r a t i o = 4 o r 12 a r e p r a c t i c a l l y c l o s e t o t h a t o f c y l i n d r i c a l c a v i t y e x p a n s i o n , e s p e c i a l l y i n s m a l l s t r a i n s . As c o m p a r e d w i t h t h e i n p u t v a l u e 0 = 4 1 ° , much h i g h e r v a l u e s o f 0 a r e p r e d i c t e d f r o m t h e p r e s s u r e m e t e r c u r v e s w i t h s m a l l e r L /D r a t i o s . H o w e v e r , i n t h e a p p l i c a t i o n o f Hughes e t a l m e t h o d t o c u r r e n t s t u d i e s , i t s h o u l d be n o t e d t h a t n o n l i n e a r e l a s t i c , s t r e s s d e p e n d e n t s o i l b e h a v i o r was e m p l o y e d i n t h e f i n i t e e l e m e n t a n a l y s e s . M o r e o v e r , t h e s h e a r - v o l u m e c o u p l i n g e f f e c t s o f g r a n u l a r m a t e r i a l was i g n o r e d . The s h e a r - v o l u m e A p p l i e d P r e s s u r e (Kpa) i n l o g 1 0 _ s c a l e 1U1 3 5 7 IO2 3 5 7 103 1 I I I » I I I I I I I I I I I I X O + G X O + G + + o OJ < m II II II X —»• T J IV) —• CO „ ^ o m (FEM) FEM) lutio + + + + m 1 39 c o u p l i n g e f f e c t s h o w e v e r , may n o t be s i g n i f i c a n t s i n c e t h e s t r a i n l e v e l e n c o u n t e r e d i n t h e a n a l y s e s was g e n e r a l l y l e s s t h a n 4% a n d t h e t e s t was s i m u l a t e d a t d e p t h where r e l a t i v e l y h i g h i n i t i a l s t r e s s s t a t e e x i s t e d . N e v e r t h e l e s s , i n t h e o r y , Hughes e t a l a n a l y t i c a l me thod may no t be e x a c t l y a p p l i c a b l e t o t h e c u r r e n t f i n i t e e l e m e n t r e s u l t s . The e r r o r i n u s i n g t h i s p r o c e d u r e may be p a r t l y r e f l e c t e d i n t h e f o u r t h c o l u m n o f T a b l e 6 . 7 , i n w h i c h i t i s shown t h a t b a s e d on Hughes e t a l p r o c e d u r e , t h e t h e o r e t i c a l f r i c t i o n a l a n g l e o f <f> = 4 1 ° w o u l d be o v e r p r e d i c t e d by a b o u t 11% f o r t h e c a v i t y e x p a n s i o n c a s e . In o r d e r t o e l i m i n a t e t h e p o s s i b l e e r r o r i n v o l v e d i n t h e e v a l u a t i o n p r o c e d u r e i t s e l f a n d t o i l l u s t r a t e t h e i n f l u e n c e o f t h e L/D r a t i o o n l y , t h e p r e d i c t e d f r i c t i o n a l a n g l e s <t> f o r d i f f e r e n t L/D r a t i o s a r e c o m p a r e d w i t h t h a t f o r c a v i t y e x p a n s i o n ( i . e . p l a n e s t r a i n ) c a s e . The r e s u l t s a r e i n c l u d e d i n t h e l a s t c o l u m n o f T a b l e 6 . 7 . A f t e r t h i s c o r r e c t i o n , i t i s shown t h a t t h e p r e s s u r e m e t e r t e s t s w i t h L /D r a t i o o f 1, 4, 12 w o u l d p r o b a b l y o v e r p r e d i c t t h e f r i c t i o n a l a n g l e <t> a b o u t 26%, 10% a n d 2% r e s p e c t i v e l y due t o t h e f i n i t e l e n g t h e f f e c t s . In o t h e r w o r d s , b a s e d on t h e a x i s y m m e t r i c a l p l a n e s t r a i n a s s u m p t i o n , t h e p r e s s u r e m e t e r w i t h membrane l e n g t h o f L/D r a t i o e q u a l t o o r g r e a t e r t h a n 4 may p r o d u c e g o o d r e s u l t s f o r t h e m a t e r i a l f r i c t i o n a l a n g l e w i t h an e r r o r l e s s t h a n 10%. S i m i l a r e v i d e n c e s were a l s o f o u n d i n t h e t r i a x i a l chamber c a l i b a r a t i o n t e s t . J e w e l l e t a l ( 1980 ) c o n d u c t e d 1 40 ' c a s t i n s i t u ' s e l f - b o r i n g t e s t s w i t h two d i f f e r e n t L/D r a t i o s i n t h e t r i a x i a l chamber u n d e r t h e same i n i t i a l v o i d r a t i o a n d i n s i t u s t r e s s s t a t e . I t was f o u n d t h a t r e d u c t i o n o f L /D r a t i o f r o m 6.2 t o 4 r e s u l t e d i n an o v e r p r e d i c t i o n o f t h e d e r i v e d f r i c t i o n a l a n g l e f r o m 4 4 . 5 ° t o 5 0 . 6 ° ( i n c r e a s i n g by a b o u t 1 4 % ) . 6 . 4 . 3 SUMMARY As shown e a r l i e r , t h e L/D r a t i o o f p r e s s u r e m e t e r membrane l e n g t h i s an i m p o r t a n t f a c t o r t o be c o n s i d e r e d i n t h e i n t e r p r e t a t i o n o f t h e p r e s s u r e m e t e r t e s t r e s u l t s u s i n g t h e a x i s y m m e t r i c a l p l a n e s t r a i n a s s u m p t i o n . B a s e d on t h e f o r e g o i n g a n a l y s e s , t h e L/D r a t i o seems t o h a v e d i f f e r e n t e f f e c t s on t h e t e s t r e s u l t s f o r c o h e s i v e s o i l s a n d c o h e s i o n l e s s s o i l s . F o r t h e p r e s s u r e m e t e r t e s t s i n s o f t c o h e s i v e s o i l s , t h e L/D r a t i o h a s l i m i t e d e f f e c t s on t h e s h a p e o f p r e s s u r e m e t e r c u r v e , a n d t h e e l a s t i c m o d u l u s d e t e r m i n a t i o n . When t h e p r e s s u r e m e t e r L /D r a t i o i s g r e a t e r t h a n 4, t h e g e n e r a l s h a p e o f p r e s s u r e m e t e r c u r v e i s c l o s e t o t h a t f o r p l a n e s t r a i n ^ c o n d i t i o n s , a n d t h e d e r i v e d e l a s t i c m o d u l u s i s a l s o c l o s e t o t h e t h e o r e t i c a l v a l u e . H o w e v e r , t h e d e r i v e d u n d r a i n e d s h e a r s t r e n g t h i s much more a f f e c t e d by t h e L/D r a t i o . The u n d r a i n e d s h e a r s t r e n g t h p r e d i c t e d f o r L /D = 4 w i l l be 15% h i g h e r t h a n t h a t o b t a i n e d f r o m t h e c y l i n d r i c a l c a v i t y e x p a n s i o n s o l u t i o n s . 141 For c o h e s i o n l e s s s o i l s , t he e f f e c t of L/D r a t i o i s more pronounced. The p r e s s u r e e x p a n s i o n c u r v e s p r o v i d e d by the c o n v e n t i o n a l p r e s s u r e m e t e r of L/D r a t i o = 8 w i l l be c l o s e t o the p l a n e s t r a i n c o n d i t i o n i n the r e l a t i v e l y s m a l l s t r a i n l e v e l (say l e s s than 4%). The l i m i t p r e s s u r e s p r e d i c t e d w i t h a f i n i t e membrane l e n g t h w i l l be s i g n i f i c a n t l y h i g h e r than the p l a n e s t r a i n v a l u e s . I t was a l s o found t h a t a l t h o u g h l a r g e d i f f e r e n c e between the l i m i t p r e s s u r e s i n t h e f i n i t e l e n g t h p r e s s u r e m e t e r t e s t i s caused by 'end e f f e c t s ' , the e l a s t i c modulus and the f r i c t i o n a l a n g l e s d e r i v e d from the e a r l i e r p a r t of the c u r v e ( l e s s than a s t r a i n of 4% i n t h i s s t u d i e s ) a r e l e s s a f f e c t e d (see F i g . 6.9(b) and T a b l e 6.7). T h i s may t h e r e f o r e suggest t h a t the i n t e r p r e t a t i o n of p r e s s u r e m e t e r t e s t d a t a t o o b t a i n <j> s h o u l d more count on the e a r l i e r p a r t of t e s t i f the p l a n e s t r a i n c o n d i t i o n i s g o i n g t o be used. However, i n p r a c t i c e i t i s known t h a t t h i s e a r l i e r p o r t i o n of the p r e s s u r e m e t e r c u r v e i s most v u l n e r a b l e t o the s m a l l s o i l d i s t u r b a n c e s . In f a c t , f o r the t e s t s i n c o h e s i o n l e s s s o i l s , due t o the g r a i n s i z e of sand p a r t i c l e s , the s m a l l d i s t u r b a n c e i n d u c e d d u r i n g the i n s e r t i o n of the probe i s almost i n e v i t a b l e . T h e r e f o r e , how t o o b t a i n a r e l i a b l e f r i c t i o n a l a n g l e from t h e p r e s s u r e m e t e r t e s t may s t i l l d e s e r v e f u r t h e r i n v e s t i g a t i o n . 1 42 6 .5 COMPARISONS OF CYL INDR ICAL CAV ITY EXPANSION ANALYSES AND  F I E L D PRESSUREMETER T E S T DATA To a s s e s s t h e v a l i d i t y o f t h e c y l i n d r i c a l c a v i t y e x p a n s i o n m o d e l f o r t h e p r e s s u r e m e t e r t e s t , t h e p r e s s u r e e x p a n s i o n c u r v e s f r o m f i n i t e e l e m e n t a n a l y s e s u n d e r c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n were c o m p a r e d w i t h t h o s e o b t a i n e d f r o m f i e l d p r e s s u r e m e t e r t e s t s . C o m p a r i s o n s were made f o r b o t h c o h e s i v e s o i l s a n d c o h e s i o n l e s s s o i l s . ' S e l f - b o r i n g p r e s s u r e m e t e r t e s t d a t a f r o m two d i f f e r e n t s i t e s were u s e d f o r t h i s p u r p o s e . F i n i t e e l e m e n t a n a l y s e s were p e r f o r m e d b a s e d on t h e f i n i t e e l e m e n t mesh shown i n F i g . 6 . 4 . H o w e v e r , i n e a c h c a s e , t h e mesh was s c a l e d by c e r t a i n f a c t o r s so a s t o g i v e t h e a c t u a l r a d i u s o f t h e p r e s s u r e m e t e r t h a t was u s e d i n t h e t e s t s . N o n l i n e a r h y p e r b o l i c s t r e s s s t r a i n r e l a t i o n s h i p was e m p l o y e d i n a l l t h e a n a l y s e s . 6 .5.1 COHESIVE SOILS S e l f - b o r i n g p r e s s u r e m e t e r t e s t r e s u l t s o b t a i n e d by Denby (1978) i n San F r a n c i s c o Bay Mud a t t h e s i t e o f t h e H a m i l t o n A i r F o r c e Ba se were e m p l o y e d f o r t h e c o m p a r i s o n . T h i s p a r t i c u l a r l o c a t i o n was s e l e c t e d b e c a u s e o f t h e l a r g e amount o f s o i l i n f o r m a t i o n a v a i l a b l e f r o m p r e v i o u s f i e l d a n d l a b o r a t o r y i n v e s t i g a t i o n s . In p a r t i c u l a r , Denby (1978) p e r f o r m e d a c o m p r e h e n s i v e t e s t p r o g r a m , and d e v e l o p e d an i n t e r p r e t a t i o n method t o o b t a i n a s e t o f s o i l p a r a m e t e r s t h a t a r e p a r t i c u l a r l y u s e f u l f o r t h e h y p e r b o l i c f i n i t e 143 e l e m e n t a n a l y s e s . The s e l f - b o r i n g p r e s s u r e m e t e r u s e d i n t h e t e s t i n g c o n s i s t s o f a b r o n z e c y l i n d e r a p p r o x i m a t e 120mm i n l e n g t h a n d 7.5mm i n d i a m e t e r , w h i c h i s p a r t i a l l y c o v e r e d by a f l e x i b l e r u b b e r membrane. The f l e x i b l e p r o b e has t h e L/D r a t i o a p p r o x i m a t e t o 6. The p r e s s u r e m e t e r t e s t d a t a a t d e p t h o f 6.8m was s e l e c t e d f o r t h e a n a l y s i s . T h e p r e s s u r e e x p a n s i o n c u r v e m e a s u r e d a t t h i s d e p t h i s shown i n F i g . 6 . 1 0 . The s o i l p a r a m e t e r s r e q u i r e d f o r t h e f i n i t e e l e m e n t a n a l y s e s were s e l e c t e d b a s e d on t h e d a t a b a s e f r o m b o t h l a b o r a t o r y t e s t s a n d f i e l d t e s t s . A . SO IL PROPERTIES AT HAMILTON A IR FORCE BASE 1) S o i l I n f o r m a t i o n f r o m t e s t d a t a The g e o l o g y c o n d i t i o n o f t h e s i t e h a s been i n v e s t i g a t e d by a number o f a u t h o r s , i n c l u d i n g T r a s k and R o l s t o n (1951) and T r e a s h e r ( 1 9 6 1 ) . F i g . 6.11 shows a l o g o f a b o r e h o l e d r i l l e d t o a d e p t h a b o u t 2 7 . 4 m a t t h e s i t e . The s o i l o f i n t e r e s t t o t h i s a n a l y s i s l i e s a t a d e p t h o f a r o u n d 6.8 m, where s o i l i s more h o m o g e n e o u s , a n d m a i n l y c o n s i s t s o f s o f t g r e y s i l t y c l a y . The g r o u n d w a t e r was f o u n d t o f l u c t u a t e b e t w e e n 2 t o 3 m be l ow t h e g r o u n d l e v e l d u r i n g t h e f i e l d t e s t i n g . - B a s i c S o i l P r o p e r t i e s 200-150-100-Pressu-emeter C u r v e d . • IZ 5U-Stress-Strain Curv* TEST NO HPH19 DEPlm 6-8 n. BAY HUO HOMOGENEOUS RflOIRL STRAIN 7. F i g . 6.10 P r e s s u r e Expansion Curve Measured a t 6.8 m Depth, H a m i l t o n A i r Force Base S i t e ( a f t e r Denby, 1978) 1 45 On 10-15--5 20--6 X CO k. r - CU 0_ u DEI <u u. Me 25-30-35-40-45 8 9 10 II 12 ,3 Oxidation Zone Roofs extremely abundant. Lf. grey stiff silty clay Oxidised Discoloration Fissures, organics, shells Sample dropped out of tube Soft,grey, silty clay Organics, shells Open fissures, organics Soft grey silty day Silt lens, mica, organics Soft grey silty clay Completely remolded Highly fissile Numerous silt Lenses Organics, shells Silt lens 45l 50-55-60' 65" X r-CL LU Q 70 75 80-8 5 -90-16 17 his 9 -20 -21 -22 -23 -24 -25 26 r-27 Silt lens Shells, organics uj 5°l Oork greenish grey clay Shells common" Light olive brown to light yellowish brown clay Dark greenish grey clay Organics common U J la) cn: §o i i CO" F i g . 6.11 Log of Borehole at Hamilton Air Force Base (after Denby, 1978) 1 46 F i g . 6.12 g i v e s some of the e n g i n e e r i n g p r o p e r t i e s d e r i v e d f o r Bay Mud a t the s i t e from t e s t s on u n d i s t u r b e d s o i l samples as r e p o r t e d by Clough and Denby (1980). The s o i l p r o f i l e c o n s i s t s of a d e s i c c a t e d c r u s t of grey c l a y t h a t g r a d u a l l y t r a n s f o r m s t o a s o f t n o r m a l l y c o n s o l i d a t e d c l a y d e p o s i t a t a de p t h about 5.5 m. The s o f t c l a y e x t e n d s t o a depth of ap p r o x i m a t e 15m, a t which a m o d e r a t e l y o v e r c o n s o l i d a t e d Bay Mud l a y e r i s e n c o u n t e r e d . The n a t u r a l water c o n t e n t i s about 60% above 2.1 m and 90% below t h i s d e p t h . The p l a s t i c and l i q u i d l i m i t s a r e about 45 and 90 r e s p e c t i v e l y . The t o t a l u n i t weight i s 1.5.7 kN/m3 t o a d e p t h of 2.1 m and 14.8 kN/m3 t h e r e a f t e r . S e n s i t i v i t i e s of the s o i l below 2.1 m range from 6 t o 8. C o n s o l i d a t i o n t e s t s (Duncan, 1965 and Denby, 1978) showed t h a t the s o i l from a 5.5m towards the s u r f a c e i s i n c r e a s i n g l y o v e r c o n s o l i d a t e d , presumably due t o the d e s i c c a t i o n c r u s t . Below 5.5m, t h e r e s u l t s s u g g e s t e d t h a t the Bay Mud t o a depth of about 15m i s s l i g h t l y o v e r c o n s o l i d a t e d w i t h OCR = 1.1 t o 1.3. - Un d r a i n e d Shear S t r e n g t h The u n d r a i n e d shear s t r e n g t h v a l u e s d e r i v e d from t h e pr e s s u r e m e t e r t e s t s r e p o r t e d by Denby (1978) a r e p l o t t e d v e r s u s d e p t h i n F i g . 6.13. A l s o shown a r e the shear s t r e n g t h from the UU t e s t s by Duncan (1965), and the l a b o r a t o r y vane and CU t r i a x i a l t e s t s by Denby (1978). D E P T H - Meters 1 1 1 1 1 1 1 1 1 (\\\\\\\\\\\W\\\\ Soft Grey Cloy (New Boy Mud) i • i i — r i 1 Dessicotion Zone | r— - 1 1 Li l 1 48 UNDRAINED SHEAR STRENGTH- kN / m 2 0 10 20 30 40 F i g . 6.13 U n d r a i n e d S t r e n g t h s w i t h Depth ( a f t e r C l o u g h and Denby, 1980) 1 49 Both r e s u l t s i n F i g . 6.12 and F i g . 6.13 from the UU t r i a x i a l c o m p r e s s i o n and l a b vane t e s t s i n d i c a t e t h a t a r a p i d d e c r e a s e of u n d r a i n e d shear s t r e n g t h w i t h d e pth t h r o u g h the d e s i c c a t e d c r u s t up t o a d e pth of 5.5m, f o l l o w i n g by a r e g i o n where the r a t i o of the u n d r a i n e d shear s t r e n g t h , C^r t o the e f f e c t i v e o verburden p r e s s u r e , P, i s of 0.32. CU t r i a x i a l t e s t r e s u l t s p erformed by Denby (1978) based on SHANSEP ( t h e S t r e s s H i s t o r y And N o n l i n e a r S o i l E n g i n e e r i n g P r o p e r t i e s ) method (Ladd and F o o t t , 1974) i n d i c a t e t h a t t h e i n c r e a s e of shear s t r e n g t h w i t h OCR = 1 f o l l o w s C u/P = 0.35. However, C l o u g h and Denby (1980) p o i n t e d out t h a t as the Bay Mud between 5.5 and 15 m i s s l i g h t l y o v e r c o n s o l i d a t e d w i t h OCR v a l u e s of 1.1 t o 1.3, the a c t u a l C^/P r a t i o of t h e i n - s i t u s o i l may be s l i g h t l y h i g h e r than t h a t i n d i c a t e d by the t e s t s p erformed on samples t h a t a r e r e c o n s o l i d a t e d t o v i r g i n c o n d i t i o n s (OCR =1) or l a b vane t e s t s on samples t h a t may be s l i g h t l y d i s t u r b e d . T h e r e f o r e , based on the t e s t d a t a r e p o r t e d by Ladd and F o o t t ( 1 9 7 4 ) , they s u g g e s t e d t h a t the C u/P r a t i o s h o u l d be i n c r e a s e d by 10% t o 20% as t h e OCR v a l u e s i n c r e a s e from 1.0 t o 1.3. Thus, the r e p o r t e d C^/P r a t i o f o r Bay Mud below the d e s s i c c a t e d zone c o u l d be around 0.4 t o 0.45 r a t h e r than the 0.32 t o 0.37 v a l u e s i n d i c a t e d by the t e s t s . The e s t i m a t e d C u/P f o r the c a s e of OCR = 1.2 was shown i n F i g 6.13. 150 In view of the f o r e g o i n g a n a l y s i s , two u n d r a i n e d shear s t r e n g t h v a l u e s were chosen f o r the f i n i t e element a n a l y s i s . One was d i r e c t l y adopted from the v a l u e d e t e r m i n e d from SBPMT u s i n g Denby's method ( 1 9 7 8 ) , which i s 24.1 Kpa. Another was s e l e c t e d from the average v a l u e s of t h e u n d r a i n e d shear s t r e n g t h r e s u l t s a v a i l a b l e from d i f f e r e n t t e s t s r e p o r t e d by Denby (1978), which i s 22.5 Kpa. - C o e f f i c i e n t of E a r t h P r e s s u r e a t Rest The c o e f f i c i e n t of e a r t h p r e s s u r e a t r e s t , K 0, i s d e f i n e d as the r a t i o of e f f e c t i v e h o r i z o n t a l t o e f f e c t i v e v e r t i c a l s t r e s s . F i e l d measurements of K 0 w i t h d epth r e p o r t e d by Denby (1978) a r e shown i n F i g . 6.14. Based on Denby's s e l f - b o r i n g p r e s s u r e m e t e r t e s t d a t a , K 0 v a l u e s were found t o d e c r e a s e from 0.8 i n the d e s i c c a t e d zone a t 2.9m t o 0.5 - 0.6 f o r depths of 3.5 t o 9.1m. However, d a t a from G l o t z l l o a d c e l l t e s t s show t h a t K 0 has v a l u e s of 1.0 and 0.8 f o r the d e s s i c c a t e d zone and of 0.7 f o r t h e s l i g h t l y O.C. zone a t 7.0 m. T h e r e f o r e , s i g n i f i c a n t d i f f e r e n c e e x i s t s between the K 0 v a l u e s from t h e s e two t y p e s of t e s t s . The d e t e r m i n e d K 0 v a l u e s a r e much dependent upon the ground water t a b l e p o s i t i o n , the u n i t weight of s o i l s , and the measurements of l i f t - o f f p r e s s u r e on the p r e s s u r e m e t e r c u r v e s . In c o n s i d e r a t i o n of the u n c e r t a i n t y i n the f i e l d measurement of i n - s i t u h o r i z o n t a l s t r e s s ( o r K 0 v a l u e s ) , two v a l u e s of K 0, i . e . K o=0.5, 1.0, were used t o c a l c u l a t e the h o r i z o n t a l t o t a l s t r e s s i n the a n a l y s i s . The h o r i z o n t a l T o t a l H o r i z o n t a l S t r e s s KN/m2 F i g . 6.14 C o e f f i c i e n t of E a r t h P r e s s u r e K ( a f t e r Denby, 1978) 1 52 t o t a l s t r e s s was c a l c u l a t e d f r o m t h e v e r t i c a l e f f e c t i v e s t r e s s w i t h t h e a s sumed u n i t w e i g h t o f s o i l a n d t h e a s sumed g r o u n d w a t e r t a b l e p o s i t i o n . T h a t i s au = K 0 - a ' + U 0 ( 6 . 5 . 1 ) r i v where ov, i s t h e i n - s i t u h o r i z o n t a l t o t a l s t r e s s , a ' i s t h e r i v i n - s i t u v e r t i c a l e f f e c t i v e s t r e s s , a n d U 0 i s t h e h y d r o s t a t i c p r e s s u r e a t d e p t h o f i n t e r e s t . - S h e a r M o d u l u s The i n i t i a l t a n g e n t s h e a r m o d u l u s c a n be o b t a i n e d f r o m t h e i n i t i a l s l o p e o f p r e s s u r e e x p a n s i o n c u r v e . The d e t e r m i n e d i n i t i a l s h e a r m o d u l u s f r o m D e n b y ' s s e l f - b o r i n g p r e s s u r e m e t e r t e s t s a r e shown i n T a b l e . 6 . 8 . A l s o shown i n t h e t a b l e a r e t h e d e r i v e d u n d r a i n e d s h e a r s t r e n g t h and t h e s h e a r m o d u l u s m u l t i p l i e r , M. As shown i n t h e t a b l e , t h e t e s t a t d e p t h o f 6.8m ( 2 2 . 2 f t ) g ave a s h e a r m o d u l u s o f 6250 KN/m 2 and M =780. T h e s e v a l u e s were u s e d i n t h e a n a l y s i s . T h e y a r e d e r i v e d u s i n g D e n b y ' s i n t e r p r e t a t i o n ( D e n b y , 1 9 7 8 ) , and a r e c o r r e s p o n d e n t t o C u = 24.1 K p a . A n o t h e r s e t o f v a l u e s , G = 7425 K N / m 2 , M = 9 9 0 , were c h o s e n i n t h e a n a l y s i s , t h e y a r e f r o m t h e a v e r a g e v a l u e s o f s h e a r m o d u l u s and m o d u l u s m u l t i p l i e r o b t a i n e d by D e n b y , a n d a r e c o r r e s p o n d e n t t o C u = 2 2 . 5 K p a . 2) S o i l P a r a m e t e r s f o r F i n i t e E l e m e n t A n a l y s e s T a b l e 6.8 P r e s s u r e m e t e r T e s t R e s u l t s a t H a m i l t o n T e s t S i t e ( a f t e r Denby, 1978) DEPTH FT. TEST NO. kN/m2 V kN/m2 M 9.5 14 28.2 5250 560 12.4 15 25.5 5400 640 15.0 16 22.1 5250 710 17.3 17 23.1 6800 880 20.0 18 21.8 5000 690 22.2 19 24.1 6250 780 24.9 20 * 22.7 4100 540 27.5 21 24.1 10000 1200 31.1 22 24.1 8400 1040 36.0 23 26.8 9500 1060 40.9 24 31.4 12800 1220 45.8 25 33.3 11600 1040 49.1 26 37.2 19100 1500 E = 2(1 + e)G * E f f e c t s of d i s t u r b a n c e e v i d e n t 1 54 B a s e d on t h e a b o v e r e v i e w on t h e a v a i l a b l e s o i l i n f o r m a t i o n a t t h e t e s t s i t e , t h e s o i l p a r a m e t e r s c a n be a s s u m e d f o r t h e f i n i t e e l e m e n t c a v i t y e x p a n s i o n a n a l y s e s . Two s e t s o f s o i l p a r a m e t e r s were u s e d i n t h e a n a l y s e s . T h e y a r e shown i n T a b l e 6 . 9 . In t h e c o m p u t e r a n a l y s e s o f c a s e ( 1 ) , t h e s o i l p a r a m e t e r s a r e t h o s e i n t e r p r e t e d f r o m t h e s e l f - b o r i n g t e s t r e s u l t s u s i n g t h e p r o c e d u r e d e v e l o p e d by Denby ( 1 9 7 8 ) . In t h e a n a l y s e s o f c a s e ( 2 ) , a s e t o f g e n e r a l s o i l p a r a m e t e r s were s e l e c t e d f r o m t h e a v e r a g e v a l u e s o f p r e s s u r e m e t e r d a t a w i t h d i f f e r e n t i n t e r p r e t a t i o n p r o c e d u r e s ( D e n b y , 1 9 7 8 ) , and t h e l a b d a t a . In t h e a n a l y s e s , t h e s o i l medium were a s sumed t o be o f i s o t r o p i c a n d homogeneous N o r m a l l y C o n s o l i d a t e d c l a y . As b e f o r e , t h e a n a l y s e s were b a s e d on a t o t a l s t r e s s a p p r o a c h . The u n d r a i n e d t e s t i n g c o n d i t i o n was a s s u m e d , and s i m u l a t e d u s i n g a P o i s s o n ' s r a t i o o f 0 . 4 9 9 . The i n i t i a l s t r e s s u s e d i n t h e a n a l y s e s was a s sumed t o be i s o t r o p i c and homogeneous , a n d i t was e q u a l t o t h e i n - s i t u h o r i z o n t a l s t r e s s w h i c h was c a l c u l a t e d f r o m t h e v e r t i c a l o v e r b u r d e n s t r e s s w i t h g i v e n v a l u e s o f K 0 . The K 0 c o n s o l i d a t i o n e f f e c t o f t h e r e a l s o i l d e p o s i t was n e g l e c t e d i n t h e a n a l y s e s . T h e r e f o r e , f o r e a c h s e t o f s o i l p a r a m e t e r s , two d i f f e r e n t i n i t i a l s t r e s s v a l u e s were a s sumed f o r s e n s i t i v i t y s t u d y . The e f f e c t s o f n e g l e c t i n g K 0 c o n d i t i o n i n t h e a n a l y s i s w i l l be d i s c u s s e d l a t e r . 155 T a b l e 6.9 S o i l Parameters adopted f o r San F r a n c i s c o Bay Mud i n F i n i t e Element A n a l y s e s S o i l P arameters C o h e s i v e (1) S o i l s (2) C u (Kpa) Gi (Kpa) M 24. 1 6250 780 0.499 75 or 110 22.5 7425 990 0.499 75 or 110 v a H (Kpa) K 0 7 (KN/m 3) Rf 0.5 or 1.0 16 0.87 0.5 or 1.0 16 0.9 2 6.8 G.W.T. (m) Depth (m) 2 6.8 M - Modulus M u l t i p l i e r ; G.W.T. - Ground Water T a b l e . B. RESULTS AND DISCUSSION The r e s u l t s from p l a n e s t r a i n c a v i t y f i n i t e element a n a l y s e s a re shown i n F i g . 6.15 and F i g . 6.16 r e s p e c t i v e l y f o r the f i r s t and the second s e t s of s o i l p arameters shown i n T a b l e 6.9. The f i e l d e x p a n s i o n c u r v e i s a l s o i n c l u d e d i n the f i g u r e s f o r co m p a r i s o n . G e n e r a l l y , the r e s u l t s of th e f i n i t e element a n a l y s e s a r e i n good agreement w i t h the f i e l d d a t a , c o n s i d e r i n g the u n c e r t a i n t y i n the s o i l p a r a m e t e r s . In a l l the a n a l y s e s , t he p r e d i c t e d p r e s s u r e e x p a n s i o n c u r v e s based on the c a v i t y e x p a n s i o n c o n d i t i o n s f i t the f i e l d e x p a n s i o n c u r v e almost p e r f e c t l y , w i t h o n l y minor d e v i a t i o n . Comparing the r e s u l t s from SBPMT parameters and thos e from the average s o i l p a r a m e t e r s , i t was found t h a t the 99 I o o . CM © 1 0 i 1 <$ i 0 FIELD MEASUREMENTS FEM(AVERAGE DATA) K o=1.0 FEM(AVERAGE DATA) K o=0.5 LJ-r\j _ ~ r i i 1 1 i 1 1 1 1 1 1 1 1 1 1 1 r 1.0 2.0 3.0 4 .0 5.0 6 .0 7.0 8 .0 9 .0 CIRCUM. STRAIN{%) SAN FRANSICO BAY M U D ( C u = 2 2 . 5 K P A ) 10.0 F i g . 6.16 Comparison w i t h F i e l d Measurements 158 a n a l y s e s w i t h g e n e r a l s o i l p a r a m e t e r s p r e d i c t a s t i f f e r r e s p o n s e a t t h e e a r l i e r s t a g e , and a s o f t e r r e s p o n s e l a t e r a t l a r g e s t r a i n s . T h i s i s r e a s o n a b l e a s i n t h e s e c o n d s e t o f s o i l p a r a m e t e r s , t h e i n i t i a l m o d u l u s i s h i g h e r , b u t t h e u n d r a i n e d s h e a r s t r e n g t h i s l o w e r t h a n t h e f i r s t o n e . H o w e v e r , t h e r e s u l t s c o m p a r e d w i t h f i e l d d a t a showed t h a t t h e d i f f e r e n c e r e s u l t e d f r o m t h e d i f f e r e n t s e t o f s o i l p a r a m e t e r s i s n e a r l y i n v i s i b l e . C o m p a r i n g t h e r e s u l t s w i t h d i f f e r e n t i n i t i a l s t r e s s e s , i t was f o u n d t h a t t h e h i g h e r i n i t i a l s t r e s s r e s u l t e d i n s m a l l e r d i s p l a c e m e n t s , b u t d i d n o t g e n e r a t e s i g n i f i c a n t d i f f e r e n c e i n t h e g e n e r a l s h a p e o f t h e p r e d i c t e d c u r v e s . A l t h o u g h t h e r e e x i s t some m i n o r d e v i a t i o n s , t h e c o m p a r i s o n o f a l l t h e f i n i t e e l e m e n t a n a l y s e s r e s u l t s w i t h f i e l d d a t a seem t o c o n f i r m t h e p r e v i o u s c o n c l u s i o n t h a t i n s o f t c o h e s i v e s o i l s , t h e p r e s s u r e m e t e r t e s t w i t h L/D r a t i o l a r g e r t h a n 4 may p r o d u c e t h e t e s t r e s u l t s t h a t a r e c l o s e t o t h o s e o f c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n . In a d d i t i o n , a l l t h e r e s u l t s a l s o t e n d t o s u g g e s t t h a t t h e h y p e r b o l i c s t r e s s s t r a i n r e l a t i o n may be a b l e t o p r o v i d e good f i t s w i t h f i e l d d a t a i n s o f t c o h e s i v e s o i l s , s u c h a s i n San F r a n c i s c o Bay Mud. The r e a s o n s f o r t h e c o n s i s t e n t r e s u l t s p r e d i c t e d by t h e f i n i t e e l e m e n t a n a l y s i s u n d e r t h e c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n , may be p o s t u l a t e d , i n a d d i t i o n t o t h e s u f f i c i e n t L/D r a t i o , t h a t t h e f o l l o w i n g f a c t o r s p l a y an a c t i v e r o l e : 1 59 1. T h e s o i l p a r a m e t e r s d e r i v e d f r o m b o t h SBPMT a n d l a b o r a t o r y t e s t s a r e o f h i g h q u a l i t y . 2. T h e s t r e s s s t r a i n r e l a t i o n o f San F r a n c i s c o Bay Mud may a p p e a r more h y p e r b o l i c . Duncan (1965) r e p o r t e d t h a t t h e s t r e n g t h d i f f e r e n c e i n e x t e n s i o n a n d c o m p r e s s i o n t e s t s a r e low r e l a t i v e t o o t h e r s o i l s . B a s e d on t h e d a t a r e p o r t e d by L a d d e t a l ( 1 9 7 7 ) , P r e v o s t ( 1 9 7 6 ) , San F r a n c i s c o Bay Mud i s o f o n l y a m i l d d e g r e e o f a n i s o t r o p y . T h e r e f o r e , i n c r e m e n t a l l i n e a r e l a s t i c , i s o t r o p i c a n a l y s i s may be more a p p r o p r i a t e t o t h i s t y p e o f s o i l s t h a n o t h e r s . 3. K 0 c o n s o l i d a t i o n e f f e c t may n o t be i m p o r t a n t . In t h e f i e l d t e s t i n g , t h r e e p o s s i b l e f a i l u r e modes o f s o i l s may o c c u r i n t h e s o i l a r o u n d t h e p r e s s u r e m e t e r , d e p e n d i n g upon t h e i n - s i t u r e l a t i v e m a g n i t u d e s o f t h r e e p r i n i c i p a l s t r e s s e s , i . e . K 0 c o n d i t i o n . T h e s e p o s s i b l e f a i l u r e modes a r e shown i n F i g . 