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UBC Theses and Dissertations

Problems in nonlinear analysis of movements in soils Wedge, Neil Edward 1977

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PROBLEMS IN NONLINEAR ANALYSIS OP MOVEMENTS IN SOILS NEIL EDWARD WEDGE B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, 197^ A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of C i v i l Engineering) We accept t h i s t h e s i s as co n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March, 1977 fcT) N e i l Edward Wedge, 1977 . In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 fo r reference and study. I f u r t h e r agree tha t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or 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 ga in s h a l l not be a l lowed without my wr i t ten pe rm i ss i o n . Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e March, 1977 s ABSTRACT The problems a s s o c i a t e d w i t h n o n l i n e a r a n a l y s i s o f the l o a d - d e f o r m a t i o n re sponse o f s o i l s and s o i l s t r u c -t u r e s a r e i n v e s t i g a t e d . Methods o f i n c r e m e n t a l n o n l i n e a r a n a l y s i s are r e v i e w e d and t h e i r r e l a t i v e advantages and d i s -a d v a n t a g e s d i s c u s s e d . S t r e s s - s t r a i n r e l a t i o n s commonly used f o r s o i l s are c r i t i c a l l y examined and t h e i r l i m i t a t i o n s d i s -c u s s e d . These s t r e s s - s t r a i n r e l a t i o n s are based on the a s s u m p t i o n t h a t s o i l s a r e i s o t r o p i c , i n c r e m e n t a l l y e l a s t i c m a t e r i a l s . E v i d e n c e r e p o r t e d by o t h e r a u t h o r s and r e v i e w e d i n t h i s s tudy shows t h a t the s t r e s s - s t r a i n r e l a t i o n s . commonly used f o r s o i l s have two major s o u r c e s o f e r r o r , the a n i s o t r o p y o f s o i l s and the e f f e c t s o f s t r e s s - p a t h are n e g -l e c t e d . The r e p r e s e n t a t i o n o f s o i l s t r e s s - s t r a i n b e h a v i o u r a f t e r y i e l d i s d i s c u s s e d . A l t h o u g h s o i l s ac t as p l a s t i c m a t e r i a l s a f t e r y i e l d , i t i s common p r a c t i c e to r e p r e s e n t p o s t - y i e l d b e h a v i o u r by models o f e l a s t i c m a t e r i a l s . Many r e s e a r c h e r s use a c o n s t a n t v a l u e o f P o i s s o n ' s r a t i o and m e r e l y r e d u c e the v a l u e o f Young ' s modulus at y i e l d . I t i s shown, w i t h n u m e r i c a l examples , t h a t t h i s p r a c t i c e r e s u l t s i n y i e l d e d s o i l e lements b e i n g u n r e a l i s t i c a l l y c o m p r e s s i b l e a f t e r y i e l d . I t i s shown, w i t h f u r t h e r n u m e r i c a l examples , t h a t the p r e d i c t e d b e h a v i o u r o f y i e l d e d s o i l e lements i s more r e a l i s t i c i f the v a l u e o f the shear modulus i s r e d u c e d at y i e l d and the b u l k modulus i s not r e d u c e d . i i TABLE OF CONTENTS Page ABSTRACT TABLE OF CONTENTS i i i LIST OF FIGURES v NOTATION v i i ACKNOWLEDGEMENTS x i i CHAPTER 1 INTRODUCTION 1 2 METHODS OF INCREMENTAL ANALYSIS FOR NONLINEAR MATERIALS 3 2.1 G e n e r a l D i s c u s s i o n o f I n c r e m e n t a l A n a l y s i s 3 2.2. I t e r a t i v e Methods 2.2 .1 One I t e r a t i o n per Load Increment \7 ; 2.2.2 I t e r a t i o n s U s i n g a Con-vergence C r i t e r i o n 10 2.2 .3 A p p l i c a t i o n o f C o r r e c t i o n F o r c e s 11 2.2.4 M o d i f i e d Newton-Raphson Method 12 3. STRESS-STRAIN RELATIONS 18 3 .1 G e n e r a l D i s c u s s i o n o f S t r e s s -S t r a i n R e l a t i o n s 18 3.2 E x p e r i m e n t a l S t u d i e s 24 3-3 L i m i t a t i o n s o f S t r e s s - S t r a i n R e l a t i o n s 33 3.4 M a t e r i a l B e h a v i o u r A f t e r Y i e l d 38 4 PROBLEMS IN NONLINEAR FINITE ELEMENT ANALYSIS OF DEFORMATIONS IN SOIL BODIES 42 I i i i v CHAPTER Page 4 . 1 G e n e r a l D i s c u s s i o n o f N o n l i n e a r ' .'. .'. F i n i t e Element A n a l y s i s o f De-f o r m a t i o n s i n S o i l Bodies 42 4 . 1 . 1 Common Assumptions 42 4 . 1 . 2 Method o f o f I n c r e m e n t a l A n a l y s i s 44 4 .2 N u m e r i c a l S t u d i e s 47 5 CONCLUSION 66 BIBLIOGRAPHY 70 APPENDIX 1: DERIVATION -OF THE STIFFNESS MATRIX FOR THE LINEAR STRAIN TRIANGULAR ELEMENT 72 APPENDIX 2 : STRESS-STRAIN SUBROUTINES j . 79 LIST OF FIGURES FIGURE Page 1 I n c r e m e n t a l A p p r o x i m a t i o n o f a Non-l i n e a r L o a d - D e f l e c t i o n Curve 5 2 One I t e r a t i o n p e r Load Increment Method 9 3 Newton-Raphson Method f o r the One-d i m e n s i o n a l Case 14 4 H y p e r b o l i c S t r e s s - S t r a i n Curve 25 5 Mohr-Coulomb F a i l u r e C r i t e r i o n 27 6 Two S t r e s s - P a t h s which Y i e l d D i f f e r e n t S t r a i n s f o r t h e Same S t r e s s Increment 35 7 F i n i t e Element Mesh 52 8 L o a d - S e t t l e m e n t Curves f o r A n a l y s e s i n which t h e Shear Modulus was Reduced t o One-Hundred Pounds per Square Foot at Y i e l d and the B u l k Modulus was not Reduced at Y i e l d 5^ 9 L o a d - S e t t l e m e n t Curves f o r A n a l y s e s i n which t h e Shear Modulus was Reduced t o F i v e Pounds per Square Foot at Y i e l d and the B u l k Modulus was not Reduced at Y i e l d 55 10 L o c a t i o n s o f F a i l e d Elements f o r t h e A n a l y s i s i n which t h e Shear Modulus was Reduced t o F i v e Pounds per Square Foot at Y i e l d , the B u l k Modulus was not Reduced at Y i e l d , and Twenty Pound Load Increments Were Used 58 11 Movements o f Nodes D u r i n g the Load I n c r e -ment from F i v e Hundred and Twenty Pounds t o F i v e Hundred and F o r t y Pounds i n the A n a l y s i s i n which the Shear Modulus was Reduced t o F i v e Pounds per Square Foot a t Y i e l d , t h e B u l k Modulus was not Reduced at Y i e l d , and Twenty Pound Increments Were Used 59 v L o a d - S e t t l e m e n t Curves f o r A n a l y s e s i n which P o i s s o n ' s R a t i o had a C o n -s tant Va lue o f 0.42 and Young's Modulu was Reduced to One-Hundred Pounds per Square Foot at Y i e l d L o c a t i o n s o f F a i l e d Elements f o r the A n a l y s i s i n which P o i s s o n ' s R a t i o had a Constant V a l u e o f 0 .42 , Young's Modulus was Reduced to F i v e Pounds per Square Foot at Y i e l d , and F i f t e e n Pound Load Increments Were Used Movements o f Nodes D u r i n g the Load Increment from One-Hundred and Twenty Pounds to One-Hundred and T h i r t y - f i v e Pounds i n the A n a l y s i s i n which P o i s s o n ' s R a t i o had a Constant Va lue o f 0 .42 , Young's Modulus was Reduced to One-Hundred Pounds per Square Foot at Y i e l d , and F i f t e e n Pound Load Increments Were Used L i n e a r S t r a i n T r i a n g u l a r Element Areas f o r N a t u r a l C o o r d i n a t e s NOTATION The f o l l o w i n g i s a l i s t o f the symbols used i n t h i s t h e s i s and t h e i r d e f i n i t i o n s the i n v e r s e o f a s y m p t o t i c v a l u e o f the r e s u l t a n t d e v i a t o r i c s t r e s s B = the r a t i o o f the shear modulus a f t e r y i e l d to tha t b e f o r e y i e l d c = the cohes ion i n t e r c e p t o f Mohr-Coulomb f a i l u r e envelope d = the r a t e o f i n c r e a s e o f the tangent v a l u e o f P o i s s o n ' s r a t i o w i t h s t r a i n dV = an i n f i n i t e s i m a l change i n volume E ^ = the i n i t i a l tangent v a l u e o f Young's modulus = the tangent v a l u e o f Young's modulus P = the decrease i n the tangent v a l u e o f P o i s s o n ' s r a t i o f o r a t e n f o l d i n c r e a s e i n the minor p r i n c i p a l s t r e s s G = the tangent va lue ..of P o i s s o n ' s r a t i o f o r an I s o t r o p i c s t r e s s equa l to a tmospher ic p r e s -sure G. = the i n i t i a l tangent v a l u e o f the shear 1 modulus G^ = the tangent shear modulus h = the depth below the s o i l s u r f a c e K = Young's modulus number K. = the i n i t i a l tangent v a l u e o f the b u l k 1 modulus = the tangent v a l u e o f the b u l k modulus K = the c o e f f i c i e n t o f l a t e r a l e a r t h p r e s s u r e ° at r e s t v i i v i i i the s lope o f s t r e s s - s t r a i n curve f o r the o n e - d i m e n s i o n a l case at the end of the l a s t l o a d Increment the r a t e o f change o f the tangent b u l k modulus w i t h the mean normal- s t r e s s the i n i t i a l tangent va lue o f the c o n s t r a i n e d modulus the tangent va lue o f the c o n s t r a i n e d modulus Young's modulus exponent a number t h a t i n d i c a t e s the r a t e o f i n c r e a s e o f the tangent b u l k modulus w i t h r e s p e c t to a n o r m a l i z e d v o l u m e t r i c s t r a i n a tmospher ic p r e s s u r e the d i s p l increment i sp lacement at the end of the i t h l o a d the l o a d at the end o f the i t h l o a d increment t h the n a p p r o x i m a t i o n o f the d i s p l a c e m e n t at the end of the i t h l o a d increment the average l o a d f o r the i ^ h l o a d increment the r a t i o o f Young's modulus b e f o r e y i e l d to t h a t a f t e r y i e l d the f a i l u r e r a t i o the s t r e s s - l e v e l the r e s u l t a n t d e v i a t o r i c s t r e s s volume the r e s u l t a n t d e v i a t o r i c s t r a i n the s t r a i increment the n a increment s t r a i n at the end of the i t h l o a d t h p p r o x i m a t i o n o f the i t h l o a d the v o l u m e t r i c s t r a i n i x e = the c h a r a c t e r i s t i c v o l u m e t r i c s t r a i n vc e = the n o r m a l i z e d v o l u m e t r i c s t r a i n vn y = the u n i t weight Y . . = the shear s t r a i n f o r the i p lane i n the 1 J j d i r e c t i o n A q n . = the change i n d i sp lacement i n the i ^ h l o a d increment A Q . = the i ^ increment o f l o a d 1 x,. = the 1 ^ n a t u r a l c o o r d i n a t e 1 l t h a_. = the s t r e s s at the end o f the I l o a d increment = the major p r i n c i p a l s t r e s s = the minor p r i n c i p a l s t r e s s ^aj~a2^ult = t h e a s y m P t o t x C d e v i a t o r s t r e s s (a^-a^)| . = the d e v i a t o r s t r e s s at f a i l u r e a | = the major p r i n c i p a l e f f e c t i v e s t r e s s al, = the minor p r i n c i p a l e f f e c t i v e s t r e s s e a the e r r o r i n a c a l c u l a t e d s t r e s s a = the mean normal s t r e s s m = the v a l u e o f P o i s s o n ' s r a t i o b e f o r e y i e l d v 2 = the v a l u e o f P o i s s o n ' s r a t i o a f t e r y i e l d v t the tangent v a l u e o f P o i s s o n ' s r a t i o T . . = the shear s t r e s s on the i p lane i n the ± ^ j d i r e c t i o n <j> = the ang le o f i n t e r n a l f r i c t i o n <j)' = the e f f e c t i v e ang le o f i n t e r n a l f r i c t i o n "Vectors: {q} = the v e c t o r o f i n i t i a l n o d a l d i s p l a c e m e n t s X { q } . = the v e c t o r o f n o d a l d i s p l a c e m e n t s at the end 1 - o f the i t h l o a d increment t h { q } . = the n a p p r o x i m a t i o n o f the n o d a l d i s p l a c e -1 , n ment v e c t o r at the end o f the i t h l o a d increment ( Q } q = the i n i t i a l n o d a l f o r c e v e c t o r { Q } . = the n o d a l f o r c e v e c t o r at the end o f the 1 i t h l o a d increment e t h {Q }. = the n v e c t o r o f n o d a l c o r r e c t i o n f o r c e s 5 f o r the i t h l o a d increment {e}Q = the v e c t o r o f i n i t i a l element s t r a i n s { e } . = the v e c t o r o f element s t r a i n s at the end o f 1 the i t h l o a d increment {e}. = the n ^ a p p r o x i m a t i o n of the v e c t o r o f 1 ' element s t r a i n s at the end o f the i t h l o a d increment 6 { q } = a v e c t o r o f v i r t u a l n o d a l d i s p l a c e m e n t s <5{e} = a v e c t o r o f v i r t u a l element s t r a i n s { A q } . = the change i n the v e c t o r o f n o d a l d i s p l a c e -1 ments d u r i n g the i t h l o a d increment { A q } . = the n t h a p p r o x i m a t i o n o f the change i n the 1 , n v e c t o r o f n o d a l d i s p l a c e m e n t s f o r the i t h l o a d increment { A q } ! _ = the n ^ c o r r e c t i o n to the v e c t o r o f n o d a l d i s p l a c e m e n t s at the end o f the l z n l o a d increment { A Q } . = the i t h increment o f n o d a l f o r c e s I { A e } . = the change i n the v e c t o r o f element s t r a i n s 1 f o r the i t h l o a d increment t h { A e } . = the n a p p r o x i m a t i o n o f the change i n the l j n v e c t o r o f element s t r a i n s f o r the i ^ h l o a d increment t h { A e } ! = the n c o r r e c t i o n to the v e c t o r o f element ' s t r a i n s at the end o f the i t h l o a d Increment x i {Aa}. = the change i n the v e c t o r o f element s t r e s s e s 1 f o r the i t h l o a d increment t h {Aa}. = the n a p p r o x i m a t i o n o f the change i n the l j n v e c t o r o f n o d a l d i s p l a c e m e n t s f o r the i t h l o a d increment {a}Q = the v e c t o r o f i n i t i a l element s t r e s s e s {a}. = the v e c t o r o f element s t r e s s e s at the end 1 o f the i ^ h l o a d increment {a}. ! = the v e c t o r o f average element s t r e s s e s """'^  f o r the i t h l o a d increment {a}.. ._^= the a p p r o x i m a t i o n o f the v e c t o r o f ^ average element s t r e s s e s f o r the i t h l o a d increment {a }. = the v e c t o r o f e r r o r s i n element s t r e s s e s f o r the end o f the i t h l o a d increment [B] = the element s t r a i n - d i s p l a c e m e n t m a t r i x [D] = the s t r e s s - s t r a i n m a t r i x [D] = the element s t r e s s - s t r a i n m a t r i x [D]^.. = the element s t r e s s - s t r a i n m a t r i x a p p r o p r i a t e to the i l o a d increment ei [D] . = the n t h a p p r o x i m a t i o n . o f the element s t r e s s -5 1 s t r a i n m a t r i x a p p r o p r i a t e to the i ^ n l o a d increment [ K r p ] = the g l o b a l tangent s t i f f n e s s m a t r i x [ K T ] . = the g l o b a l tangent s t i f f n e s s m a t r i x a p p r o -1 p r i a t e to the i ^ h l o a d increment [ K m ] = the element tangent s t i f f n e s s m a t r i x T e [Krp] . = the element tangent s t i f f n e s s m a t r i x a p p r o -6 1 p r i a t e to the i ^ h l o a d increment [K,p] . = the n t h a p p r o x i m a t i o n o f the element tangent e i ' n s t i f f n e s s m a t r i x a p p r o p r i a t e to the i ^ h l o a d increment ACKNOWLEDGEMENTS The w r i t e r g r a t e f u l l y acknowledges the a d v i c e and guidance g i v e n by h i s s u p e r v i s o r . D r . W.D. Liam F i n n , the a d v i c e o f D r . N . D . Nathan, and the a d v i c e and generous a s s i s t a n c e g i v e n by D r . K. L e e . S p e c i a l thanks are due to Mis s D e s i r e e Cheung f o r t y p i n g the m a n u s c r i p t . x i i CHAPTER 1 INTRODUCTION The f i n i t e element method o f a n a l y s i s has been used e x t e n s i v e l y ( 5 , 8 , 9 , 1 1 , 1 2 , 1 8 ) to p r e d i c t de format ions i n s o i l s t r u c t u r e s and movements o f f o u n d a t i o n s under v a r -i ous k i n d s o f a p p l i e d l o a d i n g s . T h i s method p r o v i d e s the most conven ient means o f r e p r e s e n t i n g the g e o m e t r i c a l , k i n e -m a t i c a l and e q u i l i b r i u m r e l a t i o n s o f s o i l s t r u c t u r e s . I t a l s o a l l o w s the use o f s t r e s s - s t r a i n r e l a t i o n s i n the form of e i t h e r a n a l y t i c a l or d i s c r e t e d a t a . The e f f e c t i v e n e s s o f the method i n p r e d i c t i n g the de format ions o f s o i l s t r u c t u r e s i s l i m i t e d m a i n l y by the adequacy w i t h which the s t r e s s - s t r a i n r e l a t i o n s and f a i l u r e c o n d i t i o n s f o r s o i l s can be r e p r e s e n t e d . The m o d e l l i n g o f s o i l s t r e s s - s t r a i n b e h a v i o u r p r i o r to f a i l u r e i s not a s imple p r o b l e m . The s t r e s s - s t r