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A non linear incremental finite element program for the analysis of shafts and tunnels in oilsands Gunaratne, Manjriker 1981

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A NON LINEAR INCREMENTAL FINITE ELEMENT PROGRAM FOR THE ANALYSIS OF SHAFTS AND TUNNELS IN OILSANDS by MANJRIKER GUNARATNE B.Sc., (Engineering) The U n i v e r s i t y of Peradeniya , SRI-LANKA , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in 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 conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1981 Man j r i k e r Gunaratne , 1981 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 of 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 gran t e d by the head of my department o r 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 c o p y i n g 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 of C i v i l Engineering  The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date i-q8l Apr.) 1<\. n T T _ f i I 9 /7Q \ ii ABSTRACT A method f o r a n a l y s i n g the d e f o r m a t i o n b e h a v i o u r of o i l s a n d a d j a c e n t t o s h a f t s and t u n n e l s i s p r e s e n t e d . O i l s a n d i s co m p r i s e d of a dense sand m a t r i x w i t h i t s pore spaces f i l l e d w i t h bitumen , water and f r e e or d i s s o l v e d gases. The E n g i n e e r i n g b e h a v i o u r of o i l s a n d i s governed by the s t r e s s e s i n the sand m a t r i x . The bitumen does not c o n t r i b u t e d i r e c t l y t o the s t r e n g t h of the sand. However , i n d i r e c t l y the presence of bitumen may g r e a t l y a f f e c t i t s b e h a v i o u r . T h i s i s because the presence of bitumen reduces the e f f e c t i v e p e r m e a b i l i t y of the o i l s a n d and v e r y o f t e n u n d r a i n e d c o n d i t i o n s o c c u r . Then the p r e s s u r e of the pore gases remain h i g h r e d u c i n g the e f f e c t i v e s t r e s s e s f o r u n l o a d i n g c o n d i t i o n s . A n o n l i n e a r i n c r e m e n t a l f i n i t e element model i s used t o a n a l y s e the o i l s a n d s k e l e t o n b e h a v i o u r . D i l a t i o n or shear induced volume change i s an i m p o r t a n t c h a r a c t e r i s t i c of a dense sand and t h i s i s i n c l u d e d i n the a n a l y s i s u s i n g a m o d i f i e d form of Rowe's s t r e s s d i l a t a n c y t h e o r y . The u n l o a d i n g c o n d i t i o n a t the f a c e of a t u n n e l or s h a f t can l e a d t o a v i o l a t i o n of the f a i l u r e c r i t e r i o n and t h i s c o n d i t i o n i s r e c t i f i e d by a s t r e s s r e d i s t r i b u t i o n t e c h n i q u e . The c o m p r e s s i b i l i t i e s of the o i l and water phases a r e n e g l e c t e d i n comparison w i t h t h a t of the gas phase and pore p r e s s u r e changes are p r e d i c t e d by the i d e a l gas laws. Under u n d r a i n e d c o n d i t i o n s the pore p r e s s u r e i s c o u p l e d i n t o the s k e l e t o n s t r e s s e s by m a i n t a i n i n g v o l u m e t r i c s t r a i n c o m p a t i b i l i t y between the s k e l e t o n and pore f l u i d phases. The r e s u l t s have been checked w i t h d r a i n e d and u n d r a i n e d c l o s e d form s o l u t i o n s . The s o l u t i o n f o r the u n l o a d i n g of a t u n n e l i n o i l s a n d i s p r e s e n t e d and i t shows t h a t the l i m i t i n g support p r e s s u r e s can be reduced by v e n t i n g elements t o a r e a s o n a b l e d i s t a n c e from the t u n n e l . I t i s a l s o found t h a t the e f f e c t s of shear d i l a t i o n a r e s i g n i f i c a n t o n l y when the l i m i t i n g support p r e s s u r e i s approached. ir i TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v ACKNOWLEDGEMENTS v i CHAPTER 1. INTRODUCTION 1 1.1 Background 1 1.2 Scope and Purpose 2 11. OILSAND BEHAVIOUR 5 111. EFFECTIVE STRESS MODEL ...8 3.1 S t r e s s e s 8 3.2 D e f o r m a t i o n s and S t r a i n s -....8 3.3 C o n s t i t u t i v e R e l a t i o n s 8 3.3.1 L i n e a r E l a s t i c i t y 10 3.3.2 Non L i n e a r E l a s t i c i t y 10 3.3.2.1 E q u i v a l e n t E l a s t i c Method 11 3.3.2.2 I n c r e m e n t a l Method 11 3.3.3 H y p e r b o l i c R e l a t i o n 11 3.3.4 Dependency of on Og 12 3.3.5 F a i l u r e R a t i o 12 3.3.6 Y i e l d C r i t e r i o n 12 3.3.7 R e l a t i o n between E t and s t r e s s e s 15 3.3.8 Other Non L i n e a r R e l a t i o n s 15 3.3.8.1 U n l o a d i n g R e l o a d i n g Modulus 15 3.3.8.2 E l a s t i c Volume Change 17 3.3.9 C o n s t i t u t i v e r e l a t i o n s f o r the f a i l u r e r e g i o n 19 3.4 Comment on the Non L i n e a r E l a s t i c P arameters 20 3.5 D e t e r m i n a t i o n of the Non L i n e a r Parameters 20 3.5.1 E v a l u a t i o n of Kg and m 20 3.6 Shear Volume C o u p l i n g 24 3.7 B a s i c Concepts of Rowe's S t r e s s D i l a t a n c y Theory 25 3.7.1 P l a s t i c P o t e n t i a l s 29 3.8 S t r e s s D i l a t a n c y R e l a t i o n s h i p f o r P l a n e S t r a i n C o n d i t i o n s 29 3.8.1 D i l a t i o n A n gle 32 3.8.2 Gur and P l a s t i c Shear S t r a i n s 34 3.8.3 D i l a t i o n A n gle and C o n f i n i n g P r e s s u r e ....36 3.9 M o d i f i e d NLSSIP and Shear D i l a t i o n 37 IV. FINITE ELEMENT FORMULATION 40 4.1 C o n s t i t u t i v e M a t r i x 40 4.2 S t i f f n e s s M a t r i x 41 4.3 S o l u t i o n 43 4.4 Shear D i l a t i o n 43 i V 4.5 S t r e s s R e d i s t r i b u t i o n Technique 45 4.6 I n c l u s i o n of E l a s t i c S p r i n g s c 49 V. PORE PRESSURE MODEL 52 5.1 I d e a l i z a t i o n of the Pore F l u i d B e h a v i o u r 52 5.2 Free Gas Be h a v i o u r 52 5.3 P r e s s u r e S o l u b i l i t y 52 5.4 Temperature S o l u b i l i t y 53 5.5 V o l u m e t r i c S t r a i n C o m p a t i b i l i t y For U n d r a i n e d C o n d i t i o n s 53 5.6 Comment on the use of Eqn. 110 54 V I . SOLUTION METHOD 55 6.1 Dr a i n a g e C o n d i t i o n s 55 6.2 I t e r a t i v e P r o c e d u r e s .55 6.2.1 D r a i n e d A n a l y s i s ..55 6.2.2 Und r a i n e d A n a l y s i s 56 6.2.3 Undrained Mode f o l l o w e d by V e n t i n g 59 6.3 L i m i t a t i o n s 59 6.4 S i g n C o n v e n t i o n s 59 6.4.1 Loads and D i s p l a c e m e n t s 59 6.4.2 S t r e s s e s and S t r a i n s 60. V I I . COMPARISON WITH EXISTING SOLUTIONS 62 7.1 E l a s t i c C l o s e d form S o l u t i o n s 62 7.2 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 63 7.3 One D i m e n s i o n a l U n l o a d i n g of O i l s a n d 65 7.4 D u s s e a u l t ' s S o l u t i o n 67 7.5 U n l o a d i n g of a C y l i n d e r i c a l S h a f t i n O i l s a n d 77 V I I I . SUMMARY AND CONCLUSIONS 78 REFERENCES 79 APPENDIX A 81 APPENDIX B 86 APPENDIX C 90 APPENDIX D 95 APPENDIX E 104 APPENDIX F 105 APPENDIX G 109 APPENDIX H 113 APPENDIX I 114 V LIST OF FIGURES FIG. TITLE PAGE 1. P o s s i b l e I n s t e r s t i t i a l C o n d i t i o n s of O i l s a n d 4 2. U n l o a d i n g of o i l s a n d 6 3. N o n l i n e a r E l a s t i c Curve 9 4. H y p e r b o l i c Curve 9 5. V a r i a t i o n of <f> w i t h CT3 13 6. U n l o a d i n g - r e l o a d i n g Curve 14 7. Volume Change Curves 16 8. S t r e s s - s t r a i n Curves 18 9. V i o l a t i o n of F a i l u r e C r i t e r i o n 18 10. I s o t r o p i c C o n s o l i d a t i o n 21 11. D e t e r m i n a t i o n of K g 21 12. Shear D i l a t i o n 23 13. P l a s t i c P o t e n t i a l 28 14. Mohr C i r c l e f o r s t r a i n s 30 15. Mohr C i r c l e f o r s t r e s s e s 30 16. D i l a t i o n A n gle 31 17. Comparison of Dense and Loose sand B e h a v i o u r 33 18. U n l o a d i n g - r e l o a d i n g i n shear 33 19. V a r i a t i o n of V w i t h ( J v o 35 20. Simple shear t e s t r e s u l t s 35 21. I d e a l i z e d volume change c u r v e 38 22. S t r e s s R e d i s t r i b u t i o n 46 23. I n c l u s i o n of S p r i n g s ." 48 24. Time e f f e c t s i n Pore p r e s s u r e changes 51 25. S i g n c o n v e n t i o n f o r l o a d s and d i s p l a c e m e n t s 58 26. S i g n c o n v e n t i o n f o r s t r e s s e s and s t r a i n s 58 27. Some p l a n e s t r a i n s i t u a t i o n s 61 28. L i n e a r E l a s t i c i t y i n NLSSIP 64 29. One d i m e n s i o n a l u n l o a d i n g of o i l s a n d 66 30. T h i c k w a l l e d c y l i n d e r 68 31. E l a s t i c s t r e s s e s and d i s p l a c e m e n t s around a s h a f t 70 32. E l a s t o - P l a s t i c s t r e s s e s and d i s p l a c e m e n t s around a s h a f t 72 33. One D i m e n s i o n a l U n l o a d i n g ( p r e d i c t i o n s ) 74 34. Comparison w i t h D u s s e a u l t ' s r e s u l t s 75 35. U n l o a d i n g of a c y l i n d e r i c a l s h a f t i n o i l s a n d 76 36. Mohr C i r c l e a t f a i l u r e 83 37. P a r t i c l e s l i p 85 38. E f f e c t s of P r e s s u r e and Temperature on o i l s a n d 89 39. I s o p a r a m e t r i c element 98 40. A p p r o x i m a t i o n of an i n f i n i t e problem 103 41. E l a s t o - P l a s t i c s o l u t i o n 108 * ACKNOWLEDGEMENTS The a u t h o r would l i k e t o e x p r e s s h i s g r a t i t u d e t o r e s e a r c h s u p e r v i s o r , Dr. P e t e r M. Byrne f o r h i s guidance v a l u a b l e s u g g e s t i o n s throughout t h i s r e s e a r c h . The a u t h e r a l s o wishes t o thank Dr. Y. P. V a i d f o r c o n t i n u e d i n t e r e s t and a d v i c e . F i n a l l y the Res e a r c h A s s i s t a n t s h i p awarded by Department of C i v i l E n g i n e e r i n g i s g r a t e f u l l y acknowledged. CHAPTER 1 INTRODUCTION 1.1 Background Most of the methods that have been e s t a b l i s h e d f o r the recovery of heavy o i l from deep l y i n g o i l s a n d d e p o s i t s , i n v o l v e c o n s t r u c t i o n of s h a f t s and tunnels i n o i l s a n d s i t s e l f . Thus the design of a s u i t a b l e support system f o r them turns out to be a c h i e f concern i n recovery p r o c e s s e s . In order f o r the supports to be designed, the l i m i t s t a t e loads to which they w i l l be s u b j e c t e d , w i l l have to be estimated. In the c o n v e n t i o n a l design of support systems reasonable estimates of loads on tunnel support are made by reviewing p r e v i o u s t u n n e l l i n g experience i n m a t e r i a l s with s i m i l a r p r o p e r t i e s . O i l s a n d i s comprised of a dense sand matrix with i t s pore spaces f i l l e d with bitumen , water and f r e e or d i s s o l v e d gases. Unloading a s s o c i a t e d with the exca v a t i o n s f o r tunnels and s h a f t s in o i l s a n d , causes these gases to evolve . The presence of bitumen reduces the e f f e c t i v e p e r m e a b i l i t y of the o i l s a n d and very o f t e n undrained c o n d i t i o n s occur. Then the e f f e c t i v e s t r e s s e s w i l l drop l e a d i n g to i n s t a b i l i t y at the face of the ex c a v a t i o n . Thus o i l s a n d e x h i b i t s unusual behaviour. Moreover, t u n n e l l i n g experience i n o i l s a n d , i s a l s o s c a r c e . Therefore the e m p i r i c a l design approach may not be r e l i a b l e in t h i s p a r t i c u l a r s i t u a t i o n . In view of t h i s , a n a n a l y t i c a l method f o r computing s t r e s s e s and deformations i n o i l s a n d s adjacent to tunnel openings i s in v a l u a b l e . A s the pore gases p l a y a dominant r o l e i n the behaviour of o i l s a n d s , such a method should c o n t a i n a s u i t a b l e pore pressure model. Once the convergence cu r v e ( t h e inward movement of the tunnel face p l o t t e d a g a i n s t the support presssure) i s obtained,the convergence-confinement method as d e s c r i b e d i n Smith and Byrne(1980) c o u l d be used to design the confinement system. H a r r i s and Sobkowicz(1977) presented a method of p r e d i c t i n g the gas p o r o s i t y changes i n o i l s a n d s , due to the changes i n t o t a l s t r e s s or temperature,based on i d e a l gas laws.The sand s k e l e t o n was c o n s i d e r e d to be l i n e a r e l a s t i c , and by maint a i n i n g the v o l u m e t r i c c o m p a t i b i l i t y between the pore f l u i d and the skeleton,pore pressure changes were e x t r a c t e d . In 1979, Thurber c o n s u l t a n t s , i n c o l l a b o r a t i o n with Byrne ixtended t h i s to a more r e a l i s t i c case o f , a non l i n e a r e l a s t i c sand s k e l e t o n . Byrne and Grigg(1980) f u r t h e r m o d i f i e d the model by i n t r o d u c i n g p l a s t i c v o l u m e t r i c s t r a i n s , ( d u e to shear d i l a t i o n ) , i n t o the dense sand s k e l e t o n . In t h e i r f i n i t e element 2 f o r m u l a t i o n , a n i t e r a t i v e p r o c e s s i s used t o a c h i e v e v o l u m e t r i c s t r a i n c o m p a t i b i l i t y . The c o n s t i t u t i v e r e l a t i o n s used, a r e of the e q u i v a l e n t e l a s t i c type , where the moduli a r e computed from the s e c a n t s of the non l i n e a r s t r e s s - s t r a i n c u r v e s . In the r e g i o n where the s o i l s t r e n g t h i s f u l l y m o b i l i z e d , which v e r y o f t e n i s the case of t u n n e l o p e n i n g s , t h i s s o l u t i o n i s s t a b l e . The l o a d i s a p p l i e d i n a s i n g l e s t e p and the t o t a l l o a d e q u i l i b r i u m i s s a t i s f i e d . A s o l u t i o n method, whereby e q u i l i b r i u m i s s a t i s f i e d f o r each l o a d increment up t o the f i n a l l o a d , u s i n g moduli o b t a i n e d from the t a n g e n t s of the 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 s a t any s t a g e , i s p r e f e r r e d t o the above one. The s i m u l a t i o n of the c o n s t r u t i o n p r o c e d u r e ( w h i c h e s s e n t i a l l y i s a s t e p by s t e p e x c a v a t i o n ) , a n d the p o s s i b i l i t y of i n c o r p o r a t i n g the i n f l u e n c e of the su p p o r t system,are c o n s i d e r e d t o be advantages of t h i s i n c r e m e n t a l t e c h n i q u e . 1. 2 Purpose and Scope The purpose of t h i s t h e s i s i s t o d e v e l o p a n o n l i n e a r i n c r e m e n t a l f i n i t e element method f o r the a n a l y s i s of the d e f o r m a t i o n b e h a v i o u r of o i l s a n d i n s h a f t s and t u n n e l s . The E n g i n e e r i n g b e h a v i o u r of o i l s a n d i s governed by the s t r e s s e s i n the sand s k e l e t o n . T h i s i s t o be m o d e l l e d by an e f f e c t i v e s t r e s s model which i n c o r p o r a t e s the d i l a t a n t p r o p e r t i e s of the dense o i l s a n d m a t r i x . The Non L i n e a r S o i l S t r u c t u r e I n t e r a c t i o n Program (NLSSIP) d e v e l o p e d by Byrne and Duncan(1979),adopts an i n c r e m e n t a l t e c h n i q u e , a n d i s c a p a b l e of a n a l y s i n g s t a t i c s o i l - s t r u c t u r e i n t e r a c t i o n problems such as f l e x i b l e r e t a i n i n g w a l l s and t u n n e l l i n i n g s . I t i s p o s s i b l e t o a c h i e v e the above purpose by i n t r o d u c i n g the f o l l o w i n g m o d i f i c a t i o n s t o the NLSSIP. The c o n s t i t u t i v e parameters used are inadequate t o cope up w i t h the m o b i l i z a t i o n of the f u l l s o i l s t r e n g t h , and the y i e l d c r i t e r i o n ( M o h r - C o u l o m b ) i s v i o l a t e d by the r e s u l t s under the u n l o a d i n g c o n d i t i o n s . Futhermore, the v o l u m e t r i c s t r a i n s due t o shear d i l a t i o n a r e not c o n s i d e r e d even i n the case of a dense sand. F i r s t phase of the t h e s i s i s t o overcome the v i o l a t i o n of the y i e l d c r i t e r i o n as mentioned above,by employing a s t r e s s r e d i s t r i b u t i o n p r o c e d u r e , and t o improve the v o l u m e t r i c s t r a i n s of the sand s k e l e t o n by ad d i n g a shear d i l a t i o n component , when sands w i t h d i l a t a n t p r o p e r t i e s a re a n a l y s e d . Rowe's s t r e s s d i l a t a n c y t h e o r y i s used t o a s s e s s the a p p r o p r i a t e d i l a t i o n and t h i s i s i n t r o d u c e d i n t o the f i n i t e element f o r m u l a t i o n by u s i n g a t e m p e r a t u r e a n a l o g y . T h i s v e r s i o n of the NLSSIP program, c o u l d be u t i l i z e d i n the a n a l y s i s of any p l a n e s t r a i n problem of a sand, e x h i b i t i n g d i l a t a n t b e h a v i o u r . S e c o n d l y the NLSSIP w i l l be m o d i f i e d t o i n c o r p o r a t e the gas laws r e p r e s e n t i n g the b e h a v i o u r of o i l s a n d p o r e f l u i d . T h i s would enable the program t o p r e d i c t pore p r e s s u r e changes i n each l o a d s t e p under u n d r a i n e d c o n d i t i o n s . The f i n a l s o l u t i o n i s o b t a i n e d by m a i n t a i n i n g the v o l u m e t r i c s t r a i n c o m p a t i b i l i t y between the sand s k e l e t o n and the pore f l u i d . In essence , the work p r e s e n t e d h e r e i n c o n s i s t s of m o d i f i c a t i o n of an i n c r e m e n t a l e f f e c t i v e s t r e s s model , and a method of c o u p l i n g i t w i t h a pore p r e s s u r e model f o r s o i l s c o n t a i n i n g f r e e or d i s s o l v e d gases i n t h e i r pore f l u i d s . F i g . 1 Possible I n s t e r s t i t i a l Conditions of Oilsand. (After Dusseault, 1979) 5 CHAPTER 2 OILSAND BEHAVIOUR B e f o r e a t t e m p t i n g t o p r e d i c t the d e f o r m a t i o n b e h a v i o u r of o i l s a n d , o n u n l o a d i n g , i t i s i m p o r t a n t t o v i s u a l i z e the s t r u c t u r e of o i l s a n d t h a t l e a d s t o i t s c h a r a c t e r i s t i c b e h a v i o u r . G e n e r a l l y o i l s a n d s a r e ve r y dense sands w i t h h i g h s t r e n g t h and low c o m p r e s s i b i l i t y . T h e i n t e r s t i t i a l spaces may c o n t a i n any c o m b i n a t i o n of f l u i d s of the f o l l o w i n g . ( F i g . 1 ) . a. Water o n l y b. Water+oi1(bitumen) c. Water+free gas d. Water+oi1+free gas Another i m p o r t a n t p r o p e r t y of the bitumen r i c h o i l s a n d i s i t ' s v e r y low e f f e c t i v e p e r m e a b i l i t y . W i t h t h i s p i c t u r e i n mind i t i s easy t o p e r c e i v e the o i l s a n d b e h a v i o u r on u n d r a i n e d u n l o a d i n g . In the f i e l d the un u s u a l p r o p e r t i e s of o i l s a n d are m a n i f e s t e d i n a number of w a y s . S w e l l i n g r a n g i n g between 5-15% of the o r i g i n a l volume has been o b s e r v e d , w h e n - o i l s a n d samples are brought t o the s u r f a c e from d e p o s i t s and l e f t i n an u n c o n f i n e d s t a t e . T h e p r o p e r t i e s such as the i r r e c o v e r a b l e r e d u c t i o n i n s t r e n g t h and d e n s i t y a s s o c i a t e d w i t h such s w e l l i n g have a l s o been n o t i c e d . U n l o a d i n g a s s o c i a t e d w i t h the e x c a v a t i o n s f o r t u n n e l s and s h a f t s i n o i l s a n d , causes the d i s s o l v e d gases to e v o l v e and i f the r a t e of l o a d i n g i s h i g h enough , u n d r a i n e d c o n d i t i o n s w i l l r e s u l t due t o the low p e r m e a b i l i t y of the pore f l u i d s . As e x p l a i n e d below the pore p r e s s u r e w i l l then remain h i g h a l l o w i n g the e f f e c t i v e s t r e s s e s t o drop. T h i s l e a d s t o i n s t a b i l i t y a t the fac e of the e x c a v a t i o n . S u r f i c i a l s l a b b i n g of o i l s a n d s i s another consequence of the i n s t a b i l i t y reached under u n d r a i n e d u n l o a d i n g . T h i s happens t o be a v e r y n a t u r a l phenomenon i n the e x c a v a t i o n s f o r open p i t mines , s h a f t s and t u n n e l s i n deep o i l s a n d d e p o s i t s . A t some stage the s l a b s break o f f and f a l l . On the o t h e r hand , n a t u r a l u n l o a d i n g of o i l s a n d d e p o s i t s , s u c h as the e r o s i o n of o i l s a n d s t h a t t a k e p l a c e over y e a r s , l e a v e competent sandstone w i t h no or l i t t l e v i s u a l s l a b b i n g . T h i s p roves how the p r o l o n g e d u n l o a d i n g , f a c i l i t a t e s the gas d r a i n a g e ,thereby p r e v e n t i n g the u n d r a i n e d c o n d i t i o n s . When the t o t a l a l l round s t r e s s a c t i n g on an o i l s a n d specimen w i t h a pore s t a t e d e f i n e d by F i g . I d , i s reduced the (I) INSITU TAR SAND NO GAS P R E S E N T (2) EXPANDED TAR SAND OCCLUDED GAS B U B B L E S (3) EXPANDED TAR SANO GAS B U B B L E S BECOME INTERCONNECTEC L E G E N D V ^ ^ / - Sand Groins o -Occluded Gaj Bubble juum^ -Woter yS? - Gas Void F i g . 