6 . 1 7 . B a s e d on t h e i r t r u e t r i a x i a l e x p e r i m e n t a l d a t a , Wood a n d W r o t h (1977 ) c o n c l u d e d t h a t f o r t h e i n i t i a l l y n o r m a l l y o r s l i g h t l y o v e r c o n s o l i d a t e d c l a y , where t h e v e r t i c a l p r i n c i p a l s t r e s s (o'z o r a\) i s much g r e a t e r t h a n t h e h o r i z o n t a l s t r e s s e s ( K 0 C T ' , K 0 < 1 ) , r e a d j u s t m e n t o f t h e r e l a t i v e m a g n i t u d e s o f t h e s t r e s s e s o c c u r s v e r y r a p i d l y a f t e r t h e u n d r a i n e d s h e a r i n g ha s s t a r t e d , a n d t h e f a i l u r e o f s o i l s c a n be c o n s i d e r e d t o be g o v e r n e d by t h e d i f f e r e n c e o f h o r i z o n t a l s t r e s s e s a l o n e , a n d i s i n t h e mode o f rid p l a n e a s shown i n F i g . 6 . 1 7 . The i n i t i a l v e r t i c a l s t r e s s 160 F i g . 6.17 S o i l F a i l u r e Modes a s s o c i a t e d w i t h P r e s s u r e m e t e r T e s t s i n C o h e s i v e S o i l s ( a f t e r Wood and Wroth, 1977) 161 q u i c k l y becomes i n t e r m e d i a t e p r i n c i p a l s t r e s s , and has l i t t l e e f f e c t on the whole t e s t r e s u l t s . In l i g h t of t h e s e e x p e r i m e n t a l e v i d e n c e s , the i n i t i a l i s o t r o p i c s o i l s t r e s s s t a t e assumed i n the a n a l y s i s , w h i c h was o b t a i n e d from the i n - s i t u h o r i z o n t a l s t r e s s w i l l not g e n e r a t e r e s u l t s t h a t a r e s i g n i f i c a n t l y d i f f e r e n t from the f i e l d r e s p o n s e . 6.5.2 COHESIONLESS SOILS S e l f - b o r i n g p r e s s u r e m e t e r t e s t (SBPMT) r e s u l t s o b t a i n e d by Hughes and R o b e r t s o n (1984) a t McDonald Farm, a UBC r e s e a r c h s i t e l o c a t e d on an abandoned farm near the Vancouver I n t e r n a t i o n a l A i r p o r t , were employed f o r the comparison i n c o h e s i o n l e s s s o i l s . A summary of the s o i l p r o f i l e based on s a m p l i n g , l a b o r a t o r y and cone p e n e t r a t i o n t e s t i n g (CPT) i s shown i n F i g . 6.18. The upper 2 m of s o i l c o n s i s t s of s o f t , c o m p r e s s i b l e c l a y s and s i l t s , f o l l o w e d by a sand l a y e r up t o 13 m. The sand has a medium t o c o a r s e g r a i n s i z e w i t h l a y e r s of f i n e sand. The ground water i s g e n e r a l l y 1 t o 2 m below the ground s u r f a c e , and the ground water p r e s s u r e s a r e a p p r o x i m a t e l y h y d r o s t a t i c . The t e s t d a t a o b t a i n e d a t depth of 7 m was chosen f o r c o m p a r i s o n , where s o i l m a i n l y c o n s i s t s of c o a r s e sand w i t h a few t h i n l a y e r s of f i n e sand. The p r e s s u r e m e t e r used f o r the t e s t i n g was a 76 mm i n d i a m e t e r w i t h a f l e x i b l e membrane of L/D r a t i o = 6. The p r e s s u r e e x p a n s i o n measured a t the depth PORE PRESSURE FRICTION RESISTANCE U (BAR) „ fC (BAR) 0 BEARING RESISTANCE QT (BAR) 2 00 FRICTION RATIO RF = FC/QT 1%) 0 2 20 30 0, = 60% (Boldi el ol ,1982) -I r i i 1 i I 0-DIFFERENTIAL P.P. RATIO AU/QT„ 0 . Bo 0 SOIL PROFILE 10 "Equilibrium pore pressure I BAR = lOOkPa - l k g f / c m2 = I Ton/ft. 2 Soft CLAY S SILT Coarse SAN 0 Loose to Dense with layers ot line Sand Fine SAND, some si 11 Soft, normally consolidated clayey SILT Sand = 10% Sill = 70% Cloy •• 2 0 % L.L. 3 8 % P.I. • 15% wn •• 3 5 % k=s8X|0"7cmA«c Ce = 0.3 F i g . 6.18 S o i l P r o f i l e f o r Research S i t e at McDonald Farm, Sea Island 163 i s shown i n F i g . 6.19. A. SOIL PARAMETERS Comprehensive s o i l t e s t i n g programs have been conducted a t the McDonald Farm, g e n e r a t i n g l a r g e amounts of s o i l i n f o r m a t i o n , from which c o r r e l a t i o n s f o r b a s i c s o i l p a r ameters have been e s t a b l i s h e d ( R o b e r t s o n and Campanella, 1 984) From t h e cone b e a r i n g p r o f i l e , t he r e l a t i v e d e n s i t y , D , and f r i c t i o n a n g l e , <t>, of the sand d e p o s i t can be d e t e r m i n e d ( R o b e r t s o n and Campanella, 1984). A r e l a t i v e d e n s i t y of D r = 60% and a f r i c t i o n a n g l e of <f> = 42° were o b t a i n e d a t the depth of 7 m f o r t h e a n a l y s i s . The i n i t i a l e l a s t i c modulus G or E used was de t e r m i n e d from the s e l f - b o r i n g p r e s s u r e m e t e r t e s t (SBPMT) d a t a . At dep t h of 7 m, the i n i t i a l e l a s t i c shear modulus, G, was found t o be 42 Mpa. Then the e l a s t i c Young's modulus, E, was o b t a i n e d t h r o u g h E = 2G (1 + u) (6.5.1) w i t h an as s u m p t i o n t h a t the d r a i n e d P o i s s o n ' s r a t i o , u, f o r sand i s e q u a l t o 0.2. Two s e t s of s o i l parameters were used f o r the a n a l y s e s . The f i r s t s e t of parameters was s e l e c t e d from CPT and SBPMT d a t a . They a r e shown i n the f i r s t column of T a b l e 6.10. Another s e t was from the d a t a r e p o r t e d by Byrne and Cheung 1 6 4 I l O O r -IOOO -9 0 0 -~ 8 0 0 -o O. 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 100 Rodiol Dtsplocement (%) R F i g . 6.19 P r e s s u r e E x p a n s i o n C u r v e measured a t 7 m Depth, McDonald's Farm ( a f t e r Hudges and R o b e r t s o n , 1984) 165 T a b l e 6.10 S o i l P a r amters adopted f o r C o h e s i o n l e s s S o i l s a t McDonald Farm S i t e i n F i n i t e Element A n a l y s e s S o i l Parameters SBPMT & CPT Byrne & Cheung (1) (2) D r (%) 60 % 60 % <t> (°) 42° 38° (°) 2° 2° 0 C V (°) 33° 33° Gi (Mpa) 42 40.5 K E 995 960 K B 587 576 n 0.5 0.5 m 0.25 0.25 Rf 0.8 0.8 v 0.2 0.2 7 (KN/m 3) 20.0 20.0 a v (Kpa) 90 90 aii (Kpa) 90 or 65 90 or 65 K 0 1.0 or 0.9 1.0 or 0.9 G.W.T. (m) 2 2 Depth (m) 7.0 7.0 1. G.W.T. - Ground Water T a b l e . (1984). T h e i r d a t a a r e shown i n T a b l e 6.11, which were d e t e r m i n e d from the f i n i t e element back a n a n l y s i s of the f i e l d s e t t l e m e n t o b s e r v a t i o n s , and from a comprehensive r e v i e w of the d a t a base from t h e l a b o r a t o r y and the f i e l d t e s t s . The s o i l p r o p e r t i e s a r e r e l a t e d t o the r e l a t i v e d e n s i t y of sand, and the r e l a t i v e d e n s i t y D r i s d e t e r m i n e d from t h e SPT blow c o u n t s . H e r e i n , the r e l a t i v e d e n s i t y of the sand a t the t e s t d e p t h of 7 m can be e v a l u a t e d d i r e c t l y from the cone b e a r i n g p r o f i l e . As s t a t e d e a r l i e r , the e s t i m a t e d r e l a t i v e d e n s i t y a t t h i s d e p t h i s about 60%. T h e r e f o r e , a complete s e t of Table 6.11 S o i l Parameters proposed by Byrne and Cheung (1984) D r N l k E n kB m • l A<t> 4>cv R F 25 5 300 .5 180 .25 33 0 33 0.9 50 10 600 .5 360 .25 36 2 33 0.8 75 25 1500 .5 900 .25 41 4 33 0.7 100 > 50 3000 .5 1500 .25 50 9 33 0.6 167 s o i l p a r a m e t e r s r e q u i r e d f o r t h e h y p e r b o l i c f i n i t e e l e m e n t a n a l y s i s was o b t a i n e d f r o m T a b l e 6 . 1 1 . T h i s s e t o f s o i l p a r a m e t e r s i s a l s o shown i n T a b l e 6 . 1 0 . As i n t h e c a s e o f c o h e s i v e s o i l s , t h e i n i t i a l s t r e s s i n t h e s o i l medium was a s s u m e d t o be i s o t r o p i c , h o m o g e n e o u s , a n d c o r r e s p o n d t o t h e i n - s i t u h o r i z o n t a l e f f e c t i v e s t r e s s . B a s e d on t h e SBPMT d a t a , t h e i n - s i t u h o r i z o n t a l e f f e c t i v e s t r e s s c a n be e s t i m a t e d f r o m t h e l i f t - o f f p r e s s u r e . ' F rom t h e p r e s s u r e e x p a n s i o n c u r v e o b t a i n e d a t d e p t h o f 7 m ( s e e F i g . 6 . 1 9 ) , a l i f t - o f f p r e s s u r e o f 170 Kpa was o b t a i n e d , and t h e r e f o r e a v a l u e o f K 0 = 0 .9 was e s t i m a t e d b a s e d on t h e e s t i m a t e d u n i t w e i g h t o f s o i l and t h e a s sumed g r o u n d w a t e r t a b l e p o s i t i o n . As i n c o h e s i v e s o i l s , c o n s i d e r i n g t h a t t h e m e a s u r e m e n t s o f l i f t - o f f p r e s s u r e i s v e r y s e n s i t i v e t o t h e s o i l d i s t u r b a n c e , two d i f f e r e n t v a l u e s o f i n i t i a l s t a t e were u s e d i n t h e a n a l y s e s , w h i c h c o r r e s p o n d s t o t h e i n - s i t u h o r i z o n t a l s t r e s s w i t h K 0 = 0 . 9 a n d 1.0. B. RESULTS AND COMPARISONS F i n i t e e l e m e n t a n a l y s i s r e s u l t s w i t h two d i f f e r e n t s e t o f s o i l p a r a m e t e r s i n T a b l e 6 .10 a r e shown i n F i g . 6 . 2 0 , F i g 6 . 2 1 . F o r e a c h s e t o f p a r a m e t e r s , r e s u l t s f r o m d i f f e r e n t i n i t i a l s t r e s s s t a t e a r e c o m p a r e d . In T a b l e 6 . 1 0 , t h e e l a s t i c m o d u l i ( i . e , E a n d B) a r e e s t i m a t e d w i t h P o i s s o n ' s r a t i o o f 0 . 2 . In o r d e r t o i n v e s t i g a t e t h e i n f l u e n c e o f t h e a s sumed P o i s s o n ' s r a t i o o f s a n d on t h e a n a l y s i s r e s u l t s , 168 a n o t h e r f i n i t e element a n a l y s i s was performed w i t h t h e e l a s t i c modulus c a l c u l a t e d from SBPMT and CPT d a t a u s i n g P o i s s o n ' s r a t i o of 0.3 (see Eq. ( 6 . 5 . 1 ) ) . The r e s u l t s a r e shown i n F i g 6.22 w i t h c o mparison of those from the f i r s t s e t of s o i l p a rameters i n T a b l e 6.10. R e s u l t s from the f i r s t s e t of s o i l p arameters a r e shown i n F i g 6.20. The f i n i t e element a n a l y s i s w i t h c a v i t y e x p a n s i o n c o n d i t i o n p r e d i c t s p r e s s u r e e x p a n s i o n c u r v e s s i m i l a r t o the f i e l d measurements i n shape. I t i s i n t e r e s t i n g t o note t h a t t h e c u r v e p r e d i c t e d w i t h K 0 = 0.9 which i s the v a l u e measured i n f i e l d g i v e s a v e r y good agreement w i t h the f i e l d c u r v e i n the i n i t i a l p o r t i o n , but the c u r v e becomes s o f t e r a t l a r g e s t r a i n s . L o o k i n g a t the c u r v e from the same s e t of s o i l p arameters h i g h e r i n i t i a l s t r e s s ( K 0 =1.0), i t i s o b s e r v e d t h a t the c u r v e i s s h i f t e d upwards due t o the h i g h e r i n i t i a l s t r e s s s t a t e . Such a r e s u l t can a l s o be o b s e r v e d i n F i g 6.21 w i t h the second s e t of s o i l p a r a m e t e r s . T h e r e f o r e , the above r e s u l t s seem t o suggest t h a t the f i e l d measurements of i n i t i a l shear modulus, and the h o r i z o n t a l e f f e c t i v e s t r e s s from the s e l f - b o r i n g p r e s s u r e m e t e r t e s t a t t h i s d e p th a r e p r o b a b l y r e a s o n a b l e . I t i s shown i n F i g 6.21 t h a t the a n a l y s e s w i t h t h e s o i l p a rameters from Byrne and Cheung (1984) a l s o p r e d i c t t h e same t r e n d as t h o s e w i t h t h e parameters d i r e c t l y from SBPMT. In s m a l l s t r a i n range up t o 1%, the c u r v e p r e d i c t e d w i t h K 0 = 0.9 i s a l s o v e r y c l o s e t o the f i e l d measurement, but F i g . 6.20 Comparison o f F i n i t e E l e m e n t P r e d i c t i o n w i t h F i e l d Measurements (SBPMT and CPT d a t a ) vo o o . X _ a CD - o . •oo ©— I —© 1 1 <3>— 1 0 © FIELD MEASUREMENTS f FEM(Byrne & Cheung d a t a ) K 0=t.O 4> FEM(Byrne & Cheung d a t a ) K o=0.9 I I I I I I I I 1 I I 1 I I 1 1 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 CIRCUM. STRAIN {%) MCDONALD FARM SAND(DR=60X) DEPTH=7M F i g . 6.21 Comparison of F i n i t e Element P r e d i c t i o n w i t h F i e l d Measurements ( B y r n e and Cheung d a t a ) 0.0 12.0 o 171 becomes s o f t e r i n l a r g e r s t r a i n s . S i m i l a r t o the e a r l i e r r e s u l t s , the c u r v e p r e d i c t e d w i t h K 0 = 1.0 i s s t i f f e r , i t o v e r p r e d i c t s the f i e l d c u r v e i n the s t r a i n l e s s than 2.4%, but u n d e r p r e d i c t s t h e r e a f t e r . Compared w i t h the c u r v e p r e d i c t e d from the f i r s t s e t of p a r a m e t e r s , the c u r v e w i t h Byrne and Cheung par a m e t e r s i s s l i g h t l y s o f t e r . T h i s i s j u s t i f i a b l e as the i n i t i a l modulus and the f r i c t i o n a l a n g l e ar e s l i g h t l y l e s s than the f i r s t ones (see T a b l e 6.10). As f o r the i n f l u e n c e of the d i f f e r e n t assumed i n i t i a l P o i s s o n ' s r a t i o , F i g . 6.22 shows t h a t the h i g h e r P o i s s o n ' s r a t i o would g i v e a s l i g h t l y s t i f f e r c u r v e . However, the i n f l u e n c e i s g e n e r a l l y i n s i g n i f i c a n t . For the r e s u l t s shown above, the p r e s s u r e e x p a n s i o n c u r v e s p r e d i c t e d from f i n i t e element a n a l y s e s under c a v i t y e x p a n s i o n c o n d i t i o n a r e g e n e r a l l y i n good agreement w i t h the f i e l d d a t a i n s m a l l s t r a i n s , but they a l l become s o f t e r than the f i e l d d a t a a t l a r g e s t r a i n s , e x cept one t h a t was o b t a i n e d u s i n g h i g h e r i n i t i a l s t r e s s and h i g h e r P o i s s o n ' s r a t i o . The d e v i a t i o n from the f i e l d d a t a a t l a r g e s t r a i n s may be a t t r i b u t e d t o many f a c t o r s . F i r s t of a l l , i t would be i n t e r e s t i n g t o note t h a t the t r e n d shown i n F i g 6.20, 6.21, where f i e l d measurements a r e compared w i t h c a v i t y e x p a n s i o n p r e d i c t i o n i s s i m i l a r t o the t r e n d shown i n F i g 6.8 where the a x i s y m m e t r i c a l p r e s s u r e m e t e r s i m u l a t i o n was compared w i t h the c a v i t y e x p a n s i o n p l a n e s t r a i n a n a l y s i s . T h i s o b s e r v a t i o n seems t o suggest t h a t the s t i f f e r f i e l d F i g . 6.22 I n f l u e n c e of P o i s s o n ' s R a t i o on t h e F i n i t e E lement P r e d i c t i o n to 173 p r e s s u r e m e t e r c u r v e a t l a r g e s t r a i n , d e v i a t i n g the c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n , may be due t o the L/D r a t i o e f f e c t , as the p r e s s u r e m e t e r used i n the f i e l d t e s t i n g was a probe of L/D = 6. In a d d i t i o n t o the p r e s s u r e m e t e r L/D r a t i o e f f e c t , the d e v i a t i o n of f i n i t e element p r e d i c t i o n from f i e l d d a t a a t l a r g e s t r a i n s may a l s o be due t o the d i l a t a n c y e f f e c t of s o i l s a t l a r g e s t r a i n s . E l d r i d g e (1983) showed t h a t the d i l a t a n c y e f f e c t w i l l s t i f f e n the s o i l r e s p o n s e . T h e r e f o r e , i n the f i e l d measurements, the d i l a t i o n of the sand may have o c c u r r e d , and then s t i f f e n e d the measured p r e s s u r e m e t e r c u r v e a t l a r g e s t r a i n s . Such an e f f e c t was not a c c o u n t e d f o r i n the f i n i t e element a n a l y s e s . I f t h i s e f f e c t was i n c o r p o r a t e d i n the a n a l y s e s , the p r e d i c t e d c u r v e would have become c l o s e r t o the f i e l d c u r v e a t l a r g e s t r a i n s . From the above r e s u l t s , i t may be c o n c l u d e d t h a t the p r e s s u r e m e t e r w i t h L/D r a t i o of 6 can p r o v i d e the f i e l d p r e s s u r e e x p a n s i o n c u r v e t h a t i s c l o s e t o the a x i s y m m e t r i c a l p l a n e s t r a i n c o n d i t i o n . The i n c o r p o r a t i o n of d i l a t a n t s o i l b e h a v i o r i n the n o n l i n e a r h y p e r b o l i c f i n i t e element a n a l y s i s has been o b t a i n e d by Byrne and E l d r i d g e (1984). A l l o w i n g f o r the n e g l e c t of t h e d i l a t a n t s o i l b e h a v i o r i n the a n a l y s e s and the v a r i a b i l i t y of n a t u r a l s o i l s , the r e s u l t s shown i n F i g 6.20, 6.21 t e n d t o suggest t h a t a n a l y s i s w i t h h y p e r b o l i c s t r a i n r e l a t i o n can p r o v i d e r e a s o n a b l e f i t s w i t h f i e l d measurements of p r e s s u r e m e t e r t e s t . However, f u t h e r i n v e s t i g a t i o n i s n e c e s s a r y t o c o n f i r m t h i s s t a t e m e n t . 7. SOIL-PILE INTERFACE ELEMENTS 7.1 INTRODUCTION The a n a l y s i s of the l a t e r a l l y l o a d e d p i l e s i s one of the s o i l - s t r u c t u r e i n t e r a c t i o n problems r e q u i r i n g the c o n s i d e r a t i o n of the s o i l - s t r u c t u r e i n t e r f a c e b e h a v i o u r . I t has been r e c o g n i z e d t h a t the i n t e r f a c e c h a r a c t e r i s t i c s i n f l u e n c e the p i l e l a t e r a l r e s i s t a n c e s i g n i f i c a n t l y , e s p e c i a l l y on the u l t i m a t e s o i l r e s i s t a n c e ( Y e g i a n and W r i g h t , 1973). In r e a l i t y , as a v e r t i c a l p i l e i s s u b j e c t e d t o l a t e r a l l o a d i n g s , mean normal s t r e s s e s i n the s o i l w i l l i n c r e a s e i n f r o n t of the p i l e , and d e c r e a s e b e h i n d the p i l e . D i s p l a c e m e n t s i n the s o i l w i l l t e n d t o be r a d i a l l y away from the p i l e i n f r o n t of the p i l e , and r a d i a l l y towards the p i l e b e h i n d i t . At some s t a g e s , near the ground s u r f a c e , a gap w i l l p r o b a b l y open up a t the i n t e r f a c e between the p i l e and the s o i l b e h i n d i t , w i t h s o i l i n f r o n t of p i l e f a i l i n g i n a wedge t y p e of mechanism, as shown i n F i g . 7.1. F u r t h e r down the p i l e s h a f t , s o i l w i l l e v e n t u a l l y f a i l by f l o w i n g h o r i z o n t a l l y around the p i l e . In t h i s c a s e , the i n t e r f a c e between p i l e and s o i l e x p e r i e n c e s no gapping but r e l a t i v e c y l i n d r i c a l s l i p p a g e around the p i l e s h a f t , as shown i n F i g . 7.2. T h e r e f o r e , i n o r d e r t o p r e d i c t the p i l e l a t e r a l r e s i s t a n c e s a c c u r a t e l y u s i n g the f i n i t e element p r o c e d u r e , i t i s d e s i r a b l e t o have an e f f i c i e n t but s i m p l e model b u i l t i n t h e program so t h a t the i n t e r f a c e c h a r a c t e r i s t i c s between 1 75 176 F i g . 7.1 S o i l Movement a t Shallow Depth ( a f t e r Broms, 1964) fan zone 7 \ / / / V - , f ( I 1 t pile 'sliding concentric cylindrical shells F i g 7.2 S o i l Flows around L a t e r a l P i l e a t Depth ( a f t e r Randolph & Houl s b y , 1984) 1 78 s o i l and p i l e can be a p p r o x i m a t e l y a c c o u n t e d f o r i n the a n a l y s i s . In t h i s c h a p t e r , the p o s s i b l e i n t e r f a c e d e f o r m a t i o n modes w i l l be d e s c r i b e d f i r s t , and then f o l l o w e d by a b r i e f r e v i e w of d i f f e r e n t i n t e r f a c e element models a v a i l a b l e a t p r e s e n t t i m e . F i n a l l y , a s i m p l e but r a t i o n a l i n t e r f a c e element model i s p r e s e n t e d . T h i s model w i l l be used i n the f i n i t e element a n a l y s i s t o s i m u l a t e the i n t e r f a c e b e h a v i o u r of l a t e r a l l y l o a d e d p i l e s a p p r o x i m a t e l y . 7.2 DEFORMATION MODES AT INTERFACE Under s t a t i c l o a d i n g c o n d i t i o n s , the s o i l - p i l e i n t e r f a c e may undergo v a r i o u s t y p e s of r e l a t i v e movements. These r e l a t i v e d e f o r m a t i o n c h a r a c t e r i s t i c s can be g e n e r a l l y c l a s s i f i e d i n t o t h r e e modes as i l l u s t r a t e d i n F i g . 7.3 f o r the two d i m e n s i o n a l c o n d i t i o n : a) S t i c k or no s l i p - when normal s t r e s s an i s co m p r e s s i v e and shear s t r e s s T s d e v e l o p e d on the i n t e r f a c e remains l e s s than the shear s t r e n g t h , e.g. d e f i n e d by Mohr-Coloumb c r i t e r i o n . I n t h i s c a s e , i . e . T < C + a tantf. (7.3.1) s a n I where C , a r e the a d h e s i o n , and f r i c t i o n a l a n g l e a t a I i n t e r f a c e , r e s p e c t i v e l y ; b) S l i p - when the shear s t r e s s r s d e v e l o p e d on the i n t e r f a c e i s g r e a t e r than t h e shear s t r e n g t h but normal W - -Jifi-(a) A = T o t a l A r e a (b) ///^ A F i g . 7.3 Schematic of Modes at I n t e r f a c e a) S t i c k or no s l i p b) S l i p c) Gapping ( a f t e r Desai et a l , 1984) 180 s t r e s s a n remains c o m p r e s s i v e ; c) Gapping - when normal s t r e s s a n becomes t e n s i l e . In g e n e r a l , t h e above d e f o r m a t i o n c h a r a c t e r i s t i c s a r e dependent upon the m e c h a n i c a l and g e o m e t r i c a l p r o p e r t i e s of i n t e r f a c e as w e l l as the s t a t e of s t r e s s e s a t i n t e r f a c e . 7.3 REVIEW ON INTERFACE ELEMENTS A v a r i e t y of i n t e r f a c e element models have been proposed by many r e s e a r c h e r s t o account a p p r o x i m a t e l y f o r the s p e c i a l d e f o r m a t i o n b e h a v i o u r a t s o i l - s t r u c t u r e i n t e r f a c e s , p a r t i c u l a r l y f o r s t a t i c l o a d i n g s . These models have i n c l u d e d c h a r a c t e r i z a t i o n of b e h a v i o u r of j o i n t s i n r o c k s and i n t e r f a c e s of s o i l - s t r u c t u r e i n t e r a c t i o n systems. The major d i f f e r e n c e among tho s e models l i e s i n t h e i r d i f f e r e n t t r e a t m e n t s of t h e c o n s t i t u t i v e laws f o r the i n t e r f a c e e l e m e n t s . A comprehensive r e v i e w on the s u b j e c t s of j o i n t or i n t e r f a c e b e h a v i o u r and t h e i r m o d e l l i n g s has been g i v e n by D e s a i (1981) b o t h f o r s t a t i c and dynamic l o a d i n g s . H e r e i n , o n l y some of i n t e r f a c e element models which have been used i n t h e a n a l y s i s of l a t e r a l l y l o a d e d p i l e s a r e p r e s e n t e d f o r t h e sake of completeness and l o g i c development. 7.3.1 JOINT ELEMENTS WITH ZERO THICKNESS J o i n t element model was f i r s t proposed by Goodman, T a y l o r and Brekke ( 1 9 6 8 ) . A sche m a t i c of t h i s model i s shown i n F i g . 7.4. In t h i s model, t h e element s t i f f n e s s i s \ \ \ S o l i d Element / S o l i d Element -fc s o J o i n t element F i g . 7.4 J o i n t Element with Zero T h i c k n e s s co 182 f o r m u l a t e d based on the r e l a t i v e n o d a l d i s p l a c e m e n t s of the s o l i d e lements s u r r o u n d i n g the i n t e r f a c e element which has z e r o t h i c k n e s s . The element s t i f f n e s s m a t r i x i s w r i t t e n as [K] = ; L [ B ' ] T [ C ] . [ B ' ] dx (7.3.2) where [ B ' ] i s the t r a n s f o r m a t i o n m a t r i x r e l a t i n g the r e l a t i v e d i s p l a c e m e n t s of the i n t e r f a c e element t o the n o d a l d i s p l a c e m e n t s of t h e a d j a c e n t s o l i d e l e m e n t s , [ C ] ^ i s c o n s t i t u t i v e m a t r i x and the i n t e g r a t i o n i s over the l e n g t h of the i n t e r f a c e element. For the two d i m e n s i o n a l a n a l y s i s , the c o n s t i t u t i v e m a t r i x i s e x p r e s s e d a s : f \ k 0 u n • = n • r T 0 k V S S r (7.3.3) where a„ = normal s t r e s s , r = shear s t r e s s , k„ = normal n s n s t i f f n e s s , k = shear s t i f f n e s s and u and v a r e r e l a t i v e s r r normal and shear d i s p l a c e m e n t s r e s p e c t i v e l y . Then the element s t i f f n e s s i s t r a n s f o r m e d from the l o c a l c o o r d i n a t e system t o the g l o b a l c o o r d i n a t e system, and the s t i f f n e s s m a t r i x f o r the i n t e r f a c e element i n t h e g l o b a l c o o r d i n a t e system i s f i n a l l y assembled i n t o the e n t i r e s t r u c t u r e s t i f f n e s s m a t r i x by s t a n d a r d manner a c c o r d i n g t o the i n t e r f a c e n o d a l c o n n e c t i o n w i t h the a d j a c e n t s o l i d e l e m e n t s . 183 Such a model has been i n c o r p o r a t e d i n t o l i n e a r and n o n l i n e a r i n t e r a c t i o n a n a l y s i s by a number of i n v e s t i g a t o r s . C l o u g h and Duncan (1971) used i t w i t h a g r e a t s u c c e s s f o r the p l a n e s t r a i n a n a l y s i s of r e t a i n i n g w a l l s , i n which they d e t e r m i n e d t h e shear s t i f f n e s s , k , from d i r e c t shear box s t e s t s and e x p r e s s e d i t i n terms of h y p e r b o l i c r e l a t i o n s h i p . Based on the concept s i m i l a r t o t h a t by Goodman e t a l , Y e g i a n and W r i g h t (1973) d e v e l o p e d a c y l i n d r i c a l i n t e r f a c e element as shown i n F i g . 7.5 t o s i m u l a t e the c y l i n d r i c a l r e l a t i v e movement between s o i l and p i l e s e c t i o n , i n t h e i r p r e d i c t i o n s of P-Y c u r v e s f o r t h e l a t e r a l l y l o a d e d p i l e s . The i n t e r f a c e element i s of z e r o t h i c k n e s s i n t h e r a d i a l d i r e c t i o n but p o s s e s s e s a f i n i t e l e n g t h encompassing a segment of t h e p i l e p e r i m e t e r . The c y l i n d r i c a l ( o r shear) d e f o r m a t i o n s around the p i l e s h a f t were a l l o w e d f o r and e x p r e s s e d by a b i l i n e a r e q u a t i o n . However, s i n c e the i n t e r f a c e d e f o r m a t i o n s were r e s t r i c t e d i n t h e r a d i a l (or normal) d i r e c t i o n , the s o i l - p i l e s e p a r a t i o n was not a c c o u n t e d f o r i n t h e i r i n t e r f a c e element model. I n c o n v e n i e n c e s w i t h the above z e r o - t h i c k n e s s j o i n t element models a r e t h e l a c k of a p h y s i c a l b a s i s f o r a d o p t i n g a r b i t r a r y v a l u e s of k n and k g and the need t o f o r m u l a t e a s p e c i a l element s t i f f n e s s m a t r i x based on t h e r e l a t i v e d i s p l a c e m e n t s . -Interface Elements ( a ) - Interface element at soll-plle boundary. (K) " Displacement pattern for • x ' Interface elements. F i g . 7.5 C y l i n d r i c a l I n t e r f a c e Element ( a f t e r Yegian & Wright, 1973) co 185 7.3.2 THIN LAYER INTERFACE ELEMENT E s s e n t i a l l y , the i n t e r f a c e between s o i l and s t r u c t u r a l member i n v o l v e s a t h i n l a y e r of s o l i d m a t e r i a l r a t h e r than a l a y e r of z e r o t h i c k n e s s . The b e h a v i o u r of t h i s t h i n i n t e r f a c e l a y e r may be d i f f e r e n t from the b e h a v i o u r of the s u r r o u n d i n g s t r u c t u r a l and s o i l m a t e r i a l s , due t o d i f f e r e n t d e f o r m a t i o n c h a r a c t e r i s t i c s between the s o i l and s t r u c t u r e . Based on t h e s e i d e a s , a t h i n l a y e r i n t e r f a c e element was proposed by D e s a i (1981) t o s i m u l a t e the s p e c i f i c b e h a v i o u r of the i n t e r f a c e , as shown i n F i g . 7.6. The s t i f f n e s s m a t r i x f o r t h e t h i n l a y e r i n t e r f a c e element i s f o r m u l a t e d i n the same way as f o r the o t h e r s u r r o u n d i n g s o l i d e l e m e n t s . The d i s t i n g u i s h i n g f e a t u r e s of the t h i n l a y e r element a r e the s p e c i a l t r e a t m e n t of i t s c o n s t i t u t i v e law, c h o i c e of i t s t h i c k n e s s , and the i n c o r p o r a t i o n of i n t e r f a c e d e f o r m a t i o n modes. U n l i k e the j o i n t element, the c o n s t i t u t i v e law f o r the t h i n l a y e r element i s s i m i l a r t o t h a t f o r t h e normal s o l i d e lement, e x p r e s s e d i n terms of the i n c r e m e n t a l s t r e s s and s t r a i n , i . e . [Aa] = [ C ] i [Ae] (7.3.4) The i n c r e m e n t a l s t r e s s and s t r a i n a r e s p e c i f i e d i n d i r e c t i o n of normal and t a n g e n t i a l t o t h e i n t e r f a c e , r e s p e c t i v e l y , i . e . Interface (a) Two-dimensional 8 • (average) contact dimension F i g . 