a i n r e l a t i o n s o f s o i l s are n o n l i n e a r , s t r e s s dependent and s t r e s s - p a t h dependent ( 9 , 1 0 , 1 3 , 1 7 , 1 8 ) . The f a i l u r e c r i t e r i a f o r s o i l s are not f i r m l y e s t a b l i s h e d and the p o s t - f a i l u r e b e h a v i o u r i s open to a v a r i e t y o f i n t e r p r e t a t i o n s . Once a model f o r the s t r e s s - s t r a i n b e h a v i o u r o f s o i l has been a d o p t e d , one o f two g e n e r a l methods o f n o n -l i n e a r a n a l y s i s may be used to p r e d i c t the de format ions o f 1 2 s o i l s t r u c t u r e s under a p p l i e d l o a d i n g s . One method, known as i t e r a t i v e e l a s t i c a n a l y s i s , c o n s i s t s o f a p p l y i n g the e n t i r e l o a d i n g i n one s tep and I t e r a t i n g on the s t r e s s -s t r a i n p r o p e r t i e s used i n o r d e r to o b t a i n s t r a i n s which are compat ib le w i t h the a p p l i e d loads and the s t r e s s - s t r a i n model o f the s o i l . In the second method, which i s c a l l e d i n c r e m e n t a l a n a l y s i s , the l o a d i n g i s a p p l i e d i n s u i t a b l e increments and the s t r e s s - s t r a i n b e h a v i o u r o f the s o i l i s assumed to be l i n e a r w i t h i n each l o a d i n c r e m e n t . In the a n a l y s i s o f each l o a d increment one may i t e r a t e on the s t r e s s - s t r a i n p r o p e r t i e s used i n o r d e r to o b t a i n s t r a i n s which at the end o f the i n c r e m e n t , are compat ib l e w i t h the a p p l i e d l o a d s and the s t r e s s - s t r a i n model o f the s o i l . The i n c r e m e n t a l t e c h n i q u e has the advantage t h a t i t a l l o w s the d e t e r m i n a t i o n o f the l o a d - d i s p l a c e m e n t h i s t o r y o f the. s o i l body up to the l o a d i n g o f i n t e r e s t , r a t h e r than j u s t the de format ions c o r r e s p o n d i n g to the f i n a l l o a d i n g as i n the case o f i t e r a t i v e e l a s t i c a n a l y s i s . The purpose o f t h i s s tudy i s to i n v e s t i g a t e the problems a s s o c i a t e d w i t h the use o f e x i s t i n g s t r e s s - s t r a i n r e l a t i o n s i n the i n c r e m e n t a l n o n l i n e a r a n a l y s i s o f s o i l d e f o r m a t i o n s . CHAPTER 2 METHODS OF INCREMENTAL ANALYSIS FOR NONLINEAR MATERIALS 2.1 G e n e r a l Discuss ion" o f Incrementa l A n a l y s i s A n o n l i n e a r m a t e r i a l i s one f o r which the r e l a -t i o n s h i p between the a p p l i e d s t r e s s e s and the r e s u l t i n g s t r a i n s i s n o n l i n e a r . The tangent m o d u l i f o r such a m a t e r i a l are the i n s t a n t a n e o u s s lopes o f the s t r e s s - s t r a i n curves and they may be expressed i n terms o f the s t r e s s e s or s t r a i n s at the p o i n t o f i n t e r e s t . F o r example, E t = E t ( { a } ) or E t ( { e } ) (2-1) and v f c = v t ( { o } ) or v t ( { e } ) (2-2) where E ^ i s the tangent va lue o f Young's modulus , i s the tangent v a l u e o f P o i s s o n ' s r a t i o , {a} i s the s t a t e o f s t r e s s , and {e} i s the s t a t e o f s t r a i n . S ince the s t i f f n e s s m a t r i x o f a f i n i t e element depends upon the v a l u e s o f the m o d u l i , i t i s a l s o a f u n c t i o n o f the s t a t e o f s t r e s s o r s t r a i n e x i s t i n g i n t h a t e lement . Hence, the tangent s t i f f n e s s m a t r i x o f the element based upon the tangent m o d u l i i s [ K T ] g , [ K T ] e = [ K T ( { 0 } ) ] e or [ K T ( { e } ) ] e (2-3) 3 The s t a t e s o f s t r e s s and s t r a i n change d u r i n g the l o a d i n g o f the f i n i t e element and , as a r e s u l t , no s i n g l e va lue o f the tangent s t i f f n e s s m a t r i x i s a p p l i c a b l e d u r i n g the e n t i r e a n a l y s i s . One method o f d e a l i n g w i t h the s t r e s s or s t r a i n dependency o f the tangent s t i f f n e s s m a t r i x i s to l o a d the s t r u c t u r e i n increments and to assume t h a t the m a t e r i a l behaves l i n e a r l y w i t h i n each l o a d i n c r e m e n t . T h i s a p p r o x i -mat ion to the t r u e s t r e s s - s t r a i n response i s shown i n F i g u r e 1 f o r the o n e - d i m e n s i o n a l case . F o r a body which has been d i v i d e d i n t o a number o f f i n i t e e lements the g e n e r a l approach i s as f o l l o w s : The t o t a l l o a d v e c t o r , {Q}, to be a p p l i e d i s d i v i d e d i n t o n l o a d increments {AQK . "If {Q}Q i s the i n i -t i a l l o a d v e c t o r a c t i n g on the body, then the l o a d v e c t o r , t h { Q K , a c t i n g at the end o f the i l o a d increment i s g i v e n by i {Q}. = {Q} + E {AQ}. ( 2 - 4 ) 1 ° j = l 3 t h F o r the i l o a d i n c r e m e n t , { A Q K , [ K T ] {Aq}. = {AQ}. ( 2 - 5 ) where [K^] i s the g l o b a l s t i f f n e s s m a t r i x based on the t a n -gent modul i a p p r o p r i a t e to the element s t r e s s e s i n the i ^ * 1 l o a d increment and {Aq}^ i s the i n c r e m e n t a l n o d a l d i s p l a c e -ment v e c t o r r e s u l t i n g from the a p p l i c a t i o n o f {AQ}^. The increments o f element s t r e s s r e s u l t i n g from P i e c e w i s e L i n e a r A p p r o x i m a t i o n -Correct Load-Di s p l a c e m e n t Curve q i - l q i + i D i s p l a c e m e n t , q FIGURE 1 INCREMENTAL APPROXIMATION OF A NONLINEAR LOAD-DEFLECTION CURVE the a p p l i c a t i o n o f the i t h l o a d increment are not known when one beg ins the a n a l y s i s o f the i n c r e m e n t . T h e r e f o r e , both the modul i arid the g l o b a l s t i f f n e s s m a t r i x a p p r o p r i a t e to the l o a d increment are unknown when one beg ins the a n a l y s i s o f the i n c r e m e n t . An i t e r a t i v e procedure must be used i n o r d e r to f i n d t h e s e . The t o t a l n o d a l d i sp lacement v e c t o r , { q K , at t h the end o f the i l o a d increment i s g i v e n by i ' { q K = {q} Q + { A q K (2-6) where { q } Q Is the v e c t o r o f i n i t i a l n o d a l d i s p l a c e m e n t s . F o r a p a r t i c u l a r f i n i t e e lement , the i n c r e m e n t a l s t r a i n v e c t o r , { A e K , r e s u l t i n g from the a p p l i c a t i o n o f the i ^ l o a d increment i s g i v e n by {Ae} . = [ 3 ] e { A q K (2-7) where [B] i s 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 the element which depends on ly upon the i n i t i a l geometry o f the element i f g e o m e t r i c a l n o n l i n e a r i t y i s n e g l i g i b l e . The t o t a l s t r a i n v e c t o r , { e K , f o r the e lement , at the end o f the i ^ l o a d increment i s g i v e n by i { e K = {e} + £ { A e } . (2-8) ° J - 1 3 where {e} i s the i n i t i a l s t r a i n v e c t o r , o The i n c r e m e n t a l s t r e s s v e c t o r , {Aa}^, f o r the t h e lement , f o r the i l o a d increment i s g i v e n by {Aa} . = [ D ] e ' { A e } 1 (2-9) where [D] i s the element s t r e s s - s t r a i n m a t r i x a p p r o p r i a t e t h to the i l o a d i n c r e m e n t , which depends upon the element s t r e s s e s In the increment and must be found by an I t e r a t i v e p r o c e d u r e . The t o t a l element s t r e s s v e c t o r , { a K , at the end o f the I t h l o a d increment i s g i v e n by i {a}. = {a} + E {Aa} . (2-10) ° J = l 3 where {a} i s the i n i t i a l element s t r e s s v e c t o r , o 2.2 I t e r a t i v e Methods 2.2.1 One I t e r a t i o n per Load Increment The f i r s t method to be c o n s i d e r e d i s an attempt to improve the e s t imates o f the s t r e s s - s t r a i n m a t r i x , [D] ^ , t h and the s t i f f n e s s m a t r i x , [ K T ] ^ , used i n the i l o a d i n c r e -ment wi thout u s i n g a convergence c r i t e r i o n . The method was suggested by Duncan ( 8 ) . Each l o a d increment i s a n a l y s e d t w i c e . The f i r s t t i m e , the tangent m o d u l i f o r the f i n i t e elements are c a l c u l a t e d from the i n i t i a l s t r e s s c o n d i t i o n f o r the i n c r e -ment. These modul i are used to c a l c u l a t e the element s t r e s s -s t r a i n m a t r i c e s , [D] . and the element s t i f f n e s s m a t r i c e s , t h [ K ] g i ^ , f o r the i l o a d i n c r e m e n t . That i s [ D ] e l j l = [ D ( { a } . _ 1 ) ] e (2-11) and [ K T ] e i , l = CV { 0 } i - l } ] e (2-12) f o r a p a r t i c u l a r e lement . The element s t i f f n e s s m a t r i c e s are used t o c a l c u l a t e t h e g l o b a l s t i f f n e s s m a t r i x , [ K T ] ^ which i s the n used t o c a l c u l a t e t h e i n c r e m e n t a l n o d a l de-f l e c t i o n s , ( A q K -j^, from which the i n c r e m e n t a l element s t r a i n s , {Ae}. ,, are de t e r m i n e d . The i n c r e m e n t a l element i J i s t r e s s e s , {Aa}. , are th e n computed u s i n g [D] . , i n i , l e 1 , i E q u a t i o n (2-9). S i n c e the b e h a v i o u r o f the m a t e r i a l i s assumed t o be l i n e a r w i t h i n each l o a d i n c r e m e n t , t h e a v e r -age s t a t e o f s t r e s s i n a g i v e n element d u r i n g the l o a d i n c r e m e n t , {a}. 1 S i s g i v e n by {a}. , = {a}. 1 + Js{Aa}, , (2-13) 1 , 'S 1 — 1 1 ) 1 i n which {a}. , i s the element s t r e s s v e c t o r at the end o f i - l t h e p r e c e d i n g l o a d i n c r e m e n t . Improved e s t i m a t e s o f the mo d u l i a re o b t a i n e d by u s i n g the average s t r e s s s t a t e de-f i n e d by E q u a t i o n ( 2 - 1 3 ) . The c o r r e s p o n d i n g element s t r e s s -s t r a i n and s t i f f n e s s m a t r i c e s a re the n computed as [ D ] e . j 2 = [ D ( ( c } . j } J ] e (2-14) and C K T ] e ± j 2 = [ K T U a } l j 3 5 ) ] e (2-15) The response t o the i ^ * 1 l o a d increment i s a n a l y s e d once more u s i n g t h e improved e s t i m a t e s o f the s t r e s s - s t r a i n and s t i f f -ness m a t r i c e s . The p r o c e s s i s i l l u s t r a t e d i n F i g u r e 2. S i n c e no convergence c r i t e r i o n i s used and o n l y one i t e r a t i o n i s pe r f o r m e d , t h e f a c t o r c o n t r o l l i n g t h e ac c u r a c y w i t h which the t r u e s t r e s s - s t r a i n curve o f the m a t e r i a l i s f o l l o w e d i s t h e s i z e o f t h e l o a d i n c r e m e n t s . D i s p l a c e m e n t , q FIGURE 2 ONE ITERATION PER LOAD INCREMENT METHOD 10 2 . 2 . 2 . I t e r a t i o n s U s i n g a Convergence C r i t e r i o n The a c c u r a c y o f the method d e s c r i b e d above can be i n c r e a s e d by c o n t i n u i n g i t e r a t i o n s u n t i l some convergence c r i t e r i o n , such as a s u f f i c i e n t l y s m a l l change i n the m o d u l i between two i t e r a t i o n s , i s s a t i s f i e d . t h t h From the j a n a l y s i s o f the i l o a d Increment {a}. . , = {a} . , + %{Aa}. , (2 -16) and f o r the ( j + l ) t h a n a l y s i s o f the increment [ D ] e l > . + 1 - L B ( i a } . t . ^ n e ( 2 - 1 7 ) [ K T ] e i , j + l = [ V { « } i . j V ] e ( 2 " 1 8 ) ( A e ) . j J + 1 = [ B ] e ( A q ) l j ] + 1 ( 2 - 2 0 ) and { A a } 1 ) 3 + 1 = [ D ] e l j j + 1 < ^ > i , j + l < 2" 2 1> f o r j >_ 1. The p r o c e s s i s c o n t i n u e d u n t i l the change i n modul i from one i t e r a t i o n to the next s a t i s f i e s the c o n v e r -gence c r i t e r i o n . I f t h i s o c c u r s on the n^*1 a n a l y s i s o f the l o a d i n c r e m e n t , then i^}. = { q } . ^ + U q } . ) n (2-22) {e}. = {e>. n + {Ae}. n (2 -23) I l - l i , n and {a}. = {a}. . + {Aa}. ^ (2-24) 1 i - l i , n 11 2 . 2 . 3 A p p l i c a t i o n o f C o r r e c t i o n F o r c e s In both o f the methods d i s c u s s e d above one at tempts t o ach ieve agreement between the p r e d i c t e d s t r e s s e s t h and s t r a i n s at the end o f the i l o a d increment and the t r u e s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l by a d j u s t i n g the s t r e s s - s t r a i n p r o p e r t i e s used i n the a n a l y s i s o f the i n c r e -ment . An a l t e r n a t i v e method o f o b t a i n i n g agreement between the p r e d i c t e d s t r e s s e s and s t r a i n s and the t r u e s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l which might be used i n c o n j u n c t i o n wi th the methods d i s c u s s e d above i s g i v e n h e r e . t h At the end o f the i l o a d i n c r e m e n t , i n any i n c r e m e n t a l l o a d i n g p r o c e d u r e , the c a l c u l a t e d s t r e s s v e c t o r , {aK, and c a l c u l a t e d s t r a i n v e c t o r , {eK, f o r a p a r t i c u l a r f i n i t e e lement , may not be compat ib le w i t h the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l . I f the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l i s expressed by {a} = f ({e}) ( 2 -25) t h e n , i f the s t r a i n s are assumed to be c o r r e c t , the e r r o r i n the c a l c u l a t e d s t r e s s v e c t o r , {aK, i s g i v e n by {aeK = {ah - f ( { £ } ± ) (2 -26) where {a }^  i s the e r r o r i n the c a l c u l a t e d s t r e s s v e c t o r and f({eK) i s the t r u e s t r e s s . T h i s e r r o r may be b a l a n c e d by a p p l y i n g n o d a l c o r r e c t i o n f o r c e s , {Q }, to the element at 12 the b e g i n n i n g o f the ( i + l ) t h l o a d i n c r e m e n t . By the p r i n c i p l e o f v i r t u a l work 6 { e } T { a e } 1 d V = 6 { q } T {Q 6} (2 -27) T T where 6{e} i s a v e c t o r o f v i r t u a l s t r a i n s and 6{q} i s a v e c t o r o f v i r t u a l n o d a l d i s p l a c e m e n t s . T T T I n t r o d u c i n g the r e l a t i o n 6{e} = 6{q} E B ^ e i n E q u a t i o n ( 2 -27) y i e l d s 6 { q } T [ B ] ^ { a e } 1 d V = 6{q} T {Q e } (2 -28) T and s i n c e <5{q} i s a r b i t r a r y and not a f u n c t i o n o f p o s i t i o n {Q } = [ B ] ^ { a e } . d V (2 -29 ) F u r t h e r m o r e , i f {a }^  i s not a f u n c t i o n o f p o s i t i o n {Q e} [ B ] T dV e {ae>. (2 -30) 2 . 2 . 4 M o d i f i e d Newton-Raphson Method In the methods d e s c r i b e d i n s e c t i o n s ( 2 . 2 . 1 ) and ( 2 . 2 . 2 ) one at tempts to make the element s t r e s s e s , ( a K , and t h element s t r a i n s , {e}^, at the end o f the I l o a d increment c o m p a t i b l e w i t h the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l by a d j u s t i n g the s t r e s s - s t r a i n p r o p e r t i e s used i n the a n a -l y s i s o f the response to the i n c r e m e n t / b y : r e b u i l d i n g the element and g l o b a l s t i f f n e s s m a t r i c e s and r e p e a t i n g the a n a l y s i s o f the response to the i n c r e m e n t . The subsequent a n a l y s e s o f the response to the l o a d increment y i e l d new 13 es t imates o f the n o d a l d i s p l a c e m e n t s , { q K , the element s t r a i n s , {e}. , and the element s t r e s s e s , {a} . , at the end o f the i n c r e m e n t . I t i s p o s s i b l e to make the element s t r e s s e s , {a}. , and the element s t r a i n s , {e}. , agree w i t h the s t r e s s -s t r a i n r e l a t i o n o f the m a t e r i a l wi thout a d j u s t i n g the s t r e s s -s t r a i n p r o p e r t i e s used i n the a n a l y s i s o f the response to the i ^ * 1 l o a d i n c r e m e n t . C o n s i d e r the o n e - d i m e n s i o n a l case ( F i g u r e 3 ) • The e q u i l i b r i u m s t r e s s , and the s t r a i n ^ , o b t a i n e d i n the f i r s t a n a l y s i s o f the response to the 1 ^ l o a d i n c r e -ment, do not agree w i t h the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l . I f one assumes t h a t the e q u i l i b r i u m s t r e s s , ^ , t h i s the c o r r e c t s t r e s s f o r the end o f the i l o a d Increment , one may o b t a i n b e t t e r agreement, wh i l e m a i n t a i n i n g e q u i l i -b r i u m , by a d j u s t i n g the e s t imate o f the s t r a i n u s i n g the Newton-Raphson method. I f K i s the s lope o f the s t r e s s -th s t r a i n curve o f the m a t e r i a l at the b e g i n n i n g o f the i l o a d i n c r e m e n t , and n i s the n^*1 a p p r o x i m a t i o n o f the c o r r e c t s t r a i n c o r r e s p o n d i n g t o and a i s d e f i n e d by a 6 = a . ^ - f U . j n ) ( 2 -31) i n which f (e^ n ) i s the s t r e s s p r e s c r i b e d by the s t r e s s -s t r a i n r e l a t i o n o f the m a t e r i a l f o r n , then an improved e s t i m a t e , n+]_> ° ? the c o r r e c t s t r a i n i s g i v e n by e e. ^ = e. + ( 2 -32 ) i , n + l i , n K 14 FIGURE 3 NEWTON-RAPHSON METHOD FOR THE ONE-DIMENSIONAL CASE 15 e The p r o c e s s i s c o n t i n u e d u n t i l a i s s m a l l enough to be a c c e p t a b l e . The Newton-Raphson method i s r e a d i l y m o d i f i e d f o r use i n n o n l i n e a r a n a l y s e s o f the response o f bod ie s to a p p l i e d l o a d i n g s . The i n i t i a l e s t imates o f the i n c r e m e n t a l n o d a l d i s p l a c e m e n t s , {Aq}. , , i n c r e m e n t a l s t r a i n s , {Ae}. .. ^ 5 M i , l i , l and i n c r e m e n t a l element s t r e s s e s , {Aa}. -, , are found as i n s e c t i o n ( 2 . 