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Pressuremeter tests in sand : effects of dilation Eldridge, Terry Lewis 1983

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c. /. P R E S S U R E M E T E R T E S T S I N S A N D : E F F E C T S O F D I L A T I O N b y T E R R Y L E W I S E L D R I D G E B . A . S c , T h e 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 , 1980 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D e p a r t m e n t o f C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s a s c o 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 T H E 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 F e b r u a r y , 1983 © T e r r y L e w i s E l d r i d g e , 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g 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 g r a n t e d by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s _understood_that. c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date A p r i l 19. 1983 D'E-6 (2/79) ABSTRACT An a n a l y s i s o f the response o f sand t o the p r e s s u r e m e t e r i s p r e s e n t e d . An i n c r e m e n t a l l i n e a r e l a s t i c f i n i t e e lement program i n c o r p o r a t i n g shear-volume c o u p l i n g was m o d i f i e d such t h a t i t c o u l d e f f i c i e n t l y h a n d l e a x i s y m m e t r i c p l a n e s t r a i n s t r e s s - d e f o r m a t i o n problems. The shear-volume model was m o d i f i e d so t h a t i t gave a r e a s o n a b l e a p p r o x i m a t i o n t o t r i a x i a l t e s t d a t a . A m o d i f i e d form o f Rowe's s t r e s s d i l a t a n c y t h e o r y was used i n the shear-volume c o u p l i n g model. The f i n i t e element program was used t o a n a l y z e the p r e s s u r e m e t e r t e s t c o n d i t i o n s . The d i s t r i b u t i o n o f d i s -p l a c e m e n t s , s t r a i n s and s t r e s s e s around the p r e s s u r e m e t e r were d e t e r m i n e d f o r s o i l s w i t h v a r i o u s v o l u m e t r i c r e s p o n s e s t o s h e a r . Upon f u l l m o b i l i z a t i o n o f the s o i l s s t r e n g t h , volume change c h a r a c t e r i s t i c s have no e f f e c t on the s t r e s s d i s t r i b u t i o n i n t h e f a i l e d zone. The d i s t r i b u t i o n o f s t r a i n does depend upon the volume change c h a r a c t e r i s t i c s , b u t beyond about t h r e e c y l i n d e r r a d i i , such e f f e c t s are s m a l l . The assumed shape o f the s o i l s t r e s s - s t r a i n r e l a t i o n s i g n i f i c a n t l y a f f e c t s the computed response t o p r e s s u r e -meter t y p e l o a d i n g . An e l a s t i c - p l a s t i c m a t e r i a l i s much s t i f f e r t h a n a n o n - l i n e a r e l a s t i c m a t e r i a l w i t h the same i n i t i a l modulus and s t r e n g t h . D i l a t i o n s t i f f e n s a m a t e r i a l ' s r e s p o n s e , b u t even a h i g h l y d i l a t a n t m a t e r i a l i s l e s s s t i f f t h an an e l a s t i c - p l a s t i c m a t e r i a l . The method of determining the f r i c t i o n angle and d i l a t i o n angle of f r i c t i o n a l materials from pressuremeter data proposed by Hughes, Wroth and Windle was checked i n the analysis. The method gives results i n good agreement with the f i n i t e element r e s u l t s . A method of determining the i n i t i a l shear modulus and the i n s i t u horizontal s o i l stress i s presented. This method gives good agreement with the f i n i t e element r