7.6 Thin Layer I n t e r f a c e Element ( a f t e r Desai et a l , 1984) co CTi 187 [C] ns (7.3.5) where [ C n ] , [ C g ] a r e the normal s t i f f n e s s and shear s t i f f n e s s s u b m a t r i c e s of the t h i n l a y e r element, and [C ], [C ] r e p r e s e n t the c o u p l i n g e f f e c t s of the normal and shear b e h a v i o u r , which a r e not c o n s i d e r e d a t p r e s e n t time s i n c e t hey a r e d i f f i c u l t t o e v a l u a t e from t h e e x p e r i m e n t a l d a t a . The normal s t i f f n e s s [C ] of t h e t h i n i n t e r f a c e element n a r e o f t e n assumed t o be t h e same as t h o s e of the s u r r o u n d i n g s o i l e l ements ( D e s a i , 1981 and D e s a i e t a l , 1984), w h i l e the shear s t i f f n e s s [ C g ] of the t h i n l a y e r i n t e r f a c e element i s u s u a l l y o b t a i n e d from d i r e c t shear t e s t s , and presumed t o be composed of a shear modulus f o r t h e i n t e r f a c e . The shear modulus G^ can be i n t e r p r e t e d from d i r e c t shear t e s t r e s u l t s , and e x p r e s s e d i n form o f : G i " 9r 9u f o r the g i v e n normal s t r e s s a n (7.3.6) where t = t h i c k n e s s of the element, and u r = r e l a t i v e d i s p l a c e m e n t , a n = the normal s t r e s s a c t i n g on the shear box, and r = the shear s t r e s s a c t i n g on the shear box. For t h e l i n e a r e l a s t i c , i s o t r o p i c b e h a v i o u r , the shear component i s uncoupled from the normal component i n the c o n s t i t u t i v e l a w s , and the c o n s t i t u t i v e m a t r i x f o r the i n t e r f a c e i s g i v e n based on the g e n e r a l i z e d Hooke's Law, 188 i . e . f o r the two d i m e n s i o n a l c o n d i t i o n C 0 [ C ] . - c2 c 0 (7.3.7) 0 0 where: E (1 - u) (1+u) 0-2u) E v ( 1 + y ) ( 1 - 2 U ) where E i s the e l a s t i c Young's modulus, u i s P o i s s o n ' s r a t i o , and i s the shear modulus which i s d e t e r m i n e d from th e d i r e c t shear t e s t . The v a l u e s of C, and C 2 however, a r e e x a c t l y the same as f o r the s u r r o u n d i n g s o i l e l e m e n t s . I f the b e h a v i o u r of the t h i n i n t e r f a c e element i s assumed t o be n o n l i n e a r e l a s t i c and i s o t r o p i c , then the e l a s t i c m o d u l i , E, u and G. i n the c o n s t i t u t i v e m a t r i x [ C ] . • l I a r e s t r e s s dependent, and can be e x p r e s s e d i n terms of h y p e r b o l i c f u n c t i o n , as d i s c u s s e d i n Chapter 4. However, i n t h i s c a s e , the i n t e r f a c e p a r a m e t e r s r e q u i r e d f o r the h y p e r b o l i c f u n c t i o n a r e d e t e r m i n e d from a p p r o p r i a t e t r i a x i a l and d i r e c t shear t e s t s . As f o r the a p p r o p r i a t e t h i c k n e s s of the t h i n i n t e r f a c e element i n the a n a l y s i s , D e s a i e t a l (1984) s u g g e s t e d t h a t the q u a d r i l a t e r a l i n t e r f a c e element w i t h t h i c k n e s s of 0.1 - 0.01 L would p r o v i d e s a t i s f a c t o r y r e s u l t s , where L i s th e average l e n g t h of a d j o i n i n g e l e m e n t s . 189 The c o n c e p t o f u s i n g t h e t h i n l a y e r e l e m e n t a s i n t e r f a c e e l e m e n t r a t h e r t h a n z e r o t h i c k n e s s h a s b e e n a t t e m p t e d by many r e s e a r c h e r s i n c l u d i n g Z i e n k i e w c z e t a l ( 1970) a n d D e s a i ( 1 9 8 1 ) . The c o n c e p t has been s y s t e m a t i c a l l y s t u d i e d and a p p l i e d i n t h e i m p l e m e n t a t i o n o f t h e f i n i t e e l e m e n t a n a l y s i s t o many p r o b l e m s by D e s a i and h i s a s s o c i a t e s ( D e s a i , 1981, a n d D e s a i e t a l , 1984a , b ) . The a p p l i c a t i o n i s p r o m i s i n g a n d o f p r a c t i c a l i n t e r e s t . H o w e v e r , i t s u s a g e i n t h e a n a l y s i s o f t h e l a t e r a l l y l o a d e d p i l e s ha s n o t y e t b e e n e x p l o r e d . I t i s b e l i e v e d t h a t t h e c o n c e p t o f t h i n l a y e r i n t e r f a c e e l e m e n t i s s i m p l e and u s e f u l f o r t h e a n a l y s i s o f l a t e r a l l y l o a d e d p i l e s , a n d some o f t h e r e s e a r c h e s a r e c a r r i e d o u t i n t h i s t h e s i s . 7.4 THE PROPOSED MODEL FOR S O I L - P I L E INTERFACE U n l i k e t h e Y e g i a n and W r i g h t a p p r o a c h w h i c h u s e d t h e j o i n t e l e m e n t b a s e d on t h e r e l a t i v e d i s p l a c e m e n t , a s i m i l a r c o n c e p t t o t h e f o r e g o i n g t h i n l a y e r i n t e r f a c e e l e m e n t i s e m p l o y e d i n t h e t h e s i s t o s t u d y t h e i n t e r f a c e b e h a v i o u r o f t h e l a t e r a l l y l o a d e d p i l e s . T h e b a s i s o f t h e p r o p o s e d e l e m e n t i s t h e c o n c e p t t h a t t h e s o i l - p i l e i n t e r f a c e i n v o l v e s a t h i n l a y e r o f s o i l r a t h e r t h a n a l a y e r o f z e r o t h i c k n e s s . The c h a r a c t e r i s t i c s o f t h e t h i n i n t e r f a c e l a y e r a r e c o n s i d e r e d t o be m a i n l y a t t r i b u t e d t o t h e d i f f e r e n c e s i n t h e d e f o r m a t i o n c h a r a c t e r i s t i c s a s s o c i a t e d w i t h t h e p i l e a n d t h e s u r r o u n d i n g s o i l . 190 In r e a l i t y , t h e c o n c r e t e p i l e s e c t i o n e s s e n t i a l l y u n d e r g o e s n e g l i g i b l e d e f o r m a t i o n , a s c o m p a r e d w i t h i t s l a t e r a l d e f l e c t i o n a n d s o i l d e f o r m a t i o n . T h u s t h e p i l e i s u s u a l l y t r e a t e d a s r i g i d - p l a s t i c m a t e r i a l i n t h e a n a l y s i s . T h e r e f o r e , i t may be r e a s o n a b l e t o p r o p o s e a t h i n r i n g o f n o r m a l s o i l e l e m e n t s w i t h g i v e n s t r e s s b o u n d a r y a s i n t e r f a c e e l e m e n t s e m c o m p a s s i n g t h e p i l e p e r i m e t e r , so t h a t t h e r e l a t i v e d e f o r m a t i o n c h a r a c t e r i s t i c s a t t h e s o i l - p i l e i n t e r f a c e c a n be a p p r o x i m a t e l y s i m u l a t e d . Some f e a t u r e s o f t h e p r o p o s e d i n t e r f a c e e l e m e n t and t h e m o d i f i c a t i o n s a r e d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s . 7.4.1 FORMULATION OF ST IFFNESS MATRIX - CONST ITUT IVE LAWS S i m i l a r t o t h e D e s a i ' s t h i n l a y e r i n t e r f a c e e l e m e n t , t h e f o r m u l a t i o n o f t h e s t i f f n e s s m a t r i x f o r t h e p r o p o s e d i n t e r f a c e e l e m e n t s i s e x a c t l y t h e same a s f o r t h e a d j a c e n t s o i l e l e m e n t s . H o w e v e r , i t i s n o t i c e d t h a t t h e r e e x i s t s a d i f f e r e n c e i n t h e f o r m o f c o n s t i t u t i v e l a w s e m p l o y e d i n CONOIL a n d i n t h e D e s a i ' s t h i n l a y e r i n t e r f a c e e l e m e n t . The c o n s t i t u t i v e l a w , i . e . t h e g e n e r a l i z e d H o o k e ' s law u s e d i n CONOIL e m p l o y s e l a s t i c b u l k m o d u l u s , B, and Y o u n g ' s m o d u l u s , E , t o e x p r e s s t h e e l a s t i c v o l u m e c h a n g e a n d e l a s t i c d i s t o r t i o n , r a t h e r t h a n Y o u n g ' s m o d u l u s , E , a n d s h e a r m o d u l u s , , a s p r o p o s e d by D e s a i e t a l ( s e e E q . ( 7 . 3 . 7 ) ) . H o w e v e r , t h e a b o v e f o u r e l a s t i c m o d u l i a r e r e l a t e d t o e a c h o t h e r by P o i s s o n ' s r a t i o , u, a n d o n l y two o f them a r e i n d e p e n d e n t i f t h e m a t e r i a l i s e l a s t i c and i s o t r o p i c 191 (Timoshenko and G o o d i e r , 1951). T h e r e f o r e , i t i s b e l i e v e d t h a t , i n p r i n c i p l e , the form of c o n s t i t u t i v e m a t r i x used i n CONOIL i s e q u i v a l e n t , and i n t e r c h a n g e a b l e t o the form used i n the D e s a i ' s t h i n l a y e r i n t e r f a c e element. In p r a c t i c e , i f the i n t e r f a c e p r o p e r t i e s i n terms of b u l k modulus, and Young's modulus can be p r o p e r l y e v a l u a t e d f o r the i n t e r f a c e element by c e r t a i n t y p e s of e x p e r i m e n t a l t e s t s , then i t i s p o s s i b l e t o u t i l i z e the same c o n s t i t u t i v e m a t r i x d i r e c t l y f o r the i n t e r f a c e element w i t h o u t any m o d i f i c a t i o n . In t h i s c a s e , the normal and shear b e h a v i o u r of the i n t e r f a c e element a r e e x p r e s s e d i n terms of e l a s t i c b u l k modulus, B, Young's modulus, E, and the P o i s s o n ' s r a t i o , u. For the two d i m e n s i o n a l c o n d i t i o n , Eq. (7.3.7) t h e r e f o r e becomes : 0 0 ( 7 . 3 . 7 ) ' 0 0 where 3B + E 2(1+I>) 3B - E 2(1+o) E 2(1+y) The v a l u e s of B and E can be e i t h e r c o n s t a n t s f o r l i n e a r 1 92 e l a s t i c b e h a v i o u r or f u n c t i o n s of s t r e s s l e v e l f o r the n o n l i n e a r e l a s t i c b e h a v i o u r as d e f i n e d by u s i n g h y p e r b o l i c f u n c t i o n . For t h e subsequent a n a l y s e s of l a t e r a l l y l o a d e d p i l e s , CONOIL has been m o d i f i e d so t h a t the proposed i n t e r f a c e element can be i n c o r p o r a t e d and the i n t e r f a c e b e h a v i o u r can be e a s i l y s p e c i f i e d as e i t h e r b i l i n e a r e l a s t o - p l a s t i c model or n o n l i n e a r e l a s t o - p l a s t i c model by a s s i g n i n g a c o r r e s p o n d i n g n e g a t i v e v a l u e of Rf (-1 < Rf < 0) i n t h e i n p u t d a t a . 7.4.2 DEFORMATION AND STRENGTH CHARACTERISTICS S i n c e t h e proposed i n t e r f a c e elements a r e r e l a t i v e l y t h i n as compared t o the o t h e r s o i l e l e m e n t s , the s t r e n g t h or f a i l u r e c r i t e r i o n of the i n t e r f a c e element can be b a s i c a l l y r e g a r d e d as governed by the a d h e s i o n and f r i c t i o n a l r e s i s t a n c e a t the s o i l - p i l e i n t e r f a c e . T h e r e f o r e , i t can be g e n e r a l l y proposed t h a t d u r i n g the l o a d i n g p r o c e s s the i n t e r f a c e elements have the same d e f o r m a t i o n c h a r a c t e r i s t i c s as the a d j o i n i n g s o i l e l e m e n t s , u n t i l the s o i l - p i l e i n t e r f a c e a d h e s i o n and f r i c t i o n a l r e s i s t a n c e a r e f u l l y m o b i l i z e d . Such a t r e a t m e n t may be a c c u r a t e enough f o r the p r e s e n t a n a l y s i s of l a t e r a l l y l o a d e d p i l e s where th e i n t e r f a c e between s o i l and p i l e o n l y i n v o l v e s s i m p l e and monotonic r e l a t i v e d i s p l a c e m e n t s under the s h o r t term s t a t i c l o a d i n g . F u r t h e r m o r e , as the element i s r e l a t i v e l y t h i n , the l i n e a r shear s t r a i n a c r o s s the element can be a good 1 93 r e p r e s e n t a t i o n of t h e r e l a t i v e d i s p l a c e m e n t (or r e l a t i v e s l i p ) between t h e r i g i d p i l e elements and the a d j o i n i n g s o i l e l e m e n t s . The s t r e n g t h c h a r a c t e r i s t i c s of i n t e r f a c e elements a r e u s u a l l y r e l a t e d t o the s t r e n g t h c h a r a c t e r i s t i c s of the s u r r o u n d i n g s o i l by c e r t a i n c o e f f i c i e n t s (Potyondy, 1961). For the p i l e i n s t a l l e d i n the s a t u r a t e d c l a y , the u n d r a i n e d c o n d i t i o n i s u s u a l l y assumed i n an a n a l y s i s under the s h o r t term s t a t i c l o a d i n g . I n t h i s c a s e , the maximum shear r e s i s t a n c e a t the s o i l - p i l e i n t e r f a c e i s o f t e n e x p r e s s e d as a f r a c t i o n of the u n d r a i n e d shear s t r e n g t h of the c l a y d e p o s i t , i . e . C = a.C (7.4.1) where C i s the maximum i n t e r f a c e shear r e s i s t a n c e , or t h e a s o i l - p i l e a d h e s i o n , a i s an a d h e s i o n f a c t o r , and C u i s the u n d r a i n e d shear s t r e n g t h of the s a t u r a t e d c l a y . In o t h e r c a s e s where the p i l e i s surrounded by c o h e s i o n l e s s s o i l s , the f r i c t i o n a l r e s i s t a n c e a t the s o i l - p i l e i n t e r f a c e i s o f t e n r e l a t e d t o t h e i n t e r n a l f r i c t i o n a n g l e of the c o h e s i o n l e s s s o i l , such as : 4>i = 0-0 (7.4.2) where i s the f r i c t i o n a l r e s i s t a n c e a n g l e a t s o i l - p i l e i n t e r f a c e , ^ i s a f r i c t i o n a n g l e f a c t o r , and <j> i s the 194 i n t e r n a l f r i c t i o n a n g l e of the c o h e s i o n l e s s s o i l . In p r a c t i c e , the above a d h e s i o n f a c t o r and the f r i c t i o n a n g l e f a c t o r a r e b o t h dependent i n v a l u e upon the p i l e c o n s t r u c t i o n m a t e r i a l , p i l e s u r f a c e roughness and the l o a d i n g c o n d i t i o n as w e l l . A comprehensive c o m p i l a t i o n of the a d h e s i o n f a c t o r and the f r i c t i o n a n g l e f a c t o r f o r v a r i o u s c o n s t r u c t i o n m a t e r i a l s was r e p o r t e d by Potyondy (1961), and i s shown i n T a b l e 7.1. Some of t h e v a l u e s proposed by Potyondy w i l l be used i n t h i s t h e s i s . 7.4.3 INCORPORATION OF DEFORMATION MODES M o d i f i c a t i o n has been made i n CONOIL so t h a t t h e above d e f o r m a t i o n and s t r e n g t h b e h a v i o u r of the i n t e r f a c e element can be implemented i n the f i n i t e element p r o c e d u r e . V a r i o u s d e f o r m a t i o n modes a t the i n t e r f a c e can be i n c o r p o r a t e d i n the a n a l y s i s by examing the s t r e s s s t a t e s a t t h e i n t e r f a c e element and by a s s i g n i n g d i f f e r e n t d e f a u l t v a l u e s t o the i n t e r f a c e e l a s t i c m o duli when tho s e d e f o r m a t i o n modes o c c u r . As shown i n the Mohr s t r e s s d i agram, i . e . F i g . 7 . 7 ( a ) , when a l l t h e p r i n c i p a l s t r e s s e s i n the i n t e r f a c e element a r e c o m p r e s s i v e , and the shear s t r e s s , r , d e v e l o p e d i n t h e element remains l e s s than the f r i c t i o n a l r e s i s t a n c e of the i n t e r f a c e , then the d e f o r m a t i o n of the element i s i n the mode of ' s t i c k ' and the s o i l i s s t i l l bonded on t h e p i l e s u r f a c e . In t h i s c a s e , no d e f a u l t v a l u e s a r e a s s i g n e d t o shear or b u l k m o d u l i . T a b l e 7.1 Proposed coefficients of s k i n f r i c t i o n between so i l s a n d co n s t r u c t i o n m a t e r i a l s y^/Ccs-^-fc"UlX=~^-: without factor of safety] Construction material Sand Cohesionless silt Cohesive granular soil Clay 006 <D< 2-0 mm 0-002 <£><0-06 5 0 % Clay + 5 0 % Sand £><006 mm Surface finish of construction material Dry Sat. Dry Sat. Consist. I. = 1-0-0-5 Consist. Index: 10-0-73 Dense Dense Loose Dense f* J* J* f* ft f* fc f* fc Steel Smooth Polished 0-54 0-64 0-79 0-40 0-68 0-40 — 0-50 0-25 0-50 Rough Rusted 0-76 0-80 0-95 0-48 0-75 065 0-35 0-50 0-50 0-80 Wood | Parallel to grain 0-76 0-85 0-92 0-55 0-87 0-80 0-20 0-60 0-4 0-85 At right angles to grain 0-88 0-89 0-98 0-63 0-95 0-90 0-40 0-70 0-50 0-85 Concrete-<• Smooth Made in iron form 0-76 0-80 0-92 0-50 0-87 0-84 0-42 068 0-40 100 Grained Made in wood form 0-88 0-88 0-98 0-62 0-96 0-90 0-58 0-80 0-50 100 Rough Made on adjusted ground 0-98 0-90 1-00 0-79 1-00 0-95 0-80 095 0-60 100 Oj a , 0 3 " i a b) S l i p c ) G a p p i n g F i g . 7.7 S t r e s s C o n d i t i o n s w i t h V a r i o u s I n t e r f a c e D e f o r m a t i o n Modes CTi 1 97 When the shear s t r e s s d e v e l o p e d , T , r e a c h e s the f r i c t i o n a l r e s i s t a n c e but the p r i n c i p a l s t r e s s e s s t i l l remain c o m p r e s s i v e , the r e l a t i v e s l i p a t the s o i l - p i l e i n t e r f a c e w i l l o c c u r . That i s , as shown i n F i g . 7 . 7 ( b ) , r > C + o- tan0. and a 1 f 2 > 0 (7.4.3) S a. n 1 Thus, i f t h e c o n d i t i o n of Eq. (7.4.3) i s r e a c h e d i n some el e m e n t s , then the shear m o d u l i of t h o s e elements a r e d e f a u l t e d by a f a c t o r of 1000 t o a s m a l l v a l u e t o s i m u l a t e the p l a s t i c a c t i o n , and the l o a d s h e d d i n g i t e r a t i o n t e c h n i q u e i s t r i g g e r e d t o r e d i s t r i b u t e the e x t r a s t r e s s i n f a i l e d i n t e r f a c e elements t o the a d j a c e n t e l e m e n t s . Such a p r o c e d u r e i m p l i e s t h a t the i n t e r f a c e element has no f u r t h e r r e s i s t a n c e t o s h e a r i n g and the r e l a t i v e s l i p i s o c c u r r i n g . In t h e o r y , c o h e s i o n l e s s s o i l s a r e u s u a l l y c o m p l e t e l y i n c a p a b l e of t e n s i l e s t r e s s , w h i l e c o h e s i v e s o i l s may s u s t a i n a s m a l l amount of t e n s i l e s t r e s s u n t i l the t e n s i l e s t r e s s r e a c h e s the t e n s i l e s t r e n g t h . H e r e i n , the t e n s i l e s t r e n g t h i s a s s i g n e d t o be e q u a l t o the s o i l - p i l e i n t e r f a c e a d h e s i o n C , and the s o i l - p i l e s e p a r a t i o n i s assumed t o have o c c u r r e d when the minor p r i n c i p a l s t r e s s i s i n t e n s i o n and i s g r e a t e r t h a n the s o i l - p i l e a d h e s i o n i n a b s o l u t e v a l u e , i . e . t h e s e p e r a t i o n c r i t e r i o n : a 3 < 0 f o r c o h e s i o n l e s s s o i l s , or (7.4.4) 198 o 3 < 0 and I<r3 I > C f o r c o h e s i v e s o i l s T h i s c o n d i t i o n i s shown i n F i g . 7 . 7 ( c ) . S t r i c k l y s p e a k i n g , such a s o i l - p i l e s e p a r a t i o n c r i t e r i o n o n l y a p p r o x i m a t e l y r e p r e s e n t s r e a l s i t u a t i o n . The s t r e s s examined s h o u l d be the s t r e s s normal t o the s o i l - p i l e i n t e r f a c e , i . e . a n , r a t h e r than t h e minor p r i n c i p a l s t r e s s , o 3 . However, i n the case of l a t e r a l l y l o a d e d p i l e s the c r i t i c a l a r e a s where the gapping would p r o b a b l y occur a r e r i g h t b e h i n d the p i l e s e c t i o n . I n t h o s e a r e a s , d i r e c t i o n of the minor p r i n c i p a l s t r e s s i s c l o s e t o t h e d i r e c t i o n p e r p e n d i c u l a r t o the s o i l - p i l e i n t e r f a c e , or i n o t h e r words, the minor p r i n c i p a l s t r e s s i s a p p r o x i m a t e i n v a l u e t o the normal s t r e s s , i . e . a3 a^. T h e r e f o r e , the c r i t e r i o n of Eq. (7.4.4) may be a good a p p r o x i m a t i o n . On the o t h e r hand, s i n c e the s t r e s s , an, p e r p e n d i c u l a r t o t h e s o i l - p i l e i n t e r f a c e i s n o r m a l l y l a r g e r than t h e minor p r i n c i p a l s t r e s s , the a p p l i c a t i o n of the minor p r i n c i p a l s t r e s s as the gapping c r i t e r i o n would p r o v i d e the r e s u l t s t h a t have l a r g e r p i l e d e f l e c t i o n . In view of t h e s e c o n s i d e r a t i o n s , the minor p r i n c i p a l s t r e s s , a 3 , i s used, r a t h e r than the normal s t r e s s , a n , t o i d e n t i f y t he s o i l - p i l e s e p a r a t i o n f o r the sake of s i m p l i c i t y . When Eq. (7.4.4) i s s a t i s f i e d i n a element, then both shear m o d u l i and b u l k m o d u l i i n t h e element a re d e f a u l t e d by a f a c t o r of 1000 t o s m a l l v a l u e s . The lower v a l u e s of shear modulus w i l l p r o h i b i t the s t r e s s changes i n the element i n 199 t h e s u b s e q u e n t l o a d i n g p r o c e s s , w h i l e t h e l o w e r v a l u e s o f b u l k m o d u l u s w i l l p r o v i d e l a r g e v o l u m e c h a n g e s , w h i c h s i m u l a t e s t h e f o r m a t i o n o f c a v i t y o r g a p p i n g b e h i n d t h e p i l e . The a b o v e p r o c e d u r e t o i n c o r p o r a t e t h e d i f f e r e n t modes o f i n t e r f a c e d e f o r m a t i o n i s s i m p l e i n p r o g r a m m i n g b u t r a t i o n a l f o r t h e a n a l y s i s o f l a t e r a l l y l o a d e d p i l e s u n d e r t h e s h o r t t e r m s t a t i c l o a d i n g s . J u s t i f i c a t i o n o f t h e a b o v e p r o c e d u r e c a n be s e e n f r o m t h e f i n i t e e l e m e n t r e s u l t s p r e s e n t e d i n t h e s e c t i o n s w h i c h f o l l o w . 7.4.4 INTERFACE ELEMENT - MESH LAYOUT AND ITS THICKNESS The s i g n i f i c a n t f e a t u r e o f t h e p r o p o s e d s o i l - p i l e i n t e r f a c e mode l l i e s i n t h e f a c t t h a t t h e p r o p o s e d i n t e r f a c e e l e m e n t e s s e n t i a l l y r e p r e s e n t s a s o l i d e l e m e n t o f s m a l l f i n i t e t h i c k n e s s , a n d i t r e p r e s e n t s a t h i n l a y e r o f m a t e r i a l b e t w e e n two b o d i e s . By v i r t u e o f t h i s f e a t u r e , t h e i n t e r f a c e e l e m e n t c a n be f o r m u l a t e d i n t h e same way a s o t h e r s o l i d e l e m e n t s , a n d i t i s e a s y t o programme and i m p l e m e n t . W i t h a p p r o p r i a t e s t r e s s b o u u n d a r y a n d i n c o r p o r a t i o n o f l o a d s h e d d i n g i t e r a t i o n t e c h n i q u e s , i t i s p o s s i b l e t o h a n d l e v a r i o u s d e f o r m a t i o n modes a t s o i l - p i l e i n t e r f a c e . I t s o t h e r d i s t i n c t f e a t u r e s w h i c h d e s e r v e d i s c u s s i o n a r e t h e f i n i t e e l e m e n t mesh l a y o u t a n d t h e c h o i c e o f e l e m e n t t h i c k n e s s . 200 Mesh layout U n l i k e Desai's q u a d r i l a t e r a l t h i n l a y e r i n t e r f a c e element, CONOIL employs t r i a n g u l a r elements as d i s c u s s e d i n Chapter 3. T h e r e f o r e at l e a s t two t r i a n g u l a r elements should be used to represent the uniform i n t e r f a c e l a y e r , such as the 'cross element' shown i n F i g . 7.8(a) or the 'diagonal element' i n F i g . 7.8(b). For the c r o s s element, four t r i a n g u l a r elements are used to r e p l a c e the areas which are u s u a l l y covered by one q u a d r i l a t e r a l i n t e r f a c e element so that the u n i f o r m i t y of the i n t e r f a c e l a y e r can be maintained at the expense of more elements i n v o l v e d i n an a n a l y s i s . As a consequence, the computing c o s t may be s l i g h t l y h igher than the q u a d r i l a t e r a l i n t e r f a c e element. However, such a 'cross i n t e r f a c e element' can be expected to give b e t t e r r e s u l t s s i n c e i t s composite elements, i . e . the four l i n e a r s t r a i n t r i a n g u l a r elements, can more a c c u r a t e l y r epresent the extremely h i g h s t r e s s g r a d i e n t a c r o s s the i n t e r f a c e l a y e r than one q u a d r i l a t e r a l i n t e r f a c e element. A d d i t i o n a l numerical a n a l y s e s showed that there i s no s i g n i f i c a n t d i f f e r e n c e between the r e s u l t s from the 'cross element' and 'diagonal element' mesh l a y o u t . T h i s may i n d i c a t e that i n the a p p l i c a t i o n of t r i a n g u l a r element f o r the i n t e r f a c e s i m u l a t i o n d e t a i l s of the t r i a n g u l a r element mesh lay o u t may not be of importance. Aspect R a t i o = L / t (a) 'Cross Element' (b) 'Diagonal Element' F i g . 7.8 Mesh Layout f o r T r i a n g u l a r I n t e r f a c e Element 202 Element thickness With regard to the c h o i c e of i n t e r f a c e element t h i c k n e s s , an aspect r a t i o i s d e f i n e d as the r a t i o of the average l e n g t h , L, of the i n t e r f a c e element to i t s he i g h t , t , as shown i n F i g . 7.8. S i m i l a r to Desai's t h i n l a y e r i n t e r f a c e element, the q u a l i t y of the s i m u l a t i o n of the i n t e r f a c e behaviour u s i n g the proposed model depends upon the t h i c k n e s s of the i n t e r f a c e element. I f the thidkn e s s i s too l a r g e i n comparison with the dimension of the surrounding elements, the t h i n i n t e r f a c e element w i l l behave e s s e n t i a l l y as a s o l i d element. I f i t i s too s m a l l , computational d i f f i c u l t i e s may a r i s e , e s p e c i a l l y when a l a r g e number of elements are i n v o l v e d i n an a n a l y s i s . The ch o i c e of t h i c k n e s s i s , t h e r e f o r e , an important f a c t o r to be c o n s i d e r e d i n the a p p l i c a t i o n of the proposed i n t e r f a c e element. I d e a l l y , t h i s can be r e s o l v e d by performing parametric s t u d i e s i n which the p r e d i c t i o n s from the f i n i t e element r e s u l t s with v a r i o u s t h i c k n e s s are compared with experimental o b s e r v a t i o n s . A p r e l i m i n a r y assessment of the proposed i n t e r f a c e model and the c h o i c e of the t h i c k n e s s w i l l be d e s c r i b e d i n the next s e c t i o n . 7.4.5 PRELIMINARY ASSESSMENTS - DIRECT SHEAR CONDITION The r e l a t i v e s l i p p a g e i n the d i r e c t shear t e s t c o n d i t i o n was simulated Using the proposed i n t e r f a c e element. A schematic diagram of the d i r e c t shear t e s t i s i l l u s t r a t e d i n F i g . 7.9(a) and the corresponding i n t e r f a c e (a) Schematic of D i r e c t Shear C o n d i t i o n (b) F i n i t e Element Mesh f o r D i r e c t Shear C o n d i t i o n F i g . 7.9 S i m u l a t i o n of D i r e c t Shear T e s t i n g to o oo 204 element mesh l a y o u t i n F i g . 7 . 9 ( b ) . The bottom boundary of the element mesh was p i n - c o n n e c t e d , assuming no normal and shear d i s p l a c e m e n t s . T h e r e f o r e the r e l a t i v e s l i p a c r o s s the i n t e r f a c e l a y e r can be s i m u l a t e d by a p p l i e d an u n i f o r m shear s t r e s s , T , on the upper s u r f a c e of the i n t e r f a c e e l e m e n t s . The s t u d y was performed i n b o t h c o h e s i v e s o i l and c o h e s i o n l e s s s o i l u s i n g t w o - d i m e n s i o n a l p l a n e s t r a i n c o n d i t i o n , and the b i l i n e a r e l a s t i c - p l a s t i c m a t e r i a l b e h a v i o r was assumed f o r b o t h m a t e r i a l s . The m a t e r i a l p r o p e r t i e s employed i n the st u d y a r e p r e s e n t e d i n T a b l e 7.2. C o h e s i v e s o i l s N u m e r i c a l s t u d i e s were performed i n c o h e s i v e s o i l s under the u n d r a i n e d c o n d i t i o n s . The u n d r a i n e d shear s t r e n g t h of t he s o i l i s assumed t o be 7.5 Kpa. The i n i t i a l s t r e s s c o n d i t i o n assumed i n the i n t e r f a c e elements i s i s o t r o p i c w i t h a normal s t r e s s , a n , of 30 Kpa a c t i n g normal t o the i n t e r f a c e l a y e r , and the u n i f o r m shear s t r e s s , T , i s a p p l i e d i n c r e m e n t a l l y on the top of i n t e r f a c e e l e m e n t s , as shown i n F i g . 7 . 9 ( b ) . The a s p e c t r a t i o s s t u d i e d a r e 1, 10, 100, 1000. a) Shear s t r e s s d i s t r i b u t i o n T a b l e 7.3 shows the r e s u l t s of c a l c u l a t e d shear s t r e s s a t each i n t e g r a t i o n p o i n t of f o u r l i n e a r t r i a n g u l a r elements f o r the a p p l i e d shear s t r e s s , T , of 4 KPa under d i f f e r e n t a s p e c t r a t i o s . 