2 . 1 ) . The f i r s t e s t i m a t e s o f the n o d a l d i s p l a c e -ments , {q}. , , element s t r a i n s , {e}. , , and element s t r e s -1 , 1 i , i t h s e s , {a}. at the end o f the i l o a d increment are g i v e n ' l , 1 & by + {Aq} (2-33) ( E l x - l + { A £ } i , l (2-34) and < ° } i , i - { A 0 } i , l (2-35) S i n c e { ° } ^ T . i s a n e q u i l i b r i u m s t r e s s v e c t o r f o r the end o f the i ^ l o a d i n c r e m e n t , {a}. = {a}. (2-36) 1 1 5 ± The d i f f e r e n c e , {a }. , , between {a}, and the s t r e s s e s ' i , l I s p e c i f i e d by the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l f o r the s t r a i n s , {e}. , , i s found u s i n g e q u a t i o n ( 2 - 2 6 ) . The n o d a l c o r r e c t i o n f o r c e s c o r r e s p o n d i n g to t^ 6}^ j_ a r e then found from e q u a t i o n ( 2 - 3 0 ) . A c o r r e c t i o n , {Aq}! n , t o the n o d a l d i s p l a c e m e n t v e c t o r , {q}. -, i s then found by s o l v i n g [ K T ] . {Aq}!_ x = {Q e } . ]_ (2-37) 16 i n which [K,p]^ i s the same g l o b a l s t i f f n e s s m a t r i x used to o b t a i n ( A q K ^ . An improved e s t i m a t e , { q K 2 , ° ^ t h e n o d a ± disp lacement v e c t o r i s g i v e n by ( q } l j 2 = { q } ± j l + W±al ( 2 - 3 8 ) An improved e s t i m a t e , { E K 2 , ° f the element s t r a i n s i s g i v e n by { E } , 0 = { E } , , +'{AE}1 , ( 2 - 3 9 ) i n which { A E } | ^ i s the v e c t o r o f s t r a i n s c o r r e s p o n d i n g to {Aq}! , . The d i f f e r e n c e , {a 6 } . „ , between {a}, and the s t r e s s e s s p e c i f i e d by the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l f o r the s t r a i n s , { e K 2 , i s found and the p r o c e s s t h e i s r e p e a t e d u n t i l , on the n i t e r a t i o n , {a }. i s s m a l l ' ' i , n enough to be a c c e p t a b l e . T h i s method i s c a l l e d the " i n i t i a l s t r e s s method" by Z i e n k i e w i c z (19) and the " r e s i d u a l s t r e s s method" by Nyak and Z i e n k i e w i c z (14) . T h i s method has two advantages . F i r s t , the g l o b a l s t i f f n e s s m a t r i x does not have to be changed d u r i n g the i t e r a t i v e p r o c e s s . T h i s saves c o n s i d e r a b l e computa-t i o n a l e f f o r t . S e c o n d l y , s i n c e the d i f f e r e n c e between the e q u i l i b r i u m s t r e s s e s , ( c K , and the s t r e s s e s s p e c i f i e d by the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l f o r the c a l c u l a -t e d element s t r a i n s i s e v a l u a t e d d u r i n g each I t e r a t i o n , i t i s p o s s i b l e to s p e c i f y e x p l i c i t l y the a c c u r a c y w i t h which the s t r e s s - s t r a i n r e l a t i o n o f the m a t e r i a l i s f o l l o w e d . 17 While the m o d i f i e d Newton-Raphson method i s e a s i l y a p p l i e d i n problems In which a s i n g l e known s t r e s s -s t r a i n curve i s u s e d , i t i s more d i f f i c u l t to a p p l y to problems i n which a f a m i l y o f s t r e s s - s t r a i n curves i s used and a g i v e n p o i n t , w i t h i n the body b e i n g a n a l y s e d , f o l l o w s a d i f f e r e n t s t r e s s - s t r a i n curve at d i f f e r e n t s tages o f l o a d i n g . In such c a s e s , one may have to make assumptions r e g a r d i n g the s t r e s s - s t r a i n b e h a v i o u r i n o r d e r to a l l o w the c a l c u l a t i o n of the s t r e s s e s which are compat ib l e w i t h both the s t r e s s -s t r a i n curves and the computed s t r a i n s . CHAPTER 3 STRESS-STRAIN RELATIONS 3.1 G e n e r a l D i s c u s s i o n o f S t r e s s - S t r a i n R e l a t i o n s In o r d e r to p r e d i c t the de format ions o f s o i l s t r u c t u r e s under a p p l i e d l o a d i n g s , one needs a c o n s t i t u t i v e or s t r e s s - s t r a i n model f o r the s o i l . Because o f the complex b e h a v i o u r o f s o i l s , one f i n d s i t convenient to assume the b e h a v i o u r o f s o i l s to be l i n e a r over s m a l l l o a d i n c r e m e n t s . T h e r e f o r e , one r e q u i r e s an i n c r e m e n t a l c o n s t i t u t i v e r e l a -t i o n . A p o s s i b l e c o n s t i t u t i v e r e l a t i o n f o r s o i l s Is the g e n e r a l i z e d form o f Hooke's Law f o r a homogeneous e l a s -t i c m a t e r i a l . T h i s i s , i n i n c r e m e n t a l form, A a x A a A a A x xy A x A x zx D l l D 1 2 D 1 3 D l 4 D 1 5 D 1 6 D2]_ D 2 2 D 2 3 D 2 i | D 2 5 D 2 6 D 61 D 62 D 6 3 D 64 D 6 5 ° 6 6 A e A e y A e < A y xy A y A y yz zx or (3 -D { A a } = [D] { A e } The c o n s t i t u t i v e or s t r e s s - s t r a i n m a t r i x , [ D ] , i s symmetric and c o n t a i n s twenty-one independent e l a s t i c m o d u l i . I f t h i s 18 19 g e n e r a l form o f Hooke's Law were used as the c o n s t i t u t i v e model f o r a s o i l , i t would be n e c e s s a r y to e v a l u a t e twenty-one d i f f e r e n t e l a s t i c modul i as f u n c t i o n s o f the s t a t e o f s t r e s s or s t r a i n i n the s o i l , which would be a r a t h e r formi -d i b l e t a s k . I f the s o i l i s assumed to be i s o t r o p i c as w e l l as homogeneous and e l a s t i c , e q u a t i o n ( 3 -1 ) becomes Aa Aa y Aa Ax xy Ax yz Ax zx D l l D 1 2 D 1 3 0 0 0 Ae X D 2 1 D 2 2 D 2 3 0 0 0 Ae y ° 3 1 D 3 2 D 3 3 0 0 0 < Ae Z 0 0 0 0 0 A Y 'xy 0 0 0 0 0 Ay yz 0 0 0 0 0 D 6 6 A Y ' zx (3 -2 ) i n which the c o n s t i t u t i v e m a t r i x i s symmetric and D l l = D 2 2 = D 3 3 = D 5 5 = D 6 6 and D 1 2 = ° 1 3 = D 2 3 The c o n s t i t u t i v e m a t r i x f o r the i s o t r o p i c homogeneous e l a s -t i c m a t e r i a l i s d e f i n e d by o n l y two independent e l a s t i c modul i '"since- i s a l i n e a r combinat ion o f D . ^ and T h u s , by assuming t h a t a s o i l i s i s o t r o p i c as w e l l as homo-geneous and e l a s t i c under s m a l l l o a d Increments , the number o f modul i needed to model i t s s t r e s s - s t r a i n c h a r a c t e r i s t i c s i s reduced from twenty-one to o n l y two. F o r t h i s r eason 20 s o i l s are u s u a l l y assumed to be i s o t r o p i c . Whi le the assumpt ion o f i s o t r o p y saves c o n s i d e r -ab le e f f o r t i n m o d e l l i n g the s t r e s s - s t r a i n b e h a v i o u r o f a s o i l , i t i s not a c c u r a t e f o r some s o i l s . d e f i n e d as The v o l u m e t r i c s t r a i n , e , i n any m a t e r i a l i s % - f ( 3 - 3 ) where dV i s an i n f i n i t e s i m a l change i n the volume, V , o f the element o f the m a t e r i a l . F o r an i n f i n i t e s i m a l element o f a m a t e r i a l , the v o l u m e t r i c s t r a i n may be w r i t t e n i n terms o f the normal s t r a i n s , e . e and e as ' x 5 y z e = £ + e + e (3-4) v x y z I n s p e c t i o n o f the c o n s t i t u t i v e r e l a t i o n , e q u a t i o n ( 3 - 2 ) , f o r an i s o t r o p i c homogeneous e l a s t i c m a t e r i a l r e v e a l s t h a t the normal s t r a i n s are independent o f the shear s t r e s s e s . Shear s t r e s s e s , t h e r e f o r e , do not cause v o l u m e t r i c s t r a i n s i n i s o t r o p i c m a t e r i a l s . The same i s t r u e f o r o r t h o t r o p i c m a t e r i a l s . F o r sands and o v e r c o n s o l i d a t e d c l a y s , however, shear s t r e s s e s are known to cause v o l u m e t r i c s t r a i n s . G i r i j a v a l l a b h a n and Reese (12) found t h a t , f o r a dense sand , shear s t r e s s e s may, i n some c a s e s , b-e more e f f e c t i v e i n p r o d u c i n g v o l u m e t r i c s t r a i n s than are h y d r o s t a t i c s t r e s s e s . F o r sands and o v e r c o n s o l i d a t e d c l a y s t h e n , the assumption o f i s o t r o p y i s not c o r r e c t . As a r e s u l t , the 21 deformat ions p r e d i c t e d when one uses the assumption t h a t these s o i l s are i s o t r o p i c as w e l l as homogeneous and e l a s t i c w i l l be somewhat i n e r r o r e s p e c i a l l y when l a r g e shear s t r e s -ses are i n v o l v e d . A more a c c u r a t e model f o r the s t r e s s -s t r a i n p r o p e r t i e s o f sands and o v e r c o n s o l i d a t e d c l a y s would have to take the c o u p l i n g between shear s t r e s s e s and v o l u -m e t r i c s t r a i n s i n t o a c c o u n t . T h i s would r e q u i r e more i n d e -pendent e l a s t i c m o d u l i . D e s p i t e i t s i n a c c u r a c y f o r sands and o v e r c o n s o l i d a t e d c l a y s , the assumption o f i s o t r o p y i s u s u a l l y made when m o d e l l i n g the s t r e s s - s t r a i n b e h a v i o u r o f any s o i l s i n c e i t g r e a t l y reduces the e f f o r t n e c e s s a r y to produce a mode l . geneous e l a s t i c m a t e r i a l f o r s m a l l l o a d i n c r e m e n t s , two independent modul i are needed to d e f i n e i t s s t r e s s - s t r a i n p r o p e r t i e s . Young's modulus , E f c , the tangent shear modulus , G ^ , the tangent b u l k modulus , , the tangent P o i s s o n ' s r a t i o , , and the tangent c o n s t r a i n e d modulus , ]YL_. These are d e f i n e d as f o l l o w s : I f a s o i l i s assumed to be an i s o t r o p i c homo-The m o d u l i most commonly used are the tangent 6 a . E 1 (3 -5 ) t Se. l G t (3 -6 ) K. t m v where a m (3 -7 ) 22 6e . v, = ( 3 -8 ) t o e . J 6a. and M, = - r - ^ w i t h e . = e. = 0 (3 -9 ) t 5e. j k The b u l k modulus , the shear modulus , Young's modulus and the c o n s t r a i n e d modulus must a l l be p o s i t i v e . Any two o f the modul i d e f i n e d by e q u a t i o n s ( 3 - 5 ) to ( 3 - 9 ) may be r e g a r d e d as independent . The s h e a r , b u l k and c o n s t r a i n e d m o d u l i are r e l a t e d to Young's modulus and P o i s s o n ' s r a t i o by the f o l l o w i n g e q u a t i o n s : G t = 271+77 ( 3 " 1 0 )  K t = I(Ao (3-11} E ( 1 -v ) a n d M t = ( l+v)( l-2v) ( 3 ~ 1 2 ) i f the s o i l i s i s o t r o p i c . From e q u a t i o n ( 3 - 1 1 ) , i t f o l l o w s tha t P o i s s o n ' s r a t i o may not be g r e a t e r than 0 .5 s i n c e t h i s would r e s u l t i n a n e g a t i v e b u l k modulus . D e s a i and A b e l ( 4 ) g i v e the f o l l o w i n g c o n s t i t u -t i v e m a t r i c e s , [ D ] , f o r an i s o t r o p i c homogeneous e l a s t i c m a t e r i a l : i n terms o f and v f c , 23 v t E t D 1 2 " D 1 3 " D 2 3 ( l + v t ) ( l - 2 v t ) E t D 4 4 " D 5 5 " D 6 6 " 2 ( l + v t ) and i n terms o f and , D l l = D 2 2 = D 2 3 = K t + 4 / 3 <Gt> D 1 2 = D 1 3 = ° 2 3 = K t " 2 / 3 ( V D 4 4 = D 5 5 = D 6 6 = G t D r n e v i c h (7) g i v e s the f o l l o w i n g c o n s t i t u t i v e m a t r i x i n terms o f and G^: D l l = ° 2 2 = D 3 3 = M t D 1 2 = D 1 3 = D 2 3 = M t - 2 G t D 4 4 = D 5 5 = D 6 6 = G t The c h o i c e o f which two modul i one uses to d e f i n e a s o i l ' s s t r e s s - s t r a i n p r o p e r t i e s w i l l depend upon the convenience o f the t e s t neces sary to determine them and the degree o f c e r t a i n t y w i t h which they may be d e t e r m i n e d . S m a l l u n c e r t a i n t i e s i n the va lue o f P o i s s o n ' s r a t i o may cause l a r g e u n c e r t a i n t i e s i n the elements o f the c o n s t i t u t i v e m a t r i x , [D] . For example, i f the va lue o f P o i s s o n ' s r a t i o i s 0 . 4 8 , the b u l k modulus i s e i g h t and o n e - t h i r d t imes the v a l u e o f Young's modulus . But i f the va lue o f P o i s s o n ' s r a t i o i s o n e - h a l f , the b u l k modulus i s 24 i n f i n i t e . U n c e r t a i n t i e s i n P o i s s o n ' s r a t i o a r e , however, l e s s important when i t i s s m a l l e r . S ince s o i l s are g e n e r a l l y n o n l i n e a r i n t h e i r s t r e s s - s t r a i n b e h a v i o u r , the modul i used to d e f i n e t h e i r c o n s t i t u t i v e r e l a t i o n s have to be .determined as f u n c t i o n s o f the s t r e s s or s t r a i n s t a t e . 3.2 E x p e r i m e n t a l S t u d i e s Duncan and Chang (9) proposed t h a t the c o n s t i -t u t i v e m a t r i x , [ D ] , f o r a s o i l be w r i t t e n i n terms o f the tangent v a l u e o f Young's modulus , , and the tangent va lue o f P o i s s o n ' s r a t i o , and p r e s e n t e d data on Young's modulus and P o i s s o n ' s r a t i o f o r sands and c l a y s . They found that i f d r a i n e d o r u n d r a i n e d compress ive t r i a x i a l t e s t s are conduc-t e d at cons tant v a l u e s o f the minor p r i n c i p a l s t r e s s , a^, the s t r e s s - s t r a i n r e l a t i o n s observed f o r p r i m a r y l o a d i n g o f s o i l s may be r e p r e s e n t e d by h y p e r b o l i c equat ions o f the form (a-L - <r3) = 3 — (3-13) E ± ( c?1 - o-^ ) 3 u l t In which ( - a^) i s the d e v i a t o r s t r e s s , i s the s t r a i n i n the d i r e c t i o n o f the major p r i n c i p a l s t r e s s , a^, and (.a^ - ° 2 ^ u i t "*"s a s y m p t o t i c v a l u e o f the d e v i a t o r s t r e s s ( F i g u r e 4 ) . E ^ i s the i n i t i a l v a l u e o f the modulus and can be expressed as E . = K p a ( ^ ) n (3-14) 1 a p a 25 FIGURE 4 HYPERBOLIC STRESS-STRAIN CURVE 26 i n which K i s a d i m e n s i o n l e s s q u a n t i t y c a l l e d the modulus number, p_ i s a tmospher ic p r e s s u r e , and n i s a d i m e n s i o n l e s s a number c a l l e d the modulus exponent . The va lue o f ( CT^ - ^^^ult a ^ - w a y s Sweater than the d e v i a t o r s t r e s s at f a i l u r e , ( - a^f> which may be d e f i n e d by the Mohr-Coulomb f a i l u r e c r i t e r i o n as 2 c cosij) + 2 a , sincj> ( a i - " s ' r = r^ - s iH i ( 3 " 1 5 > i n which c i s the c o h e s i o n i n t e r c e p t o f the s o i l and <b i s the angle o f i n t e r n a l f r i c t i o n o f the s o i l ( F i g u r e 5 ) • Duncan and Chang (9) expressed the r e l a t i o n s h i p between (?