2 Unloading of Oilsand (After Harnis and Sobokowicz, 1977) 7 f o l l o w i n g t h i n g s would happen. 1. The o v e r a l l volume w i l l i n c r e a s e , e s s e n t i a l l y due t o an i n c r e a s e i n the pore f l u i d volume. 2. T h i s e x p a n s i o n w i l l p u l l a p a r t the sand g r a i n s t h e r e b y r e d u c i n g the e f f e c t i v e s t r e s s , w h i c h i s p r o p a g a t e d o n l y through the g r a i n c o n t a c t s . C o n s e q u e n t l y the d e n s i t y of the sand m a t r i x as w e l l as i t ' s s t r e n g t h i s reduced. 3. The i n c r e a s e i n pore volume w i l l be accompanied by a d e c r e a s e i n pore p r e s s u r e , w h i c h i n t u r n w i l l cause e v o l u t i o n of the d i s s o l v e d gases ,and f u r t h e r e x p a n s i o n of the f r e e gases. 4. The lower pore p r e s s u r e and the e f f e c t i v e s t r e s s e s as w e l l as the l a r g e r volumes , w i l l r e s u l t i n a h i g h e r pore f l u i d and s k e l e t o n c o m p r e s s i b i l i t y , a s opposed t o i t ' s r e l a t i v e l y low c o m p r e s s i b i l i t y i n the i n i t i a l s t a t e . A l l t h e s e w i l l l e a d t o an u n s t a b l e s t r u c t u r e on u n l o a d i n g , a s o b s e r v e d i n the f i e l d . Yet t h e r e a r e l i m i t a t i o n s t o the above phenomena,depending on the l o c a t i o n of the sample and the d u r a t i o n of u n l o a d i n g . F o r example,in s h a l l o w d e p o s i t s , t h e pore p r e s s u r e a r e not h i g h enough f o r the e x i s t e n c e of p r e s s u r e d i s s o l v e d gases. F u r t h e r m o r e , d e s p i t e the low p r e m e a b i 1 i t y , i f ample time i s a l l o w e d on u n l o a d i n g f o r the e v o l v e d gases t o v e n t , t h e u n d r a i n e d c o n d i t i o n s cannot be m a i n t a i n e d . T h e r e f o r e the s i t u a t i o n w i l l be most c r i t i c a l , i f deep l y i n g o i l s a n d d e p o s i t s , w h i c h a r e r i c h i n b i t u m e n ( w i t h v e r y low p e r m e a b i l i t y ) , and p r e s s u r e d i s s o l v e d g a s e s , a r e unloaded i n a s h o r t d u r a t i o n . F i g . 2 shows how the s t r u c t u r e i s deformed,and v e n t i n g paths are formed on u n l o a d i n g of o i l s a n d s . In the e n s u i n g a n a l y s i s the o i l s a n d s k e l e t o n and the pore f l u i d a re t r e a t e d i n d e p e n d e n t l y . A n o n l i n e a r i n c r e m e n t a l e f f e c t i v e s t r e s s model d e s c r i b e d i n •the f o l l o w i n g Chapter i s used t o r e p r e s e n t the s k e l e t o n b e h a v i o u r . The above d e s c r i b e d pore f l u i d b e h a v i o u r i s r e p r e s e n t e d by a pore p r e s s u r e model based on i d e a l gas l a w s , which i s p r e s e n t e d i n Chapter 5. Under u n d r a i n e d c o n d i t i o n s the s k e l e t o n and the pore f l u i d a r e f i n a l l y c o u p l e d by the v o l u m e t r i c s t r a i n c o m p a t i b i l i t y c o n d i t i o n . 6 CHAPTER 3 EFFECTIVE STRESS MODEL The o i l s a n d s k e l e t o n i s t r e a t e d as a continuum , i n the e f f e c t i v e s t r e s s model. T h e r e f o r e continuum mechanics i s a p p l i e d i n the s t r e s s s t r a i n a n a l y s i s . 3.1 S t r e s s e s , T e r z a r g h i d e f i n e d the e f f e c t i v e s t r e s s i n the f o l l o w i n g manner. a ' = a - u ( i ) where o - e f f e c t i v e normal s t r e s s on any p l a n e 0" - t o t a l normal s t r e s s on t h a t p l a n e U - pore p r e s s u r e R e c e n t l y Skempton(1957), i n t r o d u c e d 3 t h e o r i e s on how, e f f e c t i v e s t r e s s depends on the g r a i n c o n t a c t s and the i n t e r n a l f r i c t i o n , i n s a t u r a t e d porous media.But under most p r a c t i c a l s i t u a t i o n s T e r z a g h i ' s t h e o r y was v a l i d a t e d . F u r t h e r s i n c e water c a r r i e s no shear s t r e s s e s : X' = T (2) f ' and X b e i n g the e f f e c t i v e and the t o t a l shear s t r e s s e s on the p l a n e . 3 . 2 Deformat i o n s and S t r a i n s I f u, v, and w are the d i s p l a c e m e n t s a t a p o i n t i n a c ontinuum,along the x, y, and z d i r e c t i o n s , i n a c a r t e s i a n c o o r d i n a t e system, the s i x s t r a i n s can be d e f i n e d a s , ( 3 ) dx" y dy i 2* x x dy dx 7 2 a z °y ax where o n l y the l i n e a r terms of the d i s p l a c e m e n t s a r e c o n c e rned. 3.3 C o n s t i t u t i v e R e l a t i o n s V a r i o u s forms of e l a s t i c and p l a s t i c c o n s t i t u t i v e r e l a t i o n s are used i n S o i l mechanics. Most s o i l s e x h i b i t e l a s t i c b e h a v i o u r under low s t r e s s l e v e l s . The s i m p l e s t e l a s t i c model i s the l i n e a r e l a s t i c one. o re a xi a I s t r a i n F i g . 3 Nonlinear E l a s t i c Curve F i g . 4 Hyperbolic Curve 3.3.1 L i n e a r E l a s t i c i t y L i n e a r e l a s t i c c o n s t i t u t i v e r e l a t i o n s would be i n the f o l l o w i n g form. where c o e f f i c i e n t s o = ( D ) e (A) ? - s t r e s s v e c t o r . £ - s t r a i n v e c t o r . [D] ~ c o n s t i t u t i v e m a t r i x w i t h 36 By l i m i t i n g our i n t e r e s t t o i s o t r o p i c m a t e r i a l s and assuming symmetry of (D) ,we can reduce these 36 m a t e r i a l c o n s t a n t s t o any two of the f o l l o w i n g d e f i n e d . Young's modulus (E) = / £x (5) P o i s s o n ' s r a t i o ( p ) - - £x / £y (6) Bulk modulus (B) = 0"m/£ y (j) where ( O r n = 3 ( °x + ay **" mean normal s t r e s s (8) 6V = cSx + f y t f g - v o l u m e t r i c s t r a i n . ^o) Shear modulus (G) = TXy / (jo) U n l i k e f o r most common m a t e r i a l s , v e r y o f t e n the b e h a v i o u r of s o i l cannot be r e p r e s e n t e d by l i n e a r e l a s t i c models.The s t r e s s - s t r a i n parameters a r e h i g h l y dependent on the magnitudes of the s t r e s s e s i n the s o i l . For the p r e d i c t i o n of r e a l i s t i c r e s u l t s , the c o n s t i t u t i v e r e l a t i o n s used i n the s o i l d e f o r m a t i o n a n a l y s i s s h o u l d be r e a s o n a b l e f o r t h a t s o i l . T h e non l i n e a r e l a s t i c approach encompass two s i g n i f i c a n t c h a r a c t e r i s t i c s of the . s t r e s s - s t r a i n b e h a v i o u r of s o i l s , namely, n o n - l i n e a r i t y and s t r e s s dependency. In t h i s a n a l y s i s , n o n l i n e a r e l a s t i c c o n s t i t u t i v e r e l a t i o n s a r e used. F u r t h e r the p l a s t i c volume changes a r e a l s o i n c o r p o r a t e d by u s i n g a shear volume c o u p l i n g c o n c e p t . 3.3.2 N o n l i n e a r E l a s t i c i t y The use of 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 s ( F i g . 3) i s common i n n o n l i n e a r e l a s t i c i t y . Two d i s t i n c t models can be i d e n t i f i e d depending on the method of t r a c i n g t h e s e c u r v e s . 3.3.2.1 E q u i v a l e n t E l a s t i c or Secant Method The moduli a r e d e f i n e d by the i n i t i a l and the a n t i c i p a t e d f i n a l s t a t e . E s = £ < * (II) T h i s method , works w e l l when the f a i l u r e i s approached , where the c u r v e i s f l a t t e r . 3.3.2.2 I n c r e m e n t a l or Tangent Method The c u r v e i s more c l o s e l y t r a c e d , s i n c e l o a d i n g i s done i n a s e r i e s of i n c r e m e n t s , t h e moduli f o r each one i s d e f i n e d by the tangent t o the c u r v e , a t the average s t r e s s s t a t e . T h i s method resembles the u s u a l c o n s t r u c t i o n sequence and as such i s advantageous over the p r e v i o u s o n e , a l t h o u g h i t may be a l i t t l e u n s t a b l e i n the f a i l u r e r e g i o n , w h i c h a s p e c t i s d i s c u s s e d i n d e t a i l l a t e r . T h i s a n a l y s i s i n v o l v e s an i n c r e m e n t a l approach and from here onwards i t w i l l be d i s c u s s e d i n more d e t a i l . I n c r e m e n t a l Young's modulus ( E + ) = d®> (\2) I n c r e m e n t a l Bulk modulus (B^) = d£t» (13J where O, and o~3 - a x i a l and l a t e r a l p r i n c i p a l s t r e s s e s . ^1 - a x i a l s t r a i n ( i n t r i a x i a l t e s t s ) . ° m , £y - as d e f i n e d i n Chapter 3.3.1 U s i n g l i n e a r e l a s t i c p r i n c i p l e s f o r each i n c r e m e n t , i t i s shown i n Appendix A . l t h a t the I n c r e m e n t a l P o i s s o n ' s r a t i o ( p ^ ) = 3.3.3 H y p e r b o l i c R e l a t i o n Kondner and Z e l a s k o ( 1 9 6 3 ) , h a v e shown 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 many s o i l s can be a p p r o x i m a t e d by h y p e r b o l a s of the form, (°. " °s) = I \ £ O5) Ei (°'- C Tj)u!t One advantage of using t h i s , i s that the equation parameters, Ej and (tT, -Ci)uft have p h y s i c a l s i g n i f i c a n c e , b e i n g the i n i t i a l tangent modulus and the asymptotic value of d e v i a t o r s t r e s s , r e s p e c t i v e l y , a s shown i n F i g . 4. I t should be noted here t h a t , i n the t r i a x i a l t e s t , s i n c e c 3 i s u s u a l l y kept c o n s t a n t , E j (def ined by d O d / d c ) becomes a tangent Young's modulus . (def ined by do, /<j£ ). 3.3.4 Dependency of E- on c o n f i n i n g pressure I t has been observed that f o r s o i l s (except f o r those t e s t e d under u n c o n s o l i d a t e d undrained c o n d i t i o n s ) , a n i n c r e a s e i n c o n f i n i n g pressure w i l l r e s u l t i n a steeper s t r e s s - s t r a i n curve and a higher u l t i m a t e s t r e n g t h . Consequently and (P\~0^a^ should i n c r e a s e with i n c r e a s i n g c o n f i n i n g p r e s s u r e . E m p i r i c a l formulae have been presented to represent t h i s v a r i a t i o n . Janbu presented the f o l l o w i n g equation f o r E^ where the dimensionless parameters, K E and n are r e f e r r e d to as the modulus number and the modulus exponent r e s p e c t i v e l y . Pa - the atmospheric pressure has been in t r o d u c e d to make the equation non dim e n s i o n a l . U n i t s of 0 3 , Pa and E ^ should be c o n s i s t e n t . 3.3.5 F a i l u r e R a t i o In order to determine ul-b , i t i s e m p i r i c a l l y r e l a t e d to ( 0 , - 0 3 ) C (the shear s t r e n g t h ) , by the f a i l u r e r a t i o as f o l l o w s . ' (o,-o3)u|t = (kr5»2f ( ,7) R-f - F a i l u r e r a t i o Since (CJ (-cT 3^j i s always smaller than (tT, - Oj) u ( t ,R^ . i s l e s s than unity,and v a r i e s from 0.5 to 0.9 f o r most s o i l s . 3.3.6 Y i e l d C r i t e r i o n The y i e l d c r i t e r i o n w i l l determine the maximum shear s t r e n g t h that c o u l d be developed by the s o i l . H e r ein, the Coulomb c r i t e r i o n i s used. Xj. = C -+• a ' t o n <f> C'8) F i g . 5 V a r i a t i o n of <f> with a^. Fig- 6. Unloading-reloading Curve Tj - s t r e n g t h developed on the f a i l u r e plane o' - e f f e c t i v e normal s t r e s s on the f a i l u r e plane. 0 -maximum apparent angle of f r i c t i o n C -apparent cohesion. Experimental r e s u l t s have shown that the value of 0 decreases , c o n s i d e r a b l y with the in c r e a s e i n c o n f i n i n g p r e s s u r e , as shown i n F i g . 5. Thi s i s accommodated in the a n a l y s i s by using the f o l l o w i n g equat i o n . 0 = CT<( - . ^ " l o g ^ ) (.9) 0^ 0 i - F r i c t i o n angles at conf. p r e s s u r e s of 0~3 and Pa resp. -Reduction i n 0 , for a t e n f o l d i n c r e a s e in O*, . 3.3.7 R e l a t i o n between E^and s t r e s s e s The tangent modulus can be obtained by d i f f e r e n t i a t i n g Eqn. 15 w.r.t. £ , and by the s u b s t i t u t i o n of E c, and (cr,-Cfr) U|.t , from Eqn. 16 and 17,together with the y i e l d c r i t e r i o n ( r e p r e s e n t e d by Eqn. 18). The e x p r e s s i o n ; as d e r i v e d i n the Appendix A.2 i s as f o l l o w s , E t - K E P a ( £ R j ( l - 5 , 0 0 ) ( g , - o 3 ) 2CQS(p +2a3 Sin (j) .2 (20) If the s o i l parameters given by K E , n, c,(/), and are known,this e x p r e s s i o n can be used to o b t a i n the instantaneous e l a s t i c modulus,at any d e v i a t o r s t r e s s and any c o n f i n i n g p r e s s u r e . 3.3.8 Other Non L i n e a r R e l a t i o n s 3.3.8.1 Unloading Reloading Modulus Duncan,Byrne,Wong and Mabry (1978) d e s c r i b e a way of r e p r e s e n t i n g , the i n e l a s t i c behaviour of s o i l by using d i f f e r e n t moduli values f o r l o a d i n g and unl o a d i n g . If a sample i s unloaded d u r i n g the progress of a t r i a x i a l t e s t , a n d reloaded the s t r e s s - s t r a i n curve i n F i g . 6 i s obtained. F i g . 7 Volume Change Curves The u n l o a d i n g (and r e l o a d i n g ) c u r v e , i s s t e e p e r than the pri m a r y l o a d i n g c u r v e t h e r e b y l e a v i n g an i r r e c o v e r a b l e strain(£p) on u n l o a d i n g . B y assuming t h a t u n l o a d i n g and r e l o a d i n g i s l i n e a r e l a s t i c and t h a t t h e r e a r e no h y s t e r e s i s e f f e c t s , t h e y p r e s e n t the f o l l o w i n g e x p r e s s i o n f o r E y r . E u r = "<ur Pa ( % ) " (2» From F i g . 6. e p - € - er (23; c % = e l a s t i c ( o r r e c o v e r a b l e ) a x i a l s t r a i n . £p=plastic(or i r r e c o v e r a b l e ) a x i a l s t r a i n . I t has been found o u t , t h a t the r a t i o U r/K d e c r e a s e s w i t h the i n c r e a s i n g s o i l dens i t y . Ther ef or e El«ry ^  r a t i o , wi 11 be s m a l l e r f o r the dense s a n d s . T h i s i m p l i e s t h a t a t v e r y low s t r e s s l e v e l s , £ r /e.j, r a t i o w i l l be h i g h f o r dense sands. 3.3.8.2 E l a s t i c Volume change The volume change c h a r a c t e r i s t i c s o b t a i n e d when t r i a x i a l c o mpression t e s t s a r e performed under d i f f e r e n t c o n f i n i n g p r e s s u r e s a r e shown i n F i g . 7. I t i s seen t h a t the v o l u m e t r i c c o m p r e s s i o n i s h i g h l y dependent on C3. I t can be assumed t h a t the e l a s t i c volume changes are independent of the d e v i a t o r s t r e s s , and depend o n l y on Oj . Thus, i n non l i n e a r e l a s t i c i t y , t h e b u l k modulus i s d e f i n e d a s , 6 = K a P a f i 3 ) m (24) where K B - Bulk modulus number. m - Bulk modulus exponent. Pa - Atmospheric p r e s s u r e . K &and m, j u s t as IC^and n, are d i m e n s i o n l e s s . Duncan e t a l ( 1 9 7 8 ) , s a y t h a t u s u a l l y m ranges between 0.0 and 1.0,except i n c l a y s compacted t o dry of optimum,tested under u n d r a i n e d c o n d i t i o n s , w h o s e b u l k modulus d e c r e a s e w i t h the )8 o r i g i n a l h y p e r b o l a — N L S S I P c u r v e e l a s t o p l a s t i c c u r v e (constant ) ^ C. F i g . 8 Stress S t r a i n Curves F a i l u r e l i n e T a n 0 = S i n <£> max m F i g . 9 V i o l a t i o n of the F a i l u r e C r i t e r i o n (9 i n c r e a s e i n . 3.3.9 C o n s t i t u t i v e r e l a t i o n s i n the f a i l u r e r e g i o n . Once f a i l u r e i s reached,the s h e a r i n g w i l l be e n t i r e l y governed by the Mohr-Coulomb f a i l u r e c r i t e r i o n . In the f a i l u r e r e g i o n °d = ^ c £ o s ^ t- 2 J 3 Sin 4 ^25) The v o l u m e t r i c s t r a i n s are s t i l l p r e d i c t e d by Eqns 24 The NLSSIP uses the parameters o b t a i n e d from above e q u a t i o n s ( i . e E t , B-t e t c ) , i n f o r m i n g i t ' s c o n s t i t u t i v e m a t r i x as d i s c u s s e d i n Chapter 4.In the f a i l u r e r e g i o n , i n s p e c t i o n of Eqn 25 w i l l r e v e a l t h a t E t , (which i s doa/^g) , wi 11 be zero,whereas B ^ w i l l have a f i n i t e v a l u e , f r o m Eqn.24. "t- i('-lfc) « From t h i s i t f o l l o w s t h a t , ,. E t _ Thus the t a n g e n t i a l shear modulus ( G ^ = 2(1+ pt) * r°m Appendix D ) , w i l l a l s o become z e r o a t f a i l u r e , ( s i n c e E ^ = 0 ) . I f t h i s c o n d i t i o n i s used as i t i s , Eqn. 67 w i l l produce a s i n g u l a r c o n s t i t u t i v e m a t r i x . T h i s c o u l d be p r e v e n t e d by a v o i d i n g the l i m i t i n g v a l u e of Pt of 0.5 and use 0.495 in s t e a d . T h e n i f the B v a l u e from Eqn. 24 i s b e l i e v e d i n the f a i l u r e r e g i o n , E f c w i l l get a v e r y low v a l u e compared t o B, , from Eqn. A3; i .e E -t= 0.03 B t and G ^ = 0.01 F i g . 8 shows the f i n a l s t r e s s - s t r a i n c u r v e used i n the p r e s e n t program. The i n a c c u r a c y of these u n s t a b l e c o n d i t i o n s i s r e f l e c t e d by the v i o l a t i o n of the f a i l u r e c r i t e r i o n , b y the r e s u l t s o b t a i n e d i n problems a s s o c i a t e d w i t h u n l o a d i n g of a s o i l . T h e problem of u n l o a d i n g of a t u n n e l under p l a n e s t r a i n c o n d i t i o n s , which w i l l be a n a l y s e d l a t e r i s a good example. I n i t i a l l y the program p r e d i c t e d a s t r e s s p a t h t r a v e l l i n g above the Mohr-Coulomb f a i l u r e l i n e ( a l o n g QR), when the mean normal s t r e s s of the s o i l a d j a c e n t t o the c a v i t y i s reduced due t o u n l o a d i n g , a s shown i n F i g . 9,whereas i t s h o u l d have t r a v e l l e d a l o n g QS. E x c e s s shear s t r a i n s were p r e d i c t e d f o r a p a r t i c u l a r mean normal s t r e s s . T h i s s h e a r , i n exces.s of the shear s t r e n g t h was d i s t r i b u t e d t o the s u r r o u n d i n g elements by a p p l y i n g an e q u i v a l e n t s e t of nod a l f o r c e s . T h e t e c h n i q u e used w i l l be d i s c u s s e d a l o n g w i t h the f i n i t e element f o r m u l a t i o n . 3.4 Comment on the Non l i n e a r E l a s t i c parameters The parameters have been found u s e f u l f o r many r e a s o n s . 1. They can be de t e r m i n e d from t r i a x i a l c o m p r e s s i o n t e s t r e s u l t s u s i n g s i m p l e p l o t t i n g t e c h n i q u e s . 2. They c o u l d be a p p l i e d t o e f f e c t i v e s t r e s s ( t h e parameters b e i n g o b t a i n e d from d r a i n e d t e s t s ) , a n d t o t a l s t r e s s (by u s i n g u n c o n s o l i d a t e d u n d r a i n e d t e s t r e s u l t s ) , a n a l y s e s . 3. S i n c e d a t a i s a v a i l a b l e f o r a wide v a r i e t y of s o i l s , a p p r o x i m a t e v a l u e s f o r a problem i n hand are e a s i l y a c c e s s i b l e , w h i c h i s of immense h e l p under d i f f i c u l t t e s t i n g s i t u a t i o n s . On the o t h e r hand the volume changes (or d e f o r m a t i o n s ) g i v e n by these r e l a t i o n s a r e i n c o m p l e t e as the p l a s t i c volume changes or the shear induced d i l a t i o n has not been i n t r o d u c e d . I n t h i s a n a l y s i s they have been c o n s i d e r e d a l o n g the g u i d e l i n e s of Rowe's d i l a t a n c y t h e o r y . 3.5 D e t e r m i n a t i o n of Non L i near Parameters D e t e r m i n a t i o n of K E , n , 0 , &0 , and R^ i s d e s c r i b e d i n d e t a i l i n Duncan e t a l (1978). S i n c e the volume changes are m o d i f i e d , h e r e i n a new method of e v a l u a t i n g and m i s sugges t e d . 