205 T a b l e 7.2 S o i l P r o p e r t i e s f o r I n t e r f a c e Elements Parameters C o h e s i v e S o i l s C o h e s i o n l e s s S o i l s C u (Kpa) 7.5 4> (°) 38° A0 (°) 0.0° 0cv (°) 0.0° Kg 59.21 986.9 K B 9869.0 1316.0 v 0.499 0.375 n 0.0 0.0 m 0.0 0.0 R f -1 -1 E (Kpa) 6000 10.0x10 s B (Kpa) 10 6 13.3x10" a n (Kpa) 30.0 477.0 1. Both c o h e s i v e and c o h e s i o n l e s s s o i l s a r e assumed t o be e l a s t o - p l a s t i c ; 2. N e g a t i v e Rf i s f l a g t o i n d i c a t e the i n t e r f a c e e l e m e n t s ; 3. E, B a r e the Young's and b u l k modulus r e s p e c t i v e l y f o r the s o i l s . As shown i n the t a b l e , the t h i c k n e s s of the i n t e r f a c e element i n f l u e n c e s the shear s t r e s s d i s t r i b u t i o n a c r o s s the i n t e r f a c e l a y e r s i g n i f i c a n t l y . When the a s p e c t r a t i o i s e q u a l t o u n i t y , i . e . the composi t e i n t e r f a c e element has the same t h i c k n e s s as i t s l e n g t h , the d i s t r i b u t i o n of shear s t r e s s i s e x t r e m e l y e r r a t i c i n each element. As t h e i n t e r f a c e element becomes t h i n n e r , i . e . the a s p e c t r a t i o of i n t e r f a c e element becomes l a r g e r than 10, the c o n d i t i o n of shear s t r e s s d i s t r i b u t i o n a c r o s s the i n t e r f a c e l a y e r i s improved. As an extreme c o n d i t i o n , when the a s p e c t r a t i o i s 1000, the shear s t r e s s d i s t r i b u t i o n i s e x t r e m e l y u n i f o r m , i n w hich case the v a l u e s of c a l c u l a t e d shear s t r e s s a t each 206 Ta b l e 7.3 Shear S t r e s s D i s t r i b u t i o n w i t h i n I n t e r f a c e Element a t I n t e g r a t i o n P o i n t s ( f o r C o h e s i v e S o i l s ) Element I n t e g . A spect R a t i o s No. P o i n t s L/t=1 L/t=10 L/t=100 L/t=1000 1 -2.6526 -4.2994 -4.0054 -4.0001 2 -6.7057 -3.9235 -3.9982 -4.0000 3 -2.6370 -4.0450 -4.0006 -4.0000 1 4 -4.7919 -3.9655 -3.9990 -4.0000 5 -2.4023 -4.1871 -4.0033 -4.0000 6 -4.8011 -4.1155 -4.0019 -4.0000 7 -3.9984 -4.0893 -4.0014 -4.0000 1 -2.6446 -3.9043 -3.9982 -4.0000 2 -2.6446 -3.9043 -3.9982 -4.0000 3 -6.7099 -3.7551 -3.9971 -4.0000 2 4 -4.7986 -3.8253 -3.9976 -4.0000 5 -4.7986 -3.8253 -3.9976 -4.0000 6 -2.4018 -3.9132 -3.9982 -4.0000 7 -3.9997 -3.8546 -3.9978 -4.0000 1 -6.7057 -3.9235 -3.9982 -4.0000 2 -2.6526 -4.2994 -4.0054 -4.0001 3 -2.6370 -4.0450 -4.0006 -4.0000 3 4 -2.4023 -4.1871 -4.0033 -4.0000 5 -4.7919 -3.9655 -3.9990 -4.0000 6 -4.8011 -4.1155 -4.0019 -4.0000 7 -3.9984 -4.0893 -4.0014 -4.0000 1 -2.6535 -3.9585 -4.0002 -4.0000 2 -6.7033 -3.9834 -3.9977 -4.0000 3 -2.6535 -3.9585 -4.0002 -4.0000 4 4 -4.7993 -3.9717 -3.9989 -4.0000 5 -2.4117 -3.9570 -4.0004 -4.0000 6 -4.7993 -3.9717 -3.9989 -4.0000 7 -4.0034 -3.9668 -3.9994 -4.0000 1. The a p p l i e d shear s t r e s s a t the s u r f a c e of i n t e r f a c e element i s -4.0 Kpa. i n t e g r a t i o n p o i n t of f o u r t r i a n g u l a r elements a r e e q u a l , and they a l l c o r r e s p o n d t o the e x a c t v a l u e of shear s t r e s s a p p l i e d on t h e t o p of composite i n t e r f a c e element. T h i s may, t h e r e f o r e , s u ggest t h a t the shear s t r e s s c o n d i t i o n a c r o s s 207 t h e i n t e r f a c e l a y e r o f d i r e c t s h e a r c a n be p r o p e r l y s i m u l a t e d by t h e p r o p o s e d m o d e l , p r o v i d e d t h a t t h e a s p e c t r a t i o o f t h e i n t e r f a c e e l e m e n t i s s u f f i c i e n t l y l a r g e , f o r i n s t a n c e , l a r g e r t h a n 10 f o r c o h e s i v e s o i l s . b) R e l a t i v e d i s p l a c e m e n t a n d s h e a r r e s i s t a n c e F o r c o h e s i v e s o i l s , r e s u l t s o f r e l a t i v e d i s p l a c e m e n t s o f i n t e r f a c e l a y e r a s f u n c t i o n s o f t h e a p p l i e d s h e a r s t r e s s a n d ' a s p e c t r a t i o a r e shown i n T a b l e 7 . 4 . S i n c e , f o r t h e l a r g e a s p e c t r a t i o s , t h e d i f f e r e n c e among e a c h s e t o f r e l a t i v e d i s p l a c e m e n t a s s o c i a t e d w i t h v a r i o u s a s p e c t r a t i o s become i n v i s i b l e i n a d i a g r a m , o n l y t y p i c a l r e s u l t s f o r a s p e c t r a t i o s o f 1 and 10 a r e shown i n F i g . 7 . 1 0 . As shown i n T a b l e 7 . 4 , a n d F i g . 7 . 1 0 , t h e t h i c k n e s s o f t h e p r o p o s e d i n t e r f a c e e l e m e n t , a s e x p e c t e d , h a s a s i g n i f i c a n t e f f e c t on t h e p r e d i c t e d r e l a t i v e d i s p l a c e m e n t . The t h i n n e r t h e i n t e r f a c e e l e m e n t , t h e s m a l l e r t h e r e l a t i v e d i s p l a c e m e n t w o u l d be p r e d i c t e d b e f o r e t h e s l i p o c c u r . F o r t h e a s p e c t r a t i o e q u a l t o o n e , a s shown F i g . 7 . 1 0 , t h e p r e d i c t e d u l t i m a t e s h e a r r e s i s t a n c e i s a b o u t 5 K P a , u n d e r e s t i m a t i n g t h e u n d r a i n e d s h e a r s t r e n g t h by a b o u t 30 % , and t h e i n i t i a l segment o f r e l a t i v e d i s p l a c e m e n t r e s p o n s e i s a l s o r e l a t i v e l y s o f t . The w h o l e s h e a r s t r e s s v s r e l a t i v e d i s p l a c e m e n t c u r v e i s i n a f o r m s h o w i n g g r a d u a l d e f o r m a t i o n r a t h e r t h a n a s u d d e n s l i p p a g e i n t h e i n t e r f a c e l a y e r . H o w e v e r , f o r t h e a s p e c t r a t i o l a r g e r t h a n 10, a s shown b o t h i n T a b l e 7.4 a n d F i g . 7 . 1 0 , t h e r e l a t i v e d i s p l a c e m e n t 208 T a b l e 7.4 A p p l i e d Shear S t r e s s vs R e l a t i v e D i s p l a c e m e n t of I n t e r f a c e Element under V a r i o u s A s p e c t R a t i o s Shear S t r e s s C o h e s i v e S o i l s (Kpa) L/t=1 L / t = l 0 L/t=100 L/t=1000 0.0 0.0 0.0 0.0 0.0 2.0 0. 18201 0.010540 0.00100 0.00010 4.0 0.36401 0.021085 0.00200 0.00020 5.0 0.91234 0.026357 0.00250 0.00024 6.0 69.1570 0.031628 0.00300 0.00030 7.0 0.036899 0.00350 0.00035 8.0 0.073145 0.00400 0.00040 9.0 — — 9.099900 8.79380 0.8748 Shear s t r e s s C o h e s i o n l e s s s o i l s (Kpa) L/t=1 L / t = l 0 L/t=100 L/t=1000 0.0 0.0 0.0 0.0 0.0 100.0 0.61715 0.028803 0.00275 0.00028 200.0 1.71240 0.057605 0.00550 0.00055 300.0 31.6680 0.537270 0.00826 0.00083 400.0 10.79608 0.08949 0.00862 500.0 30.3666 77.5750 1. Normal s t r e s s e s on i n t e r f a c e element s u r f a c e a r e 30 Kpa f o r C o h e s i v e s o i l s and 477 Kpa f o r c o h e s i o n l e s s s o i l s r e s p e c t i v e l y ; 2. R e l a t i v e d i s p l a c e m e n t of i n t e r f a c e element i s i n u n i t mm; 3. L / t i s the a s p e c t r a t i o of the i n t e r f a c e element. response e x h i b i t b i l i n e a r c u r v e w i t h an i n i t i a l s t e e p l i n e and an a l m o s t h o r i z o n t a l l i n e a f t e r the u l t i m a t e shear r e s i s t a n c e i s reached. T h i s i n d i c a t e s t h a t the i n i t i a l r e l a t i v e d i s p l a c e m e n t i s v e r y s m a l l , then a l a r g e r e l a t i v e s l i p p a g e o c c u r s when the u l t i m a t e r e s i s t a n c e i s f u l l y m o b i l i z e d . In t h i s c a s e , the p r e d i c t e d u l t i m a t e r e s i s t a n c e s a r e a l l about 8 KPa, which s l i g h t l y o v e r e s t i m a t e s the u n d r a i n e d shear s t r e n g t h of the i n t e r f a c e l a y e r . The e r r o r i s l e s s o —•© L / t = 1 H — h L / t = 10 - . o 1 J 1 1 1 r — — — i 1 1 1 1 1 . 1 . . r . . . 0.0 2.0 4.0 6.0 8.0 RELATIVE DISPLACEMENT(MM) 10.0 12.0 14.0 E L A S T I C - P L A S T I C CLAY r 16.0 Cu-1 18.0 5KPA 20 F i g . 7.10 R e l a t i v e Displacement vs I n t e r f a c e R e s i s t a n c e under V a r i o u s L /t R a t i o f o r Cohesive S o i l 210 t h a n 7 % . H o w e v e r , s u c h r e s u l t s a r e s a t i s f a c t o r y when c o n s i d e r i n g t h e i n c r e m e n t a l n a t u r e o f t h e n u m e r i c a l p r o c e d u r e . T h e r e f o r e , i t may be c o n c l u d e d t h a t f o r t h e c o h e s i v e s o i l t h e r e l a t i v e s l i p p a g e o f i n t e r f a c e l a y e r c a n be s i m u l a t e d s a t i s f a c t o r i l y by t h e p r o p o s e d m o d e l , a n d t h e a s p e c t r a t i o u s e d s h o u l d be s u f f i c i e n t l y l a r g e , s u c h a s a t l e a s t b e i n g 10 f o r t h e d i r e c t s h e a r t e s t c o n d i t i o n . C o h e s i o n l e s s s o i l s N u m e r i c a l t e s t s were a l s o p e r f o r m e d i n c o h e s i o n l e s s s o i l s u n d e r d r a i n e d c o n d i t i o n s . The i n t e r f a c e l a y e r was c o n s i d e r e d t o c o n s i s t o f d e n s e s a n d o f r e l a t i v e d e n s i t y Dr = 80 % , a n d f r i c t i o n a l a n g l e <t>^ = 3 8 ° . The i n i t i a l s t r e s s c o n d i t i o n a s sumed i n t h e i n t e r f a c e e l e m e n t s was i s o t r o p i c w i t h n o r m a l s t r e s s , a n , e q u a l t o 477 K p a . U n i f o r m s h e a r s t r e s s was a p p l i e d i n c r e m e n t a l l y on t h e t o p o f i n t e r f a c e e l e m e n t s . T h e r e f o r e , t h e t h e o r e t i c a l u l t i m a t e s h e a r r e s i s t a n c e c a n be c a l c u l a t e d a s : T = a t a n 0 . = 373 Kpa s n I R The a s p e c t r a t i o s o f 1, 10, 100, 1000 were u s e d i n t h e a n a l y s e s . 21 1 a) Shear s t r e s s d i s t r i b u t i o n For the i n v e s t i g a t i o n of shear s t r e s s d i s t r i b u t i o n w i t h i n the i n t e r f a c e e l e m e n t s , a s e r i e s of n u m e r i c a l s t u d i e s were performed under v a r i o u s a s p e c t r a t i o s of the i n t e r f a c e e lement. T y p i c a l r e s u l t s a r e t a b u l a t e d i n T a b l e 7.5 f o r shear s t r e s s , r , e q u a l t o 200 KPa. As shown i n the t a b l e , f o r t h e a s p e c t r a t i o e q u a l t o one, the shear s t r e s s d i s t r i b u t i o n i s v e r y random w i t h i n the f o u r l i n e a r t r i a n g u l a r e l e m e n t s , however, the c o n d i t i o n i s improved as the a s p e c t r a t i o becomes l a r g e r than 10. As an extreme c o n d i t i o n , when the a s p e c t r a t i o i s e q u a l t o 1000, t h e d i s t r i b u t i o n of shear s t r e s s a c r o s s the i n t e r f a c e e lements i s v e r y u n i f o r m , v a l u e s of the shear s t r e s s a t each i n t e g r a t i o n p o i n t of the f o u r t r i a n g u l a r elements a r e i d e n t i c a l , and e q u a l t o the a p p l i e d shear s t r e s s . b) R e l a t i v e d i s p l a c e m e n t and shear r e s i s t a n c e As f o r the c o h e s i v e s o i l , t h e p r e d i c t e d r e l a t i v e d i s p l a c e m e n t s a r e p r e s e n t e d i n T a b l e 7.4 as f u n c t i o n s of a p p l i e d shear s t r e s s e s and a s p e c t r a t i o s of i n t e r f a c e e l e m ent. I n a d d i t i o n , the r e s u l t s f o r the a s p e c t r a t i o s of 1, 10, 100 a r e a l s o i l l u s t r a t e d i n F i g . 7.11. S i m i l a r t o the r e s u l t s f o r c o h e s i v e s o i l s , the a s p e c t r a t i o has a g r e a t i n f l u e n c e on t h e r e l a t i v e d i s p l a c e m e n t r e s p o n s e . When the a s p e c t r a t i o i s s m a l l , the proposed i n t e r f a c e model p r e d i c t s lower u l t i m a t e shear r e s i s t a n c e and l a r g e i n i t i a l r e l a t i v e d i s p l a c e m e n t s , and the i n i t i a l 212 T a b l e 7.5 Shear S t r e s s D i s t r i b u t i o n w i t h i n I n t e r f a c e Element a t I n t e g r a t i o n P o i n t s ( f o r C o h e s i o n l e s s S o i l s ) Element I n t e g . A s p e c t R a t i o s No. P o i n t s L/t=1 L/t=10 L/t=100 L/t=1000 1 -77.881 -210.56 -200. 17 -200.00 2 -328.31 -197.62 -199.95 -200.00 3 -48.089 -199.95 -200.02 -200.00 1 4 -180.29 -198.08 -199.97 -200.00 5 -81.045 -205.71 -200.10 -200.00 6 -113.69 -204.34 -200.06 -200.00 t 7 -114.37 -202.71 -200.04 -200.00 1 -80.942 -195.69 -199.94 -200.00 2 -171.82 -195.69 -199.94 -200.00 3 -356.16 -195.38 -199.91 -200.00 2 4 -265.61 -195.52 -199.93 -200.00 5 -145.10 -195.52 -199.93 -200.00 6 -195.24 -195.70 -199.95 -200.00 7 -254.93 -195.58 -199.93 -200.00 1 -345.77 -197.62 -199.95 -200.00 2 -205.24 -210.56 -200.17 -200.00 3 -146.29 -199.95 -200.02 -200.00 3 4 -165.62 -205.71 -200.10 -200.00 5 -248.47 -198.08 -199.97 -200.00 6 -283.22 -204.34 -200.06 -200.00 7 -232.44 -202.71 -200.04 -200.00 1 -146.58 -200.21 -200.01 -200.00 2 -321.21 -196.57 -199.93 -200.00 3 -194.30 -200.21 -200.01 -200.00 4 4 -264.40 -198.28 -199.97 -200.00 5 -161.44 -200.42 -200.01 -200.00 6 -236.26 -198.28 -199.97 -200.00 7 -220.70 -199.00 -199.98 -200.00 1. The a p p l i e d shear s t r e s s on the s u r f a c e of i n t e r f a c e element i s -200 Kpa. p o r t i o n of r e l a t i v e d i s p l a c e m e n t c u r v e i s r e l a t i v e l y f l a t . As t h e a s p e c t r a t i o becomes l a r g e r , i . e . the i n t e r f a c e element becomes t h i n n e r , the p r e d i c t e d i n i t i a l d i s p l a c e m e n t becomes s m a l l e r , and the i n i t i a l p o r t i o n of r e l a t i v e "i i i i i i i i i i : i i i i r 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 RELATIVE DISPLACEMENT(MM) E L A S T I C - P L A S T I C SAND DR=8CU 40 .0 F i g . 7.11 R e l a t i v e Displacement vs I n t e r f a c e R e s i s t a n c e under V a r i o u s L / t R a t i o f o r C o h e s i o n l e s s S o i l ro oo 214 d i s p l a c e m e n t c u r v e becomes s t i f f e r . For the a s p e c t r a t i o l a r g e r than 100, the r e l a t i v e d i s p l a c e m e n t c u r v e i s i n form of b i l i n e a r w i t h n e g l i g i b l e d i s p l a c e m e n t s a t t h e b e g i n n i n g and enormous d i s p l a c e m e n t s a f t e r the u l t i m a t e r e s i s t a n c e i s p r e d i c t e d . For the p r e d i c t i o n of u l t i m a t e r e s i s t a n c e , the pr o p o s e d model can p r e d i c t a c o n s i s t e n t v a l u e of 400 KPa i f the a s p e c t r a t i o i s kept l a r g e r than 100, as shown i n T a b l e 7.4. The o v e r p r e d i c t i o n i s about 7% as compared w i t h the t h e o r e t i c a l v a l u e . T h e r e f o r e , i n c o n s i d e r a t i o n of the shear s t r e s s and r e l a t i v e d i s p l a c e m e n t s i m u l a t i o n , the a p p r o p r i a t e a s p e c t r a t i o may need t o be as l a r g e as 100 f o r c o h e s i o n l e s s s o i l s i n the d i r e c t shear c o n d i t i o n s . Summaries In summary, based on the above n u m e r i c a l s t u d i e s f o r d i r e c t shear c o n d i t i o n , i t i s found t h a t the proposed i n t e r f a c e model can s a t i s f a c t o r i l y p r e d i c t the u l t i m a t e r e s i s t a n c e and the r e l a t i v e s l i p of the i n t e r f a c e , p r o v i d e d t h a t the a p p r o p r i a t e a s p e c t r a t i o i s s e l e c t e d . The a s p e c t r a t i o of the i n t e r f a c e element has a s i g n i f i c a n t e f f e c t on the s i m u l a t i o n of b o t h shear s t r e s s and r e l a t i v e d i s p l a c e m e n t , i t s h o u l d be d e t e r m i n e d by p e r f o r m i n g p a r a m e t r i c s t u d i e s f o r d i f f e r e n t problems a t hand. For the d i r e c t shear c o n d i t i o n , based on the f o r e g o i n g r e s u l t s , an a p p r o p r i a t e a s p e c t r a t i o may be su g g e s t e d i n the range of 10 t o 100 The h i g h e r number seems t o be more 215 a p p r o p r i a t e f o r the c o h e s i o n l e s s s o i l s . I t i s i n t e r e s t i n g t o note t h a t the range of a s p e c t r a t i o s u g g e sted above i s the same as t h a t s u g g e s t e d by D e s a i (1984) f o r h i s t h i n l a y e r i n t e r f a c e element, i n which D e s a i s i m u l a t e d the same d i r e c t shear c o n d i t i o n and compared the r e s u l t s from f i n i t e element a n a l y s e s w i t h t h o s e from d i r e c t shear box t e s t i n g s . The above p r e l i m i n a r y assessments of the proposed i n t e r f a c e model and t h e c h o i c e of element t h i c k n e s s a r e o n l y based on the n u m e r i c a l p a r a m e t r i c s t u d i e s . A l t h o u g h such a study i s e s s e n t i a l f o r the v e r i f i c a t i o n and a p p l i c a t i o n of the proposed model, the u l t i m a t e p r o o f of the model and the a p p r o p r i a t e s e l e c t i o n of element t h i c k n e s s would r e q u i r e s t u d i e s i n which f i n i t e element r e s u l t s a r e compared w i t h e x p e r i m e n t a l o b s e r v a t i o n s , such as d i r e c t shear t e s t d a t a . Such comparisons a r e not a c c o m p l i s h e d h e r e i n due t o l a c k of the a p p r o p r i a t e e x p e r i m e n t a l d a t a , and i t i s b e l i e v e d t h a t they a r e beyond the scope of t h i s t h e s i s . T h e r e f o r e , f u r t h e r r e s e a r c h e s may be w a r r a n t e d i n t h i s d i r e c t i o n . In the f o l l o w i n g s t u d i e s of s o i l - p i l e i n t e r f a c e e f f e c t s on P-Y c u r v e s , the p r o p o s e d i n t e r f a c e model w i l l be employed t o s i m u l a t e the s o i l - p i l e i n t e r f a c e b e h a v i o r . There the i n t e r f a c e element a s p e c t r a t i o w i l l be s e l e c t e d based on the p a r a m e t r i c s t u d i e s w i t h comparison of c l o s e d form s o l u t i o n . 8. F I N I T E ELEMENT STUDIES ON LATERALLY LOADED P I L E S 8.1 INTRODUCTION A l m o s t a l l p i l e d f o u n d a t i o n s a r e s u b j e c t e d t o a t l e a s t some d e g r e e s o f l a t e r a l l o a d i n g s . Some o f them r e q u i r e c r u c i a l d e s i g n s i n w i t h s t a n d i n g t h e l a t e r a l l o a d i n g s . T h e t r a d i t i o n a l p r o b l e m s o f t h i s k i n d i n g e o t e c h n i c a l p r a c t i c e a r e p i l e d e a r t h r e t a i n i n g w a l l o r p i l e d b r i d g e a b u t m e n t , where t h e p i l e f o u n d a t i o n s a r e u s u a l l y e x p o s e d t o l a r g e amount s o f l a t e r a l e a r t h p r e s s u r e s . M o r e o v e r , r e c e n t o f f s h o r e o i l d r i l l i n g p l a t f o r m d e s i g n s h a v e a c t i v a t e d more r e s e a r c h i n t o t h i s t r a d i t i o n a l p r o b l e m . P r o b l e m s o f l a t e r a l l y l o a d e d p i l e s have r e c e i v e d more a n d more a t t e n t i o n , a n d h a v e become an i m p o r t a n t and a c t i v e r e s e a r c h a r e a i n s o i l m e c h a n i c s a n d f o u n d a t i o n e n g i n e e r i n g . So f a r , a number o f a p p r o a c h e s h a v e b e e n p r o p o s e d by many r e s e a r c h e r s . In g e n e r a l , t h e r e a r e f o u r g r o u p s o f methods a v a i l a b l e i n a n a l y s i s o f t h e p r o b l e m . A b r i e f r e v i e w o f t h e s e a p p r o a c h e s was g i v e n i n S e c . 2 . 2 . As d i s c u s s e d i n S e c . 2 . 2 , e a c h o f t h e s e a p p r o a c h e s h a s i t s a d v a n t a g e s a n d c e r t a i n d e f f i c i e n c i e s . H o w e v e r , on b a l a n c e , i t a p p e a r s t h a t t h e W i n k l e r s p r i n g a p p r o a c h , t h a t i s t h e a p p r o a c h i n w h i c h t h e s o i l d o m a i n i s r e p l a c e d by an a r r a y o f u n c o u p l e d s p r i n g s , p r o v i d e s a v e r s a t i l e and p r a c t i c a l way f o r r o u t i n e d e s i g n a n a l y s i s . T h u s , t h e r e i s a c o n t i n u o u s n e e d t o d e f i n e t h e l i n e a r o r n o n l i n e a r s p r i n g s ( o r P-Y c u r v e s ) u s e d i n t h i s k i n d o f a n a l y s i s more a c c u r a t e l y . 216 217 The c o n c e p t o f P-Y c u r v e s c a n be i l l u s t r a t e d w i t h a i d o f F i g . 8.1 ( R e e s e e t a l , 1 9 7 7 ) . A f t e r t h e p i l e h a s been i n s t a l l e d i n t o t h e g r o u n d , t h e s o i l s t r e s s s t a t e a c t i n g on a p i l e s e c t i o n a t c e r t a i n d e p t h b e f o r e any l a t e r a l l o a d i n g s w o u l d be i s o t r o p i c w i t h a r e s u l t i n g s o i l r e a c t i o n f o r c e o f z e r o . T h i s s i t u a t i o n i s shown i n F i g . 8 . 1 ( b ) . When t h e p i l e s e c t i o n e x p e r i e n c e s a l a t e r a l d e f l e c t i o n o f Y due t o t h e l a t e r a l l o a d , an u n b a l a n c e d s o i l s t r e s s s t a t e w o u l d be g e n e ' r a t e d w i t h s t r e s s i n c r e a s e i n f r o n t o f t h e p i l e a n d t h e s t r e s s d e c r e a s e b e h i n d i t . As a r e s u l t , t h e i n t e g r a t i o n o f t h i s u n b a l a n c e d s t r e s s d i s t r i b u t i o n w o u l d g i v e a r e s u l t a n t s o i l r e a c t i o n , P, e x e r t i n g on t h e f r o n t s u r f a c e o f t h e p i l e , a s i l l u s t r a t e d i n F i g . 8 . 1 ( c ) . T h i s r e s u l t a n t p r e s s u r e , P, i s d e p e n d e n t upon t h e l a t e r a l d e f l e c t i o n , Y , o f t h e p i l e s e c t i o n . The f u n c t i o n o r t h e c u r v e d e s c r i b i n g s u c h a r e l a t i o n s h i p o f s o i l r e a c t i o n i s o f t e n r e f e r r e d t o a s P -Y c u r v e s . In W i n k l e r s p r i n g a p p r o a c h , t h e p i l e i s u s u a l l y d i s c r e t e d i n t o s e v e r a l s e gment s a l o n g i t s l e n g t h , s u c h a s i n p r o g r a m COM622 ( R e e s e , 1 9 7 7 ) . A t e a c h s e g m e n t , t h e s o i l r e a c t i o n i s r e p l a c e d by s u c h a P-Y c u r v e . Due t o t h e n o n l i n e a r d e f o r m a t i o n b e h a v i o r o f s o i l s , t h e s h a p e s o f P-Y c u r v e s a r e g e n e r a l l y n o n - l i n e a r . The i m p o r t a n t e l e m e n t s c o n t r o l l i n g t h e s h a p e s o f P-Y c u r v e s a r e t h e i n i t i a l s l o p e , and t h e u l t i m a t e s o i l r e s i s t a n c e . F o r f l e x i b l e p i l e s l o a d e d a t t h e i r h e a d s w h i c h a r e u s u a l l y e n c o u n t e r e d i n p r a c t i c e , t h e l a r g e l a t e r a l F i g . 8.1 Concept of P-Y curves 219 d e f l e c t i o n o f p i l e s a r e a l w a y s i n t h e u p p e r p o r t i o n o f p i l e s , a n d t h e d e f l e c t i o n d e c r e a s e s w i t h d e p t h . As a r e s u l t , i n a r e a s n e a r t h e p i l e h e a d , t h e s o i l r e a c t i o n , P, i s i n t h e r a n g e a p p r o a c h i n g o r e x c e e d i n g t h e u l t i m a t e s o i l r e s i s t a n c e , pu]_4-f w h i l e i n a r e a s f a r b e l o w t h e p i l e h e a d , t h e s o i l r e a c t i o n s a r e m a i n l y i n t h e r a n g e o f i n i t i a l p o r t i o n o f t h e P-Y c u r v e . F o r r i g i d p i l e s , w h i c h a r e r e l a t i v e l y s h o r t , t h e u l t i m a t e s o i l r e s i s t a n c e may be f u l l y m o b i l i z e d a l o n g i t s e n t i r e l e n g t h . T h e r e f o r e , e v a l u a t i o n s o f t h e i n i t i a l s l o p e , t h e u l t i m a t e s o i l r e s i s t a n c e , P u]_ t> and t h e s h a p e o f P-Y c u r v e a r e i m p o r t a n t f a c t o r s i n t h e W i n k l e r s p r i n g a p p r o a c h . C o n v e n t i o n a l l y , t h e d e v e l o p m e n t o f P-Y c u r v e s a r e m a i n l y b a s e d on t h e s e m i - e m p i r i c a l method i n w h i c h t h e s e c u r v e s a r e b a c k c a l c u l a t e d f r o m t h e f i e l d l a t e r a l l o a d t e s t s . T h i s a p p r o a c h i s e s p e c i a l l y p o p u l a r i n t h e d e s i g n o f o f f s h o r e p l a t f o r m s (API c o d e , 1 9 7 4 ) . H o w e v e r , t h i s a p p r o a c h i s b o t h t i m e a n d c o s t c o n s u m i n g i n t h e f i e l d , a n d i s a l s o s i t e - o r i e n t e d . A g e n e r a l r e v i e w o f t h e c u r r e n t a p p r o a c h e s f o r t h e d e v e l o p m e n t o f P-Y c u r v e s was g i v e n i n S e c t i o n 2 . 3 . I t i s w e l l e s t a b l i s h e d t h a t t h e a c c u r a c y o f W i n k l e r ' s s p r i n g a p p r o a c h r e l i e s on t h e s p e c i f i e d P-Y c u r v e s . T h e s e c u r v e s must r e p r e s e n t t h e s o i l d e f o r m a t i o n m e c h a n i s m a r o u n d t h e p i l e , t h e y must i n c o r p o r a t e s o i l n o n l i n e a r i t y , i n t e r f a c e and o t h e r e f f e c t s . T h e r e f o r e , i t i s d e s i r a b l e t o compute t h e P-Y c u r v e s f r o m f i n i t e e l e m e n t a n a l y s e s w h i c h i n c o r p o r a t e s t h e s e e f f e c t s . 220 In t h i s C h a p t e r , the f i n i t e element f o r m u l a t i o n i n p l a n e s t r a i n c o n d i t i o n s w i l l be used t o compute the P-Y c u r v e , and t o examine the d i f f e r e n t f a c t o r s t h a t i n f l u e n c e t h e s e c u r v e s . 8.2 PLANE STRAIN MODEL In th e p l a n e s t r a i n model f o r the development of P-Y c u r v e s , i t i s assumed t h a t the l a t e r a l l y l o a d e d p i l e s embedded i n s o i l s a r e r i g i d and i n f i n i t e l y l o n g , and they a r e u n i f o r m l y t r a n s v e r s e e d by l a t e r a l l o a d s i n h o r i z o n t a l d i r e c t i o n so t h a t the movement of any c r o s s s e c t i o n of the s o i l and t h e p i l e i s independent of the d e f o r m a t i o n of s e c t i o n s below or above i t (Pyke and B e i k a e , 1984). A u n i t t h i c k n e s s of h o r i z o n t a l d i s k i s thus t a k e n of the s o i l and the p i l e c r o s s s e c t i o n f o r the a n a l y s i s , as shown i n F i g . 8.2. In t h i s d i s k model, a c y l i n d r i c a l p i l e s e c t i o n of r a d i u s , a, i s embraced a t the c e n t e r by a s o i l domain of o u t e r r a d i u s , R. The boundary c o n d i t i o n s a t a s u f f i c i e n t l y l a r g e o u t e r r a d i u s , R, of the s o i l domain i s f i x e d w i t h no r a d i a l and t a n g e n t i a l d i s p l a c e m e n t s , assuming t h a t the l a t e r a l p i l e movement w i l l not i n f l u e n c e t h e s o i l r e g i o n i n the f a r f i e l d ( Y e g i a n and W r i g h t , 1973, B a g u e l i n e t a l , 1977). The c o n c e n t r a t e d l a t e r a l l o a d , P, i s a p p l i e d on the p i l e s e c t i o n , and the r e s u l t e d p i l e d e f l e c t i o n i n the a x i s of l o a d i n g i s Y. I »-y F i g . 8.2 ' D i s k ' A n a l y s i s f o r P-Y C u r v e s 222 Such a 2D p l a n e s t r a i n model i s e q u i v a l e n t t o the concept of W i n k l e r ' s s p r i n g a p p r o a c h i n which each s u p p o r t i n g s p r i n g a c t s i n d e p e n d e n t l y of the o t h e r s , and i s p a r t i c u l a r l y r e a s o n a b l e a t depths where h i g h c o n f i n i n g s t r e s s e s of overburden s o i l s may r e s t r i c t the s o i l movement t o h o r i z o n t a l p l a n e ( M a t l o c k , 1970), and o u t - p l a n e d i s p l a c e m e n t may not be s i g n i f i c a n t . However, f o r t h e a r e a s near the ground s u r f a c e , c o r r e c t i o n s s h o u l d be made t o the r e s u l t s from the p l a n e s t r a i n model t o account f o r the c o n f i n i n g s t r e s s r e d u c t i o n due t o the presence of f r e e s u r f a c e and 3D d e f o r m a t i o n c o n d i t i o n s (Pyke and B i e k a e , 1984). A s i m p l e s t c o r r e c t i o n t o the s o i l r e a c t i o n near the s u r f a c e may be su g g e s t e d u s i n g l i n e a r i n t e r p o l a t i o n from the s u r f a c e where p l a n e s t r e s s c o n d i t i o n may be p r e v a i l i n g t o the d e p t h where p l a n e s t r a i n c o n d i t i o n i s more a p p r o p r i a t e . 8.3 CLOSED FORM SOLUTION For the above two d i m e n s i o n a l p l a n e s t r a i n model, a t h e o r e t i c a l s o l u t i o n can be o b t a i n e d i f the s o i l i s l i n e a r e l a s t i c ( B a g u e l i n e t a l , 1977, B a r d e t , 1979). The d i s p l a c e m e n t boundary c o n d i t i o n s f o r the s o l u t i o n a r e : a t the p i l e s u r f a c e , r = a U = Y cos0 and U„ = Y s i n e (8.3.1) r o a t the o u t e r boundary, r = R U = 0 and U f l = 0 (8.3.2) r fj and the p e r f e c t a d h e s i o n i s a l s o assumed a t the s o i l - p i l e 223 i n t e r f a c e . D i s p l a c e m e n t S o l u t i o n The r a d i a l a n d t a n g e n t i a l d i s p l a c e m e n t s , w h i c h s a t i s f y t h e a b o v e d i s p l a c e m e n t b o u n d a r y c o n d i t i o n s a r e d e r i v e d f r o m t h e A i r y f u n c t i o n a n d c a n be e x p r e s s e d i n t h e f o l l o w i n g c l o s e d f o r m s i n p o l a r c o o r d i n a t e s ( r , 6) ( B a g u e l i n e t a l , 1 977) : ° r " TZ T ± £ H 3 - 4 U ) l n ( 5 ) 2 - ? ^ [ ( ^ ) 2 - ^ ] } c o s ^ ( 8 . 3 . 3 ) r 87rE 1-U r 2 r 3 _ 4 y UB " r; T ± £ { ( 3 - 4 U ) l n ( 5 ) 2 - ^ T [ ( ^ ) 2 - ^ ] } s i n e ( 8 . 3 . 4 ) " 8irE 1-u r R 2 + r 2 r 3 -4u where E and v a r e r e s p e c t i v e l y t h e Y o u n g ' s m o d u l u s and t h e P o s s i o n ' s r a t i o o f t h e l i n e a r e l a s t i c s o i l d o m a i n . I t c a n be s e e n f r o m t h e a b o v e e q u a t i o n s t h a t t h e d i s p l a c e m e n t s o l u t i o n s f r o m t h e e l a s t i c t w o - d i m e n s i o n a l p l a n e s t r a i n m o d e l a r e much d e p e n d e n t upon t h e d i s t a n c e o f t h e o u t e r b o u n d a r y , R. As t h e d i s t a n c e o f o u t e r b o u n d a r y , R, i n c r e a s e s , b o t h r a d i a l a n d t a n g e n t i a l d i s p l a c e m e n t s d e c r e a s e . To t h e e x t r e m e c o n d i t i o n , b o t h d i s p l a c e m e n t s t e n d t o t h e i n f i n i t y when t h e o u t e r b o u n d a r y , R t e n d s t o t h e i n f i n i t y . H o w e v e r , i n r e a l i t y t h e s o i l d i s p l a c e m e n t s u n d e r l a t e r a l l o a d i n g do n o t e x t e n d t o t h e i n f i n i t y i n a s e m i - i n f i n i t e s o i l med ium. T h e r e f o r e , t h i s a n a l y t i c a l m o d e l c a n n o t p r o v i d e a r e a l i s t i c d i s p l a c e m e n t s o l u t i o n f o r t h e r e s p o n s e o f t h r e e d i m e n s i o n a l s o i l medu im. 224 L i n e a r W i n k l e r S p r i n g S t i f f n e s s When s e t t i n g 0 = 0, and r = a t o the Eq. ( 8 . 