^ - v^ult a n d ~ ° 3 ^ f ky t h e e Q u a t i o n (a± - a 3 ) f = R f ( a x - a 3 ) u l t (3-16) where R ,^ i s known as the f a i l u r e r a t i o . T y p i c a l v a l u e s o f R f range from about 0.6 to 0.95-The tangent va lue o f Young's modulus , , f o r t r i a x i a l t e s t s conducted at a f i x e d va lue o f may be d e f i n e d as 7 d (a - a ) E t = d £ l 3 ( 3 " 1 7 ) U s i n g equat ions ( 3 - 1 3 ) , ( 3 - 1 4 ) , ( 3 - 1 5 ) , ( 3 -16 ) and ( 3 - 1 7 ) , Duncan and Chang (9) expressed the tangent Young's modulus as a f u n c t i o n o f the s t r e s s e s by the e q u a t i o n FIGURE 5 MOHR-COULOMB FAILURE CRITERION 28 1 -R f (l-sin<j>) (a2_~°2-2c cos<() + 2a2Sin<)) a_ n K p Q ( - ^ ) a Pa ( 3 - 1 8 ) The tangent v a l u e o f P o i s s o n ' s r a t i o i s d e f i n e d as the i n s t a n t a n e o u s r a t e o f change o f r a d i a l s t r a i n w i t h r e s p e c t to a x i a l s t r a i n i n the t r i a x i a l t e s t . At a cons tant va lue o f the tangent va lue o f P o i s s o n ' s r a t i o n , , f o r sands and c l a y s i s g i v e n by G - F l o i ( ° 3/p„) 1 -dCo^ - a^) a R f ( a 1 - a 3 ) ( l - s i n ^ ) _ i • a Pa 2c cos(() + 2a2Sin(j) J ( 3 - 1 9 ) i n which G i s the v a l u e o f v, when Q = p , F i s the de-fy j a crease i n f o r a t e n f o l d i n c r e a s e i n and d i s the r a t e o f i n c r e a s e o f w i t h s t r a i n and a l l o t h e r parameters are the same as i n e q u a t i o n ( 3 - 1 8 ) . F o r u n d r a i n e d c o n d i t i o n s G i s o n e - h a l f and F and d are both z e r o . Domaschuk and Wade (6) proposed tha t the c o n s t i -t u t i v e m a t r i x , [ D ] , f o r a s o i l be w r i t t e n i n terms o f the tangent shear modulus , G f c , and the tangent b u l k modulus , , and they p r e s e n t e d da ta on the b u l k and shear modul i o f sands . In o r d e r to i n v e s t i g a t e the b e h a v i o u r o f a sand i n s h e a r , they performed a number o f d r a i n e d t r i a x i a l com-p r e s s i o n t e s t s at cons tant v a l u e s o f the mean normal s t r e s s , 29 a; , on sands o f v a r i o u s i n i t i a l r e l a t i v e d e n s i t i e s . These nr t e s t s y i e l d e d d a t a on the r e l a t i o n s h i p s between the r e s u l -t a n t d e v i a t o r i c s t r e s s , S^, g i v e n by S. • ( < ° 1 - V 2 + ( ° 2 - % , ) 2 + ( 0 3 - % ) 2 ) ' S ( 3 - 2 0 ) and the r e s u l t a n t d e v i a t o r i c s t r a i n , e^, g i v e n by e d i n which e i s the mean normal s t r a i n . The g e n e r a l form of m & these r e l a t i o n s h i p s was expressed by the h y p e r b o l i c s t r e s s -s t r a i n r e l a t i o n S d - 1 + ' £ 0 . « - 2 2 » d i i n which G. i s the I n i t i a l v a l u e o f the tangent shear modulus , d e f i n e d as the i n s t a n t a n e o u s r a t e o f change o f w i t h r e s p e c t to e^, and b Is the i n v e r s e o f the a sympto t i c v a l u e o f S^. Both G^ and b were found to depend upon the i n i t i a l r e l a t i v e d e n s i t y o f the sand and the mean normal s t r e s s at which the t e s t was p e r f o r m e d . By d i f f e r e n t i a t i n g e q u a t i o n (3 -22) w i t h r e s p e c t to e^, Domaschuk and Wade (6) a r r i v e d at the f o l l o w i n g e x p r e s s i o n f o r the tangent shear modulus G t = G ± ( l - b S d ) 2 (3 -23) In o r d e r to i n v e s t i g a t e the b u l k modulus o f sands , Domaschuk and Wade (6) performed a number o f i s o t r o -p i c c o n s o l i d a t i o n t e s t s on sands o f v a r i o u s i n i t i a l r e l a t i v e 30 d e n s i t i e s and p l o t t e d curves o f the mean normal s t r e s s , a , versus the v o l u m e t r i c s t r a i n , e . They found the f o l l o w i n g r e l a t i o n s h i p between the tangent b u l k modulus , , d e f i n e d as the i n s t a n t a n e o u s r a t e o f change o f a w i t h r e s p e c t to e . and the mean n o r m a l . s t r e s s , 0 v ' ' m K f c = K. + ma m (3-24) In which K. i s the v a l u e o f the tangent b u l k modulus when e i s zero and m i s the r a t e o f change o f K, w i t h a . Both v to t m K_ and m depend upon the i n i t i a l r e l a t i v e d e n s i t y o f the sand. F o r a sand of a g i v e n r e l a t i v e d e n s i t y , the r e l a t i o n s h i p between K, and a Is a b i - l i n e a r one which may ^ t m ° be expressed by two equat ions from the form o f e q u a t i o n ( 3 - 2 4 ) , each of which i s a p p r o p r i a t e to a range o f v a l u e s o f a . m Domaschuk and V a l l l a p p a n (5) extended the work o f Domaschuk and Wade (6) to c l a y s . They found t h a t the r e l a t i o n s h i p between the r e s u l t a n t d e v i a t o r i c s t r e s s and the d e v i a t o r i c s t r a i n f o r d r a i n e d t r i a x i a l compress ion t e s t s was o f the same form f o r c l a y s as t h a t f o r sands . C o n s e q u e n t l y , the r e l a t i o n s h i p between the tangent shear modulus and the r e s u l t a n t d e v i a t o r i c s t r e s s was a l s o g i v e n by e q u a t i o n ( 3 - 2 3 ) . The v a l u e s o f and b f o r a c l a y are r e l a t e d to the i n i t i a l v o i d r a t i o o f the c l a y , i t s p r e c o n -s o l i d a t i o n p r e s s u r e and the mean normal s t r e s s at which the 31 t e s t was conducted . The tangent b u l k modulus , , o f a c l a y was found to be r e l a t e d to a n o r m a l i z e d v o l u m e t r i c s t r a i n , ^ v n , by the r e l a t i o n s h i p K t = K. (1 + N £ y n n - 1 ) ( 3 -25 ) i n which i s the i n i t i a l va lue o f the tangent b u l k modulus and N i s a number which i n d i c a t e s the r a t e o f i n c r e a s e o f K, w i t h t h a t o f E . The i n i t i a l b u l k modulus Is g i v e n by t vn K. = a /e (3 -26) l mc vc i n which a and e are c h a r a c t e r i s t i c v a l u e s o f the i s o -mc vc t r o p i c s t r e s s and the v o l u m e t r i c s t r a i n at which the i n i t i a l tangent to the s t r e s s - s t r a i n curve and a s t r a i g h t l i n e a p p r o x i m a t i o n to the f i n a l p o r t i o n o f the s t r e s s - s t r a i n curve i n t e r s e c t . When n i s i n f i n i t e , e q u a t i o n ( 3 - 2 5 ) expresses an e l a s t o p l a s t i c s t r e s s - s t r a i n c u r v e . A c c o r d i n g to Domaschuk and V a l l i a p p a n ( 5 ) 5 o , E and N were found ^ mc' vc by a method o f l e a s t squares adjustment d e s c r i b e d by Deming ( 3 ) . The n o r m a l i z e d v o l u m e t r i c s t r a i n , e , i s g i v e n by 3 v n 3 ° " £ = |E /e I ( 3 -27) vn 1 v v c 1 D r n e v i c h (7) proposed the use o f the c o n s t r a i n e d modulus , M, i n n o n l i n e a r a n a l y s e s o f problems which a p p r o x i -mate o n e - d i m e n s i o n a l c o m p r e s s i o n . T h i s would be used t o g e t h e r w i t h another modulus such as the shear modulus to d e f i n e the c o n s t i t u t i v e m a t r i x , [ D ] . He performed a number 32 o f o n e - d i m e n s i o n a l compress ion t e s t s on sands and p l o t t e d the d a t a as the a x i a l s t r e s s , a , versus the a x i a l s t r a i n , e^. The r e s u l t s were r e p r e s e n t e d by h y p e r b o l i c equat ions o f the form M. e a = 1 1 (3 -28) e m i n which M. i s the i n i t i a l va lue o f the c o n s t r a i n e d modulus , I d e f i n e d as the i n s t a n t a n e o u s r a t e o f change o f w i t h r e s -pect to E^, and i s the a s y m p t o t i c v a l u e o f the s t r a i n . By d i f f e r e n t i a t i n g e q u a t i o n ( 3 -28) w i th r e s p e c t to E , D r n e v i c h (7) found the f o l l o w i n g r e l a t i o n s h i p b e -tween the tangent c o n s t r a i n e d modulus , , and £ 2 M , = M. m_ ) 2 ( 3 -29) v m 1 By. s o l v i n g e q u a t i o n ( 3 -29 ) f o r £ and s u b s t i t u t -i n g the r e s u l t i n t o e q u a t i o n (3-28) , D r n e v i c h (7) o b t a i n e d the f o l l o w i n g e x p r e s s i o n f o r the tangent c o n s t r a i n e d modulus i n terms o f a 33 3 . 3 L i m i t a t i o n s o f S t r e s s - S t r a i n R e l a t i o n s The s t r e s s - s t r a i n r e l a t i o n s d i s c u s s e d i n s e c t i o n ( 3 . 2 ) have c e r t a i n l i m i t a t i o n s . F i r s t , they are a l l based upon the assumption o f i s o t r o p i c e l a s t i c b e h a v i o u r and thus none o f them accounts f o r volume changes due to shear even though volume changes due to shear may have o c c u r r e d i n the t e s t s used to e s t a b l i s h them. The use o f these s t r e s s - s t r a i n r e l a t i o n s , t h e r e f o r e , may not be s a t i s -f a c t o r y when d e a l i n g w i t h s o i l s which develop s i g n i f i c a n t volume changes due to shear s t r a i n s . S e c o n d l y , s i n c e the e x p r e s s i o n s d i s c u s s e d i n s e c t i o n ( 3 . 2 ) , which r e l a t e the v a l u e s o f the modul i to the s t a t e o f s t r e s s , are based upon e l a s t i c t h e o r y , they are not a p p l i c a b l e beyond the p o i n t o f y i e l d when s o i l s act more l i k e p l a s t i c m a t e r i a l s . T h i r d l y , f o r s o i l s , d i f f e r e n t s t r e s s - p a t h s to the same s t a t e o f s t r e s s may l e a d to d i f f e r e n t v a l u e s o f a p a r t i c u l a r e l a s t i c modulus . U s i n g the percentage o f the s t r e n g t h o f the s o i l tha t i s m o b i l i z e d to r e s i s t an a p p l i e d l o a d i n g as the d e f i n -i t i o n o f s t r e s s - l e v e l , Lade and Duncan ( 1 3 ) have made a c l a s s i f i c a t i o n o f l o a d i n g s i n t o p r i m a r y l o a d i n g which p r o -duces m o n o t o n i c a l l y i n c r e a s i n g s t r e s s - l e v e l s above those p r e v i o u s l y e x p e r i e n c e d by the s o i l , u n l o a d i n g which produces d e c r e a s i n g s t r e s s - l e v e l s and r e l o a d i n g which produces i n c r e a s i n g s t r e s s - l e v e l s below those p r e v i o u s l y e x p e r i e n c e d 34 by the s o i l . The s t r e s s - s t r a i n b e h a v i o u r o f c o h e s i o n l e s s s o i l s i n p r i m a r y l o a d i n g i s d i f f e r e n t from t h a t i n u n l o a d i n g or r e l o a d i n g s i n c e the s t r a i n s induced d u r i n g pr imary l o a d -i n g are p a r t i a l l y i r r e c o v e r a b l e . Lade and Duncan (13) found tha t t h i s d i f f e r e n c e g e n e r a l l y i n c r e a s e d as the s t r e s s - l e v e l i n c r e a s e d . Duncan and Chang (9) found s i m i l a r r e s u l t s f o r c l a y s . Lade and Duncan (13) conc luded from a rev iew o f p u b l i s h e d d a t a t h a t , f o r c o h e s i o n l e s s s o i l s , any two s t r e s s -paths i n v o l v i n g o n l y p r i m a r y l o a d i n g would produce n e a r l y the same s t r a i n s i n g o i n g from one s t a t e o f s t r e s s to a n o t h e r . However, s t r e s s - p a t h s i n v o l v i n g any u n l o a d i n g or r e l o a d i n g would produce s m a l l e r s t r a i n s than a s t r e s s - p a t h i n v o l v i n g o n l y p r i m a r y l o a d i n g i n g o i n g from one s t a t e o f s t r e s s to a n o t h e r . For example, i n F i g u r e ( 6 ) , s t r e s s - p a t h P - l , which i n v o l v e s on ly p r i m a r y l o a d i n g , would produce l a r g e r s t r a i n s than would s t r e s s - p a t h P - 2 , which i n v o l v e s some u n l o a d i n g and r e l o a d i n g , i n g o i n g from the s t a t e o f s t r e s s r e p r e s e n t e d by p o i n t A to t h a t r e p r e s e n t e d by p o i n t B. Duncan and Chang ( 1 0 ) , i n t h e i r d i s c u s s i o n o f t h e i r e x p r e s s i o n s f o r the tangent va lues o f Young's modulus ( e q u a t i o n ( 3 - 1 8 ) ) and P o i s s o n ' s r a t i o ( e q u a t i o n ( 3 - 1 9 ) ) , s t a t e d tha t t h e i r e x p r e s s i o n s , w i t h parameters o b t a i n e d from t r i a x i a l compress ion t e s t s per formed at cons tant v a l u e s o f the minor p r i n c i p a l s t r e s s , were unable to p r e d i c t a c c u r a t e l y 35 ro b I co co cu PH -p CO U O -P cti •H > CD Q F i n a l S t r e s s - L e v e l S t r e s s - P a t h P2 -Produces S m a l l e r S t r a i n s than S t r e s s -Path PI I n i t i a l S t r e s s - L e v e l Minor P r i n c i p a l S t r e s s , FIGURE 6 TWO STRESS-PATHS WHICH YIELD DIFFERENT STRAINS FOR THE SAME STRESS INCREMENT 36 the s t r a i n s induced i n a t r i a x i a l t e s t specimen o f a u n i f o r m f i n e sand o f low r e l a t i v e d e n s i t y when the p r i n c i p a l s t r e s -ses were i n c r e a s e d i n such a way t h a t the percentage o f a v a i l a b l e s o i l s t r e n g t h m o b i l i z e d was cons tant ( that i s , so t h a t the s t r e s s - l e v e l was c o n s t a n t ) . Y u d h b i r and V a r a d a r a j a n ( 1 8 ) conducted s t r e s s c o n t r o l l e d , d r a i n e d t r i a x i a l t e s t s on n o r m a l l y c o n s o l i d a t e d c l a y s f o r s e v e r a l s t r e s s - p a t h s . The s t r e s s - p a t h s were as f o l l o w s : (a) i n c r e a s i n g w i t h c o n s t a n t ; (b) o-^ i n c r e a s i n g w i t h cons tant mean normal s t r e s s ; (c) cons tant w i t h d e c r e a s i n g ; (d) i s o t r o p i c decrease o f normal s t r e s s e s . They e v a l u a t e d the tangent v a l u e s o f Young's modulus and P o i s s o n ' s r a t i o u s i n g the a p p r o p r i a t e Hooke's Law e x p r e s s i o n f o r i s o t r o p i c m a t e r i a l s f o r each s t r e s s - p a t h . In the case o f s t r e s s - p a t h (b) ,however, o n l y the tangent shear modulus c o u l d be e v a l u a t e d d i r e c t l y and an e s t imate o f P o i s s o n ' s r a t i o was used to e s t imate the tangent va lue o f Young's modulus . Y u d h b i r and V a r a d a r a j a n ( 1 8 ) found tha t the v a r i a t i o n o f the i n i t i a l tangent va lue o f Young's modulus w i t h c o n s o l i d a t i o n p r e s s u r e c o u l d be expressed by equat ions o f the same form as e q u a t i o n ( 3 - l 4 ) but tha t the v a l u e s o f the modulus number, K , and the modulus exponent , n , were d i f f e r e n t f o r each o f the d i f f e r e n t s t r e s s - p a t h s ; K v a r i e d 37 from s i x t e e n f o r s t r e s s - p a t h (a) to one-hundred and f i v e f o r s t r e s s - p a t h (d) w h i l e n v a r i e d from about o n e - h a l f f o r s t r e s s - p a t h (c) to about n i n e - t e n t h s f o r s t r e s s - p a t h ( a ) . The tangent v a l u e o f Young's modulus , , at any stage o f a t e s t conducted w i t h a s i n g l e s t r e s s - p a t h , was expressed as a f u n c t i o n o f the s t r e s s - l e v e l , S, ( f r a c t i o n o f s t r e n g t h m o b i l i z e d ) by an e q u a t i o n o f the form E f e = (1 - R f S ) 2 E± ( 3 - 3 D i n which E ^ i s the i n i t i a l tangent Young's modulus and R^ i s d e f i n e d as i n e q u a t i o n ( 3 - 1 6 ) . The va lue o f R^ depended upon the s t r e s s - p a t h f o l l o w e d and ranged from about s i x -t e n t h s f o r s t r e s s - p a t h (a) to j u s t over n i n e - t e n t h s f o r s t r e s s - p a t h ( c ) . Y u d h b i r and V a r a d a r a j a n ( 1 8 ) conc luded that the e f f e c t o f s t r e s s - p a t h on the s t r e s s - s t r a i n r e l a t i o n s h i p and on the tangent v a l u e o f Young's modulus at a p a r t i c u l a r s t a t e o f s t r e s s i s very s i g n i f i c a n t f o r n o r m a l l y c o n s o l i -dated c l a y s . C o r o t i s , P a r z i n and K r i z e k (2) conducted t r i -a x i a l compress ion t e s t s on two s o i l s , A , a m i x t u r e o f coarse t o very f i n e sand c o n t a i n i n g about f i v e p e r c e n t s i l t , and B , a m i x t u r e o f sand , s i l t and c l a y . The t e s t s were conducted u s i n g s t r e s s - p a t h s w i t h da^/da-^ v a r y i n g from zero to one. Only one v a l u e o f da^/da^ was used i n each t e s t . They found t h a t the s t r e s s - p a t h used a f f e c t e d the r e l a t i o n -s h i p s between the s t a t e o f s t r e s s and both Young's modulus 38 and P o i s s o n ' s r a t i o f o r both o f the s o i l s t e s t e d . I t i s apparent from a l l these s t u d i e s t h a t the s t r e s s - p a t h f o l l o w e d d u r i n g l o a d i n g may have a s i g n i f i c a n t e f f e c t on the s t r e s s - s t r a i n r e l a t i o n s and e l a s t i c m o d u l i o f s o i l s . The e x p r e s s i o n s f o r the e l a s t i c modul i o f s o i l s and the s t r e s s - s t r a i n r e l a t i o n s o f s o i l s may, t h e r e f o r e , be s a i d to be c o r r e c t o n l y f o r the s t r e s s - p a t h s f o r which they were o b t a i n e d . 3 . 