3.5.1 E v a l u a t i o n of K g and m The method adopted by Duncan et a l ( 1 9 7 8 ) c o u l d be o u t l i n e d as f o l l o w s . U s i n g the volume change c u r v e s shown i n F i g . 7 , t h e Bulk modulus f o r each O ^ i s c a l c u l a t e d . From B = dcr, (27) we get = A (cr, - f az -r 3 Acfy (28) For t r i a x i a l t e s t s w i t h c o n s t a n t c o n f i n i n g p r e s s u r e , a x i a l s t r a i n F i g . 10 Isotropic Consolidation F i g . 11 Determination of Kg. or B = A (<Jt - CTa) 3 A € v ( 3 0 ) I f B i s e v a l u a t e d a t s e v e r a l v a l u e s of a x i a l s t r a i n , i t i s n o t i c e d t h a t the v a l u e s o b t a i n e d would be d i f f e r e n t from each o t h e r . I f the assumption , t h a t B i s independent of the d e v i a t o r s t r e s s , i s r e a s o n a b l e ( f o r e l a s t i c volume c h a n g e s ) , t h i s o n l y p o i n t s out the f a c t t h a t i n e l a s t i c volume changes(depending on the d e v i a t o r s t r e s s ) , a l s o c o n t r i b u t e t o the volume changes ob s e r v e d i n F i g . 7. S i n c e we take the shear volume c o u p l i n g i n t o account i n our model,we r e f r a i n from u s i n g Eqn. 11 t o c a l c u l a t e B-the modulus r e l a t e d t o the e l a s t i c volume changes. Duncan and o t h e r s , c i r c u m v e n t t h i s problem by s p e c i f y i n g the s t r e s s l e v e l at which,B i s t o be e v a l u a t e d , as f o l l o w s , l . I f the volume change c u r v e does not reach a h o r i z o n t a l tangent p r i o r t o the stage a t which 70% of the s t r e n g t h i s m o b i 1 i z e d , u s e the p o i n t s on the s t r e s s - s t r a i n and volume change c u r v e s c o r r e s p o n d i n g t o a s t r e s s l e v e l of 70%. 2 . I f the volume change c u r v e reaches a h o r i z o n t a l tangent p r i o r t o the s t a g e a t which 70% of the s t r e n g t h i s m o b i l z e d , i n which case the p o i n t s s e l e c t e d f a l l s on the d i l a t i n g p a r t , t h e p o i n t c o r r e s p o n d i n g t o the i n f l e x i o n p o i n t i n the £ y Vs.£ c u r v e , i s s e l e c t e d so as t o a v o i d the p l a s t i c volume change.This s i t u a t i o n u s u a l l y o c c u r s f o r h i g h l y d i l a t a n t sands. The method suggested h e r e i n , i s based on i s o t r o p i c c o n s o l i d a t i o n i n the t r i a x i a l a p p a r a t u s . I f a c o n s t a n t a l l round p r e s s u r e i s a p p l i e d t o a s o i l sample, the volume change c h a r a c t e r i s t i c s i n F i g . 10 are o b t a i n e d f o r d i f f e r e n t p r e s s u r e s . U n d e r the i s o t r o p i c c o n d i t i o n s , i n the absence of shear s t r e s s e s the shear d i l a t a n c y w i l l not be e x h i b i t e d . T h u s i t i s r e a s o n a b l e t o assume,that the maximum £ v o b t a i n e d i s the t o t a l e l a s t i c volume change due t o the c o r r e s p o n d i n g c o n f i n i n g p r e s s u r e . B = GO I n f a c t t h i s c o u l d be done as the f i r s t s t a g e of a l l p r e v i o u s measurements,since the sample has t o be c o n s o l i d a t e d t o 0*3 , i n any CU or CD t e s t b e f o r e s h e a r i n g i s i n t r o d u c e d . Once B v a l u e s f o r a number of v a l u e s ( i n the working r a n g e ) , a r e c a l c u l a t e d , t h e n o r m a l i s e d B ( i . e B/Pa) can be p l o t t e d a g a i n s t n o r m a l i s e d Oj ( i . e CTj/Pa) on a l o g a r i t h a m i c s c a l e . 21 F i g . 12 Shear D i l a t i o n From Eqn. 24 By t a k i n g l o g a r i t h a m s , t m The p l o t as shown i n F i g . 11. would be l i n e a r , w i t h the i n t e r c e p t and the g r a d i e n t ( p r o p e r l y c o n v e r t e d ) , i n d i c a t i n g the At t h i s s t a g e , i t s h o u l d be mentioned t h a 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 and the parameters have been w i d e l y a p p l i e d i n S o i l Dynamics problems (such as l i q u i f a c t i o n problems) as w e l l . I n t h i s c l a s s of a p p l i c a t i o n s i t i s common t o r e l a t e the moduli parameters t o the r e l a t i v e d e n s i t y of the s o i l , w h i c h i s j u s t i f i e d as r e l a t i v e d e n s i t y ( D f ) c o u l d p o s s i b l y be the dominant f a c t o r i n the moduli parameters.Such e m p i r i c a l r e l a t i o n s h i p s between the shear modulus parameter and D r have been p r e s e n t e d by H a r d i n and D r n e v i c h ( 1 9 7 2 ) . 3.6 Shear Volume C o u p l i n g A c h a r a c t e r i s t i c f e a t u r e of any l i n e a r or a non l i n e a r e l a s t i c t h e o r y i s the shear-volume d e c o u p l i n g , i . e the assumption t h a t volume changes can o n l y be caused by a change i n mean normal s t r e s s , and not by sh e a r i n g . M o s t s t r u c t u r a l m a t e r i a l s l i k e m e t a l s t i m b e r and c o n c r e t e behave i n t h i s way,with the volume changes and s h e a r i n g t a k i n g p l a c e i n d e p e n d e n t l y . On the o t h e r hand shear induced volume changes are a s a l i e n t f e a t u r e , i n sands and c l a y s . F o r i n s t a n c e , i f a s i m p l e shear t e s t i s performed on a, sample of dense or medium dense sand,volume change c h a r a c t e r i s t i c s as shown i n F i g . 12 a r e pbta i n e d . W i t h the m o b i l i z a t i o n of the shear s t r e n g t h , i n i t i a l l y a r e d u c t i o n i n volume o c c u r s . T h i s p a r t d i m i n i s h e s i f the sand i s made d e n s e r . L a t e r under h i g h e r shear s t r a i n s , a marked i n c r e a s e i n volume t a k e s p l a c e , e v e n though t h e ' e x t e r n a l normal l o a d s a re kept c o n s t a n t . T h i s phenomenon i s termed the shear induced d i l a t i o n i n S o i l mechanics, and i s predominant i n dense and medium dense sands. The maximum d i l a t i o n r a t e u s u a l l y o c c u r s i n the f a i l u r e r e g i o n , where the f u l l shear s t r e n g t h of the sand i s m o b i l i z e d . When the shear s t r a i n s r e a c h v e r y h i g h v a l u e s ( 2 0 - 3 0 % ) , d i l a t i o n c e a s e s . K B and m v a l u e s . When t r i a x i a l c ompression t e s t s a r e performed on medium dense sands,the volume, of the sample s t a r t s i n c r e a s i n g r a p i d l y a f t e r the i n i t i a l d e c r e a s e , i n s p i t e of the i n c r e a s i n g ( c o m p r e s s i v e ) mean normal s t r e s s . T h i s a g a i n p r o v i d e s a c l a s s i c example f o r the d e m o n s t r a t i o n of shear d i l a t i o n i n sands. A s i g n i f i c a n t f e a t u r e of t h i s d i l a t i o n i s t h a t u n l i k e the b u l k of the volume change due t o any change i n mean normal s t r e s s , i t i s t o t a l l y i r r e c o v e r a b l e . I t i s thought t o be due t o p l a s t i c d e f o r m a t i o n s i n the sand s k e l e t o n , such as s l i p between g r a i n s , t o assume a new arrangement. In the c a v i t y e x p a n s i o n problems i n dense sands(as t h a t t a k i n g p l a c e i n p r e s s u r e m e t e r t e s t s ) , the d i l a t i o n has an enormous e f f e c t on the boundary d e f l e c t i o n s . O n the o t h e r hand i n the u n d r a i n e d c y c l i c l o a d i n g of sands,a shear induced c o n t r a c t i o n w i l l g i v e r i s e t o r a p i d i n c r e a s e s i n pore p r e s s u r e , c a u s i n g l i q u i f a c t i o n p r o b l e m s . T h e r e f o r e i n the a n a l y s i s of medium dense or dense sands,the assumption of shear volume d e c o u p l i n g appears t o be a poor one. The fundamental r e l a t i o n s h i p i n shear volume c o u p l i n g , as used i n t h i s model, c o u l d be e x p r e s s e d a s , A£ = A £ „ ) + A £ v ) ( 3 5 ; V ' e l a s t i c ' b f a s b c U s i n g the d e f i n i t i o n of B, we can s i m p l i f y t h i s t o g i v e , B where A/=shear s t r a i n increment and f = f u n c t i o n of the d e n s i t y ,the maximum shear s t r e n g t h , o f the s o i l and the s t r e s s l e v e l . D i f f e r e n t t h e o r i e s t h a t account f o r t h i s d i l a t a n t b e h a v i o u r of sand, have been put forward.The r e v i s e d m o d i f i e d Cam-Clay model w i t h k i n e m a t i c hardening,and Rowe's s t r e s s - d i l a t a n c y t h e o r y a r e two of them. The p h y s i c a l f o r m u l a t i o n of the l a t t e r i s more a t t r a c t i v e and l o g i c a l , -while the p r e d i c t e d r e s u l t s seem to be i n good agreement w i t h the o b s e r v e d ones. Rowe's p r i n c i p l e s i n c l u d e d i n t h i s a n a l y s i s w i l l be o u t l i n e d n e x t . 3.7 B a s i c Concepts of Rowe's S t r e s s - P i l a t a n c y Theory In c o n t r a s t t o the p o p u l a r approach t o s t r e s s - s t r a i n r e l a t i o n s f o r s o i l s by assuming e l a s t i c b e h a v i o u r ( b y Duncan et a l ) , o r by a p p l y i n g c o n c e p t s of p l a s t i c i t y t h e o r y ( R o s c o e , P r e v o s t et a l ) , a study of p a r t i c u l a t e mechanics can a l s o be used t o e x p l a i n the b e h a v i o u r of s o i I s , e s p e c i a l l y i n the case of dense and medium dense sand a s s e m b l i e s . Rowe has a t t e m p t e d t o u n d e r s t a n d the mechanics g o v e r n i n g 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 by t r e a t i n g s o i l as d i s c r e t e p a r t i c u l a t e m a t t e r . S t r e s s - D i l a t a n c y t h e o r y i s a r e s u l t of h i s p r o g r e s s i n t h i s d i r e c t i o n . Rowe f i r s t i n t r o d u c e d t h i s t h e o r y i n 1963,where energy r a t i o c r i t e r i o n i s used t o . determine the c r i t i c a l a n g l e of s l i d i n g i n a random assembly of p a r t i c l e s . T h i s o r i g i n a l form was s u b j e c t e d t o c r i t i c i s m , i n the hands of Gibson and Morgenstern(1963),Roscoe and S c h o f f i e l d ( 1 9 6 4 ) , a n d o t h e r w o r k e r s . L a t e r Home (1965), Barden (1966), r e a n a l y s e d the t h e o r y behind,and e s t a b l i s h e d i t on f i r m e r s c i e n t i f i c g r ounds.In 1971 Rowe p r e s e n t e d the m o d i f i e d t h e o r y , g i v i n g s u b s t a n t i a l e x p e r i m e n t a l e v i d e n c e f o r dense and medium dense sands , i n support of i t . The f o l l o w i n g assumptions form the framework of the t h e o r y , a. The number of r o l l i n g c o n t a c t s ' i n a p a r t i c l e assembly i s n e g l i g i b l e compared t o the number of s l i d i n g c o n t a c t s , as s l i d i n g was shown t o be a more s t a b l e mechanism of d e f o r m a t i o n than r o l l i n g . b. At any i n s t a n t s l i d i n g i s c o n f i n e d t o some p r e f e r r e d a n g l e . c. D e f o r m a t i o n i s a r e s u l t of r e l a t i v e motion between temporary r i g i d group of p a r t i c l e s , which can c o n t i n u o u s l y r e f o r m by d i v i s i o n and c o a l e s c e n c e . d. The i n d i v i d u a l p a r t i c l e s a r e r i g i d , a n d c o u l d not c o n t r i b u t e an e l a s t i c component towards the d e f o r m a t i o n . As shown i n Appendix B,by b a s i c mechanics and the a p p l i c a t i o n of minimum energy p r i n c i p l e ( i . e , t h e s l i d i n g a n g l e i s chcsen such t h a t the energy absorbed f o r a g i v e n energy i n p u t , i s a minimum), r e s u l t s i n the f o l l o w i n g r e l a t i o n s h i p , a and a r e cTj have t h e i r u s u a l the i n c r e m e n t s i n meaning. volumet r i c s t r a i n , a n d the major f r i c t i o n . wnere dVand d£, p r i n c i p a l s t r a i n r e s p . (f> i s the a n g l e of i n t e r p a r t i c l e I t s h o u l d be mentioned here t h a t Eqn. 3 7 , i s v a l i d under the c o n d i t i o n s bounded by the above a s s u m p t i o n s . B u t i n p r a c t i c e r o l l i n g and r o t a t i o n occur i n a d d i t i o n t o s l i p , t h e r e b y l o w e r i n g the energy r a t i o below the one s p e c i f i e d i n the Appendix B. Moreover a t h i g h d i l a t a n c y r a t e s , t h e groups s l i d e a t v. r i o u s a n g l e s i n c o n t r a s t t o the assumption b. C o n s i d e r a t i o n of these,and o b s e r v a t i o n of t r i a x i a l t e s t r e s u l t s , h a v e l e a d t o the p r e s e n t a t i o n of the s t r e s s - d i l a t a n c y r e l a t i o n a s ; f o r c o h e s i o n l e s s s o i l s . At the c r i t i c a l s t a t e , where the s o i l deforms under c o n s t a n t volume, 4y _ o ... aj , ^ ( 4 5 . ^ ( 3 9 ) But we a l s o know t h a t at v e r y l a r g e s t r a i n s , oy N 2 / From Eqns 39 and 40, i t f o l l o w s t h a t a t the c r i t i c a l s t a t e ( w h e r e a l l the d i l a t i o n has ceased) (fii, - $<y. where 0cv =angle of i n t e r n a l f r i c t i o n measured under c o n s t a n t volume d e f o r m a t i o n . Rowe s p e c i f i e d the range of v a l u e s of ^ a s f o l l o w s . I t can be c o n c l u d e d t h a t , a lower bound f o r (pj i s p r o v i d e d by the i n t e r p a r t i c l e f r i c t i o n a n g l e ( e f f e e t i v e under i d e a l c o n d i t i o n s s p e c i f i e d by the assumptions),whereas the a n g l e of f r i c t i o n a t c o n s t a n t volume d e f o r m a t i o n , p r o v i d e s the upper bound. Very o f t e n the r e l a t i o n s h i p i s e x p r e s s e d as, / R =K D (42) where R =0T'/o3-the p r i n c i p a l s t r e s s r a t i o . P = I - dv/de, ^ F i g . 13 P l a s t i c P o t e n t i a l rt - >> fif -t>i = to, sand, seem Experimental o b s e r v a t i o n s have shown t h a t , f o r dens?, sand, at pre-peak s t r a i n s at l a r g e s t r a i n s f o r loose sands, at a l l s t r a i n s Under plane s t r a i n c o n d i t i o n s T r i a x i a l t e s t r e s u l t s f o r to be i n good agreement with the theory , a c c o r d i n g to Rowe (197.1) . 3.7.1 P l a s t i c P o t e n t i a l s A s i g n i f i c a n t f e a t u r e of Eqn 38,is that i t can be'used as a flow r u l e , i . e a r e l a t i o n s h i p governing the d i r e c t i o n of the p l a s t i c s t r a i n increment.(such as the n o r m a l i t y c o n d i t i o n used in t h e o r i e s of p l a s t i c i t y ) . dv = d£ j -»-d£^ ( f o r plane s t r a i n c o n d i t i o n s ) S u b s t i t u t i n g i n Eqn. 38, cr,' _ _ d £ 3 K r e w r i t i n g ai da, d£ , _ _ qi K (47) d £ 3 a , ' n or which i s the form of a flow r u l e i n t h e o r i e s of p l a s t i c i t y . T h e presence of a flow r u l e g i v e s r i s e to the concept of p l a s t i c  p o t e n t i a l which i s d e f i n e d as the s u r f a c e to which the p l a s t i c s t r a i n increment i s normal at the c u r r e n t s t r e s s p o i n t . In F i g . 13, PS i s the p l a s t i c p o t e n t i a l and PQ i s the s t r a i n increment v e c t o r . 3.8 S t r e s s - P i l a t a n c y R e l a t i o n s h i p f o r Plane S t r a i n C o n d i t i o n s As d i s c u s s e d p r e v i o u s l y , t h e Eqn. 38 c o u l d be r e w r i t t e n f o r the s p e c i a l case of plane s t r a i n deformation as, 0' = tanZ( 45+ & * V i - d v / \ (m) dv = d £ , + 4£i (so) d £ 3 •»£, F i g . 14 Mohr C i r c l e for Strains F i g . 15 Mohr C i r c l e for Stresses F i g . 16 D i l a t i o n Angle d £ t , d£ 3 a r e the major and minor p r i n c i p a l s t r a i n i n c r e m e n t s i n the x-y p l a n e . ( T h e t h i r d component d £ g b e i n g z e r o ) . I f d y i s the maximum shear s t r a i n increment,as" shown i n F i g . 14, d / a d £ , - <*£j O O From Eqns. 50 and 51, dv + d 1 * 2 <* £ i 62) S u b s t i t u t i o n i n Eqn. 49 g i v e s , d 2+ civ or A Mohr c i r c l e c o n s i d e r a t i o n , a s i n F i g . 1 5 , w i l l show t h a t , f o r c o h e s i o n l e s s s o i l , ql . I + Sin 0 a = t a n * ( 4 5 4 - A ) (55) I - Sin 0 d ^ 2 / 0 ^ b e i n g the d e v e l o p e d a n g l e of f r i c t i o n . 3.8.1 P i l a t i o n Angle From the volume change c h a r a c t e r i s t i c s o b t a i n e d from a s i m p l e shear t e s t , the d i l a t i o n a n g l e ( V ) i s d e f i n e d f o r a p a r t i c u l a r s t r e s s s t a t e , u n d e r c o n s t a n t c o n f i n i n g p r e s s u r e , a s f o l l o w s , S i n if = - 4 * 00 d ? w I t i s a c o n v e n i e n t measure of the r a t e of d i l a t i o n a t any s t r e s s l e v e l . ( F i g . 16) Eqns. 55 and 5 6 , c o u l d be s u b s t i t u t e d i n Eqn. 54. t o y i e l d , t a n 2 ( 4 5 + fy2) = W ( * 5 t ^ t<Ht(f*5+»fy (57) 32 F i g . 18 Unloading-reloading i n Shear I f a sand i s sheared s t a r t i n g under i s o t r o p i c (or K Q ) c o n d i t i o n s , t h e d e v e l o p e d a n g l e of f r i c t i o n (0d) i s s m a l l compared t o 0cv i n i t i a l l y , w i t h the r e s u l t t h a t the Eqn. 57 w i l l p r e d i c t a n e g a t i v e d i l a t i o n a n g l e . T y p i c a l s i m p l e shear t e s t r e s u l t s o b t a i n e d f o r l o o s e and medium dense sand,shows a c o m p r e s s i o n a t the i n i t i a l s t a g e s as shown i n F i g . 12. A f t e r reaches v a l u e , becomes p o s i t i v e and keeps on i n c r e a s i n g t i l l t he maximum f r i c t i o n i s m o b i l i z e d , w h i c h a g a i n i s ob s e r v e d e x p e r i m e n t a l l y . Even i n the case of a dense sand,Eqn. 5 7 , p r e d i c t s an i n i t i a l d e c r e a s e i n volume which i s not u s u a l l y o b s e r v e d i n the l a b o r a t o r y . I n dense sands the i n i t i a l g r a d i e n t i s s h a r p e r , a n d thus y^ cy (t h e s t a r t i n g p o i n t of d i l a t i o n ) i s m o b i l i z e d sooner. S i n c e t h i s o c c u r s over a s m a l l range of s h e a r i n g , t h e measurement of t h i s i n i t i a l c o m p r e s s i o n would be d i f f i c u l t , w h i c h may be one of the reasons f o r the above d i s c r e p a n c y . Moreover, i f the d e n s i t y of the sand i s such t h a t the a t t a i n m e n t of 0Cv i s almost i n s t a n t a n e o u s , o n l y a volume i n c r e a s e i s p r e d i c t e d by Eqn. 57. F i g . 17 d e p i c t s t h i s s i t u a t i o n , a n d the r e s u l t i n g s t r e s s -volume c h a n g e - s t r a i n c u r v e s are not v e r y f a r from the t e s t r e s u l t s f o r v e r y dense sands. 3.8.2 U n l o a d i n g - r e l o a d i n g shear modulus and p l a s t i c shear  s t r a i n s The i d e a s of an u n l o a d i n g - r e l o a d i n g e l a s t i c modulus d i s c u s s e d i n Chapter 3.3.8.1 can be e x t e n t e d t o form i t ' s shear c o u n t e r p a r t , a s shown i n F i g . 18. G u r can be e v a l u a t e d a t any c o n f i n i n g p r e s s u r e ( ), by a procedure s i m i l a r t o the d e t e r m i n a t i o n of E u r (Duncan e t a l -1978).The e l a s t i c and the p l a s t i c components of the shear s t r a i n s can be dec o u p l e d as f o l l o w s . 36 Then the p l a s t i c shear s t r a i n i n c r e m e n t s can be used i n Eqns and 5 6 , i n s t e a d of the t o t a l shear s t r a i n i n c r e m e n t . I f the same argument t h a t ^ r y ^ a t low s t r e s s l e v e l s i s h i g h f o r dense sands i s a p p l i e d t o the r a t i o of ^r/ as w e l l , i t would mean t h a t , a t low s t r e s s . e v e l s i n dense sands , a c o n s i d e r a b l e p o r t i o n of the shear s t r a i n . i s e l a s t i c . which does not produce any v o l u m e t r i c s t r a i n . T h i s f u r t h e r e x p l a i n s , t h e e x c e s s v o l u m e t r i c c o n t r a c t i o n p r e d i c t e d f o r dense sands by Eqn. F i g . 