3 . 3 ) , the l a t e r a l d i s p l a c e m e n t , Y, of the p i l e s e c t i o n r e s u l t e d from the l a t e r a l l o a d , P, i s o b t a i n e d ( B a r d e t , 1979), i . e . Y = U r ( a , 0) P 1 + o R R 2 — a 2 9 — — [ ( 3 - 4 y ) l n ( - ) 2 - ^ - ^ 7 (-T—)] (8.3.5) 87rE 1-D a R 2+a 2 3-4u T h e r e f o r e the l a t e r a l f o r c e d i s p a l c e m e n t i s r e l a t e d v i a K as f o l l o w s : P = K Y (8.3.6) where K i s the s t i f f n e s s of t h e l i n e a r W i n k l e r s p r i n g , and i t can be s i m p l y d e t e r m i n e d from Eq. ( 8 . 3 . 5 ) , i . e : K = M E (8.3.7) where LI i s c a l l e d s t i f f n e s s c o e f f i c i e n t which r e l a t e s the e m p i r i c a l ' c o e f f i c i e n t of subgrade r e a c t i o n ' , K, t o the fundamental s o i l p r o p e r t y , e l a s t i c Young's modulus, E. The v a l u e of K may be r e p r e s e n t i t i v e f o r the i n i t i a l s l o p e of the P-Y c u r v e a t v e r y s m a l l s t r a i n range. 225 Boundary and P o i s s o n ' s R a t i o E f f e c t s I t i t r e a d i l y s e e n f r o m E q s . ( 8 . 3 . 5 ) t o ( 8 . 3 . 7 ) t h a t t h e s t i f f n e s s c o e f f i c i e n t , M, f r o m t h e t w o - d i m e n s i o n a l p l a n e s t r a i n a n a l y s i s i s a f u n c t i o n o f P o i s s o n ' s r a t i o , v, a n d t h e b o u n d a r y r a t i o , a / R . F o r a g i v e n b o u n d a r y r a t i o , B a r d e t (1979) s t u d i e d t h e e f f e c t o f P o i s s o n ' s r a t i o on t h e v a l u e o f jn. The r e s u l t s showed t h a t t h e v a l u e o f s t i f f n e s s c o e f f i c i e n t , n i s maximum f o r u n d r a i n e d c o n d i t i o n s o f c o h e s i v e s o i l s , and i s min imum f o r d r a i n e d c o n d i t i o n s o f c o h e s i o n l e s s s o i l s where P o i s s o n ' s r a t i o i s c l o s e t o 0 . 2 5 . B a r d e t ' s r e s u l t s a r e shown i n F i g . 8 . 3 . The e x t r e m e v a l u e s o f t h e s t i f f n e s s c o e f f i c i e n t v e r s u s t h e b o u n d a r y r a t i o , a /R a r e shown i n F i g . 8 . 4 . The i n f l u e n c e o f b o u n d a r y r a t i o i s p r o n o u n c e d , e s p e c i a l l y f o r u n d r a i n e d c o h e s i v e s o i l where P o i s s o n ' s r a t i o i s c l o s e t o 0. 5. T h e r e f o r e , t h e p r o p e r s e l e c t i o n o f o u t e r r a d i u s i s a key p r o b l e m f o r d i s k m o d e l a n a l y s i s o f t h e s t i f f n e s s o f l i n e a r W i n k l e r ' s s p r i n g , K. T h i s c l e a r l y i n d i c a t e s t h e d i f f i c u l t i e s i n m o d e l i n g t h e t h r e e - d i m e n s i o n a l s o i l medium u s i n g two d i m e n s i o n a l i d e a l i z a t i o n . S t r e s s S o l u t i o n U n l i k e t h e s o l u t i o n f o r d i s p l a c e m e n t s a n d s t i f f n e s s , t h e s t r e s s s o l u t i o n s h a v e l i m i t v a l u e s when t h e o u t e r b o u n d a r y o f t h e e l a s t i c s o i l medium e x t e n d s t o i n f i n i t y , 1. e . f o r R » : 226 228 P [ ( 3 - 2 u ) ( a / r ) - ( a / r ) 3 ] c o s 0 ( 8 . 3 . 8 ) a r 4 na 1 - v = - a r ( 8 . 3 . 9 ) P [ ( 1 - 2 o ) ( a / r ) + ( a / r ) 3 ] s i n 0 r = -4 na 1 - u ( 8 . 3 . 1 0 ) T h e r e f o r e , t w o - d i m e n s i o n a l p l a n e s t r a i n c o n d i t i o n c a n g i v e a l i m i t e d s t r e s s s o l u t i o n f o r i n f i n i t e s o i l med ium. In f a c t , t h e l i m i t v a l u e s c a n be r e a s o n a b l l y a p p r o x i m a t e d when a/R s m a l l e r t h a n 0.1 a r e u s e d ( B a g u e l i n e t a l , 1 9 7 7 ) . I n i t i a t i o n of P l a s t i c i t y The a b o v e s o l u t i o n s a r e b a s e d upon t h e e l a s t i c t h e o r y and w i l l n o t be v a l i d when p l a s t i c i t y o f t h e s o i l medium i s i n i t i a t e d . H o w e v e r , by i n c o r p o r a t i n g t h e p l a s t i c y i e l d c r i t e r i a o f s o i l s , t h e l a t e r a l f o r c e , P^., w h i c h i n i t i a t e s t h e s o i l p l a s t i c i t y c a n be e v a l u a t e d . F o r c o h e s i v e s o i l s , t h e y i e l d c r i t e r i o n commonly u s e d i s t h e T r e s c a ' s y i e l d c r i t e r i o n , i n w h i c h s o i l u n d e r g o e s i r r e v e r s i b l e p l a s t i c d e f o r m a t i o n when t h e d i f f e r e n c e i n m a j o r a n d m i n o r p r i n c i p a l s t r e s s e s s a t i s f i e s : where a , , a 3 a r e r e s p e c t i v e l y t h e i n d u c e d m a j o r a n d m i n o r p r i n c i p a l s t r e s s e s , a n d C u i s t h e u n d r a i n e d s h e a r s t r e n g t h o f t h e c o h e s i v e s o i l . U s i n g t h i s y i e l d c r i t e r i o n , t h e = 2 C u ( 8 . 3 . 1 1 ) 229 l a t e r a l f o r c e i n i t i a t i n g the p l a s t i c d e f o r m a t i o n , P^, i s d e t e r m i n e d as ( B a r d e t , 1979) : P k = 2 ir a C J (8.3.12) where a i s p i l e r a d i u s . The p l a s t i c i t y of c o h e s i v e s o i l w i l l o c c u r i n i t i a l l y i n the s o i l s a d j a c e n t t o the p i l e s u r f a c e a t a r i g h t a n g l e t o the l o a d i n g a x i s (9 = ~ ) . For c o h e s i o n l e s s s o i l s , the y i e l d c r i t e r i o n commonly used i s the Mohr-Coulomb c r i t e r i o n , where the y i e l d i n g s t r e s s c o n d i t i o n s a r e s a t i s f i e d when the s t r e s s r a t i o ( — a 1 ~ ° 3 — ) i s e q u a l t o s i n 0 , i n which <f> i s t h e f r i c t i o n a l a x + o 3 + 2 a 0 a n g l e of s o i l s , a0 i s t h e i n - s i t u i s o t r o p i c s t r e s s , and a,, a3 a r e r e s p e c t i v e l y the i n d u c e d major and minor p r i n c i p a l s t r e s s e s . The p l a s t i c i t y i s i n i t i a t e d i n t h e s o i l a d j a c e n t t o the p i l e s u r f a c e a t a n g l e of 9 t o the a x i s of l o a d i n g : 9 = c o s - U f - ^ ( -7-4- + 0 1 , ) ] - 0 - 5 (8.3.13) 3-4u s i n z 0 3 - 4u U s i n g the Mohr-coulomb c r i t e r i o n , the l a t e r a l l o a d , P^; which i n i t i a t e s the p l a s t i c i t y i s o b t a i n e d as ( B a r d e t , 1979) : > = 2 Tr a o0 [ -7-7- + — — ]-°' 5 (8.3.14) K s m 2 0 3 - iv i n w hich v i s the P o s s i o n ' s r a t i o , a i s the r a d i u s of p i l e . 230 A l t h o u g h the r e a l s o i l b e h a v i o u r i s n e i t h e r e l a s t i c , nor p u r e l y e l a s t o - p l a s t i c , the above c l o s e d form s o l u t i o n s a r e s t i l l u s e f u l f o r b a s i c assessment of the r e s u l t s from f i n i t e element program. 8.4 FINITE ELEMENT SIMULATION As mentioned above, the r e a l s o i l b e h a v i o u r i s n e i t h e r e l a s t i c , nor even p u r e l y e l a s t o - p l a s t i c , but appears h i g h l y n o n l i n e a r , and s t r e s s l e v e l dependent. T h e r e f o r e , development of g e n e r a l c l o s e d form s o l u t i o n i s u s u a l l y o b s t r u c t e d . R e c e n t l y , many r e s e a r c h e r s have r e s o r t e d t o the f i n i t e element a n a l y s i s t o d e v e l o p the P-Y c u r v e s d i r e c t l y based on fundamental s o i l p r o p e r t i e s and b e h a v i o u r . Examples of t h i s a p p r o a c h i n c l u d e the p l a n e s t r a i n a n a l y s e s performed by Y e g i a n and W r i g h t (1973), B a r t o n and F i n n (1983), A t u k o r a l a and Byrne (1984), and more r e c e n t l y by She (1986). B a s i c c o n c e p t s employed by d i f f e r e n t r e s e a r c h e r s a r e s i m i l a r . However, t h e r e a r e s t i l l some common problems as ye t u n r e s o l v e d i n the f i n i t e element method f o r development of P-Y c u r v e s , such as o u t e r boundary e f f e c t s , i n t e r f a c e s i m u l a t i o n . Those problems d e s e r v e c a r e f u l s t u d i e s i n the c o n t e x t of the f i n i t e element m o d e l l i n g of l a t e r a l l y l o a d e d p i l e - s o i l i n t e r a c t i o n problems. 8.4.1 FINITE ELEMENT MESH LAYOUT As under the l a t e r a l l o a d i n g , s o i l and p i l e s e c t i o n move l a t e r a l l y i n the d i r e c t i o n of l o a d i n g . The a x i s of 231 l o a d i n g i s t h e r e f o r e t h e symmetry b o u n d a r y o f s o i l movement s . In v i e w o f t h i s s ymmetry b o u n d a r y c o n d i t i o n , o n l y h a l f o f t h e c i r c u l a r d i s k i n F i g . 8 .2 i s d i s c r e t e d i n t o f i n i t e e l e m e n t d o m a i n . A s e r i e s o f r o l l e r s a r e p l a c e d a t n o d e s a l o n g t h e symmetry b o u n d a r y t o e n s u r e no d i s p l a c e m e n t i n d u c e d p e r p e n d i c u l a r t o t h e b o u n d a r y . T h i s s e m i - d i s k m o d e l f o r f i n i t e e l e m e n t a n a l y s e s i s i l l u s t r a t e d i n F i g . 8 . 5 . In a d d i t i o n a l t o t h e a b o v e symmetry d i s p l a c e m e n t b o u n d a r y , t h e r e a l s o e x i s t s an a n t i s y m m e t r y s t r e s s b o u n d a r y , p r o v i d e d t h a t c e r t a i n a s s u m p t i o n s a r e made a b o u t t h e s o i l s t r e s s a n d s t r a i n c h a r a c t e r i s t i c s . T h e s e a s s u m p t i o n s a r e t h a t t h e s o i l m o d u l u s a n d s t r a i n a r e n o t s t r e s s l e v e l d e p e n d e n t , t h e s o i l s t r e s s w i l l n o t a r r i v e a t t h e l e v e l where a t e n s i l e f a i l u r e i n s o i l a n d f u r t h e r s o i l - p i l e s e p e r a t i o n o c c u r ( Y e g i a n a n d W r i g h t , 1 9 7 3 ) . I f s u c h an a n t i s y m m e t r y b o u n d a r y i s e m p l o y e d , a q u a r t e r o f c i r c u l a r d i s k c a n be u s e d a s shown i n F i g . 2 . 1 0 , a n d t h e t o t a l number o f f i n i t e e l e m e n t s u s e d i s r e d u c e d . H o w e v e r , a s d i s c u s s e d i n S e c t i o n 2 . 3 . 4 , t h i s a n t i s y m m e t r y boundary , w i l l n o t e x i s t f o r t h e n o n l i n e a r s o i l s , and t h e i r i m p o s i t i o n on t h e f i n i t e e l e m e n t a n a l y s i s w i l l u n j u s t i f i a b l y d e g e n e r a t e t h e a d v a n t a g e s o f t h e m e t h o d . T h e r e f o r e , t h e s e m i - d i s k m o d e l i s u s e d f o r p r e s e n t s t u d i e s . As shown i n F i g . 8 . 5 , b o t h s o i l r e g i o n a n d p i l e s e c t i o n a r e d i s c r e t e d i n t o l i n e a r s t r a i n t r i a n g u l a r e l e m e n t s . Due t o t h e symmetry c o n d i t i o n o f t h e d i s p l a c e m e n t , a c o n c e n t r a t e d l o a d , P/2 i s a p p l i e d a t t h e p i l e c e n t e r , t h e r e s u l t i n g 233 d i s p l a c e m e n t a t t h i s node i s use t o produce P-Y c u r v e s . T h i s p r o c e d u r e w i l l be a c c u r a t e when the p i l e elements a r e r e l a t i v e l y r i g i d . 8.4.2 OUTER BOUNDARY CONSIDERATION In r e a l i t y , the s o i l medium i s s e m i - i n f i n i t e , and extend s t o the i n f i n i t y i n h o r i z o n t a l d i r e c t i o n . For the purpose of f i n i t e element a n a l y s i s , a f i n i t e medium i s always chosen t o r e p r e s e n t t h e i n f i n i t e s o i l medium. As shown i n the f o r e g o i n g e l a s t i c c l o s e d form s o l u t i o n , the r a d i u s of the o u t e r boundary, R, has a g r e a t i n f l u e n c e on the t w o - d i m e n s i o n a l p l a n e - s t r a i n s o l u t i o n f o r d i s p l a c e m e n t , and c o n s e q u e n t l y on the c a l c u l a t e d c o e f f i c i e n t of subgrade r e a c t i o n , K. T h e r e f o r e , the prope r s e l e c t i o n of the o u t e r r a d i u s , R, i s a key problem i n the a n a l y s i s u s i n g t w o - d i m e n s i o n a l f i n i t e element a n a l y s i s , and has been examined by s e v e r a l r e s e a r c h e r s ( B a g u e l i n e t a l , 1977, S c o t t , 1981 and She, 1986). B a g u e l i n e t a l (1977) i n t e r c o n n e c t e d the s t r a i n f i e l d s o b t a i n e d from 2D and 3D a n a l y s e s . The 2D s t r a i n f i e l d used near the p i l e was o b t a i n e d based on the e l a s t i c c l o s e d form s o l u t i o n , w h i l e the 3D s t r a i n f i e l d a t f a r f i e l d was from the M i n d l i n (1936) p o i n t f o r c e s o l u t i o n . And they d e t e r m i n e d the o u t e r r a d i u s i n such a way t h a t t h e d i s p l a c e m e n t s c a l c u l a t e d a t the i n t e r s e c t i o n p o i n t of 2D, and 3D s t r a i n f i e l d s a r e the same. From the co m p a r i s o n of W i n k l e r ' s s o l u t i o n w i t h the e l a s t i c continuum s o l u t i o n , B a g u e l i n e t a l 234 recommended t h a t the o u t e r boundary r a d i u s f o r the ' d i s k ' a n a l y s i s i n u n d r a i n e d c o h e s i v e s o i l s : a) P i l e s w i t h f r e e heads s u b j e c t t o l o a d s a t the head R = 7 1 0 f o r f l e x i b l e p i l e s ( h / l 0 > 7 / 3 ) (8.4.1) R = 3 h f o r r i g i d p i l e s ( h / l 0 < 7 / 3 ) (8.4.2) b) P i l e s w i t h f i x e d heads s u b j e c t e d t o l o a d s a t the head R = 12 1 0 f o r f l e x i b l e p i l e s ( h / l 0 > 1 . 5 ) (8.4.3) R = 8 h f o r r i g i d p i l e s ( h / l 0 < 1 . 5 ) (8.4.4) where h i s the embeded l e n g t h of p i l e s , and 1 0 i s the r e l a t i v e s t i f f n e s s f a c t o r d e f i n e d as [ 4 ( E I ) /E ] 0 ' 2 5 , (EI) p s p i s the r i g i d i t y of the p i l e s e c t i o n , E g i s the c o e f f i c i e n t of s o i l subgrade r e a c t i o n . For the c o h e s i o n l e s s d r a i n e d s o i l s , t h ey s u g g e s t e d t h a t h a l f of the above v a l u e s s h o u l d be used. T h e r e f o r e , s e l e c t i o n of o u t e r r a d i u s depends upon s e v e r a l f a c t o r s , and may need t o be examined f o r each problem. I t i s a l s o a p p a r e n t from t h e i r r e s u l t s t h a t t h e a n a l y s i s f o r f i x e d - h e a d f l e x i b l e p i l e s i n u n d r a i n e d c o h e s i v e s o i l s r e q u i r e s l a r g e s t o u t e r boundary r a d i u s . For f i n i t e e l a s t i c medium of sands, based on B a r d e t ' s r e s u l t s (see F i g . 8.3 or F i g . 8.4), S c o t t (1981) compared W i n k l e r s p r i n g s o l u t i o n u s i n g K = E ( i . e . (i = I i n 235 Eq. (8.3.7)) w i t h e l a s t i c continuum s o l u t i o n , and r e p o r t e d t h a t the o u t e r boundary of 50 p i l e r a d i u s would p r o v i d e r e s u l t s t h a t a g r e e d w i t h h i s c e n t r i f u g e t e s t d a t a i n sands. More r e c e n t l y , She (1986) s t u d i e d the o u t e r boundary e f f e c t s u s i n g f i n i t e element a n a l y s i s . Under p l a n e s t r a i n c o n d i t i o n s , She showed t h a t f o r u n d r a i n e d c o h e s i v e s o i l s the e f f e c t s of o u t e r boundary r a d i u s a r e s i g n i f i c a n t on the i n i t i a l s l o p e of t h e p r e d i c t e d P-Y c u r v e s , but not so g r e a t on the u l t i m a t e s o i l r e s i s t a n c e , P ,.. W h i l e f o r u l t c o h e s i o n l e s s s o i l s , the o p p o s i t e r e s u l t s were o b s e r v e d . The e f f e c t s of mesh r a d i u s on both the i n i t i a l s l o p e and the lower p o r t i o n shape of the p r e d i c t e d P-Y c u r v e s a r e not as s i g n i f i c a n t as i n c o h e s i v e s o i l s , but the u l t i m a t e s o i l r e s i s t a n c e , p u i t / was found t o be s e n s i t i v e t o the v a r i a t i o n of mesh r a d i u s . However, the c o n c l u s i o n f o r c o h e s i o n l e s s s o i l s may need f u r t h e r v e r i f i c a t i o n s . Based on t h e above r e s u l t s , s e l e c t i o n of t h e o u t e r r a d i u s of the 2D f i n i t e element mesh domain i s c r u c i a l f o r a m e a n i n g f u l p r e d i c t i o n of i n i t i a l s l o p e s of the P-Y c u r v e s i n c l a y s , but not t h a t i m p o r t a n t f o r the o v e r a l l shapes and t h e u l t i m a t e v a l u e s of the c u r v e s . F o r t u n a t e l y , t h e d e s i g n of the l a t e r a l l y l o a d e d p i l e s i n c l a y s i s m a i n l y governed by moderate s t r a i n s a t wo r k i n g l o a d s r a t h e r than i n i t i a l s m a l l s t r a i n s . T h e r e f o r e , the p l a n e s t r a i n model of f i n i t e element a n a l y s i s f o r P-Y c u r v e s i s s t i l l u s e f u l d e s p i t e the r e m a i n i n g d i f f i c u l t i e s i n o u t e r boundary s e l e c t i o n . 236 In the f o l l o w i n g s t u d i e s , an o u t e r r a d i u s of 50 t i m e s p i l e d i a m e t e r (D) i s s e l e c t e d f o r the f i n i t e element a n a l y s e s of p i l e s i n u n d r a i n e d c o h e s i v e s o i l s . T h i s v a l u e i s t h e average v a l u e s s u g g e s t e d by B a g u e l i n e t a l and B a r d e t ( i f K = E, R = 83 D from F i g . 8.4 f o r c o h e s i v e s o i l s ) . D e t a i l s of c a l c u l a t i o n of t h i s v a l u e a r e g i v e n i n Appendix B. W h i l e f o r the p i l e s i n c o h e s i o n l e s s s o i l s , two r a d i i of 25 and 50 p i l e d i a m e t e r a r e used t o e v a l u a t e the boundary r a d i u s e f f e c t s on the P-Y c u r v e s f o r c o h e s i o n l e s s s o i l s . 8.4.3 INTERFACE ELEMENT In o r d e r t o s i m u l a t e the d e f o r m a t i o n mechanism a t t h e s o i l - p i l e i n t e r f a c e under the l a t e r a l l o a d i n g s , a t h i n r i n g of i n t e r f a c e elements was p l a c e d around the p i l e p e r i m e t e r as shown i n F i g . 8.5. The b a s i s f o r the proposed t h i n i n t e r f a c e elements and t h e c o r r e s p o n d i n g n u m e r i c a l p r o c e d u r e s were d i s c u s s e d i n d e t a i l i n Chapter 7. The purpose of i n t e r f a c e elements i s t o r e p r e s e n t a p p r o x i m a t e l y the r e l a t i v e s l i p p a g e and s o i l f l o w around the p i l e , and the s o i l - p i l e s e p a r a t i o n under s e v e r e l a t e r a l l o a d i n g s . The s t r e n g t h c h a r a c t e r i s t i c s of t h e i n t e r f a c e elements a r e r e l a t e d t o t h o s e of the a d j a c e n t s o i l medium by t h e a d h e s i o n f a c t o r , a, and the f r i c t i o n a l a n g l e f a c t o r , |3 (see T a b l e 7.1). In t h e subsequent a n a l y s i s , v a r i o u s v a l u e s of a, and 0 i n T a b l e 7.1 a r e employed t o s t u d y the i n f l u e n c e of i n t e r f a c e p r o p e r t i e s on the P-Y c u r v e s . 237 In t h e a p p l i c a t i o n o f t h e t h i n i n t e r f a c e e l e m e n t , a s p e c i a l a t t e n t i o n s h o u l d be g i v e n t o t h e e l e m e n t t h i c k n e s s ( D e s a i , 1 9 8 4 ) . H e r e i n a p a r a m e t r i c s t u d y i s p e r f o r m e d w i t h a 0.6m d i a m e t e r p i l e i n s t a l l e d i n c o h e s i v e s o i l s u n d e r t h e u n d r a i n e d c o n d i t i o n . The u n d r a i n e d s h e a r s t r e n g t h o f t h e s o i l i s a s sumed t o be 7.5 K p a , and t h e a d h e s i o n f a c t o r u s e d f o r t h e i n t e r f a c e e l e m e n t s i s a = 0 . 5 , w h i c h a r e c o r r e s p o n d e n t t o t h e s o f t c l a y and t h e c o n c r e t e p i l e ( P o t y o n d y , 1 9 6 1 ) . And t h e p i l e e l e m e n t s a r e a s s u m e d 500 t i m e s s t r o n g e r t h a n t h e s o i l e l e m e n t s . B i l i n e a r e l a s t i c p l a s t i c s o i l b e h a v i o r i s a s s u m e d . The s t u d i e s were p e r f o r m e d w i t h d i f f e r e n t a s p e c t r a t i o o f t h e i n t e r f a c e e l e m e n t . The d e f i n i t i o n o f t h e a s p e c t r a t i o ( L / t ) i s shown i n F i g . 8 . 5 . The p r e d i c t e d P-Y c u r v e s u n d e r t h e a s p e c t r a t i o o f 2, 3, a n d 12 a r e shown i n F i g . 8 . 6 ( a ) . The u l t i m a t e s o i l r e s i s t a n c e s a r e c o m p a r e d w i t h t h e c l a s s i c p l a s t i c i t y s o l u t i o n i n T a b l e 8.1 ( R a n d o l p h a n d H o u l s b y , 1 9 8 4 ) . As shown i n F i g . 8 . 6 ( a ) , i n i t i a l e l a s t i c p o r t i o n s o f t h e p r e d i c t e d P-Y c u r v e s a r e i n s e n s i t i v e t o t h e t h i c k n e s s o f t h e t h i n i n t e r f a c e e l e m e n t s . H o w e v e r , t h e p r e d i c t e d u l t i m a t e s o i l r e s i s t e n c e s a r e v e r y s e n s i t i v e t o t h e e l e m e n t t h i c k n e s s . The b e s t a g r e e m e n t be tween t h e p r e d i c t e d v a l u e a n d t h e t h e o r e t i c a l v a l u e was f o u n d when t h e a s p e c t r a t i o e q u a l s 3. In t h i s c a s e t h e f i n i t e e l e m e n t m e t h o d p r e d i c t e d t h a t P , . / C D = 1 1 . 5 5 6 . T h i s v a l u e i s i n an e r r o r o f 6.8% a s u l t u c o m p a r e d w i t h t h e t h e o r e t i c a l v a l u e o f 10 .820 i n T a b l e 8 . 1 . 238 T a b l e 8.1 P l a s t i c i t y S o l u t i o n of U l t i m a t e S o i l R e s i s t a n c e on L a t e r a l C i r c u l a r P i l e i n C o h e s i v e S o i l s a p u l t / c u D 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 9. 1 42 9.527 9.886 0.220 0.531 0.820 1 .088 1 .336 1 .563 1 .767 1 .940 1. a i s the s o i l - p i l e a d h e s i o n f a c t o r , a = C a/C u; 2. D i s the p i l e d i a m e t e r ; 3. R e s u l t s a r e a f t e r Randolph and Houlsby ( 1 9 8 4 ) . I n c r e a s e s i n t h i c k n e s s of the i n t e r f a c e element from the a s p e c t r a t i o of 3 t o 2 r e s u l t e d i n an i n c r e a s e of 7.7% i n the p r e d i c t e d u l t i m a t e s o i l r e s i s t a n c e , but not s i g n i f i c a n t i n the i n i t i a l p o r t i o n of t h e P-Y c u r v e up t o the p i l e d e f l e c t i o n of about 1.2% p i l e d i a m e t e r . However, i n c r e a s e of the a s p e c t r a t i o from 3 t o 12, i . e . reduce t h e element t h i c k n e s s of t h e f i r s t r i n g , r e s u l t e d i n a n u m e r i c a l problem which s p o i l e d the convegence of l o a d s h e d d i n g i t e r a t i o n p e r f o r m a n c e , then a b o r t i n g the e n t i r e c a l c u l a t i o n a t the l o a d w e l l below the u l t i m a t e v a l u e g i v e n by t h e p l a s t i c s o l u t i o n , and a l s o r e s u l t e d i n a s o f t e n P-Y c u r v e s t a r t i n g from the d e f l e c t i o n a t about 0.7% p i l e d i a m e t e r . T h e r e f o r e , r e g a r d i n g the element t h i c k n e s s , a c o n t r a d i c t i o n e x i s t s between the above r e s u l t s and t h o s e o o 2 : CO Q 1 O 1 1 0 1 —e> © I.F. ASPECT RATI0=3 ( P i l e = 500«Soil) I I I I I I I I 1 1 1 1 1 1 1 1 °-° 6-0 12.0 18.0 24.0 30.0 36 0 42 0 48 0 54 0 DISPLACEMENT Y - MM D=0.6M, DEPTH=5M. COEFF oi = 0.5 (a) F i g . 8.6 E f f e c t s o f I n t e r f a c e E l e m e n t L / t r a t i o 1 60 .0 co vo 240 shown i n Chapter 7. As shown i n Chapter 7, f o r t h e s i m p l e shear c o n d i t i o n where elements were of u n i f o r m s i z e and s u b j e c t e d t o pure shear s t r e s s , the proposed i n t e r f a c e element can go as t h i n as 0.1-0.01 of i t s l e n g t h w i t h both s a t i s f a c t o r y s t r e s s and d i s p l a c e m e n t p r e d i c t i o n s . However, such a t h i n t r i a n g u l a r element w i l l g e n e r a t e n u m e r i c a l problems i n a l a r g e problem w i t h complex l o a d i n g c o n d i t i o n s , e s p e c i a l l y when mixed w i t h d i f f e r e n t s i z e of e l e m e n t s . S e v e r a l a t t e m p t s were made t o use the i n t e r f a c e element of h i g h e r a s p e c t r a t i o , i t was found t h a t the a s p e c t r a t i o would i n c r e a s e t o 5 i f r e l a t i v e r a t i o of p i l e s t i f f n e s s t o s o i l s t i f f n e s s i s reduced from 500 t o 100. And t h e a c c u r a c y of p r e d i c t i n g the u l t i m a t e r e s i s t a n c e i s m a r g i n a l l y i n c r e a s e d , as shown i n F i g . 8 . 6 ( b ) . T h i s r e s u l t i n d i c a t e s t h a t the r e l a t i v e r a t i o of the p i l e element s t i f f n e s s t o the s o i l element s t i f f n e s s has some i n f l u e n c e s on the c h o i c e of i n t e r f a c e element t h i c k n e s s . L a r g e r d i f f e r e n c e between the s o i l and p i l e element s t i f f n e s s w i l l p r o h i b i t the use of v e r y t h i n i n t e r f a c e e l e m e n t s , maybe due t o the r o u n d - o f f e r r o r and the i l l - c o n d i t i o n of the g l o b a l s t i f f n e s s m a t r i x . Cook (1981) s u g g e s t e d t h a t i n o r d e r t o have a good s o l u t i o n of s t r e s s u s i n g t r i a n g u l a r e l e m e n t s , t h e a s p e c t r a t i o of the element be e m p i r i c a l l y i n the range of 3-4. The l o a d s h e d d i n g i t e r a t i o n p r o c e d u r e i s based on a s t r e s s b a s i s . V e ry t h i n t r i a n g u l a r element may g e n e r a t e e r r a t i c s t r e s s s o l u t i o n e s p e c i a l l y i n complex pr o b l e m s , and o 2:0 ^ 00 o - -© L / t = 12 -+ L/t=5 L/t=3 -x L/t=2 V 1 ~~ ( P i l e = l O O " S o i l ) 1 1 1 1 1 1 1 1 r — 1 6.0 12.0 18.0 24.0 30.0 36 0 42 0 LATERAL DISPLACEMENT - Y (MM) CU=7.5KPA D lAM 1 1 48.0 =0.6M, (X: 54.0 = 0 . 5 60 .0 (b) F i g . 8.6 E f f e c t s of I n t e r f a c e Element L / t r a t i o 242 t h e r e f o r e engender n u m e r i c a l problems i n the convergence of l o a d s h e d d i n g i t e r a t i o n . Based on the above p a r a m e t r i c s t u d i e s , an a s p e c t r a t i o of 3 was s e l e c t e d f o r the t h i c k n e s s of the i n t e r f a c e element i n a l l the f o l l o w i n g a n a l y s e s . T h i s s e l e c t i o n i s g e n e r a l l y a g r e e a b l e t o the r u l e - o f - t h u m b f o r s e l e c t i n g the a s p e c t r a t i o of t r i a n g u l a r elements s u g g e s t e d by Cook. A l t h o u g h the i n t e r f a c e elements employed a r e not v e r y t h i n ' i n terms of a s p e c t r a t i o , the above and f o l l o w i n g s t u d i e s t e n d t o i l l u s t r a t e t h a t the a s p e c t r a t i o of the elements may not be i m p o r t a n t i n the s i m u l a t i o n . The r e a l l y i m p o r t a n t f a c t o r i s the f a c t t h a t the s o i l e lements next t o p i l e elements s h o u l d be r e l a t i v e l y s m a l l so as t o r e p r e s e n t the s o i l d e f o r m a t i o m mechanism c l o s e l y . The f o r e g o i n g and f o l l o w i n g a n a l y s e s show t h a t good agreements w i t h c l o s e d form s o l u t i o n under v a r i o u s i n t e r f a c e p r o p e r t i e s can be a c h i e v e d w i t h s a t i s f a c t o r y a c c u r a c y , when the i n t e r f a c e elements w i t h a s p e c t r a t i o of 3 a r e employed i n the a n a l y s e s . In g e n e r a l , the e r r o r i s o n l y about 7%. 8.5 COHESIVE SOILS The a n a l y s e s a r e based on the u n d r a i n e d t o t a l s t r e s s a p p r o a c h , as d i s c u s s e d i n Chapter 3. In t h e a n a l y s i s , t h r e e t y p e s of s o i l models a r e employed f o r t h e c o h e s i v e s o i l s , namely: 1. B i l i n e a r e l a s t i c p l a s t i c model, 2. B i l i n e a r e l a s t i c p l a s t i c w i t h t e n s i o n c u t - o f f model, 243 3. N o n l i n e a r h y p e r b o l i c w i t h t e n s i o n c u t - o f f model. The c o n s t i t u t i v e law of the f i r s t model i s e q u i v a l e n t t o the c l a s s i c e l a s t i c i t y and p l a s t i c i t y t h e o r i e s . The purpose of u s i n g t h i s model i s t o o b t a i n f i n i t e element s o l u t i o n s which a r e comparable t o the c l o s e d form s o l u t i o n s . The second and t h i r d models i n c o r p o r a t e the t e n s i l e f a i l u r e of s o i l s , t h e r e f o r e , by comparing the r e s u l t s w i t h the f i r s t model, the i n f l u e n c e of the s o i l c r a c k i n g and s o i l - p i l e s e p a r a t i o n on the p r e d i c t e d P-Y c u r v e s can be e v a l u a t e d f o r the c o h e s i v e s o i l s . 