4 M a t e r i a l B e hav i our A f t e r Y i e l d A l t h o u g h the b e h a v i o u r o f s o i l s a f t e r y i e l d i s g e n e r a l l y tha t o f a p l a s t i c m a t e r i a l , Duncan ( 8 ) r e p o r t s tha t many authors d e a l w i t h s o i l s a f t e r y i e l d as so f t e l a s -t i c m a t e r i a l s . T h i s s i m p l i f i e s the m o d e l l i n g o f s o i l s t r e s s -s t r a i n b e h a v i o u r . Duncan and Chang ( 9 ) used Young's modulus , as g i v e n by e q u a t i o n ( 3 - 1 8 ) t o g e t h e r w i t h a cons tant va lue o f P o i s s o n ' s r a t i o , t o d e f i n e the s t r e s s - s t r a i n b e h a v i o u r o f a sand i n an a n a l y s i s o f the l o a d - d e f o r m a t i o n response o f a b u r i e d s t r i p f o o t i n g . They s i m u l a t e d y i e l d by r e d u c i n g Young's modulus to a very s m a l l v a l u e . As may be seen from e q u a t i o n s ( 3 - 1 0 ) and ( 3 - 1 1 ) , the s imple r e d u c t i o n o f the va lue o f Young's modulus at y i e l d r e s u l t s i n a p r o p o r t i o n a t e r e d u c t i o n o f both the shear modulus and the b u l k modulus . However, a c c o r d i n g to Duncan ( p e r s o n a l c o r r e s p o n d e n c e ) , the b u l k modulus 39 immediate ly a f t e r y i e l d shou ld not be very d i f f e r e n t from t h a t Immediate ly p r i o r to y i e l d . The p r a c t i c e o f s imply r e d u c i n g Young's modulus to some a r b i t r a r y s m a l l va lue at y i e l d i s , t h e r e f o r e , not a c c e p t a b l e s i n c e i t i m p l i e s t h a t the a b i l i t y o f the s o i l t o r e s i s t changes i n h y d r o s t a t i c s t r e s s i s g r e a t l y r e d u c e d . I t i s , however, p o s s i b l e to change the v a l u e s o f bo th P o i s s o n ' s r a t i o and Young's modulus at y i e l d i n such a way t h a t the v a l u e o f the b u l k modulus remains u n -changed even though the shear modulus i s g r e a t l y r e d u c e d . C o n s i d e r the case i n which Young's modulus immedia te ly b e f o r e y i e l d i s R t imes tha t immedia te ly a f t e r y i e l d , and the va lue o f P o i s s o n ' s r a t i o immediate ly be fore y i e l d i s v^. The v a l u e o f P o i s s o n ' s r a t i o immedia te ly a f t e r y i e l d , v 2 , which w i l l m a i n t a i n the va lue o f the b u l k modulus at tha t immedia te ly p r i o r to y i e l d i s g i v e n by R - 1 + 2v., F o r example, i f Young's modulus immedia te ly a f t e r y i e l d i s set e q u a l to one- thousandth of t h a t immediate ly p r i o r to y i e l d and the v a l u e o f P o i s s o n ' s r a t i o p r i o r to y i e l d was 0 . 4 5 , then the va lue o f P o i s s o n ' s r a t i o a f t e r y i e l d would have to be i n c r e a s e d to 0 . 4 9 9 5 i n o r d e r f o r the b u l k modu-l u s to remain unchanged. I f v 2 i s chosen u s i n g e q u a t i o n ( 3 - 3 2 ) , the v a l u e o f the shear modulus immediate ly a f t e r y i e l d i s then 40 B t imes t h a t immediate ly p r i o r to y i e l d , where B i s g i v e n by 1 + V l B = ~ R - 1 + 2v ( 3 - 3 3 ) R + = -F o r the above example t h e n , c h o o s i n g a c c o r d i n g to equa-t i o n ( 3 - 3 2 ) would g i v e a v a l u e o f the shear modulus a f t e r y i e l d e q u a l to 0 . 0 0 0 9 7 t imes t h a t p r i o r to y i e l d . A c c o r d i n g t o Duncan ( 8 ) , when u s i n g a computer to p e r f o r m n o n l i n e a r a n a l y s e s , i t i s common p r a c t i c e to l i m i t P o i s s o n ' s r a t i o to v a l u e s l e s s than about 0 . 4 9 i n o r d e r to a v o i d c o m p u t a t i o n a l d i f f i c u l t i e s . I n s p e c t i o n o f e q u a t i o n ( 3 - 3 2 ) shows, however, t h a t even i f the v a l u e o f P o i s s o n ' s r a t i o were zero p r i o r to y i e l d , the va lue o f P o i s s o n ' s r a t i o a f t e r y i e l d must be 0 . 4 9 5 i n o r d e r to l eave the b u l k modulus unchanged w h i l e r e d u c i n g Young's modulus by a f a c t o r o f one-hundred at y i e l d . T h e r e f o r e , l i m i t i n g P o i s s o n ' s r a t i o to v a l u e s o f not more than 0 . 4 9 w i l l l e a d to a r e d u c t i o n o f the b u l k modulus when Young's modulus i s reduced at y i e l d . I t would be d e s i r a b l e to use e q u a t i o n ( 3 - 3 2 ) to s e l e c t v a l u e s of P o i s s o n ' s r a t i o f o r use a f t e r y i e l d . However, d o i n g t h i s may l e a d to c o m p u t a t i o n a l d i f -f i c u l t i e s s i n c e the b u l k modulus tends to i n f i n i t y as P o i s s o n ' s r a t i o approaches o n e - h a l f . I f the shear and b u l k m o d u l i are used as the independent e l a s t i c c o n s t a n t s these d i f f i c u l t i e s do not occur f o r c o m p r e s s i b l e s o i l s . One may s imply reduce the 41 shear modulus at y i e l d and m a i n t a i n the b u l k modulus at the va lue i t had p r i o r to y i e l d . CHAPTER 4 PROBLEMS IN NONLINEAR FINITE ELEMENT  ANALYSIS OF DEFORMATIONS IN SOIL BODIES 4 . 1 G e n e r a l D i s c u s s i o n o f N o n l i n e a r F i n i t e Element  A n a l y s i s o f Deformat ions i n S o i l Bod ie s 4 . 1 . 1 Common Assumptions In o r d e r to per form i n c r e m e n t a l n o n l i n e a r f i n i t e element a n a l y s e s o f s o i l b o d i e s , c e r t a i n assumptions are commonly made. The d i sp lacement f i e l d f o r the f i n i t e elements may be assumed to be l i n e a r , q u a d r a t i c , or o f a h i g h e r o r d e r . F o r any assumed d i s p l a c e m e n t f i e l d , the s t i f f n e s s o f the f i n i t e element mesh used to model a body w i l l be an upper bound to the a c t u a l s t i f f n e s s o f tha t body i f the c o r -r e c t c o n s t i t u t i v e model f o r the m a t e r i a l i s used and the f o l l o w i n g c o n d i t i o n s (Desa i and A b e l ( 4 ) ) are met: 1. The d i s p l a c e m e n t s are cont inuous w i t h i n the elements and compat ib l e between a d j a c e n t e lements . 2. The d i sp lacement f i e l d a l l o w s r i g i d body d i sp lacements , o f the f i n i t e e l ements . 3. The d i s p l a c e m e n t f i e l d a l l o w s cons tant s t r a i n s t a t e s i n the f i n i t e e lements . That i s , the d i sp lacement f i e l d must c o n t a i n l i n e a r t erms . 42 43 I f the above c o n d i t i o n s are met, the s t i f f n e s s o f the f i n i t e element mesh may be made c l o s e r to tha t o f the body i t r e p r e s e n t s by u s i n g f i n e r s u b d i v i s i o n s o f the f i n i t e element mesh (Desa i and A b e l ( 4 ) ) . B r e b b i a and Connor ( 1 ) compared the a c c u r a c y o f cons tant s t r a i n t r i a n g u l a r elements w i th t h a t o f h i g h e r o r d e r elements i n the l i n e a r a n a l y s i s o f the response o f a c a n t i l e v e r beam to l o a d i n g . I t was found t h a t , f o r meshes o f the same number o f f i n i t e e l ements , those which used h i g h e r o r d e r elements gave more a c c u r a t e s t r e s s e s and de-f l e c t i o n s . F o r meshes w i t h the same number o f unknowns, those w i t h the h i g h e r o r d e r elements gave b e t t e r r e s u l t s . The same t r e n d s were found f o r r e c t a n g u l a r e l ements . F i n n and M i l l e r ( 1 1 ) s t a t e t h a t t h e i r e x p e r i e n c e has been t h a t one l i n e a r s t r a i n t r i a n g u l a r element i s r o u g h l y e q u i v a l e n t to s i x to t en cons tant s t r a i n t r i a n g u l a r elements f o r s t r e s s c o m p u t a t i o n . They r e p o r t tha t the c o n -s tant s t r a i n t r i a n g u l a r element i s even l e s s e f f i c i e n t r e l a -t i v e to the h i g h e r o r d e r elements i n n o n l i n e a r a n a l y s e s . They found t h a t , i n e l a s t i c - p l a s t i c a n a l y s e s , the use o f cons tant s t r a i n elements c o u l d prevent s l i p de format ions a l o n g y i e l d e d e lements . Another common assumpt ion i s tha t the s t r a i n s and d i s p l a c e m e n t s o c c u r r i n g d u r i n g an a n a l y s i s are s m a l l . T h i s means tha t o n l y l i n e a r terms are needed i n the r e l a -t i o n s h i p s between s t r a i n s and d i s p l a c e m e n t s (see Desa i and 44 A b e l ( 4 ) ) . A l s o , i t means t h a t the i n i t i a l geometry o f the body may be used throughout the a n a l y s i s . I t i s a l s o commonly assumed ( 2 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 7 , 1 8 ) t h a t s o i l s are i s o t r o p i c , e l a s t i c m a t e r i a l s . T h i s means t h a t o n l y two e l a s t i c m o d u l i are needed to determine the s t r e s s - s t r a i n equat ions o f s o i l s . As e x p l a i n e d i n Chapter T h r e e , however , the assumption o f i s o t r o p y i s not c o r r e c t f o r s o i l s which are s u s c e p t i b l e to volume changes due to s h e a r . In o r d e r to a l l o w the d e t e r m i n a t i o n o f the a p p r o p r i a t e v a l u e s o f the e l a s t i c modul i at any stage o f an i n c r e m e n t a l a n a l y s i s , i t i s commonly assumed tha t the t a n -gent modul i o f s o i l s are f u n c t i o n s on ly o f the e x i s t i n g s t a t e o f s t r e s s . As p o i n t e d out i n s e c t i o n ( 3 - 3 ) , however, the modul i are a l s o f u n c t i o n s o f s t r e s s - p a t h . Duncan and Chang ( 9 ) p o i n t e d out t h a t i t i s d e s i r a b l e to r e l a t e the v a l u e s o f a s o i l ' s tangent modul i t o s t r e s s e s s i n c e these are u s u a l l y more a c c u r a t e l y known at any stage o f an a n a l y s i s than are the s t r a i n s . 4 . 1 . 2 Method of I n c r e m e n t a l A n a l y s i s As was shown i n Chapter 2 , s e v e r a l methods o f i n c r e m e n t a l n o n l i n e a r a n a l y s i s are a v a i l a b l e . Any of these methods may be used f o r s o i l s i f the a p p r o p r i a t e s t r e s s -s t r a i n r e l a t i o n s are known. In-many n o n l i n e a r p r o b l e m s , d i f f e r e n t s t r e s s - p a t h s 45 are f o l l o w e d by the s o i l i n d i f f e r e n t p a r t s o f the s o i l mass at the same t i m e . Moreover , i f l o c a l f a i l u r e s occur w i t h i n the s o i l mass, the s t r e s s - p a t h f o l l o w e d at a p a r t i c u l a r p o i n t w i t h i n the s o i l mass may be d i f f e r e n t at d i f f e r e n t s tages o f l o a d i n g . In such c a s e s , i t would be neces sary to use s t r e s s - s t r a i n r e l a t i o n s which i n c l u d e a l l p o s s i b l e s t r e s s - p a t h s i f one wished to ensure t h a t the s t r e s s - s t r a i n r e l a t i o n s used are a p p r o p r i a t e to the s t r e s s - p a t h s f o l l o w e d . Such g e n e r a l s t r e s s - s t r a i n r e l a t i o n s , however, do not e x i s t f o r s o i l s and one uses a f a m i l y o f s t r e s s - s t r a i n curves d e r i v e d from t e s t s performed w i t h s t r e s s - p a t h s o ther than those f o l l o w e d i n the problem o f i n t e r e s t . D u r i n g the a n a l y s i s o f the p r o b l e m , a p o i n t w i t h i n the s o i l mass goes from one s t r e s s - s t r a i n curve i n the f a m i l y to a n o t h e r . The use o f f a m i l i e s o f s t r e s s - s t r a i n curves as d e s c r i b e d above makes d i f f i c u l t the use o f methods such as the m o d i f i e d Newton-Raphson method or the a p p l i c a t i o n o f c o r r e c t i o n f o r c e s , which are very u s e f u l i n problems i n v o l -v i n g a unique s t r e s s - s t r a i n c u r v e . In o r d e r to use these methods, one would have to make assumptions w i t h r e g a r d to the s t r e s s - s t r a i n b e h a v i o u r o f the m a t e r i a l i n o r d e r to c a l -c u l a t e the s t r e s s e s .'which are compat ib l e w i t h both the s t r e s s - s t r a i n curves and the computed s t r a i n s . F o r example, i f the s t r e s s - s t r a i n curves used were d e s c r i b e d by e q u a t i o n ( 3 -13 ) , one might assume t h a t P o i s s o n ' s r a t i o had a constant va lue i n o r d e r to f i n d the c o r r e c t r a t i o o f the major and 46 minor p r i n c i p a l s t r e s s e s and s o l v e e q u a t i o n ( 3 - 1 3 ) f o r them. I n a n a l y s e s o f problems i n which the s t r e s s -p a t h s f o l l o w e d are not th o s e used t o e s t a b l i s h the s t r e s s -s t r a i n r e l a t i o n s a v a i l a b l e , t h e bes t t h a t one may do i s t o assume t h a t t h e e l a s t i c m o d u l i a r e f u n c t i o n s o n l y o f the e x i s t i n g s t a t e o f s t r e s s , and a r e g i v e n by e q u a t i o n s d e r i v e d from t h e s t r e s s - s t r a i n r e l a t i o n s . Methods such as t h e one i t e r a t i o n p e r l o a d i ncrement method, d e s c r i b e d i n s e c t i o n ( 2 . 2 . 1 ) , or t h e i t e r a t i o n method u s i n g a convergence c r i -t e r i o n , d e s c r i b e d i n s e c t i o n ( 2 . 2 . 2 ) , may the n be used t o pe r f o r m i n c r e m e n t a l a n a l y s e s . These methods do not r e q u i r e the c a l c u l a t i o n o f t h e c o r r e c t s t r e s s e s f o r g i v e n s t r a i n s . When u s i n g t h e s e methods, however, one may not e x e r c i s e d i r e c t c o n t r o l over the e r r o r s which a r i s e from the f a c t t h a t the s o i l ' s s t r e s s - s t r a i n b e h a v i o u r i s a c t u a l l y n o n l i n e a r over the l o a d i n c r e m e n t s used. I n t u i t i v e l y , one f e e l s t h a t b e t t e r agreement between the s t r e s s - s t r a i n c u r v e s g e n e r a t e d u s i n g the one i t e r a t i o n per l o a d increment method or i t e r a t i o n s u s i n g a convergence c r i t e r i o n and t h e t r u e s t r e s s - s t r a i n r e l a t i o n s o f the s o i l , may be o b t a i n e d by d e c r e a s i n g t h e s i z e o f t h e l o a d i n c r e m e n t s used, p r o v i d e d t h a t t h e e q u a t i o n s used t o r e l a t e t h e tangent m o d u l i t o the s t a t e o f s t r e s s a r e c o r -r e c t . However, s i n c e t h e e x p r e s s i o n s used t o r e l a t e the tangent m o d u l i o f t h e s o i l t o the s t a t e o f s t r e s s do not t a k e s t r e s s - p a t h s i n t o a c c o u n t , one does not know whether 47 or not the r e d u c t i o n o f the s i z e o f the l o a d increments w i l l cause convergence to the c o r r e c t r e s u l t i n problems wi th s t r e s s - p a t h s o t h e r than those used to e s t a b l i s h the e x p r e s -s i o n s . 4 . 2 N u m e r i c a l S t u d i e s Y u d h b i r and V a r a d a r a j a n ( 1 8 ) performed n o n l i n e a r f i n i t e element a n a l y s e s f o r a r e t a i n i n g w a l l - s o i l system i n which the w a l l r o t a t e d about i t s base away from the s o i l . The s o i l was a sedimented c l a y . Constant s t r a i n t r i a n g u l a r elements were used i n the f i n i t e element mesh which r e p r e s e n t e d the s o i l . The lower boundary o f the mesh was at the same l e v e l as the base o f the w a l l which was t en fee t h i g h . A l l nodes on the. lower boundary o f the mesh were assumed to be f i x e d . W a l l r o t a t i o n was s i m u l a t e d by the h o r i z o n t a l movement o f the nodes a d j a c e n t to i t . These nodes were assumed to be f i x e d to the w a l l . The tangent v a l u e s o f Young's modulus were r e l a t e d to the s t a t e o f s t r e s s by e q u a t i o n ( 3 - 1 8 ) . In o r d e r to assess the importance o f the e f f e c t of the s t r e s s - p a t h used i n e v a l u a t i n g the parameters i n e q u a t i o n ( 3 - 1 8 ) on the r e s u l t s o f the a n a l y s i s , s e t s o f parameters were e v a l u a t e d u s i n g t r i a x i a l t e s t s f o r two d i f f e r e n t s t r e s s - p a t h s ; (a) i n c r e a s i n g w i t h c o n s t a n t , and (b) d e c r e a s i n g w i t h c o n s t a n t . 