e s u l t s . When applied to actual pressuremeter t e s t data, the shear moduli determined by this method were i n better agreement with the rebound moduli than with the moduli determined from the i n i t i a l portion of the pressure-expansion curve. i v Table o f Contents Page ABSTRACT i i LIST OF FIGURES v i LIST OF TABLES i x ACKNOWLEDGEMENT X CHAPTER 1 INTRODUCTION 1 CHAPTER 2 PRESSUREMETER THEORIES 2.1 E l a s t i c A n a l y s i s ^ 2.2 E l a s t i c - P l a s t i c A n a l y s i s 8 2.2.1 L i m i t Pressure 1^  2.3 L i m i t a t i o n s o f E l a s t i c - P l a s t i c A n a l y s i s . . 18 2.4 L i n e a r D i l a t a n t A n a l y s i s 19 CHAPTER 3 SOIL MODEL 3.1 D e f i n i t i o n o f S t r e s s and S t r a i n 38 3.2 C o n s t i t u t i v e R e l a t i o n ^1 3.3 N o n l i n e a r i t y 3^ 3.4 P l a s t i c S t r a i n s 51 3.5 Shear Volume C o u p l i n g 51 3.5.1 Rowe's S t r e s s D i l a t a n c y Theory 53 3.5.2 V a r i a t i o n o f D i l a t i o n Angle w i t h S t r e s s L e v e l . . . . . 59 CHAPTER 4 FINITE ELEMENT PROGRAM 4.1 Incremental Program 61 4.2 Shear Induce Volume Change 69 4.3 F i n i t e Element F o r m u l a t i o n o f the Shear Induced Volume Change 74 4.4 S t r e s s R e d i s t r i b u t i o n 89 CHAPTER 5 RESULTS OF THE FINITE ELEMENT ANALYSIS 5.1 F i n i t e Element Mesh 91 5.2 P r e s s u r e Expansion Curves 91 5.3 V o l u m e t r i c Response 96 5.4 S t r e s s Paths 96 5.5 Displacement F i e l d , S t r a i n s and S t r e s s e s . . 99 5.6 D i s t r i b u t i o n o f Shear Modulus i n F i n i t e Element Domain 102 5.7 Determination o f F r i c t i o n Angle and D i l a t i o n Angle 105 5.8 S t r e s s - S t r a i n Curves 105 5.9 H y p e r b o l i c S t r e s s - S t r a i n Parameters . . . . 113 5.10 Determining I n i t i a l S t r e s s and Shear Modulus 113 J V CHAPTER 6 SUMMARY AND CONCLUSION . 1 2 6 REFERENCES 1 2 9 APPENDIX A FORMULATION OF FOUR NODE POLAR COORDINATE F I N I T E ELEMENT 1 3 3 APPENDIX B STRESS STRAIN MATRIX FOR TRIAXIAL F I N I T E ELEMENT 145 APPENDIX C PRESSUREMETER TEST DATA, MACDONALD'S FARM, SEA ISLAND 1 5 3 v i F i g u r e LIST OF FIGURES Page 1 . C o o r d i n a t e System f o r Pressuremeter A n a l y s i s 5 2 . S t r e s s e s I n P o l a r S o o r d i n a t e s 6 3. P r e s s u r e Expansion Curve f o r E l a s t i c M a t e r i a l 9 4 . Comparison o f S t r e s s D i s t r i b u t i o n i n E l a s t i c and E l a s t i c - P l a s t i c Incompressible M a t e r i a l 12 5. P r e s s u r e Expansion Curve f o r E l a s t i c - P l a s t i c M a t e r i a l 15 6. Comparison o f Displacements i n D i l a t a n t and E l a s t i c M a t e r i a l s . 22 7. Comparison o f S t r a i n s i n D i l a t a n t and E l a s t i c M a t e r i a l s 23 8. Comparison o f S t r e s s e s i n D i l a t a n t and E l a s t i c M a t e r i a l s 24 9 . Volume Change Response used i n Pressuremeter A n a l y s i s 26 1 0 . P r e s s u r e Expansion Curve f o r F i n e Sand and L o g - l o g P r e s s u r e Expansion Curve f o r F i n e Sand 29 1 1 . G r a p h i c a l Method o f S o l u t i o n f o r Values o f 4> and v 30 1 2 . S t r e s s - S t r a i n Behaviour o f Ottawa i n Drained Simple Shear 32 1 3 . Volume Change Data f o r Loose Sand . . . . 33 1 4 . L o g - l o g P r e s s u r e Expansion Curve f o r F i n e Sand wi t h Nonzero * 1 5 . V a r i a t i o n o f C a l c u l a t e d Values o f $ and v w i t h Changes i n * 35 1 6 . V a r i a t i o n o f C a l c u l a t e d V a l u e s o f $ and v w i t h Changes i n * s ' 36 1 7 . -' S t r e s s e s i n C a r t e s i a n C o o r d i n a t e s . . . . 