8.5.1 SOIL PROPERTIES  S o i l Elements In p r e s e n t a n a l y s i s , a r e l a t i v e l y l o n g and f l e x i b l e p i l e i n s t a l l e d i n a n o r m a l l y c o n s o l i d a t e d c l a y was c o n s i d e r e d . The n o r m a l l y c o n s o l i d a t e d c l a y was assumed f u l l y s a t u r a t e d w i t h a u n i t weight of 7 . = 16 KN/m3, and an S3. L u n d r a i n e d shear s t r e n g t h which i n c r e a s e s w i t h the e f f e c t i v e o v e r b u r d e n s t r e s s as f o l l o w s : C u = 0.25 a; (8.5.1) i n w h i c h i s e f f e c t i v e o v e r b u r d e n p r e s s u r e a t the d e p t h . For t h e a n a l y s i s of u n d r a i n e d c o h e s i v e s o i l , the e l a s t i c d e f o r m a t i o n modulus a r e o f t e n r e l a t e d t o t h e u n d r a i n e d shear s t r e n g t h i n terms of modulus number, M, i . e . 244 E = M C u (8.5.2) A g a i n , a v a l u e of 800 was used i n t h e a n a l y s i s . W i t h the g i v e n Young's modulus the b u l k modulus can be d e t e r m i n e d under c e r t a i n assumed v a l u e s of P o i s s o n ' s r a t i o . The e l a s t i c Young's and b u l k modulus number, K E and K g employed i n the program a r e then b a c k - c a l c u l a t e d from Eq. (4.1.5) and (4.1.6) . Under the u n d r a i n e d c o n d i t i o n , t h e i n i t i a l e l a s t i c m o d u l i a r e assumed independent of t h e t o t a l s t r e s s l e v e l , hence the above modulus exponents, m and n a r e e q u a l t o z e r o , i . e . m = n = 0. So modulus numbers a r e d e t e r m i n e d as f o l l o w s : where P a i s the atmosphere p r e s s u r e = 101.33 KPa. The s o i l p arameters employed i n the a n a l y s i s a r e p r e s e n t e d i n T a b l e 8.2. P i l e Elements In t h e a n a l y s i s , the p i l e e l ements a r e b a s i c a l l y t r e a t e d as s o i l elements but p o s s e s s i n g s t i f f n e s s and s t r e n g t h of 500 t i m e s t h o s e f o r s o i l e l e ments t o l i m i t t h e d e f o r m a t i o n and f a i l u r e i n the p i l e e l e m e n t s . In g e n e r a l , K E = E / P a (8.5.3) K B - B / P a (8.5.4) 245 T a b l e 8.2 Parameters of E l a s t o - p l a s t i c S o i l , P i l e and I n t e f a c e Elements i n the F i n i t e Element A n a l y s e s of P-Y Curves S o i l S o i l I n t e r f a c e P i l e P arameters Elements Elements Elements C u (Kpa) 7.5 7.5 a 500 x 7.5 <t> (°) K E 59.21 59.21 500 x 59.21 KB 9869.0 9869.0 500 x 9869.0 n 0.0 0.0 0.0 m 0.0 0.0 0.0 Rf 0.0 0.0 0.0 V 0.499 0.499 0.499 o0 (Kpa) 80.0 80.0 80.0 1 . The p i l e elements a r e 500 t i m e s s t r o n g e r i n s t i f f n e s s and s t r e n g t h than s o i l e l e m e n t s ; 2. a i s the s o i l - p i l e i n t e r f a c e a d h e s i o n f a c t o r . t h e c o n c r e t e p i l e i s r e g a r d e d as e l a s t i c m a t e r i a l i n a l l the a n a l y s e s . I t s parameters a r e a l s o p r e s e n t e d i n T a b l e 8.2. I n t e r f a c e Elements As d i s c u s s e d i n Chapter 7, the proposed t h i n i n t e r f a c e e l ements f o r the s o i l - p i l e i n t e r f a c e a r e b a s i c a l l y f o r m u l a t e d i n the same way as the s u r r o u n d i n g s o i l e l e m e n t s . T h e r e f o r e , they g e n e r a l l y f o l l o w the same s t r e s s - d e f o r m a t i o n c h a r a c t e r i s t i c s as the s o i l elements b e f o r e t h e i r i n t e r f a c e s t r e n g t h s a r e reached. The i n t e r f a c e s t r e n g t h s were r e l a t e d t o t h e a d j a c e n t s o i l s t r e n g t h i n terms of a d h e s i o n f a c t o r , a, and f r i c t i o n a l a n g l e f a c t o r /3. The e x p e r i m e n t a l v a l u e s of a and 0 r e p o r t e d by Potyond (1961) were used i n the a n a l y s i s (see T a b l e 7 . 1 ) . The i n t e r f a c e p r o p e r t i e s a r e a l s o p r e s e n t e d 246 i n T a b l e 8.2. 8.5.2 RESULTS AND DISCUSSIONS F a c t o r s a f f e c t i n g the p r e d i c t i o n of P-Y c u r v e s from f i n i t e element method were examined s e p a r a t e l y i n the a n a l y s i s . The p r e d i c t e d P-Y c u r v e s from f i n i t e element a n a l y s e s u s i n g the f i r s t model a r e p r e s e n t e d i n F i g . 8.7 f o r v a r i o u s v a l u e s of the i n t e r f a c e a d h e s i o n f a c t o r , a. The i n t e r f a c e a d h e s i o n f a c t o r s r e p r e s e n t t h e d i f f e r e n t roughness of the p i l e s u r f a c e , v a r y i n g from th e p e r f e c t smoothness of a = 0.0 t o the p e r f e c t roughness of a = 1.0. 1) I n i t i a l s l o p e s of the P-Y c u r v e s The i n i t i a l s l o p e s of t h e p r e d i c t e d P-Y c u r v e s under v a r i o u s a a r e compared w i t h B a r d e t ' s e l a s t i c c l o s e d form s o l u t i o n (see S e c t i o n 8.3). R e s u l t s of two e l a s t i c p l a s t i c f i n i t e element a n a l y s e s and c l o s e d form s o l u t i o n a r e t a b u l a t e d i n T a b l e 8.3. In t h e c l o s e d form s o l u t i o n , the o u t e r boundary r a d i u s of 100 p i l e r a d i u s was used, which i s the same as i n t h e f i n i t e element a n a l y s e s . I n i t i a l s l o p e s of the p r e d i c t e d P-Y c u r v e s a r e c a l c u l a t e d a t the s t r a i n l e v e l of 0.2% ( A y / a ) , where a i s the p i l e r a d i u s . I n i t i a l s l o p e a t such a s t r a i n l e v e l i s c o r r e s p o n d e n t t o the i n i t i a l subgrade modulus. From the comparison of t h e r e s u l t s from the f i r s t two models, i t i s shown t h a t t h e i n i t i a l s l o p e s of the P-Y o o. I © — 1 — © 1 1 0 — I —e> €) ADHESION COEFF.a=l.O +- ADHESION COEFF.a=0.5 • ADHESION COEFF.a=0.0 -€) i i i i i i i 1 1 1 — n 1 1 1 1— 0.0 6.0 12.0 18.0 24.0 30.0 36.0 42.P 48.0 DISPLACEMENT Y - MM,D-0 .6M,C U=7 .5KPA, <2=C«/CU F i g . 8.7 F i n i t e Element P r e d i c t i o n o f p-Y Curves u s i n g Model (1) w i t h V a r i o u s o 54.0 60.0 ro 248 T a b l e 8.3 I n i t i a l S l o p e s of P r e d i c t e d P-Y Curves under V a r i o u s Adhesion F a c t o r Adhesion F a c t o r I s o t r o p i c Model T e n s i o n C u t - o f f Model a 0.0 0.5 0.8 1 .0 1.1615 E 1 .2686 E 1 .2686 E 1 .2686 E 1 . 1468 E 1.2663 E 1.2663 E 1.2663 E 1. I n i t i a l S l o p e of P-Y c u r v e from c l o s e d form s o l u t i o n = 1.162 E, where E i s the Young's modulus of s o i l c u r v e s a r e independent of the t e n s i o n b e h a v i o r of s o i l s . That i s t o be e x p e c t e d , as a t depth the h i g h i n - s i t u c o n f i n i n g s t r e s s i n s o i l mass w i l l p r e v e n t the f o r m a t i o n of t e n s i l e s t r e s s a t s m a l l s t r a i n l e v e l . From T a b l e 8.3, i t seems t h a t except f o r v e r y smooth p i l e s , the i n i t i a l s l o p e s of the P-Y c u r v e s a r e g e n e r a l l y i n s e n s i t i v e t o t h e s o i l - p i l e i n t e r f a c e b e h a v i o r . T h i s i s because a t v e r y s m a l l s t r a i n l e v e l , the i n t e r f a c e elements have not r e a c h e d t h e i r f u l l s t r e n g t h s , they s t i l l f o l l o w t h e same s t r e n g t h c h a r a c t e r i s t i c s of the s u r r o u n d i n g s o i l e l e m e n t s . The d i s t i n c t f e a t u r e s of s t r e n g t h and d e f o r m a t i o n of the i n t e r f a c e l a y e r w i l l o n l y come i n t o e f f e c t s when the i n t e r f a c e s t r e n g t h has been f u l l y m o b i l i z e d , and t h e r e l a t i v e s l i p and c a v i t y of s o i l - p i l e i n t e r f a c e have o c c u r r e d . S i m i l a r o b s e r v a t i o n s have a l s o been r e p o r t e d by Y e g i a n and W r i g h t (1973) where t h e y used c y l i n d r i c a l j o i n t medium. 249 i n t e r f a c e element w i t h z e r o t h i c k n e s s t o s i m u l a t e t h e s o i l - p i l e i n t e r f a c e b e h a v i o r . By v a r y i n g the c o e f f i c i e n t a, they found t h a t the i n t e r f a c e p r o p e r t i e s a f f e c t the p r e d i c t e d u l t i m a t e p r e s s u r e s , but do not a f f e c t the i n i t i a l s l o p e s of P-Y c u r v e s , e x c e p t f o r the case of a e q u a l t o z e r o . For the sake of c o m p a r i s o n , t h e i r r e s u l t s a r e shown i n F i g . 8.8. In view of t h i s , the proposed i n t e r f a c e element i s a l s o a b l e t o p r e d i c t t h e same t r e n d s as the z e r o t h i c k n e s s j o i n t i n t e r f a c e e l e m e n t s , a l t h o u g h t h e r e i s a s i g n i f i c a n t d i f f e r e n c e i n the c h a r a c t e r i s t i c s w i t h t h e s e two t y p e s of e l e m e n t s . As shown i n T a b l e 8.3, from the comparison of i s o t r o p i c model r e s u l t s w i t h c l o s e d form s o l u t i o n , the b e s t agreement was o b t a i n e d i n the case of a e q u a l t o z e r o , t h a t i s , f o r the p e r f e c t smooth p i l e . For o t h e r v a l u e s of a, t h a t i s , f o r rough p i l e s , a l l f i n i t e element a n a l y s e s o v e r p r e d i c t t h e t h e o r e t i c a l v a l u e of 1.162E w i t h an e r r o r about 9%. The o v e r - p r e d i c t i o n of f i n i t e element method i s normal f o r the c o n f o r m i n g i n c r e m e n t a l n o n l i n e a r f i n i t e element f o r m u l a t i o n , as d i s c u s s e d i n Chapter 3. 2) U l t i m a t e s o i l r e s i s t a n c e s The u l t i m a t e s o i l r e s i s t a n c e s of a c i r c u l a r p i l e p r e d i c t e d from the f i r s t two models a r e p r e s e n t e d i n T a b l e 8.4 under d i f f e r e n t i n t e r f a c e a d h e s i o n f a c t o r s . The c o r r e s p o n d i n g v a l u e s from p l a s t i c i t y t h e o r y a r e a l s o i n c l u d e d i n the t a b l e . As shown i n the t a b l e , u s i n g the 200 y (Inches) 8.a I n f l u e n c e s of I n t e r f a c e B e h a v i o r ( a f t e r Y egian and W r i g h t , 1973) 251 T a b l e 8.4 U l t i m a t e S o i l R e s i s t a n c e ( P u i t / c u D ^ °^ P r e d i c t e d P-Y Curves A d h e s i o n F a c t o r I s o t r o p i c T e n s i o n C u t - o f f C l o s e d a Model Model Form 0.0 9.333 8.444 9.142 0.5 11.554 9.778 10.820 0.8 12.444 10.666 11.563 1.0 12.889 11.556 11.940 1. The c l o s e d form s o l u t i o n i s a f t e r Randolph and Houlsby (1984) shown i n T a b l e 8.1. f i r s t model, i . e . b i l i n e a r e l a s t i c p l a s t i c model, t h e f i n i t e element method p r e d i c t s the t h e o r e t i c a l l i m i t p r e s s u r e w i t h a good a c c u r a c y i n a l l c a s e s . G e n e r a l l y , the o v e r p r e d i c t i o n i s about 7%. T h i s e r r o r may be a t t r i b u t e d t o s e v e r a l r e a s o n s . F i r s t of a l l , due t o the i n c r e m e n t a l n a t u r e of the n o n l i n e a r a n a l y s i s , the c o n f o r m i n g f i n i t e element a n a l y s i s i s bound t o g i v e s t i f f e r r e s u l t s than the c l o s e d form s o l u t i o n (Cook, 1981). S e c o n d l y , t h e e r r o r r e p o r t e d here l i e s i n t h e c a p a b i l i t y of the l i n e a r s t r a i n t r i a n g u l a r element i n p r e d i c t i o n of p l a s t i c u l t i m a t e l o a d s . S l o a n and Randolph (1982) r e p o r t e d a s i m i l a r range of a c c u r a c y o b t a i n e d when u s i n g t h i s type of element t o p r e d i c t the c o l l a p s e l o a d s f o r s t r i p f o o t i n g . They s u g g e s t e d t h a t h i g h e r o r d e r elements be used t o o b t a i n b e t t e r r e s u l t s . F i n a l l y , as mentioned b e f o r e , the s i z e of f i n i t e element mesh may a l s o a f f e c t the a c c u r a c y , e s p e c i a l l y f o r 252 t h o s e e l e m e n t s n e x t t o t h e p i l e e l e m e n t s . As t h e e l e m e n t s c l o s e t o t h e p i l e s e c t i o n a r e r e f i n e d , a c l o s e r p r e d i c t i o n o f l i m i t p r e s s u r e w o u l d be e x p e c t e d ( S l o a n a n d R a n d o l p h , 1 9 8 2 ) . In v i e w o f t h e a b o v e c o n s i d e r a t i o n s , t h e f i n i t e e l e m e n t p r o c e d u r e a n d t h e i n t e r f a c e e l e m e n t e m p l o y e d h e r e a r e c a p a b l e o f p r o v i d i n g s a t i s f a c t o r y r e s u l t s . As f o r t h e i n f l u e n c e s o f i n t e r f a c e b e h a v i o r , t h e r e s u l t s shown i n i n F i g . 8 .7 c l e a r l y i n d i c a t e t h a t t h e p r e d i c t e d u l t i m a t e r e s i s t a n c e i s s i g n i f i c a n t l y a f f e c t e d by t h e s o i l - p i l e i n t e r f a c e p r o p e r t i e s . S u c h an o b s e r v a t i o n i s c o n s i s t e n t w i t h t h e c l o s e d f o r m s o l u t i o n ( R a n d o l p h and H o u s l b y , 1 9 8 4 ) , a n d t h e f i n i t e e l e m e n t r e s u l t s f r o m p r e v i o u s r e s e a r c h e r s u s i n g j o i n t i n t e r f a c e e l e m e n t s ( Y e g i a n and W r i g h t , 1 9 7 3 ) . In t h i s c o n t e x t , t h e p r o p o s e d t h i n i n t e r f a c e e l e m e n t i s a d e q u a t e f o r a p p r o x i m a t e l y s i m u l a t i n g s o i l - p i l e i n t e r f a c e b e h a v i o r . As f o r t h e e f f e c t s o f s o i l t e n s i l e f a i l u r e and s o i l - p i l e s e p a r a t i o n , f i n i t e e l e m e n t a n a l y s e s u s i n g t e n s i o n c u t - o f f m o d e l were p e r f o r m e d . R e s u l t s a r e c o m p a r e d w i t h t h o s e f r o m i s o t r o p i c m o d e l , a s shown i n F i g . 8 . 9 ( a ) , ( b ) , ( c ) , ( d ) , a n d T a b l e 8 . 4 . As shown i n t h e f i g u r e , t h e i n c o r p o r a t i o n o f t e n s i l e f a i l u r e o f s o i l s a n d t h e s o i l - p i l e s e p e r a t i o n b e h i n d t h e p i l e m a r k e d l y r e d u c e s t h e p r e d i c t e d u l t i m a t e s o i l r e s i s t a n c e s u n d e r a l l t h e i n t e r f a c e a d h e s i o n f a c t o r s c o n s i d e r e d h e r e i n . As shown i n T a b l e 8 . 4 , t h e d i f f e r e n c e s 253 -0 I s o t r o p i c Model -+ Tension C u t - o f f Model -O Adhesion F a c t o r a - 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 L A T E R A L D I S P L A C E M E N T S Y - MM (DIAMETER-0.6M. DEPTH=5M) 40.0 €) I s o t r o p i c Model + Tension C u t - o f f Model (b) -O Adhesion F a c t o r a • 0.5 I I I I I I I I 1 1 1 1 1 T 0.0 6.0 12 0. 18.0 24.0 30.0 36.0 42 0 D I S P L A C E M E N T Y - MM D=0.6M DEPTH-5M i r 48.0 -1 r 54.0 60.0 F i g . 8.9 P-* Curves from F i n i t e Element P r e d i c t i o n ( I s o t r o p i c model VB Tension C u t - o f f Model) 254 F i g . 8.9 P-Y Curves from F i n i t e Element P r e d i c t i o n ( I s o t r o p i c model vs T e n s i o n C u t - o f f Model) 255 b e t w e e n t h e two f i n i t e e l e m e n t p r e d i c t i o n s a r e a l l l a r g e r t h a n 10%. F rom t h e t a b l e , i t i s a l s o n o t e d t h a t t h e d i f f e r e n c e becomes l a r g e r a s t h e v a l u e o f a d h e s i o n f a c t o r a becomes s m a l l e r . T h i s may i n d i c a t e t h a t t h e s i m u l a t i o n o f t e n s i l e f a i l u r e o f s o i l s a n d t h e s o i l - p i l e s e p a r a t i o n becomes more i m p o r t a n t when t h e s m o o t h p i l e s a r e t o be a n a l y s e d . T h i s r e s u l t seems r e a s o n a b l e , a s a smooth p i l e i s more l i k e l y t o u n d e r g o t h e s o i l - p i l e s e p a r a t i o n b e h i n d t h e p i l e t h a n a r o u g h p i l e u n d e r t h e same l o a d i n g c o n d i t i o n . B a s e d on t h e a b o v e r e s u l t s , i n o r d e r t o have r e a s o n a b l e p r e d i c t i o n o f u l t i m a t e s o i l r e s i s t a n c e , a c c u r a t e s i m u l a t i o n o f s o i l - p i l e i n t e r f a c e b e h a v i o r ( i n c l u d i n g t h e s o i l - p i l e s e p a r a t i o n ) i s n e c e s s a r y . 3) S o i l S t r e s s D i s t r i b u t i o n The l a t e r a l p i l e movement w i l l i n t r o d u c e t h e s t r e s s c h a n g e s w i t h i n s o i l m a s s e s . The s t r e s s d i s t r i b u t i o n p r e d i c t e d f r o m f i n i t e e l e m e n t a n a l y s i s u s i n g b i l i n e a r e l a s t i c p l a s t i c p e r f e c t s o i l - p i l e a d h e s i o n m o d e l a r e p l o t t e d i n F i g . 8 . 1 0 ( a ) , ( b ) , ( c ) . The r e s u l t s a r e c o m p a r e d w i t h t h e c l o s e d f o r m s o l u t i o n ( s e e E q s . ( 8 . 3 . 8 ) - ( 8 . 3 . 1 0 ) i n S e c t . 8 . 3 ) ) . As shown i n t h e s e f i g u r e s , t h e f i n i t e e l e m e n t p r e d i c t i o n s a r e a l l i n good a g r e e m e n t w i t h c l o s e d f o r m s o l u t i o n . T h i s i n d i c a t e s t h a t t h e f i n i t e e l e m e n t p r o g r a m f u n c t i o n s w e l l i n t h e a n a l y s i s o f l a t e r a l l y l o a d e d p i l e s . 256 — c l o s e d form + f i n i t e element (a) — i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r -0.0 8.0 16.0 24.0 32.0 10.0 48.0 56.0 64.0 72.0 80.0 cr a. * -I d p ' in L U i — L O O — C l o s e d form + F i n i t e element (b) a„ = > T I I 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 "j— . 0 0 8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 80.0 cr CLin cr C O o ' in L U o _ | CU ' p— «"> CCm (X • U J 7 ' X cn C l o s e d Form + F i n i t e element (c) —i r 16.0 24.0 32.0 40.0 48.0 56.0 F i g . 8.10 S o i l S t r e s s D i s t r i b u t i o n —i r ~ i r 40.0 xy 8.0 64.0 72.0 Of.) 257 4) N o n l i n e a r S o i l Response In r e a l i t y , most c o h e s i v e s o i l s a p p e a r n o n l i n e a r s t r e s s - s t r a i n b e h a v i o r r a t h e r t h a n t h e b i l i n e a r e l a s t i c p l a s t i c b e h a v i o r a s sumed i n t h e f o r e g o i n g a n a l y s e s . K o n d e r ( 1 9 6 3 ) , Duncan a n d Chang (1970) showed t h a t most c o h e s i v e s o i l s i n t r i a x i a l t e s t c o n d i t i o n s f o l l o w h y p e r b o l i c s t r e s s s t r a i n r e l a t i o n s h i p . M o r e o v e r , a s shown i n C h a p t e r 6, p r e d i c t i o n s o f c a v i t y e x p a n s i o n u s i n g h y p e r b o l i c s t r e s s s t r a ' i n r e l a t i o n s a g r e e w e l l w i t h f i e l d o b s e r v a t i o n s . T h e r e f o r e , i t i s r e a s o n a b l e t o p r e d i c t t h e l a t e r a l l y l o a d e d p i l e r e s p o n s e s u s i n g t h e h y p e r b o l i c 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 o r c o h e s i v e s o i l s . F o r t h i s r e a s o n , f i n i t e e l e m e n t a n a l y s e s u s i n g n o n l i n e a r h y p e r b o l i c s t r e s s s t r a i n c u r v e were p e r f o r m e d t o c o m p u t e t h e P-Y c u r v e f o r n o n l i n e a r s o i l s . The s o i l p a r a m e t e r s r e q u i r e d f o r t h e h y p e r b o l i c m o d e l a r e shown i n T a b l e 8 . 5 . The p r e d i c t e d P-Y c u r v e f r o m f i n i t e e l e m e n t a n a l y s i s u s i n g n o n l i n e a r s t r e s s - s t r a i n c u r v e w i t h t e n s i o n c u t - o f f m o d e l i s shown i n F i g . 8 . 1 1 . F o r t h e s a k e o f c o m p a r i s o n , t h e P-Y c u r v e f r o m b i l i n e a r e l a s t i c p l a s t i c s t r e s s - s t r a i n c u r v e w i t h t e n s i o n c u t - o f f m o d e l i s a l s o i n c l u d e d i n t h e f i g u r e . I t i s shown i n t h e f i g u r e t h a t t h e shape o f P-Y c u r v e f o r t h e n o n l i n e a r c o h e s i v e s o i l s i s g e n e r a l l y s o f t e r t h a n t h a t f o r t h e b i l i n e a r e l a s t i c p l a s t i c s o i l s . H o w e v e r , I t i s o b s e r v e d t h a t t h e i n i t i a l s l o p e a n d t h e u l t i m a t e s o i l r e s i s t a n c e o f t h e P-Y c u r v e a r e n o t a f f e c t e d by t h e T a b l e 8.5 N o n l i n e a r S o i l Parameters f o r FE A n a l y s e s of P-Y Curves Parameters C o h e s i v e S o i l C o h e s i o n l e s s S o i l C u (Kpa) <p ( ° ) ( ° ) 7.5 41° 4° 33° 1500 0cv ( ° ) K E K B 59.21 9869.0 900 Rf a0 (Kpa) m n 0.0 0.0 0.9 80.0 0.5 0.25 0.9 50.0 1. the p i l e elements a r e 500 t i m e s s t r o n g e r i n s t i f f n e s s and s t r e n g t h than s o i l e l e m e n t s ; 2. the s o i l - p i l e i n t e r f a c e a d h e s i o n f a c t o r a i s assumed t o be 0.5. n o n l i n e a r i t y of the s o i l s . T h i s r e s u l t has i t s p r a c t i c a l s i g n i f i c a n c e , i n d i c a t i n g t h a t the P-Y c u r v e s f o r the n o n l i n e a r c o h e s i v e s o i l s can be c o n s t r u c t e d s a t i s f a c t o r i l y by s i m p l y c o n n e c t i n g the i n i t i a l s l o p e and the u l t i m a t e s o i l r e s i s t a n c e u s i n g a smooth c u r v e . The i n i t i a l s l o p e can be e s t i m a t e d from Young's modulus of the s o i l medium a c c o r d i n g t o the p l a n e s t r a i n e l a s t i c t h e o r y , w h i l e the u l t i m a t e s o i l r e s i s t a n c e of the P-Y c u r v e can be e s t i m a t e d from the u n d r a i n e d shear s t r e n g t h , C u, a c c o r d i n g t o t h e f i n i t e element a n a l y s i s u s i n g b i l i n e a r e l a s t i c p l a s t i c s t r e s s s t r a i n c u r v e which i n c o r p o r a t e s the t e n s i o n f a i l u r e of s o i l s , such as the v a l u e s shown i n T a b l e 8.4. o CM Q CJ \ I t o -LU Cd O o -r\j _ © © ELASTO-PLAST IC CLAY H K NONLINEAR CLAY 0.0 1.0 2.0 3.0 4.0 5.0 NORMALIZED LATERAL DEFLECTION T T T 6.0 7.0 8.0 9 . 0 1 0 Y/D ( C U = 7 . 5 K P A . D = 0 . 6 M ) ( « i o - 2 ) F i g . 8.11 P-Y curves f o r E l a s t i c P l a s t i c S o i l and N o n l i n e a r S o i l 260 The s i m p l e s t c u r v e d e s c r i b i n g s u c h k i n d o f P -Y c u r v e f o r n o n l i n e a r c o h e s i v e s o i l s may be t h e same h y p e r b o l i c f u n c t i o n a s u s e d f o r t h e 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 . e . P / ( C u D > - a JTil/U) ( 8 ' 5 - 7 ' where a = C u / K b = C u D / P u l t D i s t h e p i l e d i a m e t e r , K i s t h e i n i t i a l s l o p e o f t h e P-Y c u r v e , w h i c h i s c l o s e t o t h e Y o u n g ' s m o d u l u s o f t h e s o i l m e d i u m . H y p e r b o l i c c u r v e f i t t i n g f o r t h e a b o v e P-Y c u r v e f o r n o n l i n e a r c o h e s i v e s o i l s u s i n g E q . (8.5.7) i s shown i n F i g . 8 . 1 2 . As i l l u s t r a t e d i n t h e f i g u r e , t h e s i m p l e h y p e r b o l i c f u n c t i o n c a n be u s e d t o r e p r e s e n t t h e P -Y c u r v e s f o r n o n l i n e a r c o h e s i v e s o i l s . H o w e v e r , t h e a b o v e s t a t e m e n t s and t h e p r o p o s e d c o n n e c t i o n f u n c t i o n a r e s u b j e c t t o f u r t h e r v e r i f i c a t i o n . T h e b e s t p r o o f o f t h e a b o v e r e s u l t s w o u l d be t h e c o m p a r i s o n w i t h t h e f i e l d t e s t d a t a . T h e r e f o r e , f u r t h e r r e s e a r c h i s w a r r a n t e d . 8 .6 COHESIONLESS SOILS S i m i l a r f i n i t e e l e m e n t a n a l y s e s were a l s o p e r f o r m e d f o r t h e c o h e s i o n l e s s s o i l s . The s t r e n g t h c h a r a c t e r i s t i c s o f t h e s o i l s a r e c o n s i d e r e d t o be g o v e r n e d by M o h r - c o u l o m b c r i t e r i o n . Due t o l a c k o f c l o s e d f o r m s o l u t i o n f o r F i g . 8 .12 Hyperbolic Curve F i t t i n g f o r P-Y curves in Cohesive S o i l s 262 c o h e s i o n l e s s s o i l , o n l y n o n l i n e a r a n a l y s e s were p e r f o r m e d . S t u d i e s w i l l be l i m i t e d on the o u t e r boundary e f f e c t and the i n t e r f a c e p r o p e r t y e f f e c t on the P-Y c u r v e s . The n o n l i n e a r s o i l p r o p e r t i e s used i n the a n a l y s e s a r e a l s o i n c l u d e d i n T a b l e 8.5. 1) Outer Boundary E f f e c t s As f o r the o u t e r boundary e f f e c t s , f i n i t e element a n a l y s e s w i t h o u t e r boundary r a d i u s of 25D and 50D were performed, where D i s the p i l e d i a m e t e r . Dense sand w i t h r e l a t i v e d e n s i t y of 75% was assumed i n the a n a l y s e s . The r e s u l t s a r e shown i n F i g . 8.13. I t i s shown i n t h e f i g u r e t h a t the s i z e of o u t e r boundary of the mesh domain does not s i g n i f i c a n t l y a f f e c t t h e i n i t i a l p o r t i o n of the p r e d i c t e d P-Y c u r v e s f o r c o h e s i o n l e s s s o i l s . However, the f i n i t e element mesh domain does s i g n i f i c a n t l y a f f e c t the p r e d i c t e d c u r v e i n the l a r g e d i s p l a c e m e n t range. S m a l l e r s i z e of the mesh domain r e s u l t s i n a s t i f f e r r e s p o n s e . The l a r g e s t d i f f e r e n c e o b s e r v e d i n the p r e d i c t i o n of d i s p l a c e m e n t i s about 15% f o r the mesh domain v a r y i n g from 25D t o 50D. S i m i l a r r e s u l t s were r e p o r t e d by She (1986). T h e r e f o r e the above r e s u l t s a r e not l i k e l y due t o n u m e r i c a l e r r o r but due t o the s t r e n g t h c h a r a c t e r i s t i c s of c o h e s i o n l e s s s o i l s . 2) I n t e r f a c e Property E f f e c t s As f o r the i n t e r f a c e p r o p e r t y e f f e c t s , F i g . 8.14 shows the p r e d i c t e d P-Y c u r v e s f o r c o h e s i o n l e s s s o i l w i t h 264 i n t e r f a c e f r i c t i o n angle f a c t o r , 0 = 0.5, 0.8, 1.0. The range of /3 used i s g e n e r a l l y correspondent to the concrete or wood p i l e i n dry and s a t u r a t e d sands (see Table 7.1). As shown i n F i g . 8.14, the p r e d i c t e d P-Y curves are g e n e r a l l y not a f f e c t e d by the i n t e r f a c e p r o p e r t i e s up to the displacements of about 20% of the p i l e diameter. Beyond t h i s displacement l e v e l , the curves are only moderately a f f e c t e d by the i n t e r f a c e p r o p e r t i e s . T h e r e f o r e , i n g e n e r a l , u n l i k e the case of c o h e s i v e s o i l s , the s o i l - p i l e i n t e r f a c e p roperty has l e s s i n f l u e n c e on the p r e d i c t e d P-Y c u r v e s . 3) Simple Curve F i t t i n g As noted i n F i g . 8.13, 8.14, the p r e d i c t e d P-Y curves f o r n o n l i n e a r c o h e s i o n l e s s s o i l s appear to have no u l t i m a t e s o i l r e s i s t a n c e at l a r g e displacement. T h i s r e s u l t tends to lend supports to S c o t t ' s b i l i n e a r curve c o n s t r u c t i o n of P-Y curves f o r sands, as d i s c u s s e d i n Sec. 2.3. In f a c t , the above p r e d i c t e d P-Y curves f o r c o h e s i o n l e s s s o i l s can a l s o be r e p r e s e n t e d by a simple power f u n c t i o n , such as: P/ED = a (Y/D) b (8.5.8) where: a = 0.3798087, b = 0.54962493, 266 As shown i n F i g . 8.15, t h e power f u n c t i o n Eq (8.5.8) can p r o v i d e a r e a s o n a b l e f i t w i t h t h e f i n i t e element p r e d i c t e d P-Y c u r v e s f o r the n o n l i n e a r c o h e s i o n l e s s s o i l s . However the proposed s i m p l e power f u n c t i o n f o r the P-Y c u r v e d e s e r v e s f u r t h e r v e r i f i c a t i o n by p r e d i c t i n g the f i e l d l a t e r a l l o a d t e s t s . F i g . 8.15 Curve F i t t i n g f o r P-Y c u r v e s i n C o h e s i o n l e s s S o i l s 9. INSTALLATION EFFECTS ON PRESSUREMETER CURVES AND P~Y  CURVES FOR LATERALLY LOADED PILES IN COHESIVE SOILS 9.1 INTRODUCTION I t i s a w e l l known f a c t t h a t the p i l e i n s t a l l a t i o n d i s t u r b s the s o i l s around t h e p i l e and c r e a t e s a zone of s o i l s t h a t have d i f f e r e n t s o i l p r o p e r t i e s from the n a t u r a l s o i l d e p o s i t . T h i s zone of d i s t u r b e d s o i l s has s i g n i f i c a n t i n f l u e n c e on the p i l e r e s p o n s e s , i n c l u d i n g the response t o l a t e r a l l o a d i n g s . At p r e s e n t , many a n a l y t i c a l methods have been proposed f o r the a n a l y s i s of l a t e r a l l y l o a d e d p i l e s ( P o u l o s , 1971, B a n e r j e e and D r i s c o l l , 1976). However, they g e n e r a l l y assume a homogeneous s o i l d e p o s i t around the p i l e s h a f t , the s o i l d i s t u r b a n c e due t o the p i l e i n s t a l l a t i o n i s u s u a l l y i g n o r e d . F i n i t e element method and the subgrade r e a c t i o n method seem t o be the e a s i e s t approaches t o i n c o r p o r a t e the p i l e i n s t a l l a t i o n e f f e c t i n the a n a l y s i s . In the subgrade r e a c t i o n method, the i n c o r p o r a t i o n of p i l e i n s t a l l a t i o n i s v i a the p r o p e r development of n o n l i n e a r s o i l r e s p o n s e , i . e . P-Y c u r v e s . In t h e development of P-Y c u r v e s from the back c a l c u l a t i o n of f i e l d l a t e r a l l o a d t e s t d a t a , the p i l e i n s t a l l a t i o n e f f e c t s a r e i m p l i c i t l y i n c l u d e d i n t he d e r i v e d P-Y c u r v e s . However, such an approach i s c o s t l y and s i t e - o r i e n t e d , and i n a d d i t i o n , how the d i s t u r b e d s o i l zone around the p i l e s h a f t a f f e c t s the s o i l P-Y re s p o n s e s i s not c l e a r . 