48 These s t r e s s - p a t h s were chosen t o bound the s t r e s s - p a t h s f o l l o w e d i n the problem. The i n i t i a l major and minor p r i n c i p a l s t r e s s e s i n t h e c l a y at a depth h below t h e top o f t h e w a l l were assumed t o be = yh ( 4 - 1 ) and a ' = K a' ( 4 - 2 ) 3 o 1 i n w h ich i s the major p r i n c i p a l e f f e c t i v e s t r e s s , i s the minor p r i n c i p a l e f f e c t i v e s t r e s s , y i s t h e u n i t weight o f the c l a y , and K = 1 - sin<)>' ( 4 - 3 ) i n w h ich a ) 1 i s the a n g l e o f I n t e r n a l f r i c t i o n f o r e f f e c t i v e s t r e s s a n a l y s i s o f the c l a y . Two a n a l y s e s o f t h e s o i l - w a l l system were c a r -r i e d o u t . In.: one, the d a t a from t h e t r i a x i a l t e s t s p e r -formed w i t h s t r e s s - p a t h A were used t o r e l a t e Young's modulus t o the s t a t e o f s t r e s s ; i n the o t h e r , t h e d a t a from the t r i a x i a l t e s t s performed w i t h s t r e s s - p a t h B were used. I n b o t h a n a l y s e s p l a n e s t r a i n c o n d i t i o n s were assumed. Y u d h b i r and V a r a d a r a j a n (18) p l o t t e d the p r e -d i c t e d h o r i z o n t a l p r e s s u r e s at t h e s o i l - w a l l i n t e r f a c e v e r s u s depth f o r each o f the a n a l y s e s . The p r e d i c t e d h o r i -z o n t a l p r e s s u r e s at any depth were found t o v a r y s i g n i f i -c a n t l y w i t h the s t r e s s - p a t h used. The h o r i z o n t a l p r e s s u r e s p r e d i c t e d f o r a w a l l r o t a t i o n o f 0 . 0 0 6 r a d i a n s u s i n g d a t a 49 from s t r e s s - p a t h A were h i g h e r at a l l depths than those p r e d i c t e d u s i n g da ta from s t r e s s - p a t h B . The d i f f e r e n c e was as much as twenty p e r c e n t . Duncan and Chang ( 9 ) a n a l y s e d the l o a d -d e f o r m a t i o n response o f a r e c t a n g u l a r f o o t i n g measur ing 1 2 . 4 4 i n c h e s by 2 . 4 4 i n c h e s i n p l a n and b u r i e d at a depth o f twent.y i n c h e s i n Chatahoochee R i v e r sand. Constant s t r a i n q u a d r i l a t e r a l e lements were used i n a p lane s t r a i n a n a l y s i s . The r e l a t i o n s h i p between the tangent v a l u e o f Young's modulus and the s t a t e o f s t r e s s was expressed by e q u a t i o n ( 3 - 1 8 ) up to the p o i n t o f f a i l u r e as d e f i n e d by the Mohr-Coulomb f a i l u r e c r i t e r i o n . The parameters used i n e q u a t i o n ( 3 - 1 9 ) were found from t h r e e t r i a x i a l compress ion t e s t s conducted at f i x e d v a l u e s o f the minor p r i n c i p a l s t r e s s . P o i s s o n ' s r a t i o was assumed to have a cons tant v a l u e o f 0 . 3 5 - The v a l u e o f Young's modulus was a r b i t r a r i l y reduced to ten pounds per square foot a f t e r f a i l u r e . The p r e d i c t e d b e h a v i o u r o f the f o o t i n g under v e r t i c a l l o a d i n g was compared w i t h tha t observed i n a model t e s t o f the f o o t i n g . The observed and p r e d i c t e d f a i l u r e l o a d s f o r the f o o t i n g - s o i l system were n e a r l y i d e n t i c a l . However, the p r e d i c t e d average f o o t i n g p r e s s u r e s a s s o c i a t e d w i t h a g i v e n s e t t l ement were as much as f i f t y percent h i g h e r than those observed i n the model t e s t p r i o r to f a i l u r e . The p r e d i c t e d average f o o t i n g p r e s s u r e s a f t e r f a i l u r e were lower than those observed i n the model t e s t w i t h the 50 d i f f e r e n c e i n c r e a s i n g w i t h i n c r e a s i n g s e t t l e m e n t . The w r i t e r performed s e v e r a l i n c r e m e n t a l n o n -l i n e a r a n a l y s e s o f the b e h a v i o u r o f a sand beneath a s i x i n c h wide s u r f a c e s t r i p f o o t i n g s u b j e c t e d to v e r t i c a l l o a d i n g . The f i n i t e element program used was NONLIN, a program deve loped at the U n i v e r s i t y o f B r i t i s h C o l u m b i a . T h i s program i s d e s i g n e d to per form i n c r e m e n t a l n o n l i n e a r a n a l y s e s f o r problems i n v o l v i n g s m a l l s t r a i n s u s i n g any compat ib l e s t r e s s - s t r a i n s u b r o u t i n e s s u p p l i e d by the u s e r . The program uses s i x node l i n e a r s t r a i n t r i a n g u l a r e lements . The s t i f f n e s s m a t r i x f o r t h i s element Is d e r i v e d i n Appendix 1 . The s t a t e o f s t r e s s r e s u l t i n g from any l o a d i n g would n o r m a l l y v a r y a c r o s s a l i n e a r s t r a i n t r i a n g u l a r e lement . The v a l u e s o f the tangent modul i wou ld , t h e r e f o r e , v a r y a c r o s s t h i s element f o r a n o n l i n e a r m a t e r i a l . However, NONLIN uses o n l y one set o f two m o d u l i , based on the s t a t e o f s t r e s s at the element c e n t r o i d , f o r the e n t i r e e lement . The w r i t e r deve loped s u b r o u t i n e s based upon e a r l i e r work done by Roy ( 1 5 ) i n o r d e r to use the h y p e r -b o l i c s t r e s s - s t r a i n r e l a t i o n s of Duncan and Chang ( 9 ) w i t h the computer program NONLIN. These s u b r o u t i n e s are p r e -sented i n Appendix 2. The s u b r o u t i n e s use equat ions ( 3 - l 8 ) and ( 3 - 1 9 ) to r e l a t e the tangent v a l u e s of Young's modulus and P o i s s o n ' s r a t i o to the s t a t e o f s t r e s s , up to the p o i n t o f y i e l d as d e f i n e d by the Mohr-Coulomb f a i l u r e c r i t e r i o n . 51 The tangent v a l u e s o f Young's modulus and P o i s s o n ' s r a t i o are then c o n v e r t e d to tangent v a l u e s of the b u l k and shear modul i f o r use w i t h NONLIN. The parameters used i n equat ions ( 3 - l 8 ) and ( 3 - 1 9 ) f o r the a n a l y s e s performed were those p r e s e n t e d by Wong and Duncan ( 1 7 ) f o r Sacramento R i v e r sand w i t h a r e l a -t i v e d e n s i t y o f t h i r t y - e i g h t p e r c e n t . These parameters were o b t a i n e d from d r a i n e d t r i a x i a l t e s t s conducted at c o n -s tant v a l u e s o f the minor p r i n c i p a l s t r e s s and are as f o l l o w s : u n i t w e i g h t , y = 8 9 . 5 pounds per c u b i c foot angle o f i n t e r n a l f r i c t i o n , <j> = 35-0 degrees cohes ion i n t e r c e p t , c = 0.0 pounds per c u b i c foo t modulus number, k = 430 modulus exponent , n = 0.27 f a i l u r e r a t i o , R f = 0.84 P o i s s o n ' s r a t i o parameter G = 0.42 P o i s s o n ' s r a t i o parameter F = 0 . 2 1 P o i s s o n ' s r a t i o parameter d = 2 . 9 The i n c r e m e n t a l a n a l y s e s were c a r r i e d out as d e s c r i b e d i n s e c t i o n ( 2 . 2 . 1 ) u s i n g one i t e r a t i o n and no convergence c r i t e r i o n . The f i n i t e element mesh used to model the f o o t i n g -s o i l system i s shown i n F i g u r e (7 ) . O n e - h a l f o f a s i x i n c h wide r i g i d s u r f a c e s t r i p f o o t i n g was r e p r e s e n t e d by m a i n -t a i n i n g e q u a l v e r t i c a l d i s p l a c e m e n t s o f the n i n e nodes FIGURE 7 FINITE ELEMENT MESH 53 c l o s e s t t o ' t h e l e f t hand edge on the upper s u r f a c e o f the mesh, d u r i n g l o a d i n g . In o r d e r to s i m u l a t e a p e r f e c t l y rough f o o t i n g base , these nodes were not a l l o w e d to move h o r i z o n t a l l y . The nodes on the s i d e s o f the mesh were a l l o w e d on ly v e r t i c a l m o t i o n . The nodes on the base o f the f i n i t e element mesh were assumed to be f i x e d i n t h e i r i n i -t i a l p o s i t i o n s . The I n i t i a l s t r e s s e s at the c e n t r o i d s o f the f i n i t e element mesh were assumed to be g i v e n by equat ions (4-1) and (4-2) w i th h measured from the top o f the f i n i t e element mesh. The c o e f f i c i e n t o f l a t e r a l e a r t h p r e s s u r e at r e s t , K , was taken to be 0.426 f o r the Sacramento R i v e r ' o' sand. A set o f a n a l y s e s i n which the tangent shear modulus was reduced to a s m a l l v a l u e at y i e l d and the v a l u e o f the tangent b u l k modulus was kept the same a f t e r y i e l d as tha t immedia te ly p r i o r to y i e l d was p e r f o r m e d . T h i s approach i s as suggested i n s e c t i o n (3-4) by the w r i t e r . The f o o t i n g l o a d - s e t t l e m e n t curves p r e d i c t e d , u s i n g d i f f e r e n t s i z e l o a d increments i n t h i s set o f a n a l y s e s , are shown i n F i g u r e s (8) and (9 ) . Those shown i n F i g u r e (8) are f o r a n a l y s e s i n which the shear modulus o f the sand was reduced to one-hundred pounds per square foot at y i e l d w h i l e those shown i n F i g u r e (9) are f o r a n a l y s e s i n which the shear modulus o f the sand was reduced t o f i v e pounds per square foo t at y i e l d . 54 lOOOr-30 pound increments 40 pound increments 2 0 nound increments 0 . 0 5 0 . 1 0 0 . 1 5 I 0 . 2 0 F o o t i n g Set t l ement ( f e e t ) FIGURE 8 LOAD-SETTLEMENT CURVES FOR ANALYSES IN. WHICH THE SHEAR MODULUS WAS REDUCED TO ONE-HUNDRED POUNDS PER SQUARE FOOT AT YIELD AND THE BULK MODULUS'WAS NOT REDUCED AT YIELD 55 1000 0.10 0.20 F o o t i n g Set t l ement ( f e e t ) FIGURE 9 LOAD-SETTLEMENT CURVES FOR ANALYSES IN WHICH THE SHEAR MODULUS WAS REDUCED TO FIVE POUNDS PER SQUARE FOOT AT YIELD AND THE BULK MODULUS WAS NOT REDUCED AT YIELD 56 The p r e d i c t e d u l t i m a t e b e a r i n g c a p a c i t y o f the f o o t i n g - s o i l system may be taken as the asymptote suggested by the p r e d i c t e d l o a d - s e t t l e m e n t c u r v e . The asymptotes suggested by the l o a d - s e t t l e m e n t curves p r e d i c t e d i n the a n a l y s e s , i n which the shear modulus o f the sand was reduced to f i v e pounds per square foot at y i e l d , v a r i e d from about f i v e - h u n d r e d and seventy pounds to about s i x - h u n d r e d and twenty pounds. The a s y m p t o t i c l o a d i n c r e a s e d w i t h i n c r e a s -i n g l o a d increment s i z e . The l o a d - s e t t l e m e n t curves p r e -d i c t e d i n the a n a l y s e s , i n which the shear modulus was r e -duced to one-hundred pounds per square foot at y i e l d , d i d not f l a t t e n as much as those p r e d i c t e d i n the a n a l y s e s i n which the shear modulus was reduced to f i v e pounds per square foot at y i e l d . The w r i t e r can o n l y say that the asymptotes suggested by them were a l l g r e a t e r than s i x hundred pounds. A g a i n , the a s y m p t o t i c l o a d appeared to i n c r e a s e w i t h i n c r e a s -i n g l o a d increment s i z e . F o r compar i son , the t h e o r e t i c a l u l t i m a t e b e a r i n g c a p a c i t y , assuming r i g i d - p l a s t i c b e h a v i o u r o f the f o o t i n g -s o i l sys tem, f o r the h a l f o f the s i x i n c h wide f o o t i n g r e p r e s e n t e d i n the f i n i t e element mesh, as found w i t h the C a q u o t - K e r i s e l b e a r i n g c a p a c i t y f a c t o r s g i v e n by V e s i c ( 1 6 ) , i s f i v e - h u n d r e d and t h i r t y - s e v e n pounds. T h i s i s s i x p e r c e n t to f o u r t e e n p e r c e n t lower than those p r e d i c t e d i n the a n a l y s e s i n which the shear modulus was reduced to f i v e pounds per square foot at y i e l d and at l e a s t t e n percent 57 lower than any o f those p r e d i c t e d i n the a n a l y s e s i n which the shear modulus was reduced to one-hundred pounds per square f o o t at y i e l d . The p r e d i c t e d l o a d - s e t t l e m e n t curves showed abrupt i n c r e a s e s and decreases i n t h e i r s l o p e s a f t e r f l a t -t e n i n g out somewhat. These were due t o elements which had p r e v i o u s l y f a i l e d coming out o f f a i l u r e , a n d , to elements which p r e v i o u s l y had not f a i l e d , f a i l i n g w i t h f u r t h e r i n -creases i n the l o a d a p p l i e d to the f o o t i n g . The l o c a t i o n s o f f a i l e d e l ements , p r e d i c t e d by the a n a l y s i s i n which the shear modulus was reduced to f i v e pounds per square foot at y i e l d and i n which twenty pound l o a d increments were used , are shown i n F i g u r e ( 1 0 ) f o r l o a d s equa l to and g r e a t e r than tha t at which the l o a d -se t t l ement curve began to f l a t t e n o u t . The l o c a t i o n s o f f a i l e d elements do not suggest any c l e a r f a i l u r e mode. The movements o f nodes i n the v i c i n i t y o f the f o o t i n g are shown i n F i g u r e (11) f o r the l o a d increment from f i v e - h u n d r e d and twenty pounds to f i v e - h u n d r e d and f o r t y pounds, i n the a n a l y s i s i n which the shear modulus was reduced t o f i v e pounds per square foo t at y i e l d , and i n which twenty pound l o a d increments were used . The nodes appear to move i n smooth t r a j e c t o r i e s b e g i n n i n g downward at the base o f the f o o t i n g and changing to h o r i z o n t a l and , f i n a l l y , upward, away from the f o o t i n g . T h i s i s s i m i l a r to the motions r e p o r t e d by V e s i c ( 1 6 ) f o r the g e n e r a l shear s i s F I G U R E 10 L O C A T I O N S O F F A I L E D E L E M E N T S F O R T H E A N A L Y S I S I N W H I C H T H E S H E A R M O D U L U S W A S R E D U C E D T O F I V E P O U N D S P E R S Q U A R E F O O T A T Y I E L D , T H E B U L K M O D U L U S W A S N O T R E D U C E D A T Y I E L D , A N D T W E N T Y P O U N D L O A D I N C R E M E N T S W E R E U S E D (remainder o f mesh omi t ted) ( F a i l e d Elements are shaded) FIGURE 11 MOVEMENTS OF NODES DURING THE LOAD INCREMENT FROM F I V E HUNDRED AND TWENTY POUNDS TO F I V E HUNDRED AND FORTY POUNDS IN THE ANALYSIS IN WHICH THE SHEAR MODULUS WAS REDUCED TO F I V E POUNDS PER SQUARE FOOT AT Y I E L D , THE BULK MODULUS WAS NOT REDUCED AT Y I E L D , AND TWENTY POUND INCREMENTS WERE USED 60 f a i l u r e o f a s t r i p f o o t i n g on sand. As was s t a t e d i n s e c t i o n ( 3 . 