20 Simple Shear Test Results. (After Stroud) 57,(where the t o t a l shear s t r a i n increment was used),as d i s c u s s e d i n the p r e v i o u s a r t i c l e . E x p e r i m e n t a l l y , f o r t r i a x i a l t e s t s , i n the post peak region there i s a r a p i d r e d u c t i o n i n the d i l a t i o n a n g l e , b e f o r e a l l the d i l a t i o n ceases at the u l t i m a t e s t a t e ( o r c r i t i c a l s t a t e ) , reached under l a r g e s t r a i n s . T h e maximum d i l a t i o n angle ,for dense sand occurs at the peak s t r e n g t h . T h i s behaviour i s a l s o modelled very w e l l by the theory,as when c7Jj s t a r t s to reduce from ^ ^ ^ h e Eqn. 57 w i l l p r e d i c t lower values f o r V .Once the r e s i d u a l s t r e n g t h i s m o b i l i z e d (at the c r i t i c a l s t a t e ) , a constant volume deformation i s c o r r e c t l y p r e d i c t e d , b y zero V , s i n c e 0 C V i s developed under such c o n d i t i o n s f o r any sand. For very l o o s e sands on the other hand,the r e s i d u a l s t r e n g t h i s the peak strength.Thus an apparent f r i c t i o n a ngle,higher than 0cv (the r e s i d u a l s t r e n g t h of any s a n d ) , i s never developed,with the r e s u l t t h at the d i l a t a n c y theory w i l l not p r e d i c t a volume increase.Under e x c e s s i v e strains,when 0CV i s developed,a constant volume deformation,as i n the case of a dense or a medium dense sand i s p r e d i c t e d , as shown in F i g . 17.This s i t u a t i o n i s not observed i n p r a c t i c e , due to the d i f f i c u l t y i n p r e p a r i n g very loose samples.According to l a b o r a t o r y i n v e s t i g a t i o n s ,even a sand of r e l a t i v e d e n s i t y of 27%, d i l a t e a l i t t l e a f t e r the i n i t i a l compression. The f o r e g o i n g d i s c u s s i o n enables one to understand how Rowe's D i l a t a n c y theory e x p l a i n s the d i f f e r e n t r a t e s of d i l a t i o n , o f i n i t i a l l y dense and loose samples.Under the same c o n f i n i n g pressure,dense samples w i l l d i l a t e more than the i n i t i a l l y l o o s e r ones,to a t t a i n a more or l e s s the same v o i d s r a t i o ( a n d t h e r e f o r e the same u l t i m a t e s t r e n g t h ) , a t l a r g e s t r a i n s . Apart from the i n i t i a l d e n s i t y , t h e v a r i a t i o n s i n the c o n f i n i n g p r essure a l s o has a marked e f f e c t on the d i l a t i o n r a t e . 3.8.3 P i l a t i o n angle and conf in ing pressure Vaid,Byrne,and Hughes(1980),investigated the dependency of the d i l a t i o n r a t e s on the c o n f i n i n g pressure( C7 v),and they concluded that,an i n c r e a s e in Ov w i l l r e s u l t i n a decrease i n the d i l a t i o n r a t e and a l s o the u l t i m a t e voids r a t i o , f o r a sand of a p a r t i c u l a r r e l a t i v e d e n s i t y . When a d i l a t i o n angle at a s p e c i f i e d s t r a i n ( i . e 10%) i s s e l e c t e d , i t was shown t h a t , t h i s r e l a t i o n s h i p between 1/ and c o n f i n i n g pressure , i s l i n e a r as shown in F i g . 19. T h i s phenomenon can be e x p l a i n e d s a t i s f a c t o r i l y using the d i l a t a n c y theory i n the f o l l o w i n g manner. Shear t e s t s on sand has r e v e a l e d t h a t , t h e maximum s t r e n g t h m o b i l i z e d ( < $ M a < ) , d e c r e a s e s with the i n c r e a s e i n e f f e c t i v e c o n f i n i n g pressure,(Seed et a l 1966),as d e p i c t e d i n Fig.3.Thus under higher c o n f i n i n g p r e s s u r e s , i f $maK i s reduced,the Eqn. 57 w i l l p r e d i c t lower values of y ,which i m p l i e s that the rate of d i l a t i o n , t h r o u g h o u t the s h e a r i n g , w i l l be lower f o r a p a r t i c u l a r r e l a t i v e d e n s i t y . The NLSSIP can i n c o r p o r a t e t h i s behaviour,as i t can accommadate the parameter (Chapter 3.3.6),which i s the r e d u c t i o n i n 0 f o r a t e n f o l d i n c r e a s e i n the e f f e c t i v e c o n f i n i n g p r e s s u r e . Along with t h i s the theory w i l l p r e d i c t lower V at higher c o n f i n i n g p r e s s u r e s f o r a sand of any i n i t i a l d ensi t y . 3.9 Modi f ied NLSSIP and the shear d i l a t ion The e x i s t i n g NLSSIP uses non l i n e a r e l a s t i c incremental c o n s t i t u t i v e r e l a t i o n s , a n d p r e d i c t s only the deformations due to e l a s t i c s t r a i n s . I t has been m o d i f i e d to take i n t o account,the c o n t r i b u t i o n from shear induced d i l a t i o n towards the v o l u m e t r i c s t r a i n s . T h i s i s done by an analogy with the Temperature-Stress problem . The mechanics and the f i n i t e element techniques a d o p t e d , w i l l be o u t l i n e d i n d e t a i l in Chapter 4. The user of the program can s e l e c t between two o p t i o n s i n i n t r o d u c i n g shear d i l a t i o n . 1. S t a r t i n g from a c e r t a i n s p e c i f i e d shear s t r a i n , a constant d i l a t i o n angle i s used throughout the shearing process, (which i s the maximum d i l a t i o n angle ).Here the d i l a t i o n angle and the s t a r t i n g shear s t r a i n have to be entered. 2. Rowe's d i l a t a n c y equation f o r plane s t r a i n c o n d i t i o n s ( E q n . 2 6 ) , i s used,and the program c a l c u l a t e s , the rate of d i l a t i o n to be used at any shear s t r a i n , f o r any f i n i t e element,depending on the s t r e s s state.The value of c#cy has to be s p e c i f i e d by the user. The o p t i o n 1 ,an a l t e r n a t i v e to the Rowe's approach, f o l l o w s from the notable behaviour of sands under plane s t r a i n c o n d i t i o n s . A s f i r s t observed by Stroud (1971),from simple shear t e s t s on L e i g h t o n Buzzard sand(of d i f f e r e n t i n i t i a l r e l a t i v e d e n s t i e s ) , t h e deformation occurs at a constant p r i n c i p a l s t r e s s r a t i o , a f t e r the peak s t r e s s r a t i o i s reached,over a s u b s t a n t i a l range of s t r a i n . Consequently the d i l a t i o n angle a l s o remains unchanged so long as t h i s happens, as seen by F i g s . 20a and 20b. Hughes et a l (1977) made use of t h i s to determine an elegant s o l u t i o n to the c a v i t y expansion problem i n sand due to a pressuremeter,which w i l l be presented i n Appendix G. Byrne et a l (1980) used the same technique to introduce d i a l t i o n to a problem of ,unloading of a tunnel i n o i l s a n d . T h e i r volume change curves were s i m i l a r to the one shown in F i g . 20b except f o r the compression p a r t . F i g . 21 Idealized Volume Change Curve The c o n s t a n t d i l a t i o n a n g l e they used i s the ^IOQH as i t o c c u r r e d a t the peak s t r e s s - r a t i o . ( F i g . 21). The former group d e f i n e d VQ as the s t r a i n a t which f a i l u r e i s f i r s t reached, whereas the l a t t e r group of workers s e l e c t e d ~#g as 2 ? c v ,where VCV i s the shear s t r a i n a t which the f r i c t i o n a n g l e d e v e l o p e d i s 0CV . I t can be c o n c l u d e d t h a t a l t h o u g h both o p t i o n s seem t o be e q u a l l y good f o r sand,the o p t i o n 2 i s advantageous over the o t h e r i n t h a t i t p r e d i c t s the i n i t i a l c o mpression of the medium dense and l o o s e sands. CHAPTER 4 FINITE ELEMENT FORMULATION The s o i l i s mod e l l e d by 2D i s o p a r a m e t r i c q u a d r i l a t e r a l or t r i a n g u l a r elements.As t h e name i m p l i e s the same i n t e r p o l a t i o n f u n c t i o n s a r e used t o d e f i n e the element shapes as w e l l as the d i s p l a c e m e n t s w i t h i n the e l e m e n t s . 4 .1 C o n s t i t u t i v e m a t r i x Most of the e n g i n e e r i n g problems e n c o u n t e r e d i n p r a c t i c e can be i d e n t i f i e d as f a l l i n g i n t o one of the f o l l o w i n g c l a s s e s . 1. P l a n e S t r a i n -2 d i m e n s i o n a l 2. P l a n e S t r e s s -2 d i m e n s i o n a l 3. A x i s y m m e t r i c -3 d i m e n s i o n a l P l a n e s t r a i n problems a r e c h a r a c t e r i s e d by the f o l l o w i n g two p r o p e r t i e s . a. There i s no d e f l e c t i o n i n the z d i r e c t i o n b. F i r s t d e r i v a t i v e s of the o t h e r d e f l e c t i o n s w . r . t . z a r e z e r o W i t h t h e s e two c o n d i t i o n s Eqn. 3 w i l l produce £ 2 * o V = o * = o xa (.0) Thus the p l a n e s t r a i n v e c t o r w i l l be e = From the g e n e r a l i z e d Hooke's law "*2 but as shown i n the Appendix D.l T h e r e f o r e we use CT = r (Q2) (<3) but c a l c u l a t e based on Eqn. 6 3 , i f r e q u i r e d , In t h i s c a s e , t h e (D) m a t r i x w i l l be a 3 x 3 one. Based on the assumption t h a t , g e n e r a l i z e d Hooke's law can be a p p l i e d t o i n c r e m e n t a l e l a s t i c i t y i n i t ' s " o r i g i n a l form, the (D) m a t r i x has been shaped out i n Appendix D . l a s , 0+pt)('-V*) 0 o o I - 2 / 4 T (65) I t i s a l s o shown t h a t i f , e q u i v a l e n t b u l k and shear moduli are d e f i n e d f o r the p l a n e s t r a i n c o n d i t i o n , s u c h t h a t , ' , 3 B t and (M) the c o n s t i t u t i v e r e l a t i o n s h i p s reduces t o the f o l l o w i n g form,as d i r e c t l y used i n the program. — 1 B'fc - G't O B fe - 6 t o o G'fc 4.2 S t i f f n e s s m a t r i x The nodal f o r c e increment and the n o d a l d i s p l a c e m e n t v e c t o r s w i l l be 8 1 i n g e n e r a l . A f = Afg and 8 = The d i s p l a c e m e n t f i e l d can be e x p r e s s e d as U where c\ i s a se t of g e n e r a l i z e d c o o r d i n a t e s | / \ J i s a* m a t r i x c o n t a i n i n g c o e f f i c i e n t s which a r e f u n c t i o n s of 4-2 the c o o r d i n a t e s of the p o i n t c o n s i d e r e d . A p p l y i n g Eqn. 69 t o the n o d a l p o i n t s gives,-$ = [ A , ] « (*>) where \^z\ i s a m a t r i x c o n t a i n i n g c o e f f i c i e n t s which are dependent on the n o d a l c o o r d i n a t e s . Ey d i f f e r e n t i a t i n g Eqn. 69 and c o m b ining i t w i t h Eqn. 3 L\t = [ C ] <X g i v e s where A_£ i s the p l a n e s t r a i n increment v e c t o r From Eqn. 70 1 - [ A a ] " ? (72) The f o l l o w i n g e f f e c t i v e s t r e s s v e c t o r A O has been o b t a i n e d u s i n g the i d e a s about e f f e c t i v e s t r e s s e s i n Chapter 3.1 A O ' - (74) I t i s shown i n Appendix D.2 t h a t the f o r c e - d i s p l a c e m e n t r e l a t i o n f o r a g e n e r a l f i n i t e element can be w r i t t e n a s , Af ~[k*] Au = U] § where (k) - E l e m e n t • s t i f f n e s s m a t r i x dependent on the geometry of the element and (D) and (k*) -A m a t r i x dependent on the geometry of the element o n l y . F u r t h e r i t i s p o i n t e d out i n Appendix D.2 how a l l the element r e l a t i o n s h i p s c o u l d be b u i l t up t o form the n o d a l f o r c e increment - d i s p l a c e m e n t r e l a t i o n s h i p f o r the whole s t r u c t t :e,which i s GO. 43 where (K) - g l o b a l s t i f f n e s s m a t r i x and ( F) - T o t a l e x t e r n a l f o r c e increment v e c t o r m o d i f i e d by the pore p r e s s u r e f o r c e s . 4.3 S o l u t i o n Gauss' e l i m i n a t i o n t e c h n i q u e i s used t o o b t a i n the s o l u t i o n t o Eqn. 76. By knowing the n o d a l d i s p l a c e m e n t s § of each element A £ = 0 ] § C77) and A a' - [o] A £ (?8) are used t o e v a l u a t e the average s t r a i n and and s t r e s s i n c rements i n each element. 4.4. Shear d i l a t i o n I t was mentioned i n Chapter 3.9, t h a t the a d d i t i o n a l volume changes due t o shear are i n t r o d u c e d by means of tem p e r a t u r e a n a l o g y . The two e q u a t i o n s f o r v o l u m e t r i c s t r a i n i n each c a s e , p r e s e n t e d b e l o w , w i l l c l a r r i f y t h i s a n a l o g y f u r t h e r . T e m p e r a t u r e - S t r e s s problem cx AT (71) S i n V . A y <7e) A.£v,=Total v o l u m e t r i c s t r a i n 4 i ^ = e l a s t i c s t r a i n E> (X. =Temperature c o e f f i c i e n t of the m a t e r i a l AT =change i n t e m p e r a t u r e . ( i n c r e a s e ) Eqn.79 shows how the v o l u m e t r i c s t r a i n i n a temperature s t r e s s problem i s made up of the s t r a i n a s s o c i a t e d w i t h the mean normal s t r e s s change,as w e l l as t h a t due t o a temperature change . ( A T ) • I n the shear d i l a t i o n problem the change i n shear can be thought of as e q u i v a l e n t t o a temperature change as f a r as the v o l u m e t r i c s t r a i n i s concerned,and i t ' s c o n t r i b u t i o n i s added t o the t o t a l v o l u m e t r i c s t r a i n , a l o n g s i d e t h a t due t o the A<J„ A£„ r ^ -Shear d i l a t i o n 6 Acv -where changes i n the mean normal s t r e s s (the e l a s t i c component of the volumetric s t r a i n ) , y i e l d i n g Eqn. 80. Comparison of these two equations w i l l show the analogy between the r a t e of expansion( oO ,and the r a t e of d i l a t i o n . (Sin If ).In the f i n i t e element method , e f f e c t s due to a temperature change are i n c o r p o r a t e d by a set of e q u i v a l e n t nodal forces.The procedure f o r o b t a i n i n g them i n the case of shear d i l a t i o n i s d e s c r i b e d below. Suppose i s the a d d i t i o n a l s t r a i n (due to shear d i l a t i o n ) , r e q u i r e d i n one element and t h i s i s to be obtained by a p p l y i n g a f o r c e vector of L\f* on i t ' s nodes. If AV i s the maximum shear s t r a i n i n the element,from Eqn. 56,the p l a s t i c volume change i s A £ ^ = - S i n v . A y (3\) From t h i s we c o n s t r u c t the Afc^ as f o l l o w s , V -average d i l a t i o n angle d u r i n g the load increment. Now i f o i s the set of d e f l e c t i o n s produced by the a p p l i c a t i o n of A f " For e q u i l i b r i u m we can write the v i r t u a l work equation,as in Appendix D.2, But from Eqn. 77, ^ = ^ g a ^ S u b s t i t u t i o n from Eqns. 78 and 84 in Eqn. 83 y i e l d s ' - i a „ v. Ay Af d y = _ J. Sin v 2. Ai o A f d = [ B ] T [ D ] At* Vt s i n c e (B) and (D) ,consists. of c o e f f i c i e n t s which are constants for the element.( Appendix D.2). C o n s i d e r i n g a u n i t t h i c k n e s s , A f d = [B] T[D] A £ d A e (8?) E q u a t i o n s 82 and 87 ,can be used t o a s s e s s the r e q u i r e d n o d a l f o r c e s f o r each element,and the g l o b a l f o r c e v e c t o r A F D i s a p p l i e d i n a d d i t i o n t o the pore p r e s s u r e m o d i f i e d l o a d increment v e c t o r . E v a l u a t i o n of the r e s u l t s i s done as d e s c r i b e d i n Chapter .4.3 ,except t h a t Eqn. 78 has t o be m o d i f i e d , as done i n the temperature problem,by s u b s t r a c t i n g A£ dfrom the f i n a l A f , s i n c e Ac does not c r e a t e a d d i t i o n a l s t r e s s e s ( s i m i l a r t o s t r a i n s caused by t e m p e r a t u r e c h a n g e s ) . The m o d i f i e d Eqn. 78 i s t h e n , A g ' - [ P ] [ A £ - Ae d ] (3s) I t w i l l be r e a l i s e d t h a t , the A^ v a l u e from Af , would d i f f e r from the p r e v i o u s A b u s e d i n Eqn. 82),when a mesh of elements i s a n a l y s ed.Thus an i t e r a t i v e p r o c e d u r e has to be a d o p t e d , f o r a c o n v e r g e n t s o l u t i o n . In c o n s t r u c t i n g A ^ E q . 8 2 ) , i t has been assumed t h a t the shear d i l a t i o n i s i s o t r o p i c , i . e A £ d i s made up of e q u a l c o n t r i b u t i o n s from A£ x dand A ^ . T h i s i s not s t r i c t l y t r u e i n the l i g h t of the p l a s t i c f l o w r u l e i n Chapter 3.7.1(Eqn. 4 8 ) . ^ = -*/ Kb£ &3) A l t h o u g h i t i s more c o m p l i c a t e d , E q n . 82 can be b u i l t up by Eqns. 90 and 9 1 , i n more c o n c i s e work. ^ = Ho,',*,-) W A (from Eqn. 48) A * / t A £ y d „ - Sir) D. A * 0?O (f rom Eqn. 5 6 ) . 4.5 S t r e s s r e d i s t r i b u t i o n t e c h n i q u e I t was mentioned i n Chapter 3.3.9,how the s t r e s s e s can be c o r r e c t e d i n the e l e m e n t s , t h a t v i o l a t e d the f a i l u r e c r i t e r i o n , b y a p p l y i n g an a d d i t i o n a l set of n o d a l f o r c e s . H e r e i n the method of o b t a i n i n g t h e s e n o d a l f o r c e s i s d e s c r i b e d . F i g . 22 Stress R e d i s t r i b u t i o n Suppose A c i s the s t r e s s c o r r e c t i o n t o be a p p l i e d f o r a s i n g l e element, and t h i s i s t o be a c h i e v e d by a p p l y i n g a s e t of f o r c e A f t o the nodes of t h a t element. I f 8 i s the a d d i t i o n a l s e t of d e f l e c t i o n s , d u e t o the a p p l i c a t i o n of A$*-r By the p r i n c i p l e of v i r t u a l work as i n Appendix D.2, (V) A t - a d d i t i o n a l s t r a i n v e c t o r From Eqn. 77 . } S u b s t i t u t i o n i n Eqn. 92 g i v e s [ 8 ( ] T A f ! =. [ 8 l j T J [ B ] r A ? i d v which on s i m p l i f i c a t i o n as d e s c r i b e d i n the p r e v i o u s a r t i c l e , r e s u l t s i n A f 1 [ B ] r A o : l A e ( W 5 i s c o n s t r u c t e d i n the f o l l o w i n g manner. F i g 22 shows the Mohr c i r c l e f o r a t y p i c a l element t h a t v i o l a t e s the f a i l u r e c r i t e r i o n . I f i t i s to- a b i d e by the Mohr-Coulomb c r i t e r i o n , / QP 4 QP The s t r e s s s t a t e of the element i s g i v e n by A(CTX j T x y ) and B ( 0 ~ y > - t X y ) r e s p e c t i v e l y . At t h i s s t a g e an assumption i s made t h a t the mean normal s t r e s s remains c o n s t a n t . C o n s e q u e n t l y the c o r r e c t e d c i r c l e w i l l a c q u i r e the p o s i t i o n of the d o t t e d c i r c l e , w i t h the s t r e s s p o i n t s A and B l a n d i n g on A' and B' r e s p e c t i v e l y . QP' = Om t a n <x = Om S-.n j 2 ^ a x (%) Q p = X m a * C97) PP' = T WAX - a ^ S ' , n ^ m a x (3&) = AA' = BB' =the r e d u c t i o n i n the r a d i u s of the Mohr c i r c l e 4 8 F i g . 23 Inclusion of Springs Now from the A AA'R S i m i l a r l y - A I x y = - A 0"„ ~ A o\ A cr, = -A a and A T A q y *y t A A ' S i n e A A ' COS 9 BB' Cos 6 = - Ao"x ( t 'VIA* (<?9) Ooo) (joi) (to 3) ( . 0 4 ; (IO?; Then Eqn. p r e v i o u s l y 95 g e n e r a t e s the n o d a l f o r c e increment v e c t o r , a n d as _ mentioned,the g l o b a l f o r c e v e c t o r ( f o r a l l such e l e m e n t s ) , i s a p p l i e d t o g i v e the d e s i r e d s t r e s s c o n d i t i o n s , a f t e r an i t e r a t i v e p r o c e d u r e . I f an element t h a t f i r s t s t a r t s y i e l d i n g a t Q ( F i g . 9 ) , i s a s s i g n e d the c o n s t i t u t i v e parameters d e s c r i b e d i n Chapter 3.3.9 f o r f a i l u r e r e g i o n , l . e w i l l = 0.01 B, B^(from Eqn. w i l l the v a l u e of G t be v e r y low compared t o 24) w i t h the r e s u l t t h a t from then onwards,the s t r e s s p a t h be a l o n g QR on f u r t h e r u n l o a d i n g . ( w i t h a v e r y l i t t l e shear s t r e s s change compared t o the change i n mean normal s t r e s s ) . Now on the r e d i s t r i b u t i o n of s t r e s s e s , i f we r e t a i n the same moduli v a l u e s , a r e d u c t i o n i n the exce s s shear s t r e s s under c o n s t a n t mean normal s t r e s s , i s i m p o s s i b l e t o be achieved,due t o the above r e a s o n s . Thus on the s t r e s s - r e d i s t r i b u t i o n , G t has t o be upgraded w.r.t B ^ , s i n c e our aim i s t o b r i n g the s t r e s s p o i n t a l o n g RS.This would amount t o l o w e r i n g of \* c R ) . I n k e e p i n g w i t h the p r a c t i c a l l i m i t s , G t such an e x t e n t t h a t p t becomes z e r o i n r e d i s t r i b u t i o n . , from 0.495(at was i n c r e a s e d t o the p r o c e s s of 4.6 I n c l u s i o n of e l a s t i c s p r i n g s F i g . 