268 269 As s t a t e d e a r l i e r i n C h a p t e r 2, t h e r e a r e s e v e r a l a d v a n t a g e s i n t h e d e v e l o p m e n t o f P-Y c u r v e s f r o m a 2D f i n i t e e l e m e n t a n a l y s i s . One o f s u c h a d v a n t a g e s i s t h a t i t a l l o w s f o r a m a t e r i a l a n a l y s i s o f d i s t u r b a n c e e f f e c t s due t o p i l e i n s t a l l a t i o n . M o r e o v e r , i n t h e d e v e l o p m e n t o f P-Y c u r v e s f r o m t h e p r e s s u r e m e t e r c u r v e s , i t i s commonly b e l i e v e d t h a t t h e p r e s s s u r e m e t e r i s a k i n d o f p h y s i c a l m o d e l f o r t h e p i l e i n s t a l l a t i o n a s t h e y e x p e r i e n c e a s i m i l a r i n s t a l l a t i o n p r o c e s s . Thus i t ha s becomes a common b e l i e f t h a t t h e p r e s s u r e e x p a n s i o n c u r v e s o b t a i n e d f r o m p r e s s u r e m e t e r t e s t s c a n r e p r e s e n t t h e s o i l d i s t u r b a n c e e f f e c t s on t h e l a t e r a l l y l o a d e d p i l e s , a n d t h a t - t h e y c a n be u s e d t o o b t a i n t h e P-Y c u r v e s f o r l a t e r a l l y l o a d e d p i l e s by a s i m p l e s c a l i n g f a c t o r ( R o b e r t s o n e t a l , 1983, A t u k o r a l a a n d B y r n e , 1984, B r i a u d e t a l , 1980, 1 9 8 1 ) . H o w e v e r , i t i s o b v i o u s t h a t t h e r e i s a s i g n i f i c a n t d i f f e r e n c e i n t h e l o a d i n g m e c h a n i s m be tween t h e l a t e r a l l y l o a d e d p i l e s a n d t h e p r e s s u r e m e t e r t e s t s . In t h e f o r m e r , t h e l o a d i s a p p l i e d i n one d i r e c t i o n , w h i l e i n t h e l a t t e r , t h e p r e s s u r e i s a p p l i e d a x i s y m m e t r i c a l l y . A l t h o u g h t h e p i l e a n d t h e p r e s s u r e m e t e r e x p e r i e n c e a s i m i l a r i n s t a l l a t i o n p r o c e s s , t h e s u b s e q u e n t d i f f e r e n c e i n t h e l o a d i n g m e c h a n i s m may g e n e r a t e d i f f e r e n t s o i l r e s p o n s e s t o t h e p i l e a n d t h e p r e s s u r e m e t e r , e v e n u n d e r t h e same e x t e n t o f s o i l d i s t u r b a n c e . I t i s p a r t o f t h e p u r p o s e s o f t h i s c h a p t e r t o e x a m i n e t h i s d i f f e r e n c e . 270 In t h i s c h a p t e r , some of the a v a i l a b l e e v i d e n c e s on the s i z e and e x t e n t of s o i l d i s t u r b a n c e zone a f t e r p i l e i n s t a l l a t i o n a r e f i r s t r e v i e w e d , and a f i n i t e element a n a l y s i s i s then performed based on p l a n e s t r a i n f o r m u l a t i o n , i n which b o t h the p i l e and p r e s s u r e m e t e r i n s t a l l a t i o n e f f e c t s a r e s i m u l a t e d . 9.2 EXPERIMENTAL AND ANALYTICAL EVIDENCES ON EXTENT OF SOIL  DISTURBANCE AFTER PILE INSTALLATION For p i l e s i n s t a l l e d i n c l a y , a summary of some f i e l d i n v e s t i g a t i o n s i n t o the e x t e n t of d i s t u r b a n c e around a p i l e a f t e r i n s t a l l a t i o n has been g i v e n by De M e l l o (1969). I f p i l e s a r e d r i v e n i n t o a deep c l a y l a y e r , the u n d r a i n e d shear s t r e n g t h of the c l a y i s u s u a l l y reduced due t o the s e v e r e r e m o l d i n g a t the s o i l - p i l e i n t e r f a c e . Cummings et a l (1950) and F l a a t e (1972) r e p o r t e d t h a t the remolded zone measured i n t h e f i e l d extends from the p i l e s u r f a c e t o 1.5 - 2 . 0 p i l e d i a m e t e r . Some d i s t u r b a n c e s beyond t h i s zone a r e a l s o o b s e r v e d . O b v i o u s l y , the r e s u l t s a r e dependent upon the s o i l c o n d i t i o n s , p i l e t y p e s , and the d r i v i n g methods and p r o c e d u r e s . D r i v i n g a p i l e w i l l change the s t r e s s e s i n the ground. In c o h e s i v e s o i l s , Lo and Stermac (1965), B j e r r u m and Johannessen (1960), K o i z u m i and I t o (1967), O r r j e and Broms (1967) a l l r e p o r t e d t h a t h i g h pore p r e s s u r e s i n the range of 2 - 4 t i m e s the t o t a l o v e r b u r d e n p r e s s u r e or h i g h e r were measured near the t e s t p i l e s (up t o 2 d i a m e t e r p i l e from the 271 p i l e f a c e ) . T h e s e h i g h p o r e p r e s s u r e c a u s e d a r e d u c t i o n o f t h e e f f e c t i v e s t r e s s e s w i t h i n t h e s o i l a t t h e t i m e o f p i l e d r i v i n g , a n d r e d u c e d t h e s h e a r s t r e n g t h s u b s t a n t i a l l y . T h e s e h i g h e x c e s s p o r e p r e s s u r e s h a v e h i g h g r a d i e n t s a n d d i s s i p a t e m a i n l y r a d i a l l y w i t h t i m e . F i e l d m e a s u r e m e n t s show t h a t t h e i n d u c e d p o r e p r e s s u r e s v a r y i n a l i n e a r l o g a r i t h m f a s h i o n w i t h r a d i a l d i s t a n c e away f r o m t h e p i l e , a s shown i n F i g . 9 . 1 . The e x c e s s p o r e p r e s s u r e s d i s s i p a t e a f t e r p i l e d r i v i n g , a c c o m p a n i e d by t h e i n c r e a s e o f u n d r a i n e d s h e a r s t r e n g t h . In t h e l o n g t e r m ( a t e n d o f r e c o n s o l i d a t i o n ) , f o r p i l e s i n s t a l l e d i n s o f t c l a y s , t h e u n d r a i n e d s h e a r s t r e n g t h i s o f t e n f o u n d somewhat h i g h e r t h a n i t s i n i t i a l v a l u e i n t h e f i e l d ( F l a a t e , 1972, O r r j e a n d B roms , 1 9 6 7 ) . B a s e d on c o r r e l a t i o n s b e t w e e n p i l e - s o i l a d h e s i o n C a n d u n d r a i n e d c o h e s i o n o f t h e s o i l C , i t w o u l d a u a p p e a r t h a t C_ /C c a n be a s h i g h a s 1.5 f o r v e r y s o f t c l a y , and a s low a s 0 .2 f o r s t i f f c l a y s (De M e l l o , 1969 , B a l a a m e t a l , 1 9 7 5 ) . A few t h e o r e t i c a l s t u d i e s by C a r t e r e t a l ( 1 9 7 9 ) , R a n d o l p h a n d W r o t h ( 1 9 7 9 ) , a n d R a n d o l p h e t a l (1979) s i m u l a t e d t h e p i l e d r i v i n g p r o c e s s a s an e x p a n s i o n o f a c y l i n d r i c a l c a v i t y . R a n d o l p h e t a l r e s u l t s o f s h o r t t e r m s t r e s s s t a t e a f t e r p i l e d r i v i n g i n B o s t o n B l u e C l a y a r e shown i n F i g . 9 . 2 . The e x c e s s p o r e p r e s s u r e v a r i e s i n a l i n e a r l o g a r i t h m f a s h i o n w i t h r a d i a l d i s t a n c e . Zones o f p l a s t i c f a i l u r e and i n d u c e d p o r e p r e s s u r e s a r e d e p e n d e n t upon t h e s o i l c o n d i t i o n s . As t h e p o r e p r e s s u r e d i s s i p a t e s , 272 Uo (kN/m2) I201-eob Bjerrum & Johannessen (1961) c u~l5kN/m 2 ft~70kN/m2 — best fit straight line x field measurements 4 0 h Is Uo (kN/m2) I201-\ \ \ \ \ Koizumi & Ito (1967) c u ~30kN/m 2 0,'~6OkN/m2 4 0 h \ x \ \ \ \ Uo IkN/m2) I20h S O r \ x X \ Lo & Stermac (I96S) c u ~20 kN/m2, &i~!20 kN/m2 4 0 h \ \ x \ r = 2r0 r«5r0 r» IOr 0 r = 30r 0 In lr/r0) F i g . 9.1 F i e l d Measurements of Ex c e s s Pore P r e s s u r e R e s u l t i n g from p i l e d r i v i n g ( a f t e r Randolph and Wroth, 1979) Boundary of Critical Stat« Region OCR-8 F i g . 9.2 T y p i c a l r e s u l t s of s t r e s s d i s t r i b u t i o n immediately a f t e r p i l e d r i v i n g ( a f t e r Randolph et a l , 1979) 274 the shear s t r e n g t h of s o i l s r e g a i n s i t s v a l u e . Randolph e t a l s o l u t i o n of u n d r a i n e d shear s t r e n g t h v a r i a t i o n w i t h t i m e , and w i t h r a d i u s a t end of s o i l r e c o n s o l i d a t i o n a r e shown i n F i g . 9 . 3 ( a ) , ( b ) . The s t r e n g t h a t the end of c o n s o l i d a t i o n i s about 60% g r e a t e r than the i n i t i a l i n - s i t u v a l u e , and t h i s u l t i m a t e s t r e n g t h d e c r e a s e s w i t h t h e l o g a r i t h m of the r a d i u s u n t i l i t reaches the i n - s i t u v a l u e a t about 10 p i l e r a d i u s . There a p p e a r s t o be l e s s i n f o r m a t i o n a v a i l a b e f o r bored p i l e s i n c l a y on the e x t e n t of d i s t u r b a n c e , but some d a t a f o r bored p i l e s i n London c l a y s u g g e s t e d t h a t s o f t e n i n g of the c l a y d u r i n g i n s t a l l a t i o n of p i l e s reduces C & t o about 0.45 C u (Balaam e t a l , 1975). There i s even l e s s i n f o r m a t i o n on changes i n the d e f o r m a t i o n p a r a m e t e r s . Byrne e t a l (1984) r e p o r t e d based on the c y c l i c t r i a x i a l t e s t d a t a on p l a s t i c c l a y e y s i l t , t h a t the p o s t c y c l i c d e f o r m a t i o n modulus would reduce as low as by a f a c t o r of 10. However i n the a n a l y s i s , i t i s r e a s o n a b l e t o assume t h a t t h e r a t i o of s o i l modulus t o c o h e s i o n i s c o n s t a n t f o r t h e u n d r a i n e d t o t a l s t r e s s a n a l y s i s ( D ' A p p o l o n i a e t a l , 1971, Clough and Denby, 1977, Byrne e t a l , 1984). 9.3 FINITE ELEMENT ANALYSIS 275 16 cu[T*| C g l 0 ) 1-4 13 12 II IO OCR - I to 32 IO_s IO-4 IO° IO'* IO'1  T » = Kculo)t IO° IO1 F i g . 9.3(a) Typical v a r i a t i o n with time of undrained shear strength of s o i l at r=1.15r 0 ( a f t e r Randolph et a l , 1979) OCR » I to 32 1 cfl I I I I I I I I I I I I I I I I I I 115 2 IO IOO r/r0 F i g . 9.3(b) Typical v a r i a t i o n with radius of undrained shear strength of s o i l at end of cons o l i d a t i o n (after Randolph et a l , 1979) 276 9.3.1 DISTURBANCE SIMULATION  P i l e s i n C l a y On the b a s i s of the a v a i l a b l e e v i d e n c e s on the e f f e c t s of p i l e i n s t a l l a t i o n d i s c u s s e d p r e v i o u s l y , i t has been assumed f o r t h e f i n i t e element a n a l y s e s of p i l e s i n c l a y t h a t t h e r e e x i s t s a d i s t u r b e d zone t o r a d i u s , r^ around the p i l e , e x t e n d i n g t o a depth of r ^  below the p i l e t i p , as shown i n F i g . 9 . 4 ( a ) . The Young's modulus of t h i s d i s t u r b e d zone i s assumed t o v a r y l i n e a r l y from a v a l u e of a t the s o i l - p i l e i n t e r f a c e t o the v a l u e E of t h e i n t a c t s o i l a t the o u t e r l i m i t of the d i s t u r b e d zone. As b e f o r e , a ' d i s k ' a n a l y s i s , i . e . p l a n e s t r a i n c o n d i t i o n was p e r formed f o r the l a t e r a l l y l o a d e d p i l e s . T h e r e f o r e , a c r o s s s e c t i o n showing the a n a l y s i s model i s g i v e n i n F i g . 9.4(b). P o i s s o n ' s r a t i o of b o t h t h e u n d i s t u r b e d and d i s t u r b e d s o i l has been t a k e n t o be 0.499 t o r e p r e s e n t u n d r a i n e d l o a d i n g of the p i l e . The r a t i o of s o i l modulus t o the u n d r a i n e d shear s t r e n g t h i s assumed t o be the same f o r b o t h i n t a c t and d i s t u r b e d s o i l s , and e q u a l s 800. T h e r e f o r e , i n the d i s t u r b e d zone, the s t r e n g t h of the s o i l i s a l s o assumed t o v a r y l i n e a r l y from C , a t t h e s o i l - p i l e i n t e r f a c e t o C a t the o u t e r l i m i t of ud ^ u the d i s t u r b e d zone, as shown i n F i g . 9.5. The same i s o t r o p i c i n - s i t u s t r e s s s t a t e i s assumed f o r b o t h s o i l r e g i o n s . V a l u e s of r ^ / r 0 of 1, 2, 3, 5 have been c o n s i d e r e d , and the modulus r a t i o i n d i s t u r b e d zone, E^/E^, (or shear s t r e n g t h r a t i o C u c j / C u ) , of 0.2, 0.5, 1, 2, 5 have been a n a l y s e d so as t o c o v e r t h e e f f e c t s of i n s t a l l a t i o n f o r b o t h d r i v e n and " 7 — 7 T 7 7—T (a) A - A S e c t i i o n (b) F i g . 9.4 S i m u l a t i o n of I n s t a l l a t i o n f o r P i l e F i g . 9.5 Assumed modulus and shear s t r e n g t h v a r i a t i o n w i t h i n d i s t u r b e d zone ts) CO 279 b o r e d p i l e s . F u l l D i s p l a c e m e n t P r e s s u r e m e t e r s i n C l a y In t h i s c h a p t e r , f u l l d i s p l a c e m e n t p r e s s u r e m e t e r was a s sumed f o r t h e s t u d y o f d i s t u r b a n c e e f f e c t s . As t h e t o p o f p r e s s u r e m e t e r i s u s u a l l y moun ted w i t h a s o l i d c o n e t i p , t h e i n s t a l l a t i o n o f a f u l l d i s p l a c e m e n t p r e s s u r e m e t e r w o u l d g e n e r a t e s i m i l a r s o i l d i s t u r b a n c e a s t h e p i l e s . T h e r e f o r e i t g e n e r a l l y i s r e g a r d e d a s a p h y s i c a l m o d e l f o r t h e p i l e i n s t a l l a t i o n . In c o n s i d e r a t i o n o f t h i s , s i m i l a r s o i l d i s t u r b a n c e z o n e i s a s s u m e d f o r t h e p r e s s u r e m e t e r . T h i s i s i l l u s t r a t e d i n F i g . 9 . 6 ( a ) . In t h e f i n i t e e l e m e n t a n a l y s e s , a x i s y m m e t r i c , p l a n e s t r a i n c o n d i t i o n s a r e a s sumed f o r t h e p r e s s u r e m e t e r e x p a n s i o n t e s t s . As f o r t h e l a t e r a l l y l o a d e d p i l e s , t h e s i z e o f t h e a n n u l a r d i s t u r b e d zone i s v a r i e d , a n d t h e Y o u n g ' s m o d u l u s ( o r t h e u n d r a i n e d s h e a r s t r e n g t h ) i n t h e d i s t u r b e d z o n e i s v a r i e d l i n e a r l y f r o m E , ( o r C ,) a t t h e a ua p r e s s u r e m e t e r m e m b r a n e - s o i l i n t e r f a c e t o E u ( o r C ) a t t h e o u t e r l i m i t o f t h e d i s t u r b e d z o n e ( s e e F i g . 9 . 5 ) . A g a i n , v a l u e s o f r ^ / r 0 o f 1, 2, 3, 5 h a v e b e e n c o n s i d e r e d , so t h a t p r e s s u r e m e t e r c u r v e s a n d t h e P-Y c u r v e s u n d e r t h e same e x t e n t o f s o i l d i s t u r b a n c e c a n be e x a m i n e d . S u r e l y , t h e r e i s a p r o b l e m w i t h t h e f u l l d i s p l a c e m e n t p r e s s u r e m e t e r s i m u l a t i n g t h e l a t e r a l l y l o a d e d p i l e s , u n l e s s t h e e x c e s s p o r e p r e s s u r e d e v e l o p e d by i n s t a l l a t i o n h a v e d i s s i p a t e d a s t h e y w o u l d f o r t h e p i l e s . •4-4- r d Axisymmetric ++• Co Plane S t r a i n (a) F i g . 9.6 S i m u l a t i o n of I n s t a l l a t i o n f o r Pressuremeter T e s t 00 O 281 As i n p r a c t i c e , t h e p r e s s u r e m e t e r t e s t s a r e u s u a l l y p e r f o r m e d soon a f t e r t h e i n s t r u m e n t i s i n s t a l l e d a t t h e d e s i r e d d e p t h . The h i g h e x c e s s p o r e p r e s s u r e s i n d u c e d d u r i n g t h e i n s t a l l a t i o n may n o t h a v e e n o u g h t i m e t o d i s s i p a t e c o m p l e t e l y . The e f f e c t i v e s t r e s s e s i n s o i l s a r e s t i l l l o w , t h e n u n l i k e f o r p i l e s , t h e u n d r a i n e d s h e a r s t r e n g t h w i l l n o t be a b l e t o g a i n i t s o r i g i n a l v a l u e . T h e r e f o r e , t h e t e s t s a r e l i k e l y t o be p e r f o r m e d w i t h a s o f t e r d i s t u r b e d z o n e o f s o i l s a r o u n d t h e p r o b e . In c o n s i d e r a t i o n o f t h i s d i f f e r e n c e , a s o f t e r z o n e o f s o i l d i s t u r b a n c e i s c o n s i d e r e d i n t h e f i n i t e e l e m e n t a n a l y s i s . V a l u e s o f E , / E ( o r C , / C ) o f 0 . 2 , 0 . 5 , 1 1 d u ud u ' ' were a n a l y s e d . 9 . 3 . 2 F I N I T E ELEMENT MESH In t h e f o l l o w i n g f i n i t e e l e m e n t a n a l y s e s o f i n s t a l l a t i o n e f f e c t s , t h e c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n i s a s sumed f o r t h e p r e s s u r e m e t e r e x p a n s i o n t e s t s , a n d t h e a x i s y m m e t r i c a l p l a n e s t r a i n f i n i t e e l e m e n t mesh d i s c u s s e d i n C h a p t e r 6 i s e m p l o y e d ( s e e F i g . 9 . 7 ) . The r a d i u s o f t h e p r e s s u r e m e t e r , r 0 , i s a s sumed t o be 5 cm. The e l e m e n t s a d j a c e n t t o t h e w a l l o f c a v i t y p r e s e n t t h e z o n e o f d i s t u r b e d s o i l s . The s t r e n g t h a n d s t i f f n e s s i n t h o s e e l e m e n t s a r e d i f f e r e n t f r o m t h e o t h e r e l e m e n t s . F o r t h e l a t e r a l l y l o a d e d p i l e s , t h e p l a n e s t r a i n f i n i t e e l e m e n t mesh d i s c u s s e d i n C h a p t e r 8 i s e m p l o y e d f o r t h e a n a l y s e s ( s e e F i g . 9 . 8 ) . The p i l e r a d i u s , r 0 , i s a s sumed t o be 30 cm. An a n n u l a r z o n e o f s o i l e l e m e n t s n e x t t o t h e p i l e r , 4 ~ Disturbed Zone F i g . 9.7 F i n i t e Element Mesh f o r Studying I n s t a l l a t i o n E f f e c t s on Pressuremeter Curves ro co F i g . 9.8 F i n i t e Element Mesh for Studying I n s t a l l a t i o n E f f e c t s on P - Y Curves to co CO 284 s h a f t s i m u l a t e s the d i s t u r b e d zone. The s t r e n g t h and d e f o r m a t i o n modulus a s s i g n e d i n t h o s e elements a r e d i f f e r e n t from the r e s t of elements r e p r e s e n t i n g the i n t a c t s o i l medium. As i n C hapter 8, a t h i n r i n g of i n t e r f a c e e lements a r e employed around the p i l e s u r f a c e t o s i m u l a t e the s o i l - p i l e i n t e r f a c e b e h a v i o r . An a d h e s i o n f a c t o r of 0.5 i s assumed a t the s o i l - p i l e i n t e r f a c e e l e m e n t s . 9.4 RESULTS AND DISCUSSION Based on the f o r e g o i n g d i s t u r b a n c e s i m u l a t i o n and the f i n i t e element mesh, f i n i t e element p a r a m e t r i c s t u d i e s were performed. In the a n a l y s e s , b i l i n e a r e l a s t o - p l a s t i c s o i l model was employed f o r the p r e s s u r e m e t e r t e s t s , w h i l e f o r the l a t e r a l l y l o a d e d p i l e s , b i l i n e a r e l a s t o - p l a s t i c w i t h t e n s i o n c u t - o f f model was employed. The o t h e r s o i l p arameters r e q u i r e d f o r the a n a l y s e s were the same as employed i n C hapter 6, Chapter 8. The r e s u l t s a r e d i s c u s s e d i n the f o l l o w i n g s e c t i o n s . 9.4.1 PRESSUREMETER CURVES The p r e s s u r e m e t e r c u r v e s o b t a i n e d under d i f f e r e n t e x t e n t s of s o f t e r s o i l zone a r e shown i n F i g . 9 . 9 ( a ) , ( b ) , ( c ) f o r d i s t u r b e d zone of 2 r 0 , 3 r 0 , 5 r 0 r e s p e c t i v e l y . As n o t e d , s o i l d i s t u r b a n c e has s i g n i f i c a n t e f f e c t on the p r e s s u r e m e t e r c u r v e s . The degree of t h e i n f l u e n c e depends upon the e x t e n t of d i s t u r b a n c e , and the s i z e of the d i s t u r b e d zone. Severe d i s t u r b a n c e ( l o s s i n the s t r e n g t h and modulus) and l a r g e 285 F i g . 9.9 Typical pressuremeter curves under d i f f e r e n t s o i l disturbance 286 d i s t u r b e d z o n e b o t h g r e a t l y s o f t e n t h e c u r v e s , t h e i n f l u e n c e seems a l s o t o e x i s t t o some e x t e n t i n t h e l a r g e s t r a i n r e g i o n . Compared t o t h e i n i t i a l s l o p e , t h e t h e o r e t i c a l l i m i t p r e s s u r e p r e d i c t e d f r o m t h e f i n i t e e l e m e n t a n a l y s i s seems t o be l e s s a f f e c t e d by t h e amount o f s o i l d i s t u r b a n c e . The d i s t u r b a n c e e f f e c t s on t h e i n i t i a l s l o p e s o f t h e p r e s s u r e m e t e r c u r v e s c a n a l s o be o b s e r v e d f r o m F i g . 9 . 1 0 , 9 . 1 1 , where t h e d i s t u r b e d i n i t i a l s l o p e s r e l a t i v e t o t h e u n d i s t u r b e d o n e , i . e . SR = ( s l o p e ) ^ / ( s l o p e ) u a r e p l o t t e d a g a i n s t t h e s i z e o f d i s t u r b e d z o n e , r ^ / r 0 , a n d t h e modu lu s r a t i o , E ( j / E u i n t h e d i s t u r b e d z o n e . I t a p p e a r s f r o m t h e f i g u r e s t h a t t h e i n i t i a l s l o p e s a r e more a f f e c t e d by t h e e x t e n t o f t h e d i s t u r b a n c e , a s c o m p a r e d t o t h e i n f l u e n c e o f t h e s i z e o f d i s t u r b e d z o n e . The s l o p e s seem t o v a r y l i n e a r l y w i t h t h e m o d u l u s r a t i o i n d i s t u r b e d z o n e . The i n i t i a l s l o p e s c h a n g e s i g n i f i c a n t l y f o r r ^ / r 0 l e s s t h a n 2. H o w e v e r , t h e i n f l u e n c e o f t h e s i z e o f d i s t u r b e d z o n e seems t o l e v e l o f f o n c e t h e r a d i u s o f d i s t u r b e d z o n e becomes l a r g e r t h a n t w i c e t h e r a d i u s o f t h e p r e s s u r e m e t e r . As shown i n F i g . 9 . 1 1 , t h e r e i s a f a c t o r o f a b o u t 3 b e t w e e n t h e d i s t u r b e d s l o p e w i t h E ^ / E ^ = 0 . 2 c o m p a r e d t o t h e u n d i s t u r b e d o n e . S u r e l y , i f t h e d i s t u r b e d z o n e g o e s t o i n f i n i t y , t h e f a c t o r w i l l be 5. The a b o v e r e s u l t s i n d i c a t e t h a t i f t h e f u l l d i s p l a c e m e n t p r e s s u r e m e t e r t e s t s a r e p e r f o r m e d r i g h t a f t e r t h e i n s t a l l a t i o n , t h e i n i t i a l s l o p e and t h e m e a s u r e d u l t i m a t e p r e s s u r e ( w h i c h i s p r a c t i c a l l y d e f i n e d a t c e r t a i n f i n i t e s t r a i n ) o b t a i n e d f r o m t h e p r e s s u r e e x p a n s i o n c u r v e F i g . 9.10 R e l a t i v e i n i t i a l s l o p e s o f p r e s s u r e m e t e r c u r v e s v s s i z e o f d i s t u r b e d z o n e o in o o H I I I I I I 1 I I I I ~~1 I I 1 I I I 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 E d / E u - M o d u l u s R a t i o i n D i s t u r b e d Z o n e F i g . 9.11 R e l a t i v e i n i t i a l s l o p e s o f p r e s s u r e m e t e r c u r v e s v s e x t e n t o f s o i l d i s t u r b a n c e to oo co 289 w i l l be s i g n i f i c a n t l y r e d u c e d by t h e i n s t a l l a t i o n s o i l d i s t u r b a n c e . 9 . 4 . 2 P-Y CURVES As f o r t h e p r e s s u r e m e t e r c u r v e s , t h e p r e d i c t e d P-Y c u r v e s u n d e r d i f f e r e n t e x t e n t o f s o i l d i s t u r b a n c e a r e p l o t t e d i n F i g 9.12 and 9 . 1 3 . F i g . 9 .12 shows t h e P-Y c u r v e s f o r l a t e r a l l y l o a d e d p i l e w i t h E ^ / E u < 1, w h i c h may s i m u l a t e t h e p i l e i n s t i f f c l a y s , w h i l e F i g . 9 .13 shows t h e P-Y c u r v e s w i t h E ^ / E u > 1, w h i c h may c o r r e s p o n d t o t h e p i l e i n s o f t c l a y . As e x p e c t e d , a s o f t e r d i s t u r b e d z o n e o f s o i l s s o f t e n s t h e c u r v e , e s p e c i a l l y i n l a r g e d i s p l a c e m e n t r a n g e , w h i l e a s t i f f e r d i s t u r b e d z o n e r e s u l t s i n a s t i f f e r P-Y c u r v e . The s i z e o f d i s t u r b e d z o n e seems t o i n f l u e n c e t h e c u r v e s a s w e l l , t h e l a r g e r z o n e o f s o i l d i s t u r b a n c e r e s u l t s i n much s o f t e r / o r s t i f f e r P-Y c u r v e s , d e p e n d i n g upon s o i l c o n d i t i o n ( i . e . s o f t o r s t i f f c l a y ) . F rom t h e c o m p a r i s o n o f d i s t u r b a n c e e f f e c t s on t h e u l t i m a t e r e s i s t a n c e and t h e i n i t i a l s l o p e , i t i s f o u n d t h a t t h e u l t i m a t e r e s i s t a n c e i s much more a f f e c t e d by d i s t u r b a n c e t h a n i s t h e i n i t i a l s l o p e . The i n f l u e n c e o f t h e e x t e n t o f d i s t u r b a n c e a n d t h e s i z e o f d i s t u r b e d z o n e on t h e i n i t i a l s l o p e s o f t h e P-Y c u r v e s a r e shown i n F i g . 9 .14 a n d 9 . 1 5 . The i n i t i a l s l o p e s a r e c a l c u l a t e d a t end o f t h e f i r s t l o a d i n c r e m e n t . T h e r e f o r e t h e y a r e a l s o c a l l e d t h e i n i t i a l m o d u l u s o f s u b g r a d e r e a c t i o n . The r e l a t i v e m o d u l u s shown i n t h e f i g u r e s i s t h e r a t i o o f i n i t i a l s l o p e s o f t h e P-Y c u r v e s w i t h s o i l 290 -O OUturbwS I./I, 0.5 0.2 -1 1 1 1 1 1 1 r 0.0 6.0 12.0 18.0 24.0 30.0 LATERAL DISPLACEMENT Y - MM (DI5 T" 36.0^ 42_._p_ _ 4.80. _ 54.0 ?0NE=2'R0 E-P CLAY) i r—1 1,0.0 n O Undliturbcd B d/e u - 1.0 O 6 Diiturtxd I d / E u • 0.S ^ 1- Disturb** « d / t u . 0.2 0 12.0 18.0 24.0 30.0 36 0 02 0 48 0 LATERAL DISPLACEMENT Y - MM (DIST.Z0NE=3R0 E-P CLAY) 54.0 60..0 1.0 « — 0 Disturbed « d/B„ • 0.S 0.2 .0 12.0 18 0 24.0 30.0 36 0 42 0 LATERAL DISPLACEMENT Y - MM (DIST Z0NE=5R0 1 1 1 r 48.0 54 o E-P CLAY) 60.0 F i g . 9.12 T y p i c a l P-Y c u r v e s u n d e r v a r i o u s s o i l d i s t u r b a n c e ( s t i f f c l a y ) 291 i < o— —o 4 —f> d i s t u r b e d Ed/Eu=2 d i s t u r b e d Ed/Eu=5 (a) — i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 L A T E R A L D I S P L A C E M E N T Y - MM (rd=2T, S O F T C L A Y ) 36.0 40.0 2=S XL I o o —o 0 —e> d i s t u r b e d Ed/Eu=2 d i s t u r b e d Ed/Eu=5 (b) i i i i i i i i i — i — i — i — 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 L A T E R A L D I S P L A C E M E N T Y - MM (rd=3 T. S O F T C L A Y ) d i s t u r b e d Ed/Eu=2 d i s t u r b e d Ed/Eu=5 -1 i i i i i i i i r 0.0 4.0 8.0 12.0 16.0 20.0 L A T E R A L D I S P L A C E M E N T Y - MM i 1 1 1 1 1 1 r 24.0 28.0 32.0 36.0 U=5T. S O F T C L A Y ) 40.0 F i g . 9.13 T y p i c a l P-Y c u r v e s under v a r i o u s s o i l d i s t u r b a n c e ( s o f t c l a y ) 292 d i s t u r b a n c e t o t h a t w i t h no s o i l d i s t u r b a n c e . As shown i n F i g . 9.14 and 9.15, both the e x t e n t s and t h e s i z e of s o i l d i s t u r b a n c e have c e r t a i n i n f l u e n c e s on the i n i t i a l subgrade modulus, however, the i n f l u e n c e i s much l e s s as compared t o the case of p r e s s u r e m e t e r c u r v e s (see F i g . 9.10 and 9.11 w i t h the same s c a l e ) . The r e s u l t s seem t o be r e a s o n a b l e as t h e r e e x i s t s two d i f f e r e n t l o a d i n g mechanisms between the p r e s s u r e m e t e r t e s t s and the l a t e r a l l y l o a d e d p i l e s . In the c a s e of l a t e r a l l y l o a d e d p i l e s , the l o a d s a r e a p p l i e d i n one d i r e c t i o n , the d i s t u r b e d s o i l r e g i o n b e h i n d the p i l e has l e s s i n f l u e n c e s on the P-Y c u r v e s . However, f o r the p r e s s u r e m e t e r c a s e , the p r e s s u r e s a r e a p p l i e d a x i s y m m e t r i c a l l y so t h a t the whole a n n u l a r s o i l d i s t u r b a n c e zone w i l l a f f e c t the p r e s s u r e e x p a n s i o n c u r v e s . In c o n t r a s t t o the c a s e of the p r e s s u r e m e t e r e x p a n s i o n c o n d i t i o n , the s i z e of d i s t u r b a n c e zone seems t o have c o n t i n u o u s i n f l u e n c e on the i n i t i a l subgrade r e a c t i o n (see F i g . 9.10 and 9.14) i f the s i z e of t h e d i s t u r b e d zone c o n t i n u e t o i n c r e a s e . I t appears from F i g . 9.15 t h a t the d i s t u r b e d subgrade modulus i s about 75% of the u n d i s t u r b e d one, when the s t i f f n e s s of d i s t u r b e d s o i l a t s o i l - p i l e i n t e r f a c e i s reduced by a f a c t o r of 5, and t h e d i s t u r b e d zone e x t e n d s t o 2.5 p i l e d i a m e t e r from the p i l e c e n t e r . T h i s r e s u l t may r e p r e s e n t the case of p i l e s i n s t a l l e d i n s t i f f c l a y s . W h i l e f o r p i l e i n s t a l l e d i n s o f t c l a y s , w hich may be r e p r e s e n t e d by E^/E u = 5.0 a t s o i l - p i l e i n t e r f a c e , the i n i t i a l subgrade modulus w i l l o o o — 0 V E u - 0.2 H — h V E u • 0.5 0 0 V E u - 2.0 x — X Ed / Eu " 5.0 3 •O O o • _ 2 CD CI > 10 o o n i i i i i i i i i i i i i i i i i i i 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 r - / r 0 - S i z e of D i s t u r b e d Zone F i g . 9.14 R e l a t i v e i n i t i a l s l o p e o f P-Y c u r v e s v s s i z e of d i s t u r b e d zone (D co F i g . 