4 ) , i t i s common p r a c t i c e to use a constant va lue o f P o i s s o n ' s r a t i o t h r o u g h -out a n o n l i n e a r a n a l y s i s o f the l o a d - d e f o r m a t i o n response o f a s o i l body and s imply to reduce the va lue o f Young's modu-l u s at y i e l d . In o r d e r to assess the e f f e c t o f t h i s p r a c -t i c e on the r e s u l t s o f the a n a l y s i s o f the l o a d - s e t t l e m e n t response o f the s i x i n c h wide s u r f a c e s t r i p f o o t i n g , a second set of a n a l y s e s was performed i n which the va lue o f P o i s s o n ' s r a t i o f o r the sand was assumed to be 0.42 r e g a r d -l e s s o f the s t a t e o f s t r e s s , and the v a l u e of Young's modulus was reduced to one-hundred pounds per square foot at y i e l d . The e q u a t i o n , used In t h i s second set o f a n a l y s e s to r e l a t e the tangent v a l u e o f Young's modulus to the s t a t e of s t r e s s p r i o r to y i e l d , was the same as t h a t used i n the f i r s t set o f a n a l y s e s . The f i n i t e element mesh used i n the second set o f a n a l y s e s was the same as tha t used i n the f i r s t set o f a n a l y s e s . The r e s u l t s o f the second set of a n a l y s e s are p r e s e n t e d below. The f o o t i n g l o a d - s e t t l e m e n t curves p r e d i c t e d , u s i n g d i f f e r e n t s i z e l o a d increments i n the a n a l y s e s i n which P o i s s o n ' s r a t i o was assumed to have a cons tant v a l u e o f 0.42 and i n which Young's modulus was reduced to one hundred pounds per square foot at y i e l d , are shown i n F i g u r e (12) . The u l t i m a t e b e a r i n g c a p a c i t y o f the f o o t i n g suggested by these l o a d - s e t t l e m e n t curves i s l e s s than about 61 FIGURE 12 LOAD-SETTLEMENT CURVES FOR ANALYSES IN WHICH POISSON'S RATIO HAD A CONSTANT VALUE OF 0.42 AND YOUNG'S MODULUS WAS REDUCED TO ONE-HUNDRED POUNDS PER SQUARE FOOT AT YIELD 62 t h r e e hundred pounds. T h i s i s about o n e - h a l f o f tha t sug-ges ted by the l o a d - s e t t l e m e n t curves p r e d i c t e d i n the f i r s t set o f a n a l y s e s , i n which the shear modulus was reduced at y i e l d w h i l e the b u l k modulus was n o t . The l o c a t i o n s o f f a i l e d e l ements , as p r e d i c t e d i n the a n a l y s i s i n which P o i s s o n ' s r a t i o had a cons tant v a l u e of 0.42 and Young's modulus was reduced to one-hundred pounds per square foo t at y i e l d and f i f t e e n pound l o a d i n c r e -ments were u s e d , are shown i n F i g u r e ( 1 3 ) f o r l oads e q u a l to and g r e a t e r than tha t at which the l o a d - s e t t l e m e n t curve began to f l a t t e n o u t . The l o c a t i o n s o f f a i l e d elements do not suggest any c l e a r f a i l u r e mode. The movements o f nodes i n the v i c i n i t y o f the f o o t i n g are shown i n F i g u r e ( 1 4 ) f o r the l o a d Increment from one-hundred and twenty pounds to one-hundred and t h i r t y - f i v e pounds i n the a n a l y s i s i n which P o i s s o n ' s r a t i o had a c o n -s tant v a l u e o f 0 .42 , Young's modulus was reduced to one hundred pounds per square foo t at y i e l d , and f i f t e e n pound l o a d increments were used . From t h i s f i g u r e , i t i s apparent t h a t the f o o t i n g f a i l e d by compress ion o f the f a i l e d e l e -ments d i r e c t l y beneath i t . The o n l y s i g n i f i c a n t motions were v e r t i c a l motions o f the nodes d i r e c t l y beneath the f o o t i n g . T h i s b e h a v i o u r i s very d i f f e r e n t from tha t p r e d i c -t e d i n the a n a l y s e s i n which the shear modulus was reduced at y i e l d w h i l e the b u l k modulus was n o t . The d i f f e r e n c e between the l o a d - d e f o r m a t i o n 63 At 12 0 pounds s i s FIGURE 13 LOCATIONS OF FAILED ELEMENTS FOR THE ANALYSIS IN.WHICH POISSON'S RATIO HAD A CONSTANT VALUE OF 0.42, YOUNG'S MODULUS WAS REDUCED TO FIVE POUNDS PER SQUARE FOOT AT YIELD, AND FIFTEEN POUND LOAD INCREMENTS WERE USED 64 FIGURE 14 MOVEMENTS OF NODES DURING THE LOAD INCREMENT FROM ONE-HUNDRED AND TWENTY POUNDS TO ONE-HUNDRED AND T H I R T Y - F I V E POUNDS IN THE ANALYSIS IN WHICH POISSON'S RATIO HAD A CONSTANT VALUE OF 0 .42 , YOUNG'S MODULUS WAS REDUCED TO ONE-HUNDRED POUNDS PER SQUARE FOOT AT Y I E L D , AND FIFTEEN POUND LOAD INCREMENTS WERE USED response o f the f o o t i n g - s o i l system p r e d i c t e d i n the f i r s t set o f a n a l y s e s and t h a t p r e d i c t e d i n the second set o f a n a l y s e s r e s u l t e d from the d i f f e r e n c e between the way i n which y i e l d o f the sand was taken i n t o account i n each a n a l y s i s . In the f i r s t set o f a n a l y s e s , the shear modulus was reduced at y i e l d but the b u l k modulus was the same a f t e r y i e l d as i t was Immediate ly p r i o r to y i e l d . T h i s p r a c t i c e reduces the r e s i s t a n c e o f the elements to shear de format ions at y i e l d but does not reduce t h e i r r e s i s t a n c e to normal de -f o r m a t i o n s . T h e r e f o r e , i n the f i r s t set o f a n a l y s e s , the y i e l d e d elements beneath the f o o t i n g c o u l d not be e a s i l y compressed by f u r t h e r i n c r e a s e s i n the l o a d a p p l i e d to them. In the second set o f a n a l y s e s , the v a l u e o f P o i s s o n ' s r a t i o was cons tant but the v a l u e o f Young's modulus was reduced at y i e l d o f the sand. T h i s p r a c t i c e reduces the v a l u e s o f both the shear and the b u l k m o d u l i at y i e l d . T h e r e f o r e , i n the second set o f a n a l y s e s , the r e s i s t a n c e o f elements to both shear and normal de format ions was reduced at y i e l d . T h i s a l l o w e d the y i e l d e d elements d i r e c t l y beneath the f o o t i n g to be compressed e a s i l y by f u r t h e r i n c r e a s e s i n the l o a d a p p l i e d to them. S i n c e the r e s i s t a n c e o f a r e a l sand to normal s t r a i n s i s not s i g n i f i c a n t l y reduced at y i e l d , as d e f i n e d by the Mohr-Coulomb f a i l u r e c r i t e r i o n , the b e h a v i o u r o f the f o o t i n g - s o i l system p r e d i c t e d i n the f i r s t set o f a n a l y s e s i s more r e a l i s t i c than t h a t p r e d i c t e d i n the second set o f a n a l y s e s . CHAPTER 5 CONCLUSION E x i s t i n g methods o f p r e d i c t i n g the n o n l i n e a r l o a d - d e f o r m a t i o n response o f s o i l s and s o i l s t r u c t u r e s are based on i n c r e m e n t a l i t e r a t i v e e l a s t i c a n a l y s i s . A s o l u t i o n f o r a l o a d increment i s c o n s i d e r e d a c c e p t a b l e i f the s t r e s s e s and s t r a i n s computed f o r the end o f the l o a d increment c o n -form to the s t r e s s - s t r a i n r e l a t i o n s o f the m a t e r i a l w i t h i n a p r e s c r i b e d t o l e r a n c e . Convergence to the s t r e s s - s t r a i n r e l a t i o n s o f the m a t e r i a l may be a c h i e v e d i n two ways: f i r s t , the computed s t r a i n s may be c o n s i d e r e d c o r r e c t and the computed s t r e s s e s s u c c e s s i v e l y c o r r e c t e d to make them conform to the s t r e s s - s t r a i n r e l a t i o n s o f the m a t e r i a l ; and s e c o n d l y , the computed s t r e s s e s may be assumed to be c o r r e c t and the s t r a i n s s u c c e s s i v e l y c o r r e c t e d to make them compat-i b l e w i t h the s t r e s s - s t r a i n r e l a t i o n s o f the m a t e r i a l and the computed s t r e s s e s . The f i r s t o b j e c t i v e Is a c c o m p l i s h e d by i t e r a t i v e adjustment o f the e l a s t i c p r o p e r t i e s used i n the a n a l y s i s o f the response to the l o a d i n c r e m e n t ; the second by a Newton-Raphson approach i n which the e l a s t i c p r o p e r t i e s are not changed. Methods o f i n c r e m e n t a l a n a l y s i s i n which both the computed s t r e s s e s and computed s t r a i n s are made to c o n -form to the s t r e s s - s t r a i n r e l a t i o n s of the m a t e r i a l by 66 67 i t e r a t i v e adjustment o f the e l a s t i c p r o p e r t i e s have the advantage o f s i m p l i c i t y . However, they r e q u i r e a c o n s i d e r -a b l e c o m p u t a t i o n a l e f f o r t s i n c e the s t i f f n e s s m a t r i x o f the s o i l body must be r e b u i l t p r i o r to each s u c c e s s i v e a d j u s t -ment. F o r t h i s r e a s o n , i t i s common p r a c t i c e to p e r f o r m o n l y one i t e r a t i o n per l o a d increment and not to use a c o n -vergence c r i t e r i o n . A l s o , s i n c e one i t e r a t e s on the e l a s t i c p r o p e r t i e s u s e d , the a c c u r a c y w i t h which the computed s t r e s -ses and s t r a i n s conform to the s t r e s s - s t r a i n r e l a t i o n s o f the m a t e r i a l i s not d i r e c t l y c o n t r o l l e d . The m o d i f i e d Newton-Raphson method has the advantage t h a t the s t i f f n e s s m a t r i x o f the body under c o n -s i d e r a t i o n i s not r e b u i l t d u r i n g the i t e r a t i v e c o r r e c t i o n o f the computed s t r a i n s . T h i s saves c o n s i d e r a b l e c o m p u t a t i o n a l e f f o r t . The m o d i f i e d Newton-Raphson method i s r e a d i l y a p p l i e d to problems i n which a s i n g l e known s t r e s s - s t r a i n curve i s f o l l o w e d . However, i n problems i n v o l v i n g s o i l s , a f a m i l y o f s t r e s s - s t r a i n curves i s u s u a l l y used and the s t r e s s - s t r a i n curve f o l l o w e d by a p o i n t w i t h i n the s o i l mass changes w i t h the s t a t e o f s t r e s s at t h a t p o i n t i f the s t r e s s -p a t h f o l l o w e d i s not t h a t f o r which the curves were o b t a i n e d . As was e x p l a i n e d i n s e c t i o n ( 4 . 1 . 2 ) , t h i s makes the a p p l i c a -t i o n o f the m o d i f i e d Newton-Raphson method to problems i n v o l v i n g s o i l s more d i f f i c u l t . The a c c u r a c y w i t h which the l o a d - d e f o r m a t i o n 68 response o f s o i l s and s o i l s t r u c t u r e s can be p r e d i c t e d Is l i m i t e d m a i n l y by the a c c u r a c y w i t h which the s t r e s s - s t r a i n p r o p e r t i e s o f the s o i l are r e p r e s e n t e d . The s t r e s s - s t r a i n r e l a t i o n s c u r r e n t l y used are based on the assumption tha t s o i l s are i s o t r o p i c e l a s t i c m a t e r i a l s and have two major sources o f e r r o r : the e f f e c t o f s t r e s s - p a t h i s n e g l e c t e d ] and the a n i s o t r o p y o f s o i l s i s n e g l e c t e d . I d e a l l y , one s h ou l d use s t r e s s - s t r a i n r e l a t i o n s which take the s t r e s s - p a t h dependent b e h a v i o u r and a n i s o -t r o p y o f s o i l s i n t o a c c o u n t . Whi le t h i s i s p o s s i b l e , the development o f such s t r e s s - s t r a i n r e l a t i o n s would r e q u i r e the e v a l u a t i o n o f more s t r e s s - s t r a i n p r o p e r t i e s and c o n s i d -e r a b l y more e f f o r t than does the development o f those c u r -r e n t l y u s e d . A l t h o u g h the behav iour o f s o i l s a f t e r y i e l d i s g e n e r a l l y tha t o f a p l a s t i c m a t e r i a l , many authors d e a l . w i t h s o i l s a f t e r y i e l d as s o f t e l a s t i c m a t e r i a l s . They s imply reduce the v a l u e s o f the e l a s t i c modul i at y i e l d . F o r a r e a l s o i l , the r e s i s t a n c e to shear s t r e s s e s i s reduced at y i e l d whereas the r e s i s t a n c e to normal s t r e s s e s i s n o t . T h e r e f o r e , wh i l e the shear modulus i s reduced at y i e l d , the b u l k modulus i s n o t . I f one d e a l s w i t h s o i l s as so f t e l a s t i c m a t e r -i a l s a f t e r y i e l d one must be c a r e f u l tha t the va lue o f the b u l k modulus i s not reduced at y i e l d . The p r a c t i c e o f u s i n g a constant v a l u e o f P o i s s o n ' s r a t i o and r e d u c i n g the va lue 69 of Young's modulus at y i e l d i s u n a c c e p t a b l e s i n c e i t r e s u l t s i n a r e d u c t i o n o f the v a l u e s o f b o t h the shear modulus and the b u l k modulus . A n a l y s e s o f the response o f a s t r i p f o o t i n g on sand to v e r t i c a l l o a d i n g , performed by the w r i t e r , show t h a t t h i s l e a d s t o u n r e a l i s t i c b e h a v i o u r i n y i e l d e d s o i l e l ements . The y i e l d e d e lements were too c o m p r e s s i b l e . T h i s a f f e c t e d to a great extent both the modes o f f a i l u r e o f the f o o t i n g - s o i l system and f a i l u r e l oads p r e d i c t e d i n the a n a l y s e s . N o n l i n e a r a n a l y s e s o f the l o a d - d e f o r m a t i o n r e s -ponse o f s o i l s and s o i l s t r u c t u r e s shou ld be performed u s i n g the shear modulus and b u l k modulus to d e f i n e the s o i l ' s s t r e s s - s t r a i n r e l a t i o n s . At y i e l d , the v a l u e o f the shear modulus shou l d be reduced and the b u l k modulus l e f t unchan-ged. As f a r as p o s s i b l e , the e l a s t i c modul i and s t r e s s -s t r a i n r e l a t i o n s used shou ld model the e f f e c t o f s t r e s s -p a t h on the s o i l s t r e s s - s t r a i n p r o p e r t i e s . BIBLIOGRAPHY B r e b b i a , C . A . and Connor , J . J . , 'Fundamentals o f F i n i t e Element T e c h n i q u e s , H a l s t e a d P r e s s , John W i l e y and Sons L t d . , New Y o r k - T o r o n t o , 1974, p p . 1 2 9 - 1 3 8 . C o r o t i s , R . B . , F a r z i n , M . H . and K r i z e k , R . J . , " N o n l i n e a r S t r e s s - S t r a i n F o r m u l a t i o n f o r S o i l s , " J o u r n a l o f the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , A S C E , V o l . 100, No. GT9, September, 1974, pp . 993 -1008. Demlng, W . E . , S t a t i s t i c a l Adjustment o f D a t a , Dover P u b l i c a t i o n s I n c . , New Y o r k , New,York, 1 9 4 3 . D e s a l , C . S . and A b e l , J . F . , I n t r o d u c t i o n to the F i n i t e Element Method, 'Van Nos trand R e i n h o l d C o . , New Y o r k , New Y o r k , 1972. Domaschuk, L . and V a l l i a p p a n , P . , " N o n l i n e a r S e t t l e -ment A n a l y s i s by F i n i t e E l e m e n t , " J o u r n a l o f the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , A S C E , V o l . 101, No. GT7, J u l y , 1 9 7 5 , PP . 601-614. Domaschuk, L . and Wade, N . H . , "A Study o f Bu lk and Shear M o d u l i o f a S a n d , " J o u r n a l o f the S o i l Mechanics and Foundat ions D i v i s i o n , A S C E , V o l . 9 5 , No. SM2, M a r c h , 1 9 6 9 , p p . 5 6 1 - 5 8 1 . D r n e v i c h , V . P . , " C o n s t r a i n e d and Shear M o d u l i f o r F i n i t e E l e m e n t s , " J o u r n a l o f the G e o t e c h n i c a l E n g i n -e e r i n g D i v i s i o n , A S C E , V o l . 101, No. GT5-, May, 1975, p p . 4 5 9 - 4 7 3 . Duncan, J . M . , " S t a t i c F i n i t e Element A n a l y s i s , " notes f o r e x t e n s i o n course e n t i t l e d Recent Developments i n the D e s i g n , C o n s t r u c t i o n and Performance o f Embankment Dams, C o l l e g e o f E n g i n e e r i n g , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , C a l i f o r n i a , J u n e , 1975 . Duncan, J . M . and Chang, C . Y . , " N o n l i n e a r A n a l y s i s o f S t r e s s and S t r a i n i n S o i l s , " J o u r n a l o f the S o i l Mechanics and Foundat ions D i v i s i o n , A S C E , V o l . 9 6 , No. SM5, September, 1970, p p . 1629-1653. Duncan, J . M . and Chang, C . Y . , " N o n l i n e a r A n a l y s i s o f S t r e s s and S t r a i n i n S o i l s - C l o s u r e , " J o u r n a l o f the S o i l Mechanics and Foundat ions D i v i s i o n , A S C E , V o l . 9 8 , No. SM5, May, 1972, p p . 4 9 5 - 4 9 8 . 70 71 1 1 . F i n n , W . D . L . and M i l l e r , R . I . S . , " N o n l i n e a r A n a l y s i s o f E a r t h S t r u c t u r e s , " N u m e r i c a l Methods i n Geo-m e c h a n i c s , ed . C . S . D e s a i , A S C E , New Y o r k , New Y o r k , 1 9 7 6 , p p . 1 9 5 - 2 0 4 . 1 2 . G i r i j a v a l l a b h a n , C . V . and Reese, L . C . , " F i n i t e Element Method f o r Problems i n S o i l M e c h a n i c s , " J o u r n a l o f the S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , A S C E , V o l . Sk, No. S M 2 , M a r c h , 1 9 6 8 , p p . 4 7 3 - 4 9 6 . 1 3 . L a d e , P . V . and Duncan, J . M . , " S t r e s s - P a t h Dependant B e h a v i o u r o f C o h e s i o n l e s s S o i l , " J o u r n a l o f the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , A S C E , V o l . 1 0 2 , No. G T 1 , J a n u a r y , 1 9 7 6 , p p . 5 1 - 6 8 . 14. Nyak, G . C . and Z i e n k i e w i c z , O . C . , " E l a s t o - P l a s t i c S t r e s s A n a l y s i s . A G e n e r a l i z a t i o n f o r V a r i o u s C o n s t i -t u t i v e R e l a t i o n s I n c l u d i n g S t r a i n S o f t e n i n g , " I n t e r -n a t i o n a l J o u r n a l f o r N u m e r i c a l Methods i n E n g i n e e r i n g , V o l . 5 , No. 1 , S e p t e m b e r - O c t o b e r , 1 9 7 2 , p p . 1 1 3 - 1 3 5 -1 5 . Roy , J . , ' N o n l i n e a r A n a l y s i s o f the Undra ined Shear S t r e n g t h o f C l a y , ' a r e s e a r c h p r o j e c t submi t t ed i n p a r t i a l f u l f i l l m e n t o f the requirement f o r the degree o f Master o f E n g i n e e r i n g , Dept . o f 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 o f B r i t i s h C o l u m b i a , 1 9 7 4 , u n p u b l i s h e d . 1 6 . V e s i c , A . S . , " A n a l y s i s o f U l t i m a t e Loads o f Shal low F o u n d a t i o n s , " J o u r n a l o f the S o i l Mechanics and Foundat ions D i v i s i o n , A S C E , V o l . 9 9 , No. SMI, J a n u a r y , 1 9 7 3 , p p . 4 5 - 7 3 . 