23a shows the f i n i t e element domain around the s h a f t , f o r the p a r t i c u l a r problem of a n a l y s i n g the u n l o a d i n g of a s h a f t i n o i l s a n d . A problem of an i n f i n i t e space has been c o n v e r t e d t o one of a f i n i t e space,by the d i s p l a c e m e n t boundary c o n d i t i o n s a t A,B,C and D.The s o l u t i o n i s not a f f e c t e d i f the boundary i s s e t a t a d i s t a n c e which i s l a r g e i n comparison t o the d i a m e t e r of the s h a f t . T h i s i n v o l v e s e x t e n d i n g the mesh t o a r e a s which have a l i t t l e i n f l u e n c e on the s h a f t d e f o r m a t i o n . T h i s problem was overcome by f i x i n g r a d i a l e l a s t i c s p r i n g s a t A,B,C and D, t o account f o r the i n f l u e n c e of the s o i l mass from the boundary t o i n f i n i t y . The advantage i n t h i s m o d i f i c a t i o n i s t h a t the problem remains t h a t of an i n f i n i t e s o i l mass,with the o p p o r t u n i t y of u s i n g a f i n e r mesh c l o s e r t o the s h a f t w i t h o u t g e n e r a t i n g an enormous m e s h . ( F i g . 23b). With the i n c l u s i o n of the s p r i n g s ( w i t h f o r 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 i n Eqn. 38 was s t i f f n e s s ) m o d i f i e d by, the a. A s s i g n i n g b o u n d a r y ( j , j + 1 e t c ) u s u a l degrees of freedom at the s p r i n g b. S p e c i f y i n g z e r o f o r c e s a t them. c. M o d i f y i n g the (K) by add i n g the s p r i n g s t i f f n e s s t o the d i a g o n a l t e r m s , c o r r e s p o n d i n g t o the above degrees of freedom ( j , j + 1 e t c ) . • A F 2 O 0 A F„ o b t a i n i n g A '22 Jj f 8, 1 Ci The method of has been d e s c r i b e d i n the Appendix E. I t s h o u l d be emphasized here t h a t the above o u t l i n e d method i s v a l i d o n l y f o r the a x i s y m m e t r i c case,where the d e f l e c t i o n s a r e r a d i a l a t any p o i n t . 51 D U 0 - initial pore pressure \ t AU ^ | U ( - final •• It O t ime F i g . 24 Time E f f e c t i n Pore Pressure Changes CHAPTER 5 PORE PRESSURE MODEL 5.1 I d e a l i z a t i o n of the p o r e - f l u i d b e h a v i o u r One of the most i m p o r t a n t assumptions made r e g a r d i n g the p o r e - p r e s s u r e d i s t r i b u t i o n i s t h a t ,the p r e s s u r e i s the same i n a l l t h r e e f l u i d p h a s e s . T h i s i s r e a s o n a b l e as the s u r f a c e t e n s i o n a t the i n t e r f a c e s , w h i c h makes the d i f f e r e n c e i s n e g l i g i b l e . D u s s e a u l t ( 1 9 7 9 ) , s a y s t h a t the gas bubb l e s a re l i k e l y t o be i n c o n t a c t w i t h bitumen o n l y , a n d hence o n l y the s u r f a c e t e n s i o n at the gas-bitumen and water - bitumen i n t e r f a c e s s h o u l d be c o n s i d e r e d , a n d t h a t they can be n e g l e c t e d , s i n c e t h e i r magnitudes are i n s i g n i f i c a n t . The second assumption i s t h a t , t h e c o m p r e s s i b i l i t y of water and bitumen i s neglected,when compared t o t h a t of gas b u b b l e s . l t s h o u l d be borne i n mind t h a t t h i s i s s t r i c t l y v a l i d under f a i r l y h i g h i n i t i a l gas p o r o s i t i e s . Consequent t o t h i s assumption ,the volume changes of water and bitumen on u n l o a d i n g s h a l l not be taken i n t o a c c o u n t . F i n a l l y the time taken f o r gas e v o l u t i o n ( I ) i s n e g l e c t e d . ( A s shown i n F i g . 2 4 , s i n c e the gas e v o l u t i o n i s not i n s t a n t a n e o u s , a c e r t a i n time e l a p s e s b e f o r e the e q u i l i b r i u m pore p r e s s u r e i s a t t a i n e d , o n the r e d u c t i o n of e x t e r n a l s t r e s s ) . 5.2 Free gas b e h a v i o u r The volume changes o c c u r r i n g i n the f r e e gas under p r e s s u r e and t e m p e r a t u r e changes,are p r e d i c t e d by the i d e a l gas e q u a t i o n of s t a t e . P V = n R T ( 1 0 7 ) For r e a l gases l i k e the o r g a n i c gaseous compounds,present i n the o i l s a n d s , t h i s e q u a t i o n g i v e s a c c u r a t e r e s u l t s o n l y under h i g h p r e s s u r e s and low t e m p e r a t u r e s . 5.3 P r e s s u r e s o l u b i 1 i t y There i s a marked i n c r e a s e i n the s o l u b i l i t y of gases i n l i q u i d s , w i t h i n c r e a s e d p r e s s u r e . C o n v e r s e l y , as p r e v i o u s l y mentioned , o c c l u s i o n of the gas from s o l u t i o n w i l l r e s u l t , o n removal of the excess p r e s s u r e . T h i s was q u a n t i t a t i v e l y e x p r e s s e d by H a r r i s and Sobkowicz(1977),by the f o l l o w i n g e q u a t i o n ,which we would be u s i n g i n our program due t o i t ' s s i m p l i c i t y . V g = -H vAu ( 1 0 8 ) where Vg i s the gas g o i n g i n t o s o l u t i o n ( i n d i c a t e d by the - i v e sign),when the p r e s s u r e i n c r e a s e s by A u, a t c o n s t a n t t e m p e r a t u r e . ( T h i s volume i s r e f e r r e d t o a s e t of s t a n d a r d 51 c o n d i t i o n s ) . V -volume of the s o l u t i o n H -a c o e f f i c i e n t measured at the standard c o n d i t i o n s . The pressure s o l u b i l i t y . o f g a s e s , i n d i l u t e s o l u t i o n s i s governed by Henry's law,which says t h a t , a t constant temperature,the p a r t i a l pressure of a gas i n contact with a s o l u t i o n i s p r o p o r t i o n a l to the mole f r a c t i o n of that gas i n s o l u t i o n . S t a r t i n g from Henry's law,an equation s i m i l a r to Eqn. 108,has been d e r i v e d i n Appendix C.2.a and i t i s a l s o shown that the c o e f f i c i e n t H i s h i g h l y dependent on the temperature. 5.4 Temperature s o l u b i 1 i t y An i n c r e a s e i n temperature w i l l decrease the s o l u b i l i t y of gases,and t h e r e f o r e changes i n gas p o r o s i t i e s due to temperature changes a s s o c i a t e d with e x c a v a t i o n work i n o i l s a n d s , c o u l d be expected. H a r r i s and Sobokowicz (1977), use the f o l l o w i n g r e l a t i o n s h i p to take t h i s i n t o account. Vj =pVAT where Vg i s the gas coming out of the s o l u t i o n ( r e f e r r e d to the same set of standard conditions),when' the temperature i s i n c r e a s e d by AT,at constant p r e s s u r e . V - volume of the s o l u t i o n ^ - a c o e f f i c i e n t measured at the standard c o n d i t i o n s . I t c o u l d a l s o be shown t h a t , t h e c o e f f i c i e n t turns out to be a f u n c t i o n of the pressure and the temperature at which the gas i s -evolved. 5.5 V o l u m e t r i c s t r a i n c o m p a t i b i 1 i t y ( f o r undrained c o n d i t i o n s ) From the assumptions made re g a r d i n g the c o m p r e s s i b i l i t y of o i l s a n d phases,the volume change of the gas phase f o r a u n i t volume of o i l s a n d , s h o u l d be the same as the vol u m e t r i c s t r a i n of the s o i l s k e l e t o n obtained from the e f f e c t i v e s t r e s s c o n s i d e r a t i o n s , u n d e r undrained c o n d i t i o n s . T h i s c o m p a t i b i l i t y c o n d i t i o n , a l o n g with Eqns. 107,108,and 109 were combined by Byrne et al(1980),as i n Appendix C . l , t o o b t a i n the expre s s i o n f o r the pore pressure increment,under pr e s s u r e and temperature changes,as, — o= la m o ) where - P o r o s i t y of the water p h a s e ( d i m e n s i o n l e s s ) n 0 - P o r o s i t y of the o i l p h a s e ( d i m e n s i o n l e s s ) n g - P o r o s i t y of the gas p h a s e ( d i m e n s i o n l e s s ) £ w-Temperature s o l u b i l i t y c o e f f i c i e n t of water( °K~ ) A 0 -Temperature s o l u b i l i t y c o e f f i c i e n t of o i l ( "K"') H v - P r e s s u r e s o l u b i l t y c o e f f . of w a t e r ( p r e s s u r e u n i t ) H 0 - P r e s s u r e s o l u b i l i t y c o e f f . of o i K p r e s s u r e u n i t ) . - R e f e r e n c e ( a t m o s p h e r i c ) p r e s s u r e . ( p r e s s u r e u n i t ) T^ -Reference(room) temperature.(283°K) T 0 - I n i t i a l t e m p e r a t u r e ( • K ) T ( - F i n a l t e m p e r a t u r e ( *< ) A T -Change i n te m p e r a t u r e ( *K ) Up - I n i t i a l a b s o l u t e p r e s s u r e ( p r e s s u r e u n i t ) Au -Change i n pore p r e s s u r e ( p r e s s u r e u n i t ) A£ y-Volumetric s t r a i n i n c r e m e n t ( d i m e n s i o n l e s s ) 5.6 Comment on the use of Eqn. 110 In a d d i t i o n t o b e i n g s u b j e c t e d t o the v a l i d i t y of the assumptions made i n d e r i v i n g i t , t h i s e q u a t i o n has o t h e r l i m i t a t i o n s as well.Whether the b e h a v i o u r of the gases p r e s e n t i n o i l s a n d obeys the i d e a l gas or Henry's law,and i f so under what c o n d i t i o n s they do so w i t h o u t much d e v i a t i o n , s t i 1 1 l i e s i n q u e s t i o n , d u e t o the l a c k of t e s t d a t a . P o s s i b l e c h e m i c a l a c t i v i t y between the gases and the bitumen or gases and sand g r a i n s may a l s o p r e v e n t them from behaving i d e a l l y . I t i s a d v i s a b l e t o work w i t h a H v a l u e o b t a i n e d a t the wor k i n g t e m p e r a t u r e range, and a j£ v a l u e v a l i d f o r the wo r k i n g p r e s s u r e and temp e r a t u r e r a n g e , s i n c e the Henry's law r e f u t e s the f a c t t h a t H and fi are c o n s t a n t s . Y e t t h i s c o u l d o n l y be done i f d a t a i s a v a i l a b l e from t e s t s done under a wide range of p r e s s u r e s and t e m p e r a t u r e s . F u r t h e r the s i g n i f i c a n c e of the time e f f e c t s , ( s u c h as the time taken f o r gas e v o l u t i o n ) has t o be checked by a l a b o r a t o r y i n v e s t i g a t i o n , i f one i s t o r e l y on Eqn. 110 which i g n o r e s them. CHAPTER 6 SOLUTION METHOD 6.1 Drainage c o n d i t i o n s Depending on the drainage c o n d i t i o n s two types of a n a l y s i s can be done. The d r a i n e d analysis(when s u f f i c i e n t time i s allowed f o r the pore f l u i d to escape on l o a d i n g ) and the undrained a n a l y s i s ( d u e to r e s t r i c t e d or no drainage paths,or due to r a p i d l o a d i n g when the pore f l u i d i s prevented from e s c a p i n g ) . The d r a i n e d c o n d i t i o n s p r e s e n t s no problems s i n c e the pore p r e s s u r e s are known. The undrained c o n d i t i o n s can be handled i n two ways. An e f f e c t i v e s t r e s s a n a l y s i s can be performed i f the pore pressure changes are e v a l u a t e d . In numerical a n a l y s i s , c h a n g e s in p r e s s u r e ( A u ) are obtained as f r a c t i o n s of the t o t a l s t r e s s changes,using Skempton's pore pressure parameters A and B.Instead, i n t h i s program the pore p r e s s u r e s are p r e d i c t e d by Eqn. 110, and c a l c u l a t i o n of p o r o s i t i e s i s done based on Eqns. H4 to H6. The t r o u b l e of e v a l u a t i n g A u can be avoided by adopting a t o t a l s t r e s s analysis,where t o t a l s t r e s s c o n s t i t u t i v e parameters are used. The program can be used to analyse problems i n v o l v i n g deformation of sand i n both d r a i n e d and undrained modes.In the undrained case,the program p r e d i c t s the pore pressure changes as a p a r t of the i t e r a t i v e procedure. If the user has some idea of the changes i n pore pressure t a k i n g p l a c e i n a subsequent load case,the number of i t e r a t i o n s can be cut down by , i n i t i a l l y a s s i g n i n g them.Otherwise the program assumes A u to be zero in a l l elements, i n i t s i n i t i a l t r i a l . 6.2 I t e r a t i v e procedures T h i s a n a l y s i s i s comprised of a number of load increments. Each increment i n v o l v e s the f o l l o w i n g s t e p s . 6.2.1 Drained A n a l y s i s 1. The e x t e r n a l load increment i s a p p l i e d to the mesh at the nodal p o i n t s . ( A F ) . 2. The g l o b a l s t i f f n e s s matrix i s b u i l t up by u s i n g the c o n s t i t u t i v e parameters (E^_,B t ),evaluated at the i n i t i a l c o n d i t i o n . 3. The corresponding d e f l e c t i o n s are obtained by s o l v i n g Eqn. 76. 4. Using Eqns. 78 and 77 s t r e s s and s t r a i n increments are p r e d i c t e d , a n d and B-^  a r e updated t o the average s t r e s s c o n d i t i o n d u r i n g the i n c r e m e n t . 5. I f shear d i l a t i o n i s t o be i n t r o d u c e d , t h e c o r r e s p o n d i n g e q u i v a l e n t s e t of n o d a l f o r c e s ( AF ) , i s c a l c u l a t e d based on the above shear s t r a i n i n c r e m e n t , u s i n g Eqn. 87. 6. Steps 1,2,3,and 4 a r e r e p e a t e d w i t h the f o l l o w i n g changes. a. AF + A f d i s a p p l i e d i n s t e a d of A F . b. Updated E t and B± a r e u s e d , i n s t e p 2. c. Eqn. 88 i s u s e d , i n s t e a d of Eqn. 78 t o c a l c u l a t e the s t r e s s i n c r e m e n t s . 7. Then i t i s checked whether a l l the elements have d i l a t e d i n accordance w i t h t h e i r c u r r e n t shear s t r a i n increment ,and i f so the p r o c e s s of i t e r a t i n g t e r m i n a t e s . O t h e r w i s e the s t e p s 1-5 are r e t r a c e d over and over ,each time u p d a t i n g A F^E and u n t i l convergence i s reached. ~ J t-8. F i n a l l y i t i s checked whether any of t h e ' e l e m e n t s v i o l a t e the f a i l u r e c r i t e r i o n . I f they do so ,an a d d i t i o n a l s e t of nodal f o r c e s ( AF^) i s c a l c u l a t e d from Eqn. 95 and a p p l i e d s e p e r a t e l y a l o n g s t e p s 1 - 4 , u n t i l s t r e s s r e d i s t r i b u t i o n i s done so t h a t a l l the elements conform t o the y i e l d c r i t e r i o n . By making the e x t e r n a l l o a d i n c r e m e n t ( A F ) as s m a l l as p o s s i b l e , m o r e a c c u r a t e r e s u l t s can be p r e d i c t e d u s i n g a low number of i t e r a t i o n s . 6.2.2 U n d r a i n e d a n a l y s i s 1. E x t e r n a l l o a d increment i s a p p l i e d t o the mesh a t the n o d a l p o i n t s . ( A f ) 2. The s t i f f n e s s m a t r i x - ( K ) , ( i n Eqn. 7 6 ) , i s formed i n a manner i d e n t i c a l t o t h a t f o r the d r a i n e d c a s e , s i n c e an e f f e c t i v e s t r e s s a n a l y s i s i s done. 3. The s e t of a d d i t i o n a l n o d a l f o r c e s , p r e s c r i b e d by the v e c t o r ( )K*)Au(as d i s c u s s e d i n Chapter 4.2 ) , i s c a l c u l a t e d . I n i t i a l l y the program assumes Au o=0 f o r a l l e l e m e n t s , i f v a l u e s are not p r o v i d e d by the u s e r . 4. Eqn. 76 i s s o l v e d t o o b t a i n the n o d a l d e f l e c t i o n s . 5. U s i n g Eqns. 77 and 7 8 , e f f e c t i v e s t r e s s and s t r a i n i n c r e m e n t s a r e p r e d i c t e d and the i n c r e m e n t a l moduli (E^. and B^) a r e updated t o the average e f f e c t i v e s t r e s s c o n d i t i o n d u r i n g the i n c r e m e n t . 6. I f shear d i l a t i o n i s i n t r o d u c e d A F i s c a l c u l a t e d as i n the p r e v i o u s c a s e . 7. Au v a l u e s a r e e s t i m a t e d f o r each element based on the v o l u m e t r i c s t r a i n increments and Eqn. 110. Then the c u r r e n t Au, i s compared with the guessed A u 6 . i n step 3,and i f they d i f f e r by more than a s p e c i f i e d t o l e r a n c e , a new pore p r e s s u r e increment A u 2 i s prepared f o r the next i t e r a t i o n i n the f o l l o w i n g way. i . For elements i n the p r e - f a i l u r e r e g i o n ,where the c o n d i t i o n s are s t a b l e , A u 2 =Au, i i . For elements i n the f a i l u r e region,even a small r e d u c t i o n of the e x t e r n a l l o a d , w i l l cause a s e n s a t i o n a l drop i n the mean normal s t r e s s , f o l l o w e d by a s i m i l a r expansion g i v i n g r i s e to an e x c e s s i v e Au,,which i f a p p l i e d in the next i t e r a t i o n might even overbalance the e x t e r n a l l o a d e f f e e t , t h e r e b y l e a d i n g to a d i v e r g e n t s o l u t i o n . O n the other hand i f A u B was a reasonable guess(or the c o r r e c t guess),Au, would have been approximately equal to Au Q.With t h i s i n mind i t i s l o g i c a l to assume t h a t , t h e c o r r e c t Au l i e s i n between A u 0 and A u , , i f they d i f f e r from each other. Thus A u z = ^ u D + A ut) 8. In the next t r i a l , s t e p s 1,2,3,4, and 5 are c y c l e d through once again with the f o l l o w i n g m o d i f i c a t i o n s . a. AF + A F d i s a p p l i e d i n s t e a d of AF. b. (K) i s b u i l t up with updated moduli. c. A u 2 i s used to e v a l u a t e the pore pressure f o r c e v e c t o r . d. Eqn. 88 i s used to e x t r a c t the e f f e c t i v e s t r e s s changes. 9. I t i s checked whether a d i l a t i o n a p p r o p r i a t e to the present shear s t r a i n increment has been in t r o d u c e d i n t o each element.If not A F d i s updated. 10. Pore pressure increments f o r the next i t e r a t i o n are prepared (A u^).For elements in the p r e - f a i l u r e region,the same procedure as d e s c r i b e d i n step 7 i s followed.For the other elements,the next guess i s based on the f a c t that A u l i e s between i . A u Q and Au, (from the previous t r i a l ) i i . Au^ and Au^ (from the c u r r e n t t r i a l ) . (Aui, i s the one p r e d i c t e d by Eqn. 110, f o r the v o l u m e t r i c s t r a i n increment i n the c u r r e n t i t e r a t i o n , u s i n g a guessed Au^) I f f o r any i t e r a t i o n , the guessed Au,agrees with the A u e v a l u a t e d from Eqn. 110,within the s p e c i f i e d t o l e r a n c e , f o r a l l elements,the i t e r a t i v e procedure t e r m i n a t e s . 11. F i n a l l y s t r e s s - r e d i s t r i b u t i o n i s d o n e , i f nee!ed,as d e s c r i b e d in step 8 of the d r a i n e d case. F i g . 26 Sign Convention for Stresses and Strains 6.2.3 U n d r a i n e d a n a l y s i s f o l l o w e d by Vent i n g Due t o the h i g h c o m p r e s s i b i l i t y of the system a t any s t a g e , o r even o t h e r w i s e the user may d e c i d e t o vent some of the c r i t i c a l e l e m e n t s . ( I n a c e r t a i n r e g i o n ) . T h i s sudden r e d u c t i o n i n pore p r e s s u r e i s h a n d l e d by the program as a n o t h e r l o a d case a t c o n s t a n t t o t a l load.The e q u i l i b r i u m v a l u e s of the s t r e s s , s t r a i n and pore p r e s s u r e s ( i n o t h e r e l e m e n t s ) , a r e o b t a i n e d by t r a c i n g s t e p s 1-11 i n the u n d r a i n e d case w i t h the f o l l o w i n g e x c e p t i o n s i . A F i n s t e p 1 i s a z e r o v e c t o r ( c o n s t a n t t o t a l l o a d ) . i i . Au f o r v e n t e d elements are not p r e d i c t e d by Eqn. 110,but i n s t e a d a r e a s s i g n e d the sudden f a l l i n pore p r e s s u r e i n them,for a l l i t e r a t i o n s . In the subsequent a n a l y s i s , t h e v e n t e d elements are s e t t o undergo the d r a i n e d i t e r a t i v e p r o c e d u r e , w i t h no changes i n pore p r e s s u r e , w h i l e the o t h e r elements a r e c y c l e d t h r o u g h the s t e p s of the u n d r a i n e d a n a l y s i s . 6.3 L i m i t a t i o n s 1. The mean normal s t r e s s used t o c a l c u l a t e the e l a s t i c and b u l k moduli a r e l i m i t e d t o minimum of 0.1 times the a t m o s p h e r i c p r e s s u r e . 2. The i n c r e m e n t a l P o i s s o n ' s r a t i o ( U, ) i s i n the range of 0 t o 0.495. 1 3. D i l a t i o n may not be i n t r o d u c e d t o the elements which have de v e l o p e d t e n s i o n . 4. Gas laws cannot be used t o p r e d i c t pore p r e s s u r e changes where p o s i t i v e volume changes exceed the gas p o r o s i t y which would imply n e g a t i v e gas p o r o s i t i e s . i . e an u n r e a l i s t i c c o n d i t i o n . 5. Once a r e g i o n reaches f a i l u r e , subsequent l o a d i n c r e m e n t s s h o u l d be an o r d e r of magnitude s m a l l e r than the p r e v i o u s ones. 6. On v e n t i n g , t h e pore p r e s s u r e i n v e n t e d elements w i l l d rop t o z e r o i n s t a n t a n e o u s l y , a n d a f t e r t h a t they w i l l never behave i n an u n d r a i n e d manner. 