9.15 R e l a t i v e i n i t i a l s l o p e o f P-Y c u r v e s v s "extent o f s o i l d i s t u r b a n c e 295 i n c r e a s e by about 36% f o r r d / r 0 = 5. The r e l a t i v e r e d u c t i o n of u l t i m a t e r e s i s t a n c e due t o the s t r e n g t h and modulus l o s s e s i n d i s t u r b e d s o i l r e g i o n a r e shown i n F i g . 9.16 and 9.17. The r e l a t i v e u l t i m a t e r e s i s t a n c e i s d e f i n e d as the r a t i o of u l t i m a t e r e s i s t a n c e from the d i s t u r b e d P-Y c u r v e s t o t h a t from the u n d i s t u r b e d P-Y c u r v e . As shown i n F i g . 9.16, compared t o t h e i n f l u e n c e on the subgrade modulus, the s i z e of the d i s t u r b e d zone has a more pronounced but l i m i t e d e f f e c t on t h e u l t i m a t e r e s i s t a n c e . The i n f l u e n c e seems t o l e v e l o f f when r ^ / r 0 i s l a r g e r than 2. S i m i l a r l y , the e x t e n t s of s o i l d i s t u r b a n c e a l s o have l a r g e r i n f l u e n c e on the u l t i m a t e r e s i s t a n c e s than t h e subgrade modulus. For r ^ A o = 5, r e d u c t i o n of t h e modulus and shear s t r e n g t h by f a c t o r of 5 i n t h e d i s t u r b e d s o i l a t the s o i l - p i l e i n t e r f a c e r e s u l t s i n about a 73% r e d u c t i o n i n the p r e d i c t e d u l t i m a t e r e s i s t a n c e s . In the p r a c t i c e , t h i s might c o r r e s p o n d , t o the c o n d i t i o n of p i l e s i n s t a l l e d i n s t i f f c l a y s . For the p i l e s i n s o f t c l a y s however, due t o the i n c r e a s e of u n d r a i n e d shear s t r e n g t h a t t h e end of s o i l r e c o n s o l i d a t i o n , the P-Y c u r v e s would be s t i f f e r , and have h i g h e r v a l u e s of u l t i m a t e r e s i s t a n c e s (see F i g . 9 . 1 3 ( a ) , ( b ) , ( c ) ) . o 4) U c 10 Ul a 4) 4 J (0 E o o Ed/Eu = 0.2 H 1- Ed/Eu = 0.5 LO C o 4-1 3 •O 01 K 0) > o ro " « ~i 1 1 1 1 1 1 1 1 1 1 r 1.5 2.0 2.5 3.0 3.5 4.0 r d / r o - Size of the Disturbed Zone 1 .0 4.5 5.0 5.5 5.0 F i g . 9.16 R e l a t i v e u l t i m a t e r e s i s t a n c e of P-Y c u r v e s v s s i z e of d i s t u r b e d zone CT\ < cn 298 9.5 SUMMARY As s t a t e d b e f o r e , s o i l d i s t u r b a n c e e f f e c t due t o the p i l e i n s t a l l a t i o n i s i m p o r t a n t t o the p i l e l a t e r a l b e h a v i o r , i t s h o u l d be i n c l u d e d i n the P-Y c u r v e s . The s t u d y of t h i s e f f e c t i s n e c e s s a r y f o r the development of P-Y c u r v e s from f i n i t e element a n a l y s i s or from the p r e s s u r e m e t e r c u r v e s . As shown b e f o r e , t h e f i n i t e element a n a l y s i s a l l o w s an u n d e r s t a n d i n g of how d i s t u r b a n c e a f f e c t s the P-Y c u r v e s and the p r e s s u r e m e t e r c u r v e s . Based on t h e above a n a l y s e s , i t may be c o n c l u d e d t h a t : 1. f o r p r e s s u r e m e t e r expansion c u r v e s , t h e s o i l d i s t u r b a n c e due t o the p r e s s u r e m e t e r i n s t a l l a t i o n has much more e f f e c t s on t h e i n i t i a l s l o p e s than t h e l i m i t p r e s s u r e s , 2. f o r the P-Y c u r v e s i n the l a t e r a l l y l o a d e d p i l e s , i n c o n t r a s t t o the case of p r e s s u r e m e t e r t e s t s , the u l t i m a t e r e s i s t a n c e s of the c u r v e s a r e much more a f f e c t e d by t h e s o i l d i s t u r b a n c e due t o t h e p i l e i n s t a l l a t i o n than i s the i n i t i a l modulus of t h e s o i l subgrade r e a c t i o n , 3. Compared t h e e f f e c t s of s o i l d i s t u r b a n c e on t h e p r e s s u r e m e t e r c u r v e s and the P-Y c u r v e s , i t i s found t h a t under t h e same e x t e n t of s o i l d i s t u r b a n c e , the e f f e c t s a r e more pronounced i n p r e s s u r e m e t e r c u r v e s than i n the P-Y c u r v e s . In p r a c t i c e , the p r e s s u r e m e t e r i s g e n e r a l l y r e g a r d e d as a p h y s i c a l model f o r p i l e i n s t a l l a t i o n . The s e l f - b o r i n g p r e s s u r e m e t e r s i m u l a t e s the bored p i l e , w h i l e t h e f u l l 299 d i s p l a c e m e n t p r e s s u r e m e t e r s i m u l a t e s the d r i v e n p i l e . However, t h e r e i s a problem i n s i m u l a t i n g t h e p i l e r e sponses u s i n g p r e s s u r e m e t e r , as i n the f i e l d the p i l e s a r e u s u a l l y l o a d e d l o n g a f t e r the p i l e d r i v i n g , the pore p r e s s u r e g e n e r a t e d d u r i n g the p i l e d r i v i n g would d i s s i p a t e . T h e r e f o r e , u n l e s s the p r e s s u r e m e t e r t e s t s t a r t s l o n g a f t e r the pore p r e s s u r e due t o the i n s t a l l a t i o n has d i s s i p a t e d , the p r e s s u r e m e t e r e x p a n s i o n t e s t can not p r o p e r l y s i m u l a t e the i n s t a l l a t i o n e f f e c t on p i l e r e s p o n s e s . In a d d i t i o n , based on p r e s e n t s t u d i e s , t h e r e i s a d i f f e r e n t i n s t a l l a t i o n e f f e c t on p r e s s u r e m e t e r c u r v e s and P-Y c u r v e s , due t o the d i f f e r e n t l o a d i n g mechanisms a s s o c i a t e d the p r e s s u r e m e t e r and the l a t e r a l l y l o a d e d p i l e . T h e r e f o r e , i t i s d e s i r a b l e t o c o n s i d e r t h i s f a c t o r i n the development of P-Y c u r v e s from the p r e s s u r e m e t e r e x p a n s i o n c u r v e s . However, such an i n c o r p o r a t i o n w i l l be d i f f i c u l t i n p r a c t i c e , as t h e zone and the e x t e n t of s o i l d i s t u r b a n c e due t o the i n s t a l l a t i o n a r e v e r y d i f f i c u l t t o d e t e r m i n e , they depend upon v a r i o u s f a c t o r s , i n c l u d i n g s o i l c o n d i t i o n s , p i l e t y p e s , and the i n s t a l l a t i o n p r o c e d u r e . F u r t h e r s t u d i e s i n t h i s d i r e c t i o n i s w a r r a n t e d . 10. SUMMARY AND CONCLUSIONS N u m e r i c a l s t u d i e s have been performed t o examine some a s p e c t s r e l a t e d t o the p r e s s u r e m e t e r e x p a n s i o n t e s t s and the l a t e r a l l y l o a d e d p i l e s . The s t u d i e s i n c l u d e t h r e e t o p i c s : S o l u t i o n of C y l i n d r i c a l C a v i t y E x p a n s i o n , P r e s s u r e m e t e r E x p a n s i o n Curves and P-Y Curves f o r L a t e r a l l y Loaded P i l e s . The newly d e v e l o p e d f i n i t e element program CONOIL was examined and m o d i f i e d f o r the s e p u r p o s e s . On the s i m u l a t i o n of c y l i n d r i c a l c a v i t y e x p a n s i o n c o n d i t i o n , i t has been shown t h a t the m o d i f i e d program i s c a p a b l e of p r o v i d i n g the r e s u l t s t h a t a r e i n good agreement w i t h the c l o s e d form s o l u t i o n . Based on the p r e s e n t s t u d i e s , i t i s found t h a t u s i n g the s i m p l e h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n , d i f f e r e n t f a i l u r e modulus r e d u c t i o n f a c t o r s may be r e q u i r e d f o r d i f f e r e n t t y p e s of s o i l s . The r e d u c t i o n f a c t o r f o r c o h e s i o n l e s s s o i l may be lower than t h a t f o r c o h e s i v e s o i l so as t o s i m u l a t e t h e shear f a i l u r e of s o i l s and a v o i d the n u m e r i c a l i n s t a b i l i t y . The l o a d s h e d d i n g i t e r a t i o n i s found t o be a u s e f u l t o o l t o f a c i l i t a t e t he s i m u l a t i o n of s o i l f a i l u r e . On t h e p r e s s u r e m e t e r membrane l e n g t h e f f e c t s , i . e . L/D r a t i o e f f e c t on the p r e s s u r e m e t e r e x p a n s i o n r e s u l t s , i t i s found t h a t the L/D r a t i o i s an i m p o r t a n t f a c t o r i f the i n t e r p r e t a t i o n of p r e s s u r e m e t e r t e s t r e s u l t s i s based on the c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y . A s m a l l e r L/D r a t i o r e s u l t s i n a s t i f f e r p r e s s u r e m e t e r c u r v e s . A s u f f i c i e n t l y l a r g e L/D r a t i o i s r e q u i r e d i n o r d e r t o have a p l a n e s t r a i n 300 301 c o n d i t i o n i n the p r e s s u r e m e t e r t e s t s . For c o h e s i v e s o i l s , i f L/D r a t i o i s e q u a l t o 4, t h e g e n e r a l shape of t h e e x p a n s i o n c u r v e and the d e r i v e d e l a s t i c modulus a r e c l o s e t o t h o s e from the c y l i n d r i c a l c a v i t y e x p a n s i o n t h e o r y . However, the d e r i v e d u n d r a i n e d shear s t r e n g t h would be s i g n i f i c a n t l y o v e r - p r e d i c t e d by 15% even w i t h L/D r a t i o of 4. For c o h e s i o n l e s s s o i l s , the L/D r a t i o t o p r o v i d e p l a n e s t r a i n c o n d i t i o n seems t o be l a r g e r . I t i s found t h a t the p r e s s u r e m e t e r e x p a n s i o n c u r v e i s more a f f e c t e d than f o r c o h e s i v e s o i l s . V a l u e s of the d e r i v e d e l a s t i c modulus and t h e f r i c t i o n a n g l e d e r i v e d from the e a r l y p a r t of t h e c u r v e a r e g e n e r a l l y not s i g n i f i c a n t l y a f f e c t e d by the L/D r a t i o . However, such a r e s u l t can not be a p p l i e d t o the f i e l d o p t i m i s t i c a l l y , as i t i s known i n g r a n u l a r c o h e s i o n l e s s s o i l s t h a t the i n i t i a l p a r t of the c u r v e i s most v u l n e r a b l e t o a s m a l l amount of s o i l d i s t u r b a n c e . T h e r e f o r e , how t o r e l i a b l y o b t a i n the f r i c t i o n a n g l e from t h e p r e s s u r e m e t e r t e s t d e s e r v e s f u r t h e r r e s e a r c h work. Based on the comparison of p l a n e s t r a i n a n a l y s i s and f i e l d t e s t d a t a , i t i s found t h a t the f i n i t e element a n a l y s e s w i t h h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n and p l a n e s t r a i n c o n d i t i o n can p r o v i d e r e s u l t s t h a t a r e i n good agreement w i t h the f i e l d measurements. However, f o r c o h e s i o n l e s s s o i l s , the p l a n e s t r a i n s o l u t i o n i s found t o be s o f t e r than t h e f i e l d measurements. The s t i f f e r c u r v e measured i n f i e l d may be due t o the i n s u f f i c i e n t L/D r a t i o 302 of the p r e s s u r e m e t e r used. On the o t h e r hand, th e s o f t n e s s of the p r e d i c t e d c u r v e may a l s o i n d i c a t e t h e i n s u f f i c i e n c y of the h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n i n d e s c r i b i n g g r a n u l a r c o h e s i o n l e s s s o i l s as r e g a r d t o the d i l a t i o n e f f e c t . Good agreement w i t h the f i e l d measurements i n c o h e s i v e s o i l s may a l s o t e n d t o i n d i c a t e t h a t K 0 c o n s o l i d a t i o n e f f e c t i s not t h a t i m p o r t a n t f o r the n o r m a l l y c o n s o l i d a t e d c l a y s , as the r e a d j u s t m e n t of the r e l a t i v e magnitudes of t h r e e p r i n c i p a l s t r e s s e s o c c u r s v e r y r a p i d l y a f t e r t h e u n d r a i n e d s h e a r i n g has s t a r t e d , and the f a i l u r e of s o i l i s governed by the d i f f e r e n c e of h o r i z o n t a l s t r e s s e s a l o n e . The i n i t i a l v e r t i c a l s t r e s s soon becomes i n t e r m e d i a t e p r i n c i p a l s t r e s s , and has l i t t l e e f f e c t on the whole t e s t r e s u l t s . However, f u r t h e r a n a l y s e s of e x p e r i m e n t a l d a t a a r e n e c e s s a r y t o complement the f o r e g o i n g n u m e r i c a l i n v e s t i g a t i o n r e g a r d i n g the L/D r a t i o e f f e c t on t h e i n t e r p r e t a t i o n of p r e s s u r e m e t e r t e s t d a t a , s t r e s s v a r i a t i o n i n t h e s o i l element around the p r e s s u r e m e t e r and i t s e f f e c t on the t e s t r e s u l t s . Such e x p e r i m e n t a l d a t a may i n c l u d e the t r u e t r i a x i a l t e s t , and the t r i a x i a l chamber c a l i b r a t i o n t e s t d a t a . N u m e r i c a l s t u d i e s of t h e s e t e s t d a t a may a l s o p r o v i d e o p p o r t u n i t i e s t o e v a l u a t e the s t r e s s - s t r a i n model of s o i l s employed i n the a n a l y s i s . As r e g a r d t o the l a t e r a l l y l o a d e d p i l e s , a s i m p l e i n t e r f a c e element model based on the f o r m u l a t i o n of o r d i n a r y element i s p r e s e n t e d f o r the s t u d y of s o i l - p i l e i n t e r f a c e 303 b e h a v i o r . Based on the c o mparison w i t h the c l a s s i c p l a s t i c i t y s o l u t i o n , i t i s found t h a t the p r o posed i n t e r f a c e element can p r o v i d e a g r e e a b l e r e s u l t s w i t h c l o s e d form as l o n g as the i n t e r f a c e element i s s u f f i c i e n t l y s m a l l i n s i z e r e l a t i v e t o the p i l e and s o i l e l e m e n t s . However, f u r t h e r s t u d i e s a r e n e c e s s a r y t o e v a l u a t e t h i s model by comparing p a r a m e t r i c s t u d i e s w i t h e x p e r i m e n t a l r e s u l t s , such as d i r e c t shear box t e s t d a t a . Based on p r e s e n t s t u d i e s , i t i s found t h a t f o r c o h e s i v e s o i l s , the i n t e r f a c e p r o p e r t i e s s i g n i f i c a n t l y a f f e c t the u l t i m a t e s o i l r e s i s t a n c e of P-Y c u r v e s , w h i l e f o r c o h e s i o n l e s s s o i l s , the whole P-Y c u r v e s a r e i n s e n s i t i v e t o the s o i l - p i l e i n t e r f a c e p r o p e r t i e s . I t i s a l s o found t h a t f o r c o h e s i v e s o i l s , the t e n s i l e c r a c k i n g or s o i l - p i l e s e p e r a t i o n s i g n i f i c a n t l y r e duces the u l t i m a t e s o i l r e a c t i o n . T h e r e f o r e , i t i s d e s i r a b l e t o i n c o r p o r a t e t h e s e f a c t o r s i n the development of P-Y c u r v e s . I t i s shown t h a t the f i n i t e element a n a l y s i s i s a p o w e r f u l t o o l t o i n c o r p o r a t e t h e s e e f f e c t s . I t i s a l s o found t h a t the P-Y c u r v e s f o r c o h e s i v e and c o h e s i o n l e s s s o i l can be s i m p l y r e p r e s e n t e d by h y p e r b o l i c f u n c t i o n and power f u n c t i o n r e s p e c t i v e l y . However, the v a l i d i t y of t h e s e f u n c t i o n r e q u i r e s f u r t h e r e x a m i n a t i o n by a n a l y z i n g f i e l d l o a d t e s t d a t a . As f o r the d i f f e r e n t i n s t a l l a t i o n e f f e c t s on p r e s s u r e m e t e r c u r v e s and P-Y c u r v e s , a p a r a m e t r i c s t u d y was p erformed i n c o h e s i v e s o i l s . I t i s noted t h a t due t o the pore p r e s s u r e g e n e r a t e d d u r i n g the i n s t a l l a t i o n , a 304 p r e s s u r e m e t e r e x p a n s i o n t e s t r i g h t a f t e r i n s t a l l e d a t c e r t a i n depth i s u s u a l l y performed w i t h a s o f t e r d i s t u r b e d zone of s o i l s around the probe. U n l e s s the pore p r e s s u r e has d i s s i p a t e d as they would b e f o r e p i l e l o a d i n g , t h e p r e s s u r e m e t e r e x p a n s i o n t e s t can not p r o p e r l y s i m u l a t e t h e i n s t a l l a t i o n e f f e c t on p i l e r e s p o n s e s . Based on the p a r a m e t r i c s t u d i e s , i t i s found t h a t even under the same amount of s o i l d i s t u r b a n c e , i n s t a l l a t i o n has a d i f f e r e n t e f f e c t on the p r e s s u r e m e t e r c u r v e s and the P-Y c u r v e s due t o the d i f f e r e n t l o a d i n g mechanisms a s s o c i a t e d w i t h the p r e s s u r e m e t e r and the l a t e r a l l y l o a d e d p i l e s . For p r e s s u r e m e t e r e x p a n s i o n c u r v e s , i t i s found t h a t t h e i n i t i a l s l o p e of the c u r v e i s much a f f e c t e d by the s o i l d i s t u r b a n c e , but not so much f o r the u l t i m a t e p r e s s u r e . However, the o p p o s i t e e f f e c t s a r e ob s e r v e d f o r the P-Y c u r v e s , the u l t i m a t e s o i l r e s i s t a n c e i s s i g n i f i c a n t l y a f f e c t e d by the s i z e and e x t e n t of s o i l d i s t u r b a n c e w h i l e the i n i t i a l modulus of s o i l subgrade r e a c t i o n i s r e l a t i v e l y i n s e n s i t i v e t o t h e i n s t a l l a t i o n e f f e c t . In view of t h e s e , i t i s d e s i r a b l e t o i n c o r p o r a t e t h i s d i f f e r e n t s o i l d i s t u r b a n c e e f f e c t i n the development of P-Y c u r v e s from p r e s s u r e e x p a n s i o n c u r v e s . In the p r e s e n t s t u d i e s of i n s t a l l a t i o n e f f e c t s , however, the c o n d i t i o n of s t r e s s changes due t o the p i l e i n s t a l l a t i o n i s not c o n s i d e r e d i n the a n a l y s i s , and the stu d y i s a l s o l i m i t e d t o the u n d r a i n e d c o h e s i v e s o i l s . A d d i t i o n a l work i s d e s i r a b l e f o r s t u d y i n g t h e i n s t a l l a t i o n 305 e f f e c t on p i l e s d r i v e n i n c o h e s i o n l e s s s o i l s . T h i s may be a c c o m p l i s h e d by s i m u l a t i n g the p i l e d r i v i n g i n c o h e s i o n l e s s s o i l s as a c a v i t y e x p a n s i o n p r o c e s s , then l o a d i n g the p i l e i n one d i r e c t i o n , or l o a d i n g the p r e s s u r e m e t e r a x i s y m m e t r i c a l l y . In t h i s p r o c e d u r e , l a r g e s t r a i n f o r m u l a t i o n i s n e c e s s a r y . T h e r e f o r e , f u r t h e r r e s e a r c h i s w a r r a n t e d i n t h i s d i r e c t i o n . REFERENCES American P e t r o l e u m I n s t i t u t e (1976), API Recommended P r a c t i c e f o r P l a n n i n g , D e s i g n i n g , and C o n s t r u c t i n g F i x e d O f f s h o r e P l a t f o r m , API RP-2A, 7 t h ed., pp21-26 A t u k o r a l a , U. and Byrne, P.M. 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( 1 9 6 3 ) , H y p e r b o l i c S t r e s s - s t r a i n Response: C o h e s i v e S o i l s , J.S.M.F.D., ASCE, V o l . 89, No. Sm1 , Feb, 1963, p i 15 Kuhlemeyer, R.L. (1977), S t a t i c and Dynamic L a t e r a l l y Loaded F l o a t i n g P i l e s , Proc. ASCE. V o l . 105, GT2, pp289-304 Ladd, C.C., F o o t t , R., I s h i h a r a , K., S c h l o s s e r , F. and P o u l o s , H.G., S t r e s s - D e f o r m a t i o n and S t r e n g t h C h a r a c t e r i s t i c s , Proc. of 19th Inter. Conf. on SMFE, Tokyo, V o l . 2, 1977, pp.421-494 L a i e r , J.E., Schmertmann, J.H., Schaub, J.H. (19 7 5 ) , E f f e c t of F i n i t e P r e s s u r e m e t e r Length i n Dry Sand, Proc. Spec. Conf. on In-Situ Measurement of Soil Propertier, ASCE, R a l e i g h , N.C., V o l . 1, pp.241-259 L i v n e h , M., G e l l e r t , M., Uzan, S. (1971), D e t e r m i n a t i o n of the E l a s t i c Modulus of S o i l by the P r e s s u r e m e t e r T e s t , T h e o r e t i c a l Background, /. of Materials, V o l . 6, No. 2, pp.348-355 Lo, K.Y. and Stermac, A.G. 1965, Induced Pore P r e s s u r e s d u r i n g P i l e D r i v i n g O p e r a t i o n s , Proc. Sixth Int. Conf. Soil Mech. Found. Eng., M o n t r e a l 2, pp285-289 M a t l o c k , H. ( 1 9 7 0 ) , C o r r e l a t i o n s f o r D e s i g n i n g of L a t e r a l l y Loaded P i l e s i n S o f t C l a y , OTC, paper No. OTC 1204 M c C l e l l a n d , B. and F o c h t , John A. J r . ( 1 9 5 8 ) , S o i l Modulus 310 f o r L a t e r a l l y Loaded P i l e s , Trans. ASCE., V o l . 1 2 3 , paper No.2954, ppl049-1086 M i n d l i n , R.D. (19 3 6 ) , F o r c e a t A P o i n t i n t h e I n t e r i o r of A S e m i - i n f i n i t e S o l i d , Physics 7, ppl95~202 M u r c h i s o n , J.M., and O ' N e i l l , M.W. (1984), E v a l u a t i o n of P-Y R e l a t i o n s h i p s i n C o h e s i o n l e s s S o i l s , Proc. Symp. on Analysis and Design of Pile Foundations, San F r a n c i s c o , C a l i f . O c t . 1984 N a g t e g a a l , J.C., P a r k s , D.M., and R i c e , J.R. (19 7 4 ) , On N u m e r i c a l l y A c c u r a t e F i n i t e Element S o l u t i o n s i n the F u l l y P l a s t i c Range, Computer Methods in Applied Mechanics and Engineering 4 pp153-177 N a y l o r , D.J. (19 7 3 ) , D i s c u s s i o n , P r o c . of the Symp. on the Role of Plasticity in Soil Mechanics, Cambridge, 13-15, P291-294, S e p t . N a y l o r , D.J. Pande, G.N., Simpson.B, and Tabb, B., F i n i t e  Element i n G e o t e c h n i c a l E n g i n e e r i n g , P i n e r i d g e P r e s s , Swansea, U.K., 1981 Novak, M. (1 9 8 0 ) , D i s c u s s i o n , Proc. Num. Method Offshore Piling, London, 1977 O r r j e , 0. and Broms, B. (1967), E f f e c t s of P i l e D r i v i n g on S o i l P r o p e r t i e s , Proc. ASCE. V o l . 93, No SM5, PART 1, Sept. Potyondy, J.G. (1961), S k i n F r i c t i o n Between V a r i o u s S o i l s and C o n s t r u c t i o n M a t e r i a l s , Geotechnique, 11, 4, pp339-353 P o u l o s , H.G. (1971) The b e h a v i o r of L a t e r a l l y Loaded P i l e s , I - S i n g l e P i l e , Proc. ASCE, V o l . 97, SM5, pp738-75l P r e v o s t , J.H. (19 7 6 ) , U n d r a i n e d S t r e s s - S t r a i n - T i m e B e h a v i o r of C l a y s , /. of Geot. Eng. Div.,ASCE., V o l . 102, No. GT12, Dec. 1976, pp.1245-1260 Pyke, R. and B e i k a e , M. (1984), A New S o l u t i o n f o r the R e s i s t a n c e of S i n g l e P i l e s t o L a t e r a l L o a d i n g , Laterally Loaded Deep Foundations: Analysis and Performance, ASTM SPT 835, pp3-20 Randolph, M.F., C a r t e r , J.P. and Wroth, C P . (1979) D r i v e n P i l e s i n C l a y - the E f f e c t s of I n s t a l l a t i o n and Subsequent c o n s o l i d a t i o n , Geotechnique, V o l . 24, No. 4, pp36l-393 Randolph, M.F. and Wroth, C P . (1979), An A n a l y t i c a l S o l u t i o n f o r t h e C o n s o l i d a t i o n around a D r i v e n P i l e , 311 Int. J. Num. Anal. Meth. Geom., V o l 3, pp217-229 Randolph, M.F. and Houls b y , G.T. (1984), The L i m i t i n g P r e s s u r e on A C i r c u l a r P i l e Loaded L a t e r a l l y i n C o h e s i v e S o i l , Geotechnique 34, No. 4, pp613~623 Reese, L.C. (19 5 8 ) , D i s c u s s i o n of " S o i l Modulus f o r L a t e r a l l y Loaded P i l e s " by B. M c C l e l l a n d and John A. F o c h t , J r . , Trans. ASCE, V o l . 1 2 3 , pp1071-1074 Reese, L.C. (1962), U l t i m a t e R e s i s t a n c e a g a i n s t a R i g i d C y l i n d e r Moving L a t e r a l l y i n C o h e s i o n l e s s S o i l , /. Society of Petroleum Engineers, Dec. 1962 Reese, L.C. (19 7 7 ) , L a t e r a l l y Loaded P i l e s : Program Documentation, PROC. ASCE, No.GT4, A p r i l 1977 Reese, L.C, Cox, W.R., and Koop, F.D. (1 9 7 5 ) , F i e l d T e s t i n g and A n a l y s i s of L a t e r a l l y Loaded P i l e s i n S t i f f C l a y , Proc. 7th OTC, Vol.11 pp-671-690 Reese, L.C, Cox, W.R., and Koop, F.D. (1 9 7 4 ) , A n a l y s i s of L a t e r a l l y Loaded P i l e s i n Sand, OTC, Paper NO. OTC 2080 Reese, L.C. O ' N e i l l , M.W., and S m i t h , E. (19 7 0 ) , G e n e r a l i z e d A n a l y s i s of P i l e F o u n d a t i o n s , Proc. ASCE, V o l 96, SM1, pp235-250 Reese, L.C. and Welch, R.C. (1975), L a t e r a l L o a d i n g of Deep F o u n d a t i o n s i n S t i f f C l a y , Proc. ASCE, V o l . 1 0 1 , No.GT7, pp633-649 R o b e r t s o n , P.K, Hughes, J.M.O., Campanella, R.G., and Sy, A. (19 8 3 ) , P r e d i c t i o n of L a t e r a l l y Loaded P i l e s u s i n g P r e s s u r e m e t e r T e s t s , Soil Mechanics Series, No.67, Dept. of C i v l E n g i n e e r i n g , UBC, Vancouver, Canada, May 1983 Samarasekera, L a i (1982), N o n l i n e a r E l a s t i c U n d r a i n e d S t r e s s - s t r a i n Model f o r A n i s o t r o p i c C o n s o l i d a t e d C a l y , M.A.Sc T h e s i s , Dept. of C i v i l E n g i n e e r i n g , UBC, A p r i l S c o t t , R.F. A n a l y s i s of C e n t r i f u g e P i l e T e s t s : S i m u l a t i o n of P i l e D r i v i n g , Research Report, API OSAPR P r o j e c t 13, C a l t h , Pasadena, C a l i f . , June 1980 S c o t t , R.F., F o u n d a t i o n A n a l y s i s , P r e n t i c e - H a l l , I n c . Englewood C l i f f , N.J., 1981 She, J . (1 9 8 6 ) , F o r t h c o m i n g M.A.Sc T h e s i s , Dept. of C i v i l E n g i n e e r i n g , UBC, Vancouver Skempton, A.W. (1951), B e a r i n g C a p a c i t y of C l a y s , Proc. the Building Research Congress, London, ICE, D i v . 1:180 Skempton, A.W. (1954), C o e f f i c i e n t s A and B, Geotechnique, V o l 4, p143 312 Skempton, A.W. , (1960), E f f e c t i v e S t r e s s e s i n S o i l , C o n c r e t e , and Rock, Conf. Pore Pressure and Suction in Soils, London, pp4-16 S l o a n , S.W. and Randolph, M.F. (19 8 2 ) , N u m e r i c a l P r e d i c t i o n of C o l l a p s e Loads u s i n g F i n i t e Element Methods, Int. J. Num. Anal. Meth. Geom. v o l . 6 , pp47-76 Timoshenko, S. and G o o d i e r , J.N. (19 5 1 ) , Theory of  E l a s t i c i t y , 2nd ED. M c G r a w - H i l l , N.Y., 1951 V a z i r i , H. (1986), Ph.D D i s s e r t a t i o n , Dept. of C i v i l E n g i n e e r i n g , UBC, Vancouver V e s i c , A.S. (1972), E x p a n s i o n of C a v i t i e s i n I n f i n i t e S o i l Mass, /. Soil Mesh. Found. Divn., A.S.C.E., V o l . 98, •SM3, pp.265-290 Y e g i a n , M. and W r i g h t , S.G. (1973), L a t e r a l S o i l R e s i s t a n c e D i s p l a c e m e n t R e l a t i o n s h i p s f o r P i l e F o u n d a t i o n s i n S o f t C l a y , O r e , paper No. 1893, D a l l a s APPENDIX A -- R e l a t i o n between B^/n and Skempton's pore p r e s s u r e p arameters B s k e m : From the c o m p a t i b i l i t y of the u n d r a i n e d c o n d i t i o n : A e f = ^ v n From Eq. (3.3.4) i n the t e x t , t h e r e f o r e : where A e v i s the v o l u m e t r i c s t r a i n change of the pore f l u i d element under u n d r a i n e d c o n d i t i o n s , A e v i s t h e c o r r e s p o n d i n g v o l u m e t r i c s t r a i n change of the s o i l s k e l e t o n . Au = B c Ae (A2) f S i n c e ACT - Au m = B' Ae v (A3) From Eq. ( A 2 ) , Eq.(A3) becomes : ACT - Au m (A4) 313 314 T h e r e f o r e . / AB »n ACT = AU (1 + " m B r ) From the d e f i n i t i o n of the Skempton's parameter B s k e m i Au skem ~ m 1 B' -n 1 + (A5) B f R e a r r a n g i n g Eq. (A5) : B f / n " B ' B s k e n / ( 1 " B s k e m ) T h i s i s the Eq. (3.3.11) i n the Text, APPENDIX B Based on B a g u e l i n e t a l (1977) r e s u l t s (see Eqs. (8.4.1) t o ( 8 . 4 . 4 ) ) , assuming t h a t a f l e x i b l e p i l e i n c o h e s i v e s o i l s i s l o a d e d under u n d r a i n e d c o n d i t i o n s : R = 7 t o 12 1 0 . where 1 0 i s the r e l a t i v e s t i f f n e s s f a c t o r d e f i n e d as [ 4 ( E I ) _ / E ] 0 2 5 , (EI) i s the r i g i d i t y of the p i l e s e c t i o n , E g i s the c o e f f i c i e n t of s o i l subgrade r e a c t i o n . In the f i n i t e element a n a l y s i s , the p i l e e lements a r e assumed 500 t i m e s s t r o n g e r than the s o i l e l e ments i n the s t i f f n e s s . I f assuming the c o e f f i c i e n t of s o i l subgrade r e a c t i o n , E i s e q u a l t o the Young's modulus of the s o i l s ( i . e . t a k i n g u = 1 i n Eq. ( 8 . 3 . 7 ) , then the r a t i o of E /E = P s 500. S i n c e I = 7rD 4/64, where D i s the p i l e d i a m e t e r , then by the d e f i n i t i o n of 1 0, l o = [ 4 ( E I ) / E c ] ° ' 2 5 = 3.15 D. P s T h e r e f o r e , the o u t e r boundary r a d i u s R = 7 t o 12 1 0 = 22 D t o 37 D. Based on B a r d e t ' s r e s u l t s shown i n F i g . 8.4, i f t h e c o e f f i c i e n t of s o i l subgrade r e a c t i o n K i s c l o s e t o the Young's modulus, E, f o r u n d r a i n e d c o h e s i v e s o i l s , the o u t e r boundary r a t i o a/R i s e q u a l t o 0.006, i . e . R = 83 D, w h i l e 315 316 f o r d r a i n e d c o h e s i o n l e s s s o i l , a/R i s e q u a l t o 0.02, i . e . R = 25 D. Based on the above r e s u l t s , a r e p r e s e n t a t i v e v a l u e of R = 50 D i s used f o r the f i n i t e element a n a l y s e s of f l e x i b l e p i l e s i n u n d r a i n e d c o h e s i v e s o i l s . T h i s v a l u e l i e s between the v a l u e s s u g g e s t e d by B a g u e l i n e t a l and B a r d e t , and i s b e l i e v e d t o be r e a s o n a b l e f o r p r e s e n t a n a l y s i s . 

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