1 7 . Wong, K . S . and Duncan, J . M . , H y p e r b o l i c S t r e s s - S t r a i n Parameters f o r N o n l i n e a r F i n i t e Element A n a l y s e s o f S t r e s s e s and Movements i n S o i l M a s s e s , ' Report No. T E - 7 4 - 3 t o N a t i o n a l S c i e n c e F o u n d a t i o n , Dept . o f C i v i l E n g i n e e r i n g , I n s t i t u t e o f T r a n s p o r t a t i o n and T r a f f i c E n g i n e e r i n g , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , C a l i f o r n i a , 1 9 7 4 . 1 8 . Y u d h b i r and V a r a d a r a j a n , A . , " S t r e s s - P a t h Dependent Deformat ion M o d u l i o f C l a y , " J o u r n a l o f the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , A S C E , V o l . 1 0 1 , No. G T 3 , M a r c h , 1 9 7 5 , p p . 3 1 5 - 3 2 7 . 1 9 . Z i e n k i e w i c z , O . C . , The F i n i t e Element Method i n E n g i n e e r i n g S c i e n c e , M c G r a w - H i l l Book C o . , L t d . , London , E n g l a n d , 1 9 7 1 , pp . 3 6 9 - 3 7 7 . APPENDIX 1 DERIVATION OF THE STIFFNESS MATRIX FOR THE LINEAR STRAIN TRIANGULAR ELEMENT The element s t i f f n e s s m a t r i x which r e l a t e s the n o d a l f o r c e s , {Q}, and the n o d a l d i s p l a c e m e n t s , {q}, f o r the l i n e a r s t r a i n t r i a n g u l a r element shown i n F i g u r e 15 may be d e r i v e d as f o l l o w s : The d i s p l a c e m e n t s , {u}, o f a p o i n t , p , w i t h i n the l i n e a r s t r a i n t r i a n g u l a r element are r e l a t e d to the n o d a l d i s p l a c e m e n t s , ( q ) , by the e q u a t i o n {u} =4 u u y [A] {q} (1) where [A] as g i v e n by D e s a i and A b e l ( 4 ) , In terms o f n a t u r a l c o o r d i n a t e s , i s 1) 0 [A.]. = 0 ? 2 ( 2 C 2 - 1) ? 3 ( 2 C 3 - 1) 4 ? 1 ? 2 4 V s 4 V l ? 1 ( 2 ? 1 - 1) 0 ? 2 ( 2 ? 2 - 1) ? 3 ( 2 ? 3 - 1) 4 C 1 ? 2 4 C 2 ? 3 4 ? 3 ? 1 '(2) 72 FIGURE 15 LINEAR STRAIN TRIANGULAR ELEMENT 74 i n which A g 2 A, A and '1  '  A ? 3 A where A i s the a r e a o f the t r i a n g u l a r element and A^ i s the a r e a o f the p o i n t p and the nodes o p p o s i t e node I as i l l u s t r a t e d i n F i g u r e 1 6 . The r e l a t i o n s h i p between the s t r a i n s , {e}, at any p o i n t w i t h i n the element and the n o d a l d i s p l a c e m e n t s , 3 {q}, f o r p l a n e s t r a i n i s {e} i n which L xy = [L] [A] {q} [B] {q} (3) 0 5 / 6 y 0 6 / 6 ! y (4) For n a t u r a l c o o r d i n a t e s , D e s a i and A b e l (4) g i v e the f o l l o w i n g d i f f e r e n t i a t i o n formulae : 6TJ b 3.-U m x 6 X 2 A 6 r, (5) m i n i n d i c i a l n o t a t i o n where b n = Y0 - y-= y 3 - y x = y x - y 2 and 75 x - A x i s FIGURE 1 6 AREAS FOR NATURAL COORDINATES 6U x ,6y a 6U m x 2A 6? m i n i n d i c i a l n o t a t i o n where : 2 y 3 - x 3 y 2 a 2 = x 3 y l " x l y 3 a 3 = X l y 3 - x 2 Y l and s i m i l a r formulae f o r the o ther d e r i v a t i v e s . The m a t r i x [B] i n e q u a t i o n ( 3 ) i s then [B] = $ « 1 - %)b 1 0 « 1 — %)a 1 0 - %)a 1 - %)b1 - 0 - %)a 2 0 u 2 - h)a2 - %)b2 ^3 - %)b 3 0 - %)a 3 0 ( C 3 - %)a 3 ( c 3 - %)b 3 ? 2 b l + ? l b 2 0 ? 2 a l + ? l a 2 0 ? 2 a l + c l a 2 c 2 b 1 + ? l b 2 c 3 b 2 + ? 2 b 3 0 c 3 a 2 + c a 3 0 t, 3<3-2 + + ? 2 b 3 ^ b 3 + ? 3 b l 0 c l a 3 + ? 3 a l 0 c l a 3 + z 3 a l ? l b 3 + ? 3 b l 77 The s t r e s s e s , {a}, at any p o i n t i n a l i n e a r e l a s t i c m a t e r i a l are r e l a t e d to the s t r a i n s , {e}, at that p o i n t by the e q u a t i o n xy J [D] { £ } (8) f o r p lane s t r a i n c o n d i t i o n s . F o r an i s o t r o p i c e l a s t i c m a t e r i a l [D] = k + y 3 G k 2 / 3 G k - 2 / 3 G k + V 3 G 0 (9) i n which k i s the b u l k modulus and G i s the shear modulus . F o r e q u i l i b r i u m , by the p r i n c i p a l o f v i r t u a l work, the work done by the r e a l e x t e r n a l l o a d s , { Q } , a c t i n g through the v i r t u a l n o d a l d i s p l a c e m e n t s , ( q ) , must be e q u a l t o the work- done by the r e a l i n t e r n a l s t r e s s e s , {a}, a c t i n g through the v i r t u a l s t r a i n s , {e}, c o r r e s p o n d i n g to the v i r t u a l n o d a l d i s p l a c e m e n t s . T h e r e f o r e , {e} T {a} dV = {q} T {Q} (10) V w h i c h , upon the i n t r o d u c t i o n o f equat ions (3) and (8) becomes { i } T [ A ] T [ L ] T [ D ] [ L ] [ A ] { q } d V = {q} T{Q) ( I D V and s i n c e {q} 1 and {q} are not f u n c t i o n s o f p o s i t i o n , = '{Q} (12) o r [K] (q) = {Q> where [K] i s c a l l e d the element s t i f f n e s s m a t r i x . [ A ] T [ L ] T [ D ] [ L ] [ A ] d V V APPENDIX 2 STRESS-STRAIN SUBROUTINES The w r i t e r deve loped s t r e s s - s t r a i n s u b r o u t i n e s based on e a r l i e r work done by Roy ( 1 5 ) to a l l o w the use o f the h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n s o f Duncan and Chang ( 9 ) w i t h the f i n i t e element program NONLIN. These sub-r o u t i n e s c o n t r o l the i t e r a t i v e p r o c e s s and use equat ions ( 3 - 1 8 ) and ( 3 - 1 9 ) to r e l a t e the tangent v a l u e s o f Young's modulus and P o i s s o n ' s r a t i o to the s t a t e o f s t r e s s at the element c e n t r o i d s . The i t e r a t i v e p r o c e s s used i s the one i t e r a t i o n per l o a d Increment method d e s c r i b e d i n s e c t i o n ( 2 . 2 . 1 ) . The s u b r o u t i n e s were w r i t t e n i n FORTRAN and are p r e s e n t e d below. A l l FORTRAN v a r i a b l e s and a r r a y s are d e f i n e d where they a p p e a r . The f u n c t i o n s o f the s u b r o u t i n e s are e x p l a i n e d at the s t a r t o f each s u b r o u t i n e . 79 1 SUBROUTINE SETUPCX) 2 C  3 C THIS.SUBROUTINE* TOGETHER WITH SUBROUTINE EU 9 CALCULATES 4 C THE I N I T I A L BULK AND SHEAR MODULI FOR THE ELEMENTS. 5 C _. 6 I MPLICIT RE AL* 8 { A-H, D - 2 ) " " " ~ ~ 7 DIMENSION X( 50) 8 C OM MO_N_ / CN S J J T / P A TRN (1 0,10? ,NCONST ? IPLANE 9 CO MMO N / E U B L K / TsT"E"P7N3TEP ,KSTEP,IFGA : 10 CATA ISTRT/O/ 11 JC . 12 N S T F P = 0 * ~ ~ 13 KSTEP=1 14 IFGA=0 15 I" F(TSTRT.EQ.O) ISTEP=0 ' 16 ISTRT=1 " 1 8 " " "C THE ARRAY X CO N T AI N S ~ T H E~TQ V C 0 WI NG ELEMENT INFORM AT 10 NT 19 C 20 C 1 SOIL UNIT WEIGHT  21 C 2 COHESION INTERCEPT, C 22 C 3 ANGLE OF INTERNAL FRICTION IN DEGREES 23 C 4 FAILURE RATIO* RF 24 C 5 MODULUS NUMBER, K " " " ~ " 25 C 6 MODULUS EXPONENT, N _26 C 7 MODULU_S_AFTER_F A I LURE 2 7 C 8 PGISSON•S RAT 10 PARAMETER? G 28 C 9 THICKNESS(PLANE PROBLEMS ONLY) 29 C 10-15 INITIAL ELEMENT STRESSES:SIGX»SIGY,SIGZtTAUXY,TAUYZ,TAUZX 30 C 16 INITIAL ELEMENT PORE PRESSURE 31 C 17-22 ZERO STRAINS EPS X,EPSY,EPSZ,GAMXY,GAMYZ,GAMZX _32 C 34^  POI SSON* S R A T 10 _PARAMETER,F 33 C 35 POISSON'S RATIO" PARAMETER,^ 34 C 36 TENSILE STRENGTH 35 C 37 ATMOSPHERIC PRESSURE, PATM o 36 C 38 SCRATCH SPACE (NOT USED) 37 C 39 SCRATCH SPACE (NOT USED) _38 C A0 COEFFICIENT OF LATERAL EARTH PRESSURE, KO  39 C 40 C ON RETURN, X CONTAINS THE ADDITIONAL INFORMATION BELOW. A l C _ 4 2 C 23 INITIAL YIELD FUNCTION 43 C 24 I N I T I A L BULK MODULUS 44 C 25 INITIAL SHEAR MODULUS  45 C 46 X(23)=0.D0 47 C ._ 4 8 C " ' CALCULATE THE I N I T I A L BULK AND SHEAR MODULI-49 C _50 CALL E U ( E T , U T , X ( 1 0 ) , X ( 2 ) , X t 3 4 ) )  51 C 52 C COMPUTE THE I N I T I A L BULK AND SHEAR MODULI 5 3 C ' __ 54 X(24}=UT 55 X(25)=ET/(3.D0-<ET/(3.D0*UT))> 56 C . . 57 C X(24) AND XC25) CONTAIN THE I N I T I A L BULK AND SHEAR MODULI* 58 C 59 FFTURN ___ 6 0 " F N D ~ ' " " "" 61 C __6 2 S UBROUT INE ST I F LT ( X , SS S y DE PS 1, EPS V t EPS A )  63 C 64 C THIS SUBROUTINE, TOGETHER WITH SUBROUTINES EU AND EUD, 65 C CCNTROLS THE STRESS-STRAIN BEHAVIOUR OF THE F I N I T E ELEMENTS. "66 C " " " " " 67 C ARRAY X CONTAINS: _6 8 C i SOIL UNIT WEIGHT 69 C 2 COHESION I NTS RC E P T t C ' c 70 C 3 ANGLF OF INTERNAL FRICTION IN DEGREES • h 71 C 4 FAILURE_RATIO, RF 7 2 C 5 M O D U L U S NUMBER, K 7 3 C 6 MODULUS E X P O N E N T , N. 7 4 C 7 MODULUS A F T E R F A I L U R E 75 C 8 P O I S S O N ' S R A T I O P A R A M E T E R , G 7 6 C 9 T H I C K N E S S * P L A N E PROBLEMS ONLY ) 7 7 C 1 0 - 1 5 S T R E S S E S AT THE END OF L A S T S T E P ( U P D A T E D BY S T I F L T ) 7 0 C 16 POPE P R E S S U R E AT T H E END OF L A S T S T E P ( U P D A T E D BY S T I F L T ) 7 9 C 1 7 - 2 2 ZERO S T R A I N S E P S X , E P S Y , E P SZ » G A M X Y , G A M Y Z , G A M Z X _ 8 0 C 23 Y I E L D F U N C T I O N 81 C 2 4 - 3 3 MODULI USED IN P R E V I O U S S T E P , R E T U R N E D MODUL I FOR N E X T S T E P * 82 C 34 P O I S S O N ' S R A T I O P A R A M E T E R S 83 C 35 P O I S S O N ' S R A T I O P A R A M E T E R , D _ 84 C " 36 SCRATCH SPACE ( N O T U S E D ) " 85 C 37 ATMOSPHER IC P R E S S U R E , PATM 86 C 3 8 SCRATCH_SPACE_< NOT_USE_DJ .__ 87 C 39 SCRATCH S P A C E (NOT USED) 88 C AO C O E F F I C I E N T , KO 89 C X ( 4 1 ) - X ( 5 0 ) MAY BE I N I T I A L I Z E D FOR SUBSEQUENT USE IN S T I F L T . ' 9 0 C S S S IS TO RETURN THE MODULI WHICH ARE TO BE U S E D WHEN 91 C THE S T E P IS R E I T E R A T E D . 9 2 C P E P S IS THE INCREMENTAL STRAI N[S FOR 'j_HE_ L A S T S T E P . 9 3 C EPSV IS T H E S T R A I N V ' E L O C I T T E S A T ~ T H E P R E S E N T T I M E . 9 4 C EPSA IS THE S T R A I N A C C E L E R A T I O N S AT T H E P R E S E N T T I M E . 9 5 C 9 7 P E A L * 8 X ( 6 3 ) , SSS ( 10) , D E P S ( 6 ) , E P S V ( 6 ) , E P S A ( 6 ) ,D ( 1 6 ) , S D { 4 ) , D E P S 1 ( 6 ) _ 9 8 I NT EG F R * 2 I F A L ( 5 0 0 ), IAD 0 9 9 COMMON / F A L B L K / D T , DELT A T » D T N E X T , I R E S T P , I E L T , I F G O , I C H N G , I F A L , 1 0 0 * I P L A N E , I D E B U G , I S T A T , I F D T , I F D I A G , I T H A X , L S T E P , I F E R R F 101 __ COMMON / E U B L K / I ST EP ,_NST E P,_K5 T EP» I F G A _ _ _ 1 0 3 N S T E P = 1 _1 04 I F G A - I F G O  1 0 5 C 1 0 6 C / F A L B L K / C O N T A I N S GOVERN ING I N F O R M A T I O N FOR T H E PROGRAM AS A WHOLE. 1 0 7 C 1 0 8 C I P E S T P I S THE ITERATION NUMBER (1 ON THE FIRST P A S S ) . 109 C I F L T IS THE NUMBER OF THE ELEMENT BEING PROCESSED BY S T I F L T . H O C IFGO IS A FLAG SET WHEN THE_ ACCURACY CRITERION IS VIOLATED "111 C FOR -A~GI VENTSTEP- I"F NOT 0 IT CAUSIE 'SnrHE -ST'£T^p~BE 112 C REPEATED (UNLESS I RESTP>ITMAX). 113 C ICHNG IS A FLAG TO INDICATE THAT THE ELEMENT MODULI FOR 11* C THE NEXT STEP ARE THE SAME AS FOR THEPREVIOUS STEP. 115 C J 1 6 C OF THESE, ONLY IFGO AND ICHNG ARE CHANGED 8Y STIFLT • 117 C 118 C 119 C UPDATE THE ELEMENT STRESSES. _ 120 C " 121 I F ( I R E S T P . G T . l ) GO TO 100 122 IFGC=1 123 D O 101 1=1,4 '• 124 101 0 = PS( I )=DEPS1(I)/2.D0 125 99 CALL GF T D(X(24) , 0) . 1 2 6 " C A L L BL0WUP(D,D,4) " "~ 127 CALL DGMATV(D,DEPS,SD,4,4,4) 128 00 1 1 = 1,4 129 1 X'Q"+"9 "j="X"( 1+9) +SDTTJ ; 130 C 131 C CALCULATE NEW MODULI. _ 132 " C ~ ~ " I F E L E M E N T H A S Y I E L D E D , T H E BULK MODULUS I S 133 C NOT CHANGE0 FROM ITS LAST VALUE. 134 C  135 C ALL Eli ( E T , U T , X f 10) ,X( 2 ) , X ( 3 4 ) ) 136 C 137 IF(UT .EO.O.ODO) GOTO 500 _ 139 X(25)=ET/('3.D0-{ET/t3.D0*UT) J) J^4() GO_TO_ 6_0_0_ 141 500 CONTINUE 142 X(25)=ET 143 600 CONTINUE 144 C 145 C X(2A) AND X(25) ARE THE BULK AND SHEAR MODULI RESPECTIVELY. 146 c 147 SSS{1)=X(24) 148 SSS(2)=X(25) 149 ICHNG=1 -150 ; c" 151 RETURN 152 c 153 100 IFGO=0 154 DP 102 1=1,4 155 102 DEPS{IJ=DEPS1(I) 156 GO TO 99 157 c 158 END 159 c 160 SUBROUTINE EUD(ET ,UT,STRESS,PR0P1,PR0P2,CR,SI,S3, I STEP,NEB) 161 c 162 c THIS SUBROUTINE CHECK FOR FAILURE AND COMPUTES THE 163 c VALUES OF YOUNG'S MODULUS AND POISSON'S RATIO. 164 c (REF: DUNCAN AND CHANG, ASCE,JOUR. OF S.M.GF.E. DIV.,V0L.96, 165 c N0.SM5, SEPTEMBER,1970) 166 c 167 IMPLICIT REAL*8(A-H,0-Z) 168 DIMENSION STRESS(14),PROPi(7),PR0P2(7) 169 c 170 IF(NEB.GT.l)WRITE(6,60) CR,S1,S3 171 60 FORMA T( • CR = • ,F10.5,'S1='»F10.5» ,S3 = "»F10.5) 172 FA=PROP1(2)/57.28DO 173 CO=DCnS(FA) 174 S I = D SIN(FA) 175 FFF=2.D0*PR0P1(1)*C0+( S1+S3)*SI 176 IF(NEB.GT.l) WRITE(6,90) FFF 177 90 FORMAT( «FFF=« ,F10.5) 178 RCS=2.D0*(PR0P1(1 )*C0 *S3*SI)/(1.DO-SI) 179 HCS=RCS/PR0P1(3) 180 ~ S 3 0 P A = S 3 / P P 0 P 2 ( 4 ) " : 181 FS=2 . D 0*CR/FFF 182 STPESS( 14) =FS 183 C 184 C TF ST FOR FAILURE USING MOHR-COULOMB CRITERION. 185 C 186 ~ ' I F ( F S . r t 7 i . b 6 ) G O T O 3 1 " ' " " 187 W R I T E ( 6 , 7 0 ) ISTEP 188 70 FORMATC 'ELEMENT',I5t'HAS FAILED BY MQHR-CQULQMB8 )  189 C 190 C TEST FOR TENSILE FAILURE. 191 _ C ._ . 192 31 IF{S3.GT.0.DO)GOTO 33 193 WRITE(6,41) ISTEP 194 41 FOPMAT('TENSILE FAILURE IN ELEMENT 9 ,15)  19 5 G O T O 3 0 196 C 197 C CALCULATE TANGENT MODULUS FOR UNFAI LED ELEMENT . 198 C ~ 199 33 IF {FS.GE.l.DOGOTO 30 2 0 0 EI=PPCP1(4)*PR0P2(4)«( S30PA**PR0P1 ( 5 ) )  201 ET=E~l*( ( l.D0-(2.D0*CR/HCSI ) **2) 202 I F ( E T . L T . P R 0 P 1 ( 6 ) I E T = P R 0 P l i 6 ) 203 GO TO 4 0 _ _ _ _ _ _ 2 04 C ~ ~ ™ 205 C CALCULATE MODULUS AND POISSON'S RATIO FOR FAILED ELEMENT. 206 C _ _ '2 0 7 30 F T = P R 0 P l T 6 l '. : ' 208 UT=O.OD0 2 09 GO T O 5 0 210 'C " ~ " " " " " " " 211 C COMPUTE POISSON'S RATIO FOR UNFAILEC ELEMENT. 212 C ; • '213 40 STRAIN=2.D0*CR/ ( E I*t1.C0-(2.DO*CR/HCS) ) ) ~ 214 PRAT=PR0P117)-PR0P2(1)*DL0G10(S30PA) oo 215 UT=PRAT/(l.D0-PROP2(2)*STRAIN)**2 . 0 1 216 50 IF(UT.GT.0.495D0)UT=0.495DO 217 C _ 2 1 8 C ET= YOUNG'S MODULUS AND UT^ = POISSON'S RATIO.  219 C 220 RETURN 221 ENO 2 2 2 c : " " " ~ 223 SUBROUTINE EUCET,UT,STRESS,PROP1,PR0P2) _2 24 C 225 C 226 C THIS SUBROUTINE, TOGETHER WITH EUD, COMPUTES THE 227 C APPROPRIATE MODULI FOR THE F I N I T E ELEMENTS* 2 2 8 C 229 C PRCP1(1)=COHESION,C 230 C PP0P1(2)=THE ANGLE OF INTERNAL_FRICTION 231 C PPCP1(3)=THE FAILURE RATIO, RF 232 C P F 0 P 1(4)=THE MODULUS NUMBER, K 233 C PP0P1(5)=THF MODULUS EXPONENT, N 234 C PPOPl{6)-MODULUS AFTER FAILURE " 235 C PP0P1 (7)=PCISSON*S RATIO PARAMETER, G 236; C PRCP2 ( 1 )=POI SSON 8 S RATIO PARAMETER, F  237 C PP0P2(2)=POISSON'S RAT IC PARAMETER t D 238 C PR0P2I3)=SCRATCH SPACE (NOT USED) 239 C_ PR0P2(4>=ATM0SPHERIC PRESSURE 240 " C " P P 0 P 2 I 5 J = SCRATCH' SPACE (NOT USED) 241 C PR0P2(6)=SCRATCH SPACE (NOT USED) 2 4 2 C PR OP 2 (7)=C0EFFICIENT OF LATERAL EARTH PRESSURE, K3 243 C 244 C S T P E S S d - 6 ) CONTAINS THE ELEMENT STRESSES. 245 C STRE S S{7) CONTAINS THE PORE PRESSURE. ~ 2 4 6 ~ ~ C " S T R E S S ( 8-13) CONTAINTHE ELEMENT STRAINS." 247 C STRESSU4) CONTAINS THE YIELD FUNCTION. 248 C 2 49 "IMPLICIT R' ETAL * 8 V A ^ H T O - T T 250 DIMENSION ST R E S S ( 1 4 ) , P R C P 1 I 7 ) , P R 0 P 2 ( 7 ) 251 COMMON/EUBLK/ISTEP,NSTEPj^STEP,IFGA 252 " ' DATA N E N T E R / O / " 253 IF(NENTER.NE.O) GO TO 6 2 54 NFMTER=1 ; 255 C 256 READ(5,4) NN VNEB 257 C ' ' 2 5 8 " C N N I S THE NUMBER' G'f^rNTTl '_LE"ME'RTS« " 259 C NEB IS THE DEBUG OUTPUT LEVEL. _2 60 C 261 4 FORMAT(215) 262 6 CONTINUE 263 STPESS ( 14) =0.00 ; _._ , 264 C 265 C CALCULATE THE PRINCIPAL STRESSES, S I AND S 3 . _2 66 c : 267 STRE S2= —(STRE S S (2) + STRESSC 7 ) J 268 STRESl=-(STRESS(1)+STRESS(7 ) ) 269 CC=(STRES1+STRES2)/2.D0 270 BB={STR ESl-STRES2)/2.D0 271 CR = DSCRT( STRESS (4 )*STRESS (4)«-BB*BB) _2 7 2 Sl^CC+CR ; ; 2 7 3 S3=CC-CR 2 74 C 275 C UPDATE THE ELEMENT NUMBER, ISTEP. _ 2 76 C " " " ~ " ~ 277 ISTEP = ISTEP«-1 _?7 8 C^  ;  279 C GET NEW TANGENT MODULUS AND POISSON'S RATIO. 280 C 281 29 CALL EUD(ET,UT,STRESS,PROP 1,PR0P2,CR,SI,S3,1STEP,MEB) 282 C 283 I F ( N E B . G T . l ) WRITE(6,203) I STEP _28<t 203 FORMAT ( ' ELEMENT 1 » 15) . 285 C .:. 286 C COMPUTE THE BULK MODULUS AND STORE IT I N THE co 287 C LOCATION OF POISSONS RATIO, UNLESS.THE ELEMENT ^ 288 C HAS FAILED. 289 C 2 9 0 IF ( UT . E Q.O.QDO) GOTO 9100 "2 9 1 U T = E T / (3.0 0*11-DO-2.DO*UT)) 292 9100 CONTINUE 293 IF(ISTEP.NE.NN) GO TO 40 _ _ 294 IF ( N E B .GT.l)WRITEC 6 , 2 4 } N N ~ ~ : " ~ 295 24 FORMAT(•THE NUMBER OF ELEMENTS I S = e , I 4 ) _2 9 6 38 ISTEP = 0 297 40 CONTINUE 2 98 C 299 C FT IS YOUNG• S MODULUS. _ \ _ 3 0 0 C UT IS THE BULK MODULUS."" " ' ~ 3 0 1 C _30 2 C 3 0 3 CR=2.D0*CR 3 0 4 I F ( N E B . G T . l ) WRITE(6,100) CR 305 100 FORMAT (•S1-S3=•,G15„5) 30 6 " I F ( N EB.GT.i >WRITE (6,202 ) E T , U T ~ " ""' 307 202 FORMAT!'MODULUS »•,G15.5,1 OX,•BULK MODULUS^ 9,G15.5 J 3 08 C ; .  309 RETURN 310 END END O F F I L E oo oo 

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