6.4 S i g n convent i o n s The f o l l o w i n g s i g n c o n v e n t i o n s a r e used i n e v a l u a t i n g the r e s u l t s ( s t r e s s e s , s t r a i n s , a n d d i s p l a c e m e n t s ) . 6.4.1 Loads and d i s p l a c e m e n t s P o s i t i v e d i r e c t i o n s of the n o d a l l o a d s and d i s p l a c e m e n t s are i n d i c a t e d i n F i g . 25'. 6.4.2 S t r e s s e s and s t r a i n s F i g . 26 i n d i c a t e s the p o s i t i v e d i r e c t i o n s of the s t r e s s e s and s t r a i n s . The same c o n v e n t i o n s are used i n the i n p u t of s t r e s s e s and s t r a i n s f o r the p r e - e x i s t i n g s o i l e l e m e n t s . I n s t r u c t i o n s r e g a r d i n g the i n p u t of e x t e r n a l f o r c e s i s p r o v i d e d i n the u s e r ' s manual i n Appendix I . F i g . 27 Some Plane S t r a i n Situations CHAPTER 7 COMPARISON WITH EXISTING SOLUTIONS Under.both d r a i n e d and u n d r a i n e d c o n d i t i o n s the program p r e d i c t i o n s were checked 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 s and o t h e r r e s u l t s . 7.1 E l a s t i c c l o s e d form s o l u t i o n s • Deformation b e h a v i o u r of i . T h i c k w a l l e d c y 1 i n d e r , ( F i g . 27a) i i . Opening i n an i n f i n i t e s p a c e , ( F i g . 27b) are two p l a n e s t r a i n p r o b l e m s , t o which complete , l i n e a r e l a s t i c c l o s e d form s o l u t i o n s a re a v a i l a b l e . I f a and b a r e the i n t e r n a l and e x t e r n a l r a d i i of a t h i c k w a l l e d c y l i n d e r as shown,subject t o p r e s s u r e s P and P from the i n s i d e and o u t s i d e r e s p e c t i v e l y . T h e s t r e s s and the d i s p l a c e m e n t c o n d i t i o n s a t any r a d i u s r a r e g i v e n by the f o l l o w i n g e x p r e s s i o n s . b 1- a* T ( t f - a » ) r * ("0 T ( I y g = 0 due t o axisymmetry) S t r e s s e s and the d i s p l a c e m e n t s i n such a c y l i n d e r ( a=10 f t . , b=30 f t . ),at v a r i o u s l o c a t i o n s when s u b j e c t e d t o an i n t e r n a l s u c t i o n of 500 psf ,are p l o t t e d i n F i g . 30 ,along w i t h the r e s u l t s o b t a i n e d from the m o d i f i e d NLSSIP ,and the r e s u l t s are i n good agreement. I f the p r e s s u r e on the i n n e r f a c e of a s h a f t ( o f r a d i u s a) i n an i n f i n i t e medium i s changed from P 0 t o P^,the f i n a l s t r e s s s t a t e a t any p o i n t (at a r a d i u s r ) i s g i v e n by r *• CT 6 =• P 0 -(Pi'Po)^ C n 5 ) and the r a d i a l d e f l e c t i o n r e s u l t i n g . f r o m the change i s 8 = (LLW. E (The d e r i v a t i o n of the eqns. I l l and mechanics i s p r e s e n t e d i n Appendix F ) . 117,from p r i n c i p l e s of The s t r e s s e s and d i s p l a c e m e n t s i n such a s i t u a t i o n , w h e r e the s h a f t r a d i u s i s 10 f t . , i s p l o t t e d (when the i n t e r n a l p r e s s u r e has been reduced t o 1000 p s f from 2000 p s f ) , i n F i g . 31.Again the r e s u l t s are i n remark a b l y good agreement w i t h those from the m o d i f i e d NLSSIP. 7. 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 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 t e s t s i n sand,where r e l a t i v e l y h i g h s t r a i n s o c c u r , i s a good p l a n e s t r a i n problem t o c o n s i d e r . Many n o n - e l a s t i c s o l u t i o n s have been p r e s e n t e d t o i n t e r p r e t e t he p r e s s u r e m e t e r t e s t d a t a , o u t of which the one g i v e n by Hughes et a l ( 1 9 7 7 ) appears t o be adequate f o r the p a r t i c u l a r s i t u a t i o n . As s t a t e d i n Chapter 3.9,they assume the f a i l u r e of sand at a c o n s t a n t e f f e c t i v e s t r e s s r a t i o ( b a s e d on the s i m p l e shear t e s t d a t a ) , a n d t h e i r s o l u t i o n i s s u p e r i o r t o the o t h e r s i n t h a t ,shear d i l a t i o n of sand a t f a i l u r e i s taken i n t o a c c o u n t . The t h e o r y and the assumptions b e h i n d i t a re reviewed i n Appendix G,and the f o l l o w i n g r e l a t i o n s h i p s g e n e r a t e d by t h i s s o l u t i o n f o r the p l a s t i c zone a r e combined w i t h the p r e v i o u s l i n e a r e l a s t i c ones i n f o r m u l a t i n g a complete s o l u t i o n t o the problem of a c y l i n d e r i c a l opening i n an i n f i n i t e sand mass. 0 9 Or, =. N tern 4 0 \- r n a * (I i 3 ) 2 S-m (116) where a,?0,\?i,t are the same q u a n t i t i e s d e f i n e d i n the p r e v i o u s a r t i c l e ,and O g i s the r a d i a l s t r e s s a t the e l a s t i c - p l a s t i c i n t e r f a c e o c c u r r i n g at a r a d i u s R. 103 G r' - (|- N ) log & (.2.) 0' and f i n a l l y the r a d i a l d e f l e c t i o n s i n the f a i l u r e r e g i o n i s / / Non l inear e l a s t i c - p l a s t i c at R f=OOOl L i n e a r el a st ic pi a s t i c F i g . 28 Linear E l a s t i c i t y i n NLSSIP e x p r e s s e d by D + i U = r ( H \ lift T (•22) C'23) U - D e f l e c t i o n a t the i n t e r f a c e where D = ( <t 5 + ^ ) (from t h e s t r e s s - d i l a t a n c y t h e o r y ) When the i n t e r n a l p r e s s u r e of a s h a f t of r a d i u s 10 f t i s reduced from 2000 p s f t o 800 p s f f a i l u r e s e t s i n , a n d the c l o s e d form p r e d i c t i o n s ( u s i n g Eqns. 118-122) are p l o t t e d i n F i g . 32,along w i t h t h e m o d i f i e d NLSSIP r e s u l t s . The same s i t u a t i o n p r e d i c t e d u s i n g t h e l i n e a r e l a s t i c r e l a t i o n s i s a l s o p l o t t e d . I t can be seen t h a t NLSSIP r e s u l t s l i e between the e l a s t i c and e l a s t o - p l a s t i c c u r v e s . T h i s d e v i a t i o n c o u l d be re v i e w e d i n the c o n t e x t of t h e f o l l o w i n g a p p r o x i m a t i o n s made. 1. The P o i s s o n ' s r a t i o was assumed t o be 0.495,in the f a i l u r e r e g i o n , t o a v o i d i n s t a b i 1 i t y , ( C h a p t e r 3.3.9),whereas the a c t u a l p t s h o u l d be 0.5. 2. The c o a r s e n e s s of t h e f i n i t e element mesh used. 3.In the s t r e s s - r e d i s t r i b u t i o n p r o c e d u r e , a d d i t i o n a l l o a d s coming on the s h a f t b o u n d a r y , d i s t u r b s the s t r e s s boundary c o n d i t i o n s t o some e x t e n t . I n t h e above p r e d i c t i o n s a low v a l u e of- R #(=0.001) was used,as under such c o n d i t i o n s , 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 c u r v e , r e s e m b l e s a t y p i c a l l i n e a r e l a s t i c p l a s t i c c u r v e , a s i n d i c a t e d i n F i g . 28. The c o n s t i t u t i v e r e l a t i o n s a r e s e t up i n NLSSIP i n such a way t h a t , t h e parameters E t and B t do not depend on the s t r e s s l e v e l a t v e r y low R^ . v a l u e s . 7.3 One d i m e n s i o n a l u n l o a d i n g of o i l s a n d Up t o now i t was shown how w e l l the d r a i n e d b e h a v i o u r of sand was p r e d i c t e d i n a few c a s e s by the m o d i f i e d NLSSIP,which appears t o be a s a t i s f a c t o r y check on the e f f e c t i v e s t r e s s model used. The pore p r e s s u r e model f o r o i l s a n d , d e s c r i b e d i n Chapter 3 w i l l now be checked a g a i n s t some e x i s t i n g p r e d i c t i o n s . When the o i l s a n d c o r e samples a r e r e c o v e r e d i n d r i l l h o l e s and brought up i n s t e e l c o n t a i n e r s , t h e y s w e l l by 5-15% of the o r i g i n a l volume. I f the c o r e l i n e r s a r e r i g i d enough i t c o u l d be assumed t h a t , t h i s e x p a n s i o n i s t o t a l l y a x i a l . But as s t a t e d i n Byrne e t a l (1980),some r a d i a l d e f o r m a t i o n c o u l d a l s o o c c u r , s i n c e the c u t diameter of the c o r e i s s l i g h t l y l e s s than t h a t of a I L L U I L initial ef fect ive stress 7 MPa i pore pressure 3 « F i g . 29 One Dimensional Unloading of Oilsand the l i n e r . Thus i f we i g n o r e t h i s e f f e c t , t h e s i t u a t i o n c o u l d be s i m u l a t e d by c o n f i n e d a x i a l u n l o a d i n g of a c y l i n d e r i c a l o i l s a n d s a m p l e , i n i t i a l l y under a v e r y h i g h e f f e c t i v e s t r e s s and a pore p r e s s u r e . W h i l e the system shown i n F i g . 29 i s u n l o a d e d , i t ' s b e h a v i o u r i s p l o t t e d , i n F i g . 33 w i t h the p r e d i c t i o n s of the OILSTRESS program. From t h i s i t i s seen t h a t the change i n t o t a l s t r e s s i s a p r o p o r t i o n e d i n such a way between the e f f e c t i v e s t r e s s and the t o t a l s t r e s s , t h a t the e f f e c t i v e s t r e s s drops r a p i d l y to z e r o , w h i l e the pore p r e s s u r e reduces s l i g h t l y . W h e n such a s t a g e i s approached,the volume i s i n c r e a s e d a t such a h i g h r a t e t h a t the gas may be a l l o w e d t o v e n t . On t h i s assumption the pore p r e s s u r e was dropped t o a t m o s p h e r i c p r e s s u r e i n s t a n t a n e o u s l y , w h e r e b y the e f f e c t i v e s t r e s s shoots up c a u s i n g a c o n t r a c t i o n i n the s k e l e t o n . F u r t h e r u n l o a d i n g t a k e s p l a c e under d r a i n e d c o n d i t i o n s . V e n t i n g u s u a l l y o c c u r s when the s w e l l i n g i s about 5-15% of the o r i g i n a l volume.The c o r r e s p o n d i n g gas p o r o s i t y i s 2-5%. In F i g 33,under the c o n d i t i o n of z e r o e f f e c t i v e s t r e s s , t h e gas p o r o s i t y = i n i t i a l n^ + v o l u m e t r i c s t r a i n = 0.015+0.012 = 2.7% T h i s f i g u r e i s b e l i e v e d t o l i e i n the u s u a l range of v e n t i n g p o r o s i t y . As i n the OILSTRESS program ,here the e f f e c t s due t o shear d i l a t i o n were not c o n s i d e r e d . 7.4 D u s s e a u l t ' s s o l u t i o n D u s s e a u l t (1979) d e r i v e d , a r i g o r o u s o n e - d i m e n s i o n a l e q u a t i o n of s t a t e f o r an u n d r a i n e d s o i l , b a s e d on c e r t a i n a s s u m p t i o n s . l t e x p r e s s e s B( Skempton's pore p r e s s u r e parameter) a s , 6 = i y = ! (124.; dor I + K^cr) where jt^jG) c o n s t i t u t e s a r e l a t i o n s h i p between t h e - s k e l e t o n c o m p r e s s i b i l i t y , p o r e p r e s s u r e , p o r o s i t i e s of each phase and t e mperature and p r e s s u r e s o l u b i l i t y c o n s t a n t s . The assumptions made r e g a r d i n g the pore f l u i d b e h a v i o u r a r e v e r y s i m i l a r t o those i n d i c a t e d i n Chapter 5,but the s k e l e t o n c o m p r e s s i b i l i t y was e x p r e s s e d as a l o g a r i t h a m i c r e l a t i o n s h i p , ( a s I l l IC K W A I. I. I I) ( Y I. I IN l> I R 3 "lO-0 150 a o o 25 0 500 RADIAL 1)1 ST A N C I ' ( f t . ) Fig. 30.b. Displacements i n a Thick Walled Cylinder U N L O A D I N G O l A C V 1.1 N Dl R ICA L S H A H F i g . 31.a. E l a s t i c Stresses i n Around a Shaft U N L O A D I N G O l A ( V I . I N D I R I C A L S H A H i : =i05(lioo p s f p = 0-323 a = 10 f t K L A S T I C c l o s e d f o r m N I - S S M " O 18 0 12-0 KO 1 6 0 , v R A D I A L 1)1 S ' l ' A N C K ( f t . ) 31.b. E l a s t i c Displacements Around a Shaft 20 0 (psf) 3000 2500 U N L O A D I N G OF A (. V 1.1 N DI R I( A L SIIAF K = 10391 OO p |.i = 0.333 a = 10 It 'OL' 2000 as 1500 1000 - .Q 500 F L A S K ) - P L A S T I C c l o s e d forr N L S S II* L I . A S T I C c l o s e d f o r m P 0= 2000 .P i\-= 800 <f> = 30" 10 0 \'l 0 \A 0 16 0 1 8 0 R A D I A L D I S T A N C E ( i t Fig. 32.a. El a s t o - P l a s t i c Stresses Around a Shaft U N L O A D I N G O K A C Y L I N D L R I C A L S I I A K T E L A S T O - P L A S T I C c l o s e d f o r m ( w i t h d i l a t i o n ^ N I . S S I I' ( w i t h d i l a t i o n ) K L A S T I C c l o s e d f o r m 10 0 120 14 0 16 0 K = I058IOO psf p = 0.333 a = 10 f t Wo 2 0 0 R A D I A L D I S T A N C E ( f t . ) F i g . 32.b. E l a s t o - P l a s t i c Displacements Around a Shaft ONE DIMENSIONAL U N L O A D I N G OF O I L S A N D ~E 0 » 10 O I L S T R E S S c u r v e O rroo ,ram results MPo F i g . 33 One Dimensional Unloading of Oilsand UNLOADING OF A CYLINDERICAL SHAFT IN OILSAND F i g . 35 Unloading of a C y l i n d e r i c a l Shaft i n Oilsand compared t o our e x p o n e n t i a l form),as shown below, e = A -C In a ' (|25) where o - v e r t i c a l e f f e c t i v e s t r e s s e - v o i d s r a t i o and A and C a r e s o i l p a r a m e t e r s . S i n c e t h i s was the o n l y a n a l y t i c a l s o l u t i o n a v a i l a b l e , t h e p r e d i c t i o n s of our program f o r the one d i m e n s i o n a l u n l o a d i n g was compared w i t h D u s s e a u l t ' s r e s u l t s , a s shown i n F i g . 34. F i r s t and f o r e m o s t , t h e c o n s t i t u t i v e parameters were s e l e c t e d so as t o g i v e a p p r o x i m a t e l y the same system c o m p l i a n c e as Eqn. 1 2 5 ( f o r A=0.46 and C=0.005).The s o i l was unloaded from an i n i t i a l t o t a l s t r e s s of 308.9 kpa ( c o r r e s p o n d i n g to a s e l e c t e d depth) and a pore p r e s s u r e of 147.1 kpa. The v a r i a t i o n of u and B w i t h the t o t a l s t r e s s , c o m p a r e d w e l l w i t h the c o r r e s p o n d i n g c u r v e s o b t a i n e d from OILSTRESS program(Byrne and Grigg-1980) and D u s s e a u l t ( 1 9 7 9 ) . 7.5 U n l o a d i n g of a c y l i n d e r i c a l s h a f t i n o i l s a n d F i n a l l y , the neighbourhood of a c y l i n d e r i c a l opening i n o i l s a n d was m o d e l l e d by a f i n i t e element domain and a number of s p r i n g s which r e p r e s e n t e d the e f f e c t of the s o i l mass upto i n f i n i t y from the boundary of the domain. The s o i l p r o p e r t i e s as i n d i c a t e d F i g . 35 were a s s i g n e d t o the e l e m e n t s . U n l o a d i n g a t the i n n e r s u r f a c e of the opening was done s t a r t i n g w i t h an i n i t i a l o v e r a l l e f f e c t i v e s t r e s s of 1.0 MPa and a pore p r e s s u r e of 1.5 MPa. F i r s t a t o t a l l y u n d r a i n e d a n a l y s i s was performed,and the convergence of the s h a f t was p l o t t e d as the t o t a l s t r e s s was reduced.As seen from F i g . 35,as a l i m i t i n g s u p port p r e s s u r e ( t h e t o t a l e x t e r n a l s t r e s s below which the s h a f t becomes u n s t a b l e ) of a p p r o x i m a t e l y 1.2 MPa was r e a c h e d , t h e d e f l e c t i o n s grow r a p i d l y , a n d t h e s e r e s u l t s a r e i n r e m a r k a b l y good agreement w i t h those of the OILSTRESS. Ne x t , t h e a n a l y s i s was r e p e a t e d , by v e n t i n g the elements upto a r a d i u s of 5m, by r e d u c i n g t h e i r pore p r e s s u r e t o z e r o and m a i n t a i n i n g d r a i n e d c o n d i t i o n s t h e r e u p o n . T h i s way the l i m i t i n g s u p p o r t p r e s s u r e c o u l d be reduced t o a p p r o x i m a t e l y 0.15 MPa.Once a g a i n the NLSSIP and the OILSTRESS p r e d i c t i o n s were c l o s e t o each o t h e r . A c h a r a c t e r i s t i c f e a t u r e of F i g . 35 i s t h a t , t h e convergence of the s h a f t has been markedly b o o s t e d up by the shear d i l a t i o n , o n l y i n the v i c i n i t y of the l i m i t i n g s u p port p r e s s u r e s . CHAPTER 8 SUMMARY AND CONCLUSIONS A f i n i t e element method f o r a n a l y s i n g the d e f o r m a t i o n b e h a v i o u r of o i l s a n d a d j a c e n t t o s h a f t s and t u n n e l s has been d e v e l o p e d . An e x i s t i n g , n o n l i n e a r e l a s t i c i n c r e m e n t a l e f f e c t i v e s t r e s s model i s m o d i f i e d i n two a s p e c t s , t o r e p r e s e n t the dense o i l s a n d s k e l e t o n b e h a v i o u r . Shear d i l a t i o n i s i n c o r p o r a t e d i n i t u s i n g a temperature a n a l o g y . Rowe's s t r e s s d i l a t a n c y t h e o r y forms the b a s i s of t h i s . The s t r e s s e s p r e d i c t e d u s i n g the e x i s t i n g program , v i o l a t e d the f a i l u r e c r i t e r i o n a t the fa c e of the t u n n e l , under heavy u n l o a d i n g c o n d i t i o n s . A s t r e s s r e d i s t r i b u t i o n t e c h n i q u e i s used t o remedy t h i s . A pore p r e s s u r e model f o r m u l a t e d on the b a s i s of i d e a l gas laws i s used t o r e p r e s e n t the o i l s a n d pore f l u i d b e h a v i o u r . Under u n d r a i n e d c o n d i t i o n s the i n f l u e n c e of the pore f l u i d i s c o u p l e d i n t o the o i l s a n d s k e l e t o n by a c h i e v i n g v o l u m e t r i c c o m p a t i b i l i t y between them, t h r o u g h an i t e r a t i v e p r o c e s s . The p r e s e n t program i s c a p a b l e of p r e d i c t i n g the- l i m i t i n g s u p p o r t p r e s s u r e s , r e q u i r e d by underground t u n n e l s i n o i l s a n d . The p r e d i c t i o n s were checked a g a i n s t a v a i l a b l e c l o s e d form s o l u t i o n s and o t h e r r e s u l t s . T h e y a r e i n good agreement w i t h the l i n e a r e l a s t i c and l i n e a r 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 , f o r the problem of a c i r c u l a r opening i n an i n f i n i t e s p a c e , d e f o r m i n g under p l a n e s t r a i n c o n d i t i o n s . F u r t h e r m o r e , the program y i e l d e d r e s u l t s t h a t agreed w e l l w i t h t h o s e o b t a i n e d f o r the o n e - d i m e n s i o n a l u n d r a i n e d u n l o a d i n g of o i l s a n d by D u s s e a u l t ( 1 9 7 9 ) , a n d f o r the u n l o a d i n g of a c i r c u l a r s h a f t i n o i l s a n d by Byrne and G r i g g ( 1 9 8 0 ) . The s o l u t i o n f o r the u n l o a d i n g of a t u n n e l i n o i l s a n d shows t h a t the l i m i t i n g s u p p ort p r e s s u r e s can be reduced by v e n t i n g elements t o a r e a s o n a b l e d i s t a n c e from the t u n n e l . I t i s a l s o found t h a t the e f f e c t s of shear d i l a t i o n a r e s i g n i f i c a n t o n l y when the l i m i t i n g s u p p ort p r e s s u r e i s approached. REFERENCES Byrne, P.M.,Smith, L . B . , G r i g g , R.F.,and S t e w a r t , W.P.,1980. A Computer model f o r S t r e s s - S t r a i n and D e f o r m a t i o n A n a l y s i s of O i I s a n d s . P r o c e e d i n g s of A p p l i e d O i l s a n d G e o s c i e n c e C o n f e r e n c e , Edmonton. Byr n e , P.M.,Grigg, R.F.,1980.OILSTRESS-A computer program f o r A n a l y s i s of S t r e s s e s and D e f o r m a t i o n s i n O i l s a n d . S o i l Mechanics S e r i e s No. 42, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, Vancouver. B y r n e , P.M., Duncan, J.M., 1979.NLSSIP- A computer program f o r the N o n l i n e a r A n a l y s i 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 P r o b l e m s . S o i 1 Mechanics S e r i e s No. 41, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h C o l u m b i a , Vancouver. Cook, R.D., 1974. Concepts and A p p l i c a t i o n s of F i n i t e Element A n a l y s i s . John W i l e y and Sons,Inc. Duncan, J . M., Byrne, P.M., Wong, K.S., and Mabry, P.,1978. S t r e n g t h , S t r e s s - S t r a i n and B u l k Modulus Parameters f o r F i n i t e Element A n a l y s e s of S t r e s s e s and Movements i n S o i l Masses.Report No. UCB/GT/78-02 t o N a t i o n a l S c i e n c e Foundat i o n . D u s s e a l t , M.B.,1979. U n d r a i n e d Volume and S t r e s s Change Beha v i o u r of U n s a t u r a t e d Very Dense Sands.Canadian G o e t e c h n i c a l J o u r n a l , V o l . 16, No.4,pp. 627-640. H a r r i s , M.C., and Sobkowicz, J.C., 1977. E n g i n e e r i n g Behaviour of O i l s a n d . P r o c e e d i n g s of the Canada-Venezuela O i l s a n d s Sympos i urn,Edmonton. Hughes, J.M.O., Wroth, C P . , and W i n d l e , D., 1977. P r e s s u r e m e t e r T e s t s i n Sands. G e o t e c h n i q u e , V o l . X X V l l , No.4, pp. 455-477. P a u l i n g , L., 1959. G e n e r a l C h e m i s t r y . Second E d i t i o n . W.H.Freeman and Company. San F r a n c i s c o . 341 pp. P i c k e r i n g , D.J., 1969. Simple Shear Machine f o r S o i l s . PhD T h e s i s . Rowe, P.W.,1963. The S t r e s s D i l a t a n c y R e l a t i o n f o r S t a t i c E q u i l i b r i u m of an Assembly of P a r t i c l e s i n C o n t a c t . P r o c . Of R o y a l S o c i e t y . Rowe, P.W., 1971. The T h e o r i t i c a l Meaning of Observed V a l u e s of D e f o r m a t i o n P a r a m e t e r s . P r o c e e d i n g s of the Roscoe Memorial Symposium. S c o t t , C.R., 1978.Developments i n S o i l M e c h a n i c s - 1 . A p p l i e d S c i e n c e P u b l i s h e r s L t d . London. S m i t h , L.B., and Byrne, P.M., 1980. Convergence Confinement Method of Design of S h a f t s and T u nnels i n O i l s a n d s . Paper p r e s e n t e d a t the c o n f e r e n c e on A p p l i e d O i l s a n d s G e o s c i e n c e . 1980. Edmonton. Timoshenko, S.,1941. S t r e n g t h of M a t e r i a l s . V o l . 2. Van N o s t r a n d , New York, 510 pp. Thurber C o n s u l t a n t s , 1979. A Computer Model f o r S t r e s s - S t r a i n A n a l y s i s of O i l s a n d . A r e p o r t s u b m i t t e d t o the Canadian Department of Energy Mines and R e s o u r c e s , June, 1979. T h u r a i r a j a h , A., and S i t h a m p a r a p i l l a i , V., 1979. S t r e n g t h -D e f o r m a t i o n of Sand d u r i n g D r a i n e d T r i a x i a l T e s t s . P r o c e e d i n g s of the F o u r t h A s i a n R e g i o n a l C o n f e r e n c e on S o i l M echanics and F o u n d a t i o n E n g i n e e r i n g , V o l . 1 , pp. 177-182. V a i d , Y.P., B y r n e , P.M., and Hughes, J.O., 1980. D i l a t i o n Rate as a Measure of L i q u i f a c t i o n R e s i s t a n c e of S a t u r a t e d G r a n u l a r M a t e r i a l . S o i l Mechanics S e r i e s No.43, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h C o lumbia, Vancouver, B.C. Weast, R.C., 1971. Handbook of C h e m i s t r y and P h y s i c s . Chemical Rubber Company, C l e v e l a n d , Ohio. F.79 pp. Z i e n k i e w i c z , O.C., 1977. The F i n i t e Element Method. T h i r d E d i t i o n . M c G r a w - H i l l Book Company(UK) L i m i t e d . APPENDIX A 1 E t A . l P r o o f of yfc = 2 (±— / 3 B Aa I f f o r any s t r e s s increment Aa = [ A t ] , the c o r r e s -~ xz Ae X ponding s t r e s s increment i s Ae = [ . ] then u s i n g , xz G e n e r a l i z e d Hooke's law f o r the in c r e m e n t ; A e x = - | [ A a x - u t ( A a y + A a z ) ] (AO) A e y = \ U V " t ( A a x + A a z ) ] ( A 1 ) A e 7 = | [ A a z - p t ( A a y + A x ) ] (A2) By AO + A l + A2, A e x + A £ y + A e z = ! t < l - 2 u t ) < A o x + A o y + A o z ) A e v = l t ( 1 " 2 v i t ) A a m B. ^ 5 t 3 ( l - 2 y f c ) _ - M t . (A3) A M T - 2 { 1 3B, ' A.2 D e r i v a t i o n of the e x p r e s s i o n f o r E f c Kondner's 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 from eqn. 15 ( a - a ) = j - = E i ^ V V u i t -1 3 E i ( 0 ; L o 3 ) u l t (A4) 82 D i f f e r e n t i a t i n g u>. y.t e ( qi" q 3 nt vWt1 + ^ , v ° 3 ) [ ¥ ( v S ] = 1 i ( o l " ° 3 ) d e ( V 0 3 ) = ' 1 , e — U ( v V u l t E i t o 1 - o 3 ) u l t F r o m e q n . A4, e = ( ( q r ° 3 ) d e o 1 - o 3 ) u l t J d ( o 1 - o 3 ) d e [1-( q r q 3 } 3 ( a ± - o 3 ) ( a 1 - a , ) 3 u l t E ^ l - ' V ^ ' ( O - j - o , ) 3 ' u l t d ' V ° 3 » . E . U ]2 F o r t r i a x i a l t e s t s o ^ O j ) = E f c ( o, - o-) U s i n g e q n s . 16 and 17 o R _ ( a. - o_ ) E t = E P a ( P T ) U ( o 1 - o 3 ) f " J F i g . 36. Mohr C i r c l e at F a i l u r e 2h An e x p r e s s i o n f o r (°2~ a3)f c o u l d be d e r i v e d from the Mohr-Coulomb p l o t i n terms o f C, *, and In the F i g . 36, BQ = l / 2 ( a±- o 3 ) f Sine). AB = 1/2 ( a 1 - o 3 ) f Cos * OB = OQ - BQ (°l-°3 ) {al-a3]f ..A = — 2~ S i n * = _ _ _ + a 3 _ _ s i n * ( V ° 3 } f = 2 [ 1 - S i n * ] + a 3 AP AB-C t a n * - CP " ~75B~ 1/2( o 1 - o 3 ) f Cos <(> - C l / 2 ( a 1 - a 3 ) f + [l-Sin<|>J + a 3 S i n 2 * l/2( a1-a3)f t a n * - l / 2 ( V ^ ' f °3 t a n * = 1/2( o 1 - o 3 ) f Cos* - C ( 0 l - o 3 ) f 2C Cos * + 2 a 3 S i n * (1 - S i n * ) (A6) By s u b s t i t u t i n g i n eqn. A5 °3 n R f ( V ^ * ( 1 _ S i n < t > ) 2 E t = K E P a ( P ~ ) t l _ 2C Cos * + 2 a 3 S i n *] * ( A ? ) F i g . 37. P a r t i c l e S l i p A P P E N D I X B D e r i v a t i o n o f s t r e s s - d i l a t a n c y e q u a t i o n The a c t u a l p i c t u r e w i l l be made s i m p l e i n F i g . 37 by a 2 - D c o n s i d e r a t i o n w i t h t h e s l i p o r i e n t a t i o n , a l o n g t h e p l a n e o f t h e p a p e r . d x 3 t a n 0 = — d x ^ F r o m i n t e r p a r t i c l e f r i c t i o n L l t a n ( = — E n e r g y r a t i o , E i s d e f i n e d a s , i n c r e m e n t o f e n e r g y a p p l i e d t o t h e c o n t a c t by g — _ . i n c r e m e n t o f w o r k done a g a i n s t t h e e x t e r n a l L 3 - L . d x , E = ( B l ) L 3 d x 3 A b s o l u t e e n e r g y a b s o r b e d = L x 6x1 + L 3 d x 3 = L x dx1 (B2) / . F o r a g i v e n e n e r g y i n p u t ( L , d x ) , a b s o l u t e e n e r g y a b s o r b e d i s a m i n i m u m , when E i s a m i n i m u m . F r o m e q n . B l , t a n ( <|> + 6) 87 t a n g S e c 2 ( <|> + 3) - tan( <{> + 3) S e c 2 3 dE _ u v  dg t a n z 3 For minimum E, ta n 3 Sec 2 ( 4>^ +3) = tan(«J>+B) S e c 2 3 S i n 2 6 = S i n 2 ( <j^ + 3) 2 3 = n-2 • -2 3 n <t> S u b s t i t u t i n g i n eqn. B3, tan( TI/4 + 4>^2) E m i n ~ tan( IT/4— <t>l/2) E . = t a n 2 ( IT/4+ dp /2) = K min V p y (B4) I f a l l the p a r t i c l e s s a t i s f y the minimum energy c o n d i t i o n i n s l i p p i n g , combining e q u a t i o n s B l and B4 g i v e s , n -E L^dx^ 1 = K (B5) n p y E L d x , 1 where n r e f e r s t o the no: of s l i p p i n g p a r t i c l e s Under t r i a x i a l c o n d i t i o n s ( a x i - s y m m e t r i c ) n a d £ = E L.dx a a 1 1 and n 2 o d e = E L - d x o r r 3 3 From eqn. B5, o d e a a = K 2 o rd e r P y 98 I f d v i s t h e v o l u m e t r i c s t r a i n d v = d E +2d e a r o r 2d e •r^-— = 1 + — j — d e a d £ a — = (1 - 7f—) K ° r d £ a P l J a a ; , , d v a r (1 - tan2(45+<)>i/2) ( B 6 ) f ree gas water oi l sand 'w A u A T 1+8: w Ini t ia l state % T 0 Final state-F i g . 38. E f f e c t of Pressure and Temperature on Oilsand 9o A P P E N D I X C P r o o f of the e x p r e s s i o n f o r the change i n pore p r e s s u r e i n  und r a i n e d o i l s a n d . C o n s i d e r a u n i t volume of o i l s a n d under a p r e s s u r e and temperature o f U Q and TQ r e s p e c t i v e l y . ( F i g . 3 3 ) . The f i n a l s t a t e , T^ as i n d i c a t e d i s a t t a i n e d by the changes Au and AT o c c u r i n g s i m u l t a n e o u s l y , o r one a f t e r the o t h e r , o t h e r . The i n i t i a l p o r o s i t i e s n , n and n become the i n i t i a l ^ g w o volumes of the t h r e e phases a s , a u n i t t o t a l volume i s c o n s i d e r e d . Ay - the volume of gas coming out of s o l u t i o n due t o the i n c r e a s e i n p r e s s u r e Au, measured a t s t a n d a r d c o n d i t i o n s of U and T , can be o b t a i n e d by u s i n g e q u a t i o n 108. V = -H V Au g AV, = -(H n +H n ) Au (CI ) 1 v o o w w S i m i l a r l y i f AV"2 i s the volume o f gas coming out o f the s o l u t i o n due t o the i n c r e a s e i n temperature of AT, (measured a t U a. and T ). From e q u a t i o n 109, V = $ V AT g AV_ = ( 6 n + 3 n ) AT (C2) 2 o o ww where H Q, H , 3 Q, S w are the p r e s s u r e and temperature s o l u b i l i t y c o n s t a n t s f o r o i l and water phases. From eqns. CI and C2 the t o t a l volume o f gas coming out a t STP, A V 1 + A V 2 = ( B o n o + B w n w ) A T - (H n +H n ) Au v o o W W From PV = nRT t h e v olume o f t h e g a s c o m i n g o u t a t t h e f i n a l s t a t e i s , AV 3 = [( 3 o n Q + 6 w n w ) AT - ( H ^ + H ^ ) A u ] ^ ^ (C3) j. a The e x p a n s i o n o f t h e f r e e g a s a l s o t a k e s p l a c e u n d e r c h a n g e s o f Au and AT; and f r o m e q n . 107 i f AV^ i s t h e i n c r e a s e i n v o l u m e o f t h e f r e e g a s , A V 4 = V ^ 7 - 1 ] ( C 4 ) o 1 AV = AV..+ AV. g 3 4 = V ^ I 7 ' 1 ] + ^ [ ( f 3 o n o + V V A T ^ o 1 I a - (H n +H n ) Au] v o o w w = t h e n e t i n c r e a s e i n t h e gas v o l u m e . Now i f A e v i s t h e c o m p r e s s i v e s t r a i n o f t h e sa n d s k e l e t o n , The i n c r e a s e i n volume o f t h e sa n d s k e l e t o n = - A e v x 1 = -Ae v S i n c e t h e c o n d i t i o n s a r e u n d r a i n e d , and f o l l o w i n g t h e a s s u m p t i o n s r e g a r d i n g t h e c o m p r e s s i b i l i t y o f o i l w a t e r and sa n d p a r t i c l e s , ^ o 1 l a - (H n +H n ) Au] v o o W W <?2 w h i c h s i m p l i f i e s t o T n u u o A e v + u o ( f 5 o n o + f W A T + -T --° AT o AU = u T, n _ A e + a 1 (H n +H n ) g v T o o WW R e v i e w o f c o n s t a n t H H e n r y ' s l a w f o r t h e s o l u b i l i t y o f g a s e s i n l i q u i d s m e n t i o n e d i n C h a p t e r 5, c o u l d be s t a t e d as f o l l o w s . The c o n c e n t r a t i o n o f a s o l u t e i n a s o l u t i o n i s p r o p o r t i o n a l t o t h e p a r t i a l p r e s s u r e o f t h e s o l u t e a t t h e i n t e r f a c e ; a t c o n s t a n t t e m p e r a t u r e o r m a t h e m a t i c a l l y N P 1 4- = K A M ( C 5 ) s o l u t e N +N g I N - no. o f m o l e s o f t h e gas i n s o l u t i o n 9 N' - " " " " " " " s o l v e n t . i K - H e n r y ' s l a w c o n s t a n t , s p e c i f i e d f o r t h e p a r t i c u l a r t e m p e r t u r e . S i n c e H e n r y ' s l a w c o u l d be a c c u r a t e l y a p p l i e d f o r d i l u t e s o l u t i o n s F u r t h e r i f V i s t h e v olume o f t h e s o l v e n t and p i s i t ' s d e n s i t y s a t t h a t t e m p e r a t u r e , <?3 M = m o l e c u l a r w t . o f t h e s o l v e n t . S u b s t i t u t i o n i n e q n . C5 y i e l d s , kM P = — N s o l u t e pV^ g S i n c e i t was a s sumed t h a t t h e p a r t i a l p r e s s u r e a l l o f p h a s e s i n o i l s a n d i n t e r s t i c e s i s t h e s a m e , P . . = u s o l u t e U • 7 ? , N 9 ( C 6 ) C . 2 . a C h a n g e s i n p r e s s u r e u n d e r c o n s t a n t t e m p e r a t u r e (T ) The q u a n t i t i e s , k , M , p and a r e u n a f f e c t e d by t h e p r e s s u r e c h a n g e s , and by d i f f e r e n t i a t i n g e q n . C6 we g e t AN = - T - £ AU ( C 7 ) g kM w h i c h g i v e s t h e n o . o f m o l e s s e n t i n t o s o l u t i o n , on i n c r e a s i n g t h e p r e s s u r e , by A u . I f AVg i s t h e v o l u m e g o i n g i n t o s o l u t i o n ( m e a s u r e d a t t h e p r e s e n t p r e s s u r e u Q + A u and t h e c o n s t n a t t e m p e r a t u r e T ) ; u s i n g t h e i d e a l g a s e q u a t i o n pV = nRT ( u + Au) AV = AN RT o g g S u b s t i t u t i n g i n e q n . C 7 , (u +Au) pV AV_ = ^ Au RT " ' g kM AV = ( ^ ) V , -4- (C8) g v kM £ u + Au K ' ^ o I f we r e f e r t h i s v o l u m e t o t h e STP o f U and T we g e t , 3. 3 ( u + Au) T A V ) = v & u _ l _ o _ a g J u , T v k M ' i (u +Au) T u a a a o a pRT A V , _ {. £. (C9) a a » By comparison of eqns. C9 and 108, PRT = H = kMu a (CIO) I f we i g n o r e the temperature dependence of p - the d e n s i t y of the l i q u i d , and make s u r e t h a t a l l the measurements are made a t u a and T a l w a y s , then M, p, R, T and u w i l l be c o n s t a n t s ; but not k. I t s h o u l d be remembered here t h a t , a l t h o u g h the volume of gas i s r e f e r r e d t o the STP o f u and T f o r s i m p l i c i t y , the gas c o u l d be e v o l v e d a t any o t h e r temperature - T. Thus, from the p r e v i o u s d e f i n i t i o n of Henry's c o n s t a n t (k) i n eqn. C5, k i s s p e c i f i e d f o r T. .•.k = f ( T ) From t h i s r e s u l t and eqn. CIO we can conclude t h a t the c o e f f i c i e n t H i s h i g h l y dependent on the temperature a t which the gas i s e v o l v e d . 45 APPENDIX D D.l C o n s t i t u t i v e m a t r i x The g e n e r a l i z e d Hooke's l a w f o r i n c r e m e n t a l e l a s t i c i t y , c a n be s t a t e d a s , A e x = l t [ A o x - ^ t ( A o y + A a 2 ) ] ( A 0 ) 1. A £ y E t [ Ao - u t ( A a x + A a z ) ] ( A l ) 1 A e z E f c[ A o z - u t ( A a y + A o x ) ] (A2) AT - _x.Y. ( D l ) AT A Y X Z " ~G xy G 1 x z (D2) t AT t A T y z (D3) AY = N ~ T y z G T E t w here G f c - 27/1+ufc) From t h e d i s c u s s i o n i n C h a p t e r 4.1, ( f o r p l a n e s t r a i n ) Ae = AY = Ay = 0 a z x z 'yz S u b s t i t u t i o n i n e q n s . D l - D3 g i v e s AT = — (D4) A T x y • 2 ( l + u t ) AT = AT = 0 x z y z S u b s t i t u t i o n i n e q n . A2 g i v e s A o z = u f c( Ao y+ Aa x) (D5) w h i c h i s e q u i v a l e n t t o a = u, ( a + a ) z t y x Eqn. D5 along with eqns. AO and Al y i e l d A e x = l tI« 1-"t ) A°x " U t d + P t ) A o y ] A e y = | h K l - p | ) A o y - M td+V t) Aa x] In the m a t r i x form A E " A e i_ E. o r " A ex E.. A e t _ y _ "(l-u|) - u t ( l + V -v t(l+u t) (1-M£) ' ( l - u t ) - M t - u t d - u t ) A o. A o. A a on i n v e r s i o n A °x A a. ( l + H t ) ( l - 2 n t ) "I - u t u t \ 1 - u t A a. A a. y_J E q u a t i o n s D4 and D6 c o u l d be combined t o g i v e , ( l + y t ) ( l - 2 u t ) xy Now, i f we d e f i n e "1-P t  v t 0 1-vi. 0 0 ( l - 2 u t ) Ae, x Ae y A6 xy (D6) ( D 7 ) 3B, B' But B' t 2(l+u.) t 3 ( l - 2 u ) and G' t 2 ( l + y t ) t (A3) V t - 2(l+u t)(i-2u t) F u r t h e r B ' + G ' f c 2 ( l + y t ) ( l-2y f c) 2 ( l+n t ) B ' t " G ' t E f c (1+ U t ) (1+ P t ) (1-2 u f c) 2 ( l + u t ) (1-2 P t ) [ 2 U t ] (1+U t ) ( l-2u t) S u b s t i t u t i n g t h e s e v a l u e s i n t h e c o e f f i c i e n t s o f t h e [D] m a t r i x i n t h e e q n . D 7 , A o X Aa — y AT - B ' t + G ' B ' t - G ' t B ' t - G ' t B ' t + G ' t 0 0 G ' Ae A t AY xy_ D.2 F o r 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 A s m e n t i o n e d i n C h a p t e r 4, t h e same f u n c t i o n i s u s e d t o d e f i n e t h e s h a p e o f t h e e l e m e n t as w e l l as t h e d i s p l a c e m e n t s w i t h i n i t . T h i s f u n c t i o n i s e x p r e s s e d i n n a t u r a l c o - o r d i n a t e s ( E, and n) - a s y s t e m o f c o - o r d i n a t e s i n t r i n s i c t o t h e e l e m e n t . 98 X u F i g . 39. Isoparametric Element 99 Thus s i n c e the d i s p l a c e m e n t s a r e t o be r e l a t e d t o the system c o - o r d i n a t e s n and y, f i r s t o f a l l a r e l a t i o n s h i p between these two c o - o r d i n a t e s has to be found, i . e . , the shape of the element has t o be d e f i n e d by n a t u r a l c o - o r d i n a t e s ; as f o l l o w s . x y N 1 0 N 2 0 N 3 0 N 4 0 0 N± 0 N 2 0 N 3 0 N 4 (D8) where N 2 , N 3 , and N 4 are c a l l e d shape f u n c t i o n s whose v a l u e s are g i v e n by, N l = (1-5) (1 - n ) M . (1-5) d - n ) N 2 = 4 M - (1+5) (1+n)  N 3 4 and N 4 = (1-5) (1+n) T h i s t r a n s f o r m a t i o n , i n f a c t maps the q u a d r i l a t e r a l i n t o a square as shown by F i g . 39. In the f i n i t e element p r o c e d u r e , the next s t e p i s to assume a c o m p a t i b l e s e t of d i s p l a c e m e n t s . u v = [N] = [N] 6 (D9) roo u and v a r e t h e d i s p l a c e m e n t s i n x,y d i r e c t i o n s a t any p o i n t ( x , y ) o r (£,n); w h e r e a s t h e 6^  v a l u e s a r e t h e n o d a l v a l u e s o f t h e u and v . I n k e e p i n g w i t h t h e i s o p a r a m e t r i c e l e m e n t p r o p e r t i e s , - [N] h a p p e n s t o be t h e same m a t r i x i n t r o d u c e d ' i n e q n . D8. The f u n c t i o n s e l e c t e d i n e q n . D9, d e f i n e s t h e assumed d i s p l a c e m e n t f i e l d , and i n o r d e r f o r t h e s o l u t i o n o b t a i n e d t o c o n v e r g e a t t h e c o r r e c t s o l u t i o n , on s u b d i v i s i o n i n t o s m a l l e r e l e m e n t s , t h i s assumed f i e l d s h o u l d s a t i s f y two p r o p e r t i e s , n a m e l y a . a d m i s s i b i l i t y b. c o m p l e t e n e s s , w h i c h w i l l n o t be d i s c u s s e d h e r e i n . E q n s . D8 and D9 c o u l d be c o m b i n e d w i t h "x e " I x ey 3y T x y 3y ' 3x t o o b t a i n , Ae = [B] 6 (D10) where [B] - t h e s t r a i n - d i s p l a c e m e n t m a t r i x - a f u n c t i o n o f t h e n o d a l c o - o r d i n a t e s . Now f r o m e q n . D7, Ao" = [D] Ae (D7) t h e e f f e c t i v e s t r e s s has been i n t r o d u c e d h e r e as i t i s t h e s t r e s s r e s p o n s i b l e f o r t h e d e f o r m a t i o n . Ao' = Aa - Au 1 Aa - A u [ l ] = [D] Ae 0 c o m b i n i n g w i t h e q n . DlO 1 Aa - A u [ l ] = [D] [B] 5 ( D l l ) 0 1 0 1 E x t e r n a l v e r t i c a l w o r k done by n o d a l d i s p l a c e m e n t s r i d i n g t h r o u g h t h e n o d a l f o r c e i n c r e m e n t s T = 6 Af I n t e r n a l v e r t i c a l w o r k done b y t h e c o m p a t i b l e s t r a i n i n c r e m e n t s r i d i n g t h r o u g h t h e s t r e s s i n c r e m e n t s T = / Ae Aa d v F r o m t h e p r i n c i p l e o f v e r t i c a l w o r k ( s i n c e t h e f o r c e - s t r e s s e q u i l i b r i u m e x i s t s ) E x t . v i r t u a l w o r k = I n t . v i r t u a l w o r k 6 T Af = / A E T Aa d v u s i n g e q n . D l l = / Ae [ [D] [B] S + T T T F r o m e q n . D l O , Ae = 6 [B] V 1 0 A j d v 6 T Af = 7 5 T [ B ] T {[D] [B] 6 + "1 1 0 Au } d v = 6 T [ / [ B ] T [ D ] [B] dv 6 + / [ B ] T "1" 1 0 dv AU ] Af = / [ B ] T [ D ] [ B ] dv 6 + / [ B ] T "1" 1 0 d v Au I f we d e f i n e / [ B ] T [ D ] [ B ] d v = [K] = e l e m e n t s t i f f n e s s m a t r i x a n d / [ B ] "1 1 0 dv = [ K * ] t h e f o r 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 r e s u l t s Af = [K] 6 + [K] Au I f [ K * ] Au i s t r e a t e d a s a n a d d i t i o n a l s e t o f n o d a l d e f l e c t i o n s i n m o d i f i e d n o d a l f o r c e i n c r e m e n t v e c t o r A f i s e x p r e s s e d a s 1 0 2 A f = [K] 5 (D12) 8x1 8x8 8x1 I f n i s the t o t a l no. of degrees of freedom f o r the system, eqn. D12 can be a r r a n g e d , i n such a way t h a t the g e n e r a l i z e d co-o r d i n a t e s o f t h i s f i r s t element, t a l l y w i t h t h e i r c o r r e s p o n d i n g degree of freedom ( i f d.o.f. i s non-zero) then A F ^ = [ K 1 ] ^ n x l nxn n x l E A F \ = E [K^] A ^ (summed t o a l l elements) y i e l d s A F = [ K ] A n x l nxn n x l S i n c e the c o e f f i c i e n t s of [B] are f u n c t i o n s of the n o d a l co-o r d i n a t e s and those of [D] are o n l y s t r e s s - s t r a i n parameters [K] = [ B ] T [ D ] [ B ] V V - V o l . of the element o r i f a u n i t t h i c k n e s s i s c o n s i d e r e d and A g - area of the element, [K] = [ B ] T [ D ] [B] A e (D13) F i g . 40. Approximation of an I n f i n i t e Problem A p p e n d i x E S p r i n g S t i f f n e s s X ) W i t h r e f e r e n c e t o F i g . 4 0 , i f t h e u n i f o r m i n g d i s t r i b u t e d l o a d i n t h e s p r i n g b o u n d a r y i s p, and i t ' s e q u i v a l e n t p o i n t l o a d on e a c h s p r i n g i s P 2 Trap = NP ( E l ) N - no. o f s p r i n g s . R a d i a l d e f o r m a t i o n ( e l a s t i c ) r e s u l t i n g a t t h e i n s i d e b o u n d a r y o f an i n f i n i t e mass, u n d e r an i n s i d e p r e s s u r e o f p i s 6 = a.p (E2) ( T h i s i s f u r t h e r d i s c u s s e d i n A p p e n d i x F) C o m b i n i n g e q n s . E l and E2, 2 na E 6 a ( l + u ) = NP N (1+ u) By c o m p a r i s o n w i t h P = X 6 2 TT E X = N (1+y) F o r non l i n e a r i n c r e m e n t a l e l a s t i c i t y X = N (1+ u t) 4JT G f c N E^ Appendix F E l a s t i c C l o s e d Form S o l u t i o n s The e l a s t i c s t r e s s e s f o r a x i s y m m e t r i c , plane s t r a i n problems, ( u s i n g an A i r y s t r e s s f u n c t i o n ) , can be shown to be the form ar = f 2 + B ( 1 + 2 l o 9 r ) + 2 C ( F 1 ) °8 = r 2 + B ( 3 + 2 l o 9 r ) + 2 C < F 2 ) x r 9 = 0 (F3) (Timoshenko, 1941) For c o n t i n u o u s r e g i o n s , as shown i n F i g . 27 a f u l l c i r c l e s o l u t i o n e x i s t s and the term B - which g i v e s r i s e t o t a n g e n t i a l d i s p l a c e m e n t s , v a n i s h e s . .•. Eqns. F l and F2 reduce t o .-. p r = -p 2 + 2C (F4) " A + 2C (F5) °6 = r At r = a, o r = P i (comp. s t r e s s c u r ve) and a t r = b, °T = P o S u b s t i t u t i n g i n eqn. F4 P. = - + 2C 1 a 2 P = - + 2C O b 2 which g i v e s A = ( P i - P Q ) a 2 b 2 / ( b 2-a 2) and 2C = - ( a 2 P i - b 2 P Q ) / (b ^ a 2) - ( a 2 P . - b 2 P 0 ) ( P i - P Q ) a 2 b 2 *'* °^ b 2 - a 2 + ( b 2 - a 2 ) r 2 - ( a 2 P . - b 2 P Q ) ( p . - P o ) a 2 b 2 ° 6 ~ ( b 2 - a 2 ) ( b 2 - a 2 ) r 2 T _ — 0 r e U s i n g g e n e r a l i z e d H o o k e 1 s l a w s , e 9 = I [ % - ^ r + 0 z » ] ( F 6 ) ( f o r p l a n e s t r a i n c o n d i t i o n ) i ^ H - [ ( l + M ) o f l - y o l (F7) • ' • £ e = — E — " M U r S u b s t i t u t i n g e q n s . F4 and F5 i n e q n . F7 - l _ t _ . [ ( 1 - y) ( 2 C - A / r 2 ) - y ( 2 C + A / r 2 ) ] E = [ 2 c-£ 2-4Cy] E r ( l - 2 y ) ( l + u ) 2 C _ (1+y) A E E r 2 U n d e r a x i s y m m e t r i c c o n d i t i o n s , 6 e 9 = r where 6 i s t h e r a d i a l d i s p l a c e m e n t l ( l - 2 y ) ( l + y ) 2 C (1+y) a 5 E — - | S u b s , f o r 2C and A ( l - 2 y ) (1+y) ( a 2 P i - b 2 P Q ) r (1+y) < P . - P o ) a 2 b 2 E ( b 2 - a 2 ) E ( b 2 - a 2 ) r 107 F o r t h e c a s e o f a s h a f t i n an i n f i n i t e m e d i u m , t h e s o l u t i o n c o u l d be d e d u c e d , f r o m t h e l i m i t i n g c a s e o f b + % F r o m e q n s . I l l and 112 ( P . - P ) a 2 a = P + 1 ° r o r 2 ( P . - P ) a 2 a n d a Q = P -e o r 2 x = 0 H e r e P Q c o u l d be i n t e r p r e t e d a s t h e i n i t i a l s t r e s s o f t h e w h o l e s o i l m a s s . F r o m e q n . 113, t h e i n i t i a l d e f l e c t i o n due t o a c o n s t a n t p r e s s u r e P Q " i s -(1-2 p) (1+y) r P 6 i = E ( S i n c e P i = P ) S i m i l a r l y t h e t o t a l d e f l e c t i o n , when P Q and P^ a r e a p p l i e d i s 6 f = " ( l - 2 u ) ( l + y ) P P Q d + y) ( p i - p 0 ) a = 6. + 6 l .•.The a d d i t i o n a l d e f l e c t i o n due t o t h e c h a n g e o f t h e i n n e r s u r f a c e p r e s s u r e f r o m P Q t o P ^ i s 6 6 = !_+__ S _ (p._p , E r v l o ' F i g . 41. E l a s t o - P l a s t i c Solution I0<? APPENDIX G E l a s t o - p l a s t i c s o l u t i o n (Based on Hughes e t a l (1977), s o l u t i o n f o r an expanding c y l i n d e r i c a l c a v i t y i n sa n d ) . When s h e a r i n g i s i n t r o d u c e d i n t o the s o i l mass by e i t h e r i n c r e a s i n g o r d e c r e a s i n g the p r e s s u r e a t the i n s i d e f a c e , the p r i n c i p a l s t r e s s r a t i o , a t any p o i n t w i l l change i n a manner shown i n F i g . G.b. The immediate v i c i n i t y o f the c a v i t y w i l l r e a c h f a i l u r e f i r s t , and on f u r t h e r s h e a r i n g the f a i l u r e zone w i l l be p o r o p a g a t i n g away as seen i n Fig.41a. The p o s i t i o n s of the p o i n t s P, Q and R i n the s t r e s s - s t r a i n and volume change c u r v e s are marked i n the f i g u r e s . For a n a l y t i c a l convenience these c u r v e s a re i d e a l i z e d i n each c a s e , such t h a t s t a t e s of a l l p o i n t s l i e on ST. I t has a l r e a d y been d i s c u s s e d i n Chapter 3.9, how t h i s i d e a l i z a t i o n i s j u s t i f i e d by s i m p l e shear t e s t r e s u l t s . For the p l a s t i c zone d e f i n e d by (a < r < R) o' Q 1+Sin <b * 6 „ Ymax . ?,„,-. max, — = N = T ^ S l n l = t a n 2 ( 4 5 + - T - ) r Tmax E q u i l i b r i u m e q u a t i o n i n p o l a r c o - o r d i n a t e s , do' o' -o' -L + H = 0 (G2) d r d°'r „ , R — = ( N - l ) l o g -dr V l ' J r R a l s o l i e s on the i n n e r boundary of the e l a s t i c zone and on a p p l y i n g eqn. 115, 110 ° V R = V ( 0 , R - V R* ' 2 PO-°'R B u t f r o m e q n . 118, a" ) = No' /.Na' .-a *. a R R 2P - a ' o 2P R R 1+N 2P (1-Sin<(> ) o max P ( l - S i n * ) o % a x S i n c e e q n . 121 s a t i s f i e s t h e i n n e r b . c , ( i . e . o' = P. a t r = a) K 1 l o g -zr R R (1-N) l o g | S u b s t i t u t i n g o' f r o m e q n . 119 i n e q n . G3, K (G3) P ( 1 - S i n A ) o v ymax |,1 - N R a R = a (1-Sin<}> ) v Ymax Po ( 1 " S i n W ' p. v Ymax l 1-N Sind> -1 max 2 S i n <t, Tmax 1 - S i n <j> max 2 S m <() max F o r t h e s h e a r d i l a t i o n f r o m e q n . 56 S i n v = - -3— d Y HI S i n c e v i s c o n s t a n t a l o n g ST v = - ( S i n v) y + C The c o n s t a n t C r e s u l t s f r o m t h e i n i t i a l c o m p r e s s i o n t h a t o c c u r s m a i n l y i n l o o s e s a n d s . F o r d e n s e s a n d C c o u l d be i g n o r e d w i t h o u t l o s s o f a c c u r a c y /. v = - ( S i n v) y (G4) F o r p l a n e s t r a i n c o n d i t i o n s , v = e + e . (v £ = 0) Y 6 z Y = e„ - E ( s i n c e e and e n a r e 1 6 r v Y 9 p r i n c i p l e s t r a i n s w i t h Y Q = 0 due t o a x i s y m m e t r y ) r 9 e = r a d i a l c o m p r e s s i v e s t r a i n e)Q = c i r c u m f e r e n t i a l c o m p . s t r a i n F r o m e q n . G4 £ r + £ Q = - ( S i n v ) ( e Q - £ p ) _ ( 1 + S i n v)  e r " ( 1 + S i n v ) e 8 B u t ^ s i n v = t a n 2 ( 4 5 + v / 2 ) D w h e r e D i s t h e d i l a t i o n f a c t o r i n t h e s t r e s s - d i l a t a n c y r e l a t i o n .-. £ r = - D e 0 (G5) F o r a x i s y m m e t r i c p o l a r c o - o r d i n a t e s y s t e m e r d r d u ' d i u e 8 = r u - r a d i a l d e f l e c t i o n S u b s t i t u t i n g i n e q n . G5, d r r I m p o s i n g t h e o u t e r b.c o f u=u a t r=R and i n t e g r a t i n g u _ , R l o g - = D l o g -R u _ R.D ,R ° + 1 U R o r u = r ( - ) F -u c a n be o b t a i n e d f r o m a p p l y i n g e q n . 117 f o r t h e o u t e r R b o u n d a r y o f t h e e l a s t i c zone 1+u R 2 , U R = ~ E ~ ~R ( 0 R " V B u t o ' R = P 0 d - S i n * r a a x ) U R 8 8 " R P o S i n * m a x A p p e n d i x H I f a u n i t v o l u m e o f s o i l i s c o n s i d e r e d , t h e v o l u m e s o f e a c h p h a s e become t h e i r r e s p e c t i v e p o r o s i t i e s , a s shown i n F i g . C a . A f t e r a v o l u m e t r i c s t r a i n o f e v ( i n c r e a s e o f v o l u m e ) t h e s i t u a t i o n i s d e p i c t e d by F i g . C . b , i n a c c o r d a n c e w i t h t h e a s s u m p t i o n t h a t s o i l g r a i n s , w a t e r and b i t u m e n a r e i n c o m p r e s s i b l e Thus i f n ' , n ' and n ' a r e t h e new p o r o s i t i e s , g o w n + e n ' = ^ -^ ( H D g i + e v n 1 = " ° (H2) o n n w w l + s v (H3) F o l l o w i n g t h e s i g n c o n v e n t i o n t h a t c o m p r e s s i v e s t r a i n s a r e p o s i t i v e n - e N . = _2 Y. (H4) g u -e v > " o . (H5) o ( l - e v ) n w (H6) w ( l - e v ) APPENDIX I USERS' MANUAL PROGRAM ORGANISATION I.1 V a r i a b l e D e s c r i p t i o n The f o l l o w i n g a d d i t i o n a l v a r i a b l e s have been used i n the m o d i f i e d NLSSIP ,where the s t r e s s r e d i s t r i b u t i o n t e c h n i q u e and the shear d i l a t i o n a r e found. AREA - Area of any element a t the b e g i n i n g of any l o a d c a s e . BULK - Bulk modulus of any element,used i n the p r e s e n t i t e r a t i o n . BULKS Bulk modulus of any element,used i n the p r e v i o u s i t e r a t i o n . DELEPS - A d d i t i o n a l s t r a i n t o be i n t r o d u c e d i n the d i r e c t i o n s f o r a n y d i l a t i n g element. DELV - T o t a l v o l u m e t r i c s t r a i n i n the element. DILFAC = 1 f o r d i l a t i n g e l e m e n t s . = 0 f o r o t h e r e l e m e n t s . and DSIG Twice the change element. i n mean normal s t r e s s f o r any EPSIN - The i n i t i a l shear s t r a i n ( % ) at which the d i l a t i n g s t a r t s f o r any m a t e r i a l ( i f c o n s t a n t d i l a t i o n r a t e o p t i o n i s u s e d ) . ETA - C o n s t a n t d i l a t i o n a n g l e . ( i f c o n s t a n t d i l a t i o n a n g l e o p t i o n i s u s e d ) . IT = 1 f o r t r i a l s . = 2 f o r the f i n a l i t e r a t i o n , where tape r e c o r d i n g i s done and r e s u l t s a r e p r i n t e d . ITR - Counter of the no. of i t e r a t i o n s . MMM - Counter of the no. of elements w i t h o u t a s a t i s f a c t o r y d i l a t i o n . NOPT = 1 i f c o n s t a n t d i l a t i o n a n g l e o p t i o n i s used. = 2 i f s t r e s s - d i l a t a n c y t h e o r y i s used. NITR - Maximum no. of i t e r a t i o n s r e q u i r e d by the u s e r . NMI1 - Counter of the no. of elements s t i l l v i o l a t i n g the y i e l d c r i t e r i o n . US' PHYCV - The c o n s t a n t volume f r i c t i o n a n g l e f o r any m a t e r i a l . PRS3 - The a r r a y c o n t a i n i n g the maximum shear s t r a i n f o r any element. PRS5 - The a r r a y c o n t a i n i n g the p r i n c i p a l s t r e s s r a t i o f o r any element. STAR = 0 f o r the f i r s t run i n a l o a d c a s e . = 1 f o r the i t e r a t i o n s . A p a r t from the above d e s c r i b e d , the f o l l o w i n g v a r i a b l e s have been used i n programming the pore p r e s s u r e model. CAPPA - E l a s t i c s p r i n g s t i f f n e s s . BSAVE - A r r a y s a v i n g the l o a d v e c t o r f o r the i t e r a t i o n s . DELU - Pore p r e s s u r e change f o r any element. DOFAS - Degree of freedom c o n n e c t e d w i t h the f i r s t s p r i n g , ( l o c a t e d by the program i t s e l f ) . FAC = 1 f o r the elements i n the f a i l u r e r e g i o n . = 0 f o r o t h e r s . ITR1 - Counter of the elements w i t h o u t an a c c u r a t e pore p r e s s u r e change. IVENT = 1 f o r the elements t o be v e n t e d . = 0 f o r o t h e r s . MOON - Counter of the l o a d case number. NAVE = 1 For elements f o r which pore p r e s s u r e a v e r a g i n g i s done. = 0 f o r o t h e r s . NEL1 - The element no. of the element t o be v e n t e d . NREAD = 1 i f pore p r e s s u r e s a r e t o be r e a d . = 0 i f the program a s s i g n s z e r o pore p r e s s u r e changes i n i t i a l l y . NSPRIN - No. of e l a s t i c s p r i n g s . NVEL - T o t a l no. of elements t o be v e n t e d . NVENT = 1 i f v e n t i n g i s t o be done f o r a c e r t a i n l o a d c a s e . = 0 i f v e n t i n g i s not t o be done. PS1,PS2 - A r r a y s c o n t a i n i n g the p r e v i o u s l y l o c a t e d l i m i t s of the a c c u r a t e pore p r e s s u r e change. PWI - A r r a y c o n t a i n i n g the pore p r e s s u r e increment used i n ti6 the p r e v i o u s i t e r a t i o n f o r any element. PWO - Pore p r e s s u r e a t the i n c e p t i o n of l o a d i n g , o f any e l e m e n t . ( e n t e r e d by the u s e r . ) PWW - Pore p r e s s u r e c h a n g e ( g u e s s e d ) , f o r any element t o be used i n the next i t e r a t i o n . SUN = 0 a t the f i r s t e n t r y i n t o the s u b r o u t i n e ITRAN. = 1 f o r subsequent e n t r i e s . I»7 1.2 D e s c r i p t i o n of subrout i n e s The s u b r o u t i n e s of the o r i g i n a l NLSSIP a f f e c t e d by the m o d i f i c a t i o n s w i l l be d e s c r i b e d h e r e i n . LAYOUT - Reads and p r i n t s the s o i l i n p u t d a t a . ( I n c l u d i n g d i l a t i o n d a t a . ) In the program by which o i l s a n d can be a n a l y s e d i t reads the o r i g i n a l pore p r e s s u r e s i n each element,the i n i t i a l p o r o s i t i e s , and the s o l u b i l i t y c o n s t a n t s f o r each m a t e r i a l and p r i n t s them. F i n a l l y i t computes and p r i n t s the i n i t i a l s t r e s s e s and the i n i t i a l m oduli v a l u e s f o r the s o i l e lements. FVECT - C a l c u l a t e s the n o d a l p o i n t f o r c e s due t o w e i g h t s of added e l e m e n t s , reads c o n c e n t r a t e d l o a d d a t a and/or boundary p r e s s u r e d a t a , p r i n t s the n o d a l p o i n t f o r c e s , s e t s up the f o r c e v e c t o r and s t o r e s i t i n a r r a y BSAVE , f o r the use i n the f u t u r e i t e r a t i o n s . F u r t h e r i t m o d i f i e s t h i s v e c t o r i n eve r y i t e r a t i o n depending on the a d d i t i o n a l n odal f o r c e s due t o d i l a t i o n . Later,when s t r e s s r e d i s t r i b u t i o n i s done i t forms a s e p e r a t e f o r c e v e c t o r f o r each i t e r a t i o n . For the subsequent use i n the c u r r e n t run the f o r c e v e c t o r i s s t o r e d i n tape 10. POREF - Reads element i n c r e m e n t a l pore water p r e s s u r e s i n the f i r s t run i n a p a r t i c u l a r l o a d case,computes e q u i v a l e n t n o d a l f o r c e s , s t o r e s c u r r e n t a r e a s of a l l elements i n the AREA a r r a y , a n d m o d i f i e s tape 10,by ad d i n g the f o r c e v e c t o r due t o Au. I f o i l s a n d i s b e i n g a n a l y s e d , i t checks whether any element i s t o be v e n t e d , a f t e r r e a d i n g the v e n t i n g d a t a . (The m o d i f i e d A u v a l u e s f o r any element a r e used i n c a l c u l a t i n g the n o d a l f o r c e s . ) ELAW - C a l c u l a t e s the moduli v a l u e s f o r the s o i l elements i n accordance w i t h the magnitudes of the s t r e s s e s . When r e d i s t r i b u t i n g the s t r e s s e s i t uses 'J, of 0 f o r the elements i n the f a i l u r e r e g i o n . ; " ISQUAD - F o r m u l a t e s the c o n s t i t u t i v e e q u a t i o n s , f o r m s the element s t i f f n e s s m a t r i x f o r each element,and w r i t e s i t on tape 2. I f e l a s t i c s p r i n g s a r e used t o a n a l y s e an i n f i n i t e d o m a i n , i t c a l c u l a t e s the r e q u i r e d s p r i n g s t i f f n e s s based on the e l a s t i c m o d u l i i , and m o d i f i e s the d i a g o n a l terms of the s t i f f n e s s m a t r i c e s of the elements c o n c e r n e d . I t a l s o forms the s t r a i n d i s p l a c e m e n t m a t r i x f o r each element,and w r i t e s i t on tape 11. ne ISRSLT - C a l c u l a t e s s t r e s s / s t r a i n i n c r e m e n t s , average s t r e s s e s and c u m u l a t i v e s t r a i n s f o r the t r i a l i t e r a t i o n s and e v a l u a t e s the m o d u l i i f o r each element a f t e r each i t e r a t i o n . In the f i n a l i t e r a t i o n i t c a l c u l a t e s the i n c r e m e n t a l and c u m u l a t i v e s t r e s s e s and s t r a i n s , f o r the elements and a c c o r d i n g l y updates the m o d u l i i v a l u e s t o be used i n the next l o a d case.The i n t e r n a l f o r c e s i n the s t r u c t u r a l elements a r e a l s o computed and p r i n t e d . F u r t h e r i t p r i n t s out ,the 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 f o r the e l e m e n t s , b e f o r e w i n d i n g up the i t e r a t i v e procedure.' DILAT - C a l c u l a t e s the change i n maximum shear s t r a i n f o r each element i n a l o a d case ,and by u s i n g the average s t r e s s c o n d i t i o n w i t h i n the l o a d c a s e ( i f s t r e s s -d i l a t a n c y t h e o r y i s u s e d ) , d e t e r m i n e s the d i l a t i o n a n g l e t o be u s e d , i n computing the d i l a t i o n f o r each element. I t a l s o e v a l u a t e s the a d d i t i o n a l n o d a l f o r c e s t o be i n t r o d u c e d . Moreover, here a check i s imposed on each element t o ensure t h a t a p l a s t i c volume change, w i t h i n a r e a s o n a b l e t o l e r a n c e of the e x a c t v a l u e has been i n t r o d u c e d i n t o each element. ITRAN - C a l c u l a t e s the v o l u m e t r i c s t r a i n t h a t has o c c u r r e d so f a r , and by u s i n g the i n c r e m e n t a l v o l u m e t r i c s t r a i n , g e t s s u b r o u t i n e PORVOL t o p r e d i c t the A u f o r each n o n - v e n t i n g element. In a d d i t i o n t o t h i s i t p r e p a r e s A u v a l u e s t o be used i n t he next i t e r a t i o n , e m p l o y i n g an a v e r a g i n g t e c h n i q u e i f n e c c e s s a r y . . A f t e r a s c e r t a i n i n g t h a t the c o r r e c t A u i s p r e d i c t e d f o r a l l e l e m e n t s ( w i t h i n a r e a s o n a b l e l i m i t ) , i t a l s o updates the pore p r e s s u r e of each element. PORVOL - Updates the p o r o s i t i e s of each element,and u s i n g the s o l u b i l i t y c o e f f i c i e n t s and the i n c r e m e n t a l v o l u m e t r i c s t r a i n , c a l c u l a t e s A u . LSHED S e l e c t s the elements v i o l a t i n g the Mohr-Coulomb f a i l u r e c r i t e r i o n , and c a l c u l a t e s an i n c r e m e n t a l s e t of n o d a l f o r c e s t o be a p p l i e d t o such e l e m e n t s . I.3 I n s t r u c t i o n s on da t a i n p u t Input of d a t a i s done a c c o r d i n g .to NLSSIP manual (Byrne and Duncan,1979) , except f o r the below i t e m s . Modi f i e d NLSSIP 1. C o n t r o l c a r d s A f t e r the c o n t r o l c a r d 1(b) e n t e r , 1 - 5 NCHECK = 1 i f the maximum no. of i t e r a t i o n s i s s p e c i f i e d . = 2 i f the maximum no. of i t e r a t i o n s i s s e t t o 3. Next c a r d 1 - 5 NITR - Maximum no. of i t e r a t i o n s r e q u i r e d , i f NCHECK=1 2. M a t e r i a l p r o p e r t y c a r d s A f t e r the c a r d 4(b) e n t e r , 1 - NOPT = 1 i f c o n s t a n t d i l a t i o n a n g l e o p t i o n i s used. = 2 i f s t r e s s - d i l a t a n c y t h e o r y i s used, Next c a r d I f N0PT=1 e n t e r 1 - 5 L 5 - 1 5 ETA 16 - 25 EPSIN - M a t e r i a l number. - C o n s t a n t d i l a t i o n a n g l e . - The shear a t which d i l a t i o n s t a r t s . ( % ) I f NOPT=2 e n t e r 1 - 5 L 6 - 1 5 PHYCV - M a t e r i a l number. - C o n s t a n t volume f r i c t i o n a n g l e . Modi f i e d NLSSIP ( f o r a n a l y s i n g o i l s a n d )  1. C o n t r o l c a r d s In the c a r d 1 ( b ) , e n t e r NITR i n s t e a d of IPW 66 - 70 NITR - Maximum no. of i t e r a t i o n s r e q u i r e d , E n t e r 5 i f NCHECK=2. a f t e r c a r d K b ) e n t e r , 1 - NREAD 6 - 1 0 NSPRIN 1 i f pore p r e s s u r e i n c r e m e n t s are t o read f o r a l l l o a d c a s e s . 0 i f pore p r e s s u r e i n c r e m e n t s a r e not t o be re a d . - No. of e l a s t i c s p r i n g s used. Next c a r d 1 - 5 NCHECK = 1 (As p r e v i o u s l y mentioned.) «20 = 2 i f max. no. of i t e r a t i o n s i s s e t t o 5. 2. M a t e r i a l p r o p e r t y c a r d s P i l a t i o n d a t a - e n t e r d i l a t i o n d a t a as s p e c i f i e d above, a f t e r the c a r d 4 ( b ) . Pore f l u i d data - e n t e r a f t e r the d i l a t i o n d a t a . 1 - 5 KL - M a t e r i a l number. 6 - 15 PN(KL,1) - Water p o r o s i t y . 16 - 25 PN(KL,2) - O i l p o r o s i t y . 26 - 35 PN(KL,3 ) - Gas p o r o s i t y . 36 - 45 BTT(KL,1) - Temperature s o l u b i l i t y C o e f f i c i e n t f o r w a t e r . 46 - 55 BTT(KL,2) - Temperature s o l u b i l i t y C o e f f i c i e n t f o r o i l . 56 - 65 HP(KL,1) - P r e s s u r e s o l u b i l i t y C o e f f i c i e n t f o r water 1 - 10 HP(KL,1) - P r e s s u r e s o l u b i l i t y C o e f f i c i e n t f o r o i l . 3^ I n i t i a l Pore p r e s s u r e d a t a - e n t e r a f t e r 1 - 5 MN - Element number. 6 - 15 PWO(MN) - Pore p r e s s u r e . Use NUMELT c a r d s . 4. Pore p r e s s u r e i n c r e m e n t s and V e n t i n g d a t a E n t e r a f t e r each l o a d case I f NREAD=1 e n t e r the pore p r e s s u r e i n c r e m e n t s as s p e c i f i e d i n the NLSSIP w r i t e - u p ( i t e m 13) Next c a r d 1 - 5 NVENT Next c a r d I f NVENT=1 e n t e r 1 - 5 NVEL Next c a r d 1 - 5 N E L l ( l ) = 0 i f v e n t i n g i s not t o done. = 1 i f v e n t i n g i s t o be done d u r i n g t h i s l o a d c a s e . - No. of elements t o be v e n t e d . - F i r s t element t o be v e n t e d . 6 - 1 0 NEL1(2) - Second element t o be v e n t e d . Upto NELl(NVEL) ,upto a maximum of 5 elements per c a r d . 

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