40 1 8 . S t r e s s - S t r a i n Curves f o r Drained T r i a x i a l T e s t s on Sand 44 1 9 . M o d i f i e d H y p e r b o l i c S t r e s s - S t r a i n R e l a t i o n 45 2 0 . Angle o f I n t e r n a l F r i c t i o n f o r Chattahoochee R i v e r Sand 47 2 1 . I s o t r o p i c C o n s o l i d a t i o n Data 49 2 2 . L o g - l o g V o l u m e t r i c S t r a i n versus Mean Normal S t r e s s 50 2 3 . D e f i n i t i o n o f Rebound Modulus 52 2 4 . D e f i n i t i o n o f D i l a t i o n Angle 56 2 5 . Volume Change i n Constant Mean Normal S t r e s s T r i a x i a l T e s t s 58 Dependence o f D i l a t i o n Angle on R e l a t i v e D e n s i t y and C o n f i n i n g P r e s s u r e 60 27 Dependence o f D i l a t i o n Angle on Mean Normal S t r e s s Based on Rowe's E q u a t i o n . . 60 26 v i i 2 8 . Flowchart f o r F i n i t e Element Program . . . 6 2 2 9 . S p e c i f i c a t i o n of Shear Induced Volume Change 7 2 3 0 . . Comparison of Shear Induced Volume Change Models 7 3 3 1 . Flowchart f o r Subroutine DILAT 7 5 3 2 . Four Node Polar Coordinate Element . . . . 8 1 3 3 . Constant o$ T r i a x i a l Test on Medium Dense Sand 8 4 3 4 . F i n i t e Element Analysis of 03 Constant T r i a x i a l Test 8 5 3 5 . Comparison of Displacements i n Linear Dilatant Material and F i n i t e Element Solution 8 6 3 6 . Comparison of Strains i n Linear Dilatant Material and F i n i t e Element Solution . . 8 7 3 7 . Comparison of Stresses i n Linear Dilatant Material and F i n i t e Element Solution . . 8 8 3 8 . Stress Path f o r F a i l e d Element 9 0 3 9 . F i n i t e Element Mesh f o r Analysis 9 2 4 0 . Pressure Expansion Curves from F i n i t e Element Analysis 9 4 4 1 . E l a s t i c - P l a s t i c and Nonlinear S o i l Pressure Expansion Curves 9 5 4 2 . Volumetric Response from F i n i t e Element Analysis 9 7 4 3 . Stress Paths Followed by S o i l Around an Expanding Cylinder 9 8 4 4 . Comparison of Displacements Around an Expanding Cylinder 1 0 0 4 5 . Comparison of Strains Around an Expanding Cylinder 1 0 1 4 6 . Comparison of Stresses Around an Expanding Cylinder 1 0 3 4 7 . Shear Modulus i n S o i l Around an Expanding Cylinder . . . . . 1 0 4 4 8 . Log-log Pressure Expansion Curves from F i n i t e Element Analysis 1 0 6 ^ 9 . Calculation of Shear Stress From Pressure Expansion Curve 1 0 9 5 0 . Determination of Shear S t r a i n from Circumferential S t r a i n I l l 5 1 . Stress-Strain Curves Developed from Pressure Expansion Curves 1 1 2 5 2 . Transformed P l o t to Determine Hyperbolic Parameters 1 1 4 5 3 . V a r i a t i o n of Mean Normal Stress with S t r a i n Level 1 1 5 5 4 . V a r i a t i o n of Bulk Modulus with S t r a i n Level 1 1 6 5 5 . V a r i a t i o n of Shear Modulus with S t r a i n Level 1 1 7 ^6 . Calculation of I n i t i a l Shear Modulus From Pressuremeter Data 1 1 9 5 7 . S e n s i t i v i t y of Calculated Shear Modulus to I n i t i a l Estimate of Horizontal Stress . 1 2 1 v i i i 5 8. D e t e r m i n a t i o n o f S o i l S t i f f n e s s Parameters f o r H y p e r b o l i c S t r e s s - S t r a i n R e l a t i o n . . . 1 2 5 5 9 . Four Node P o l a r C o o r dinate F i n i t e Element. . 1 3 5 6 0. F i n i t e Element Mesh 1 4 0 6 1. S t r a i n s Around Expanding C y l i n d e r i n I n f i n i t e E l a s t i c Medium . 1 4 1 6 2. S t r e s s e s Around Expanding C y l i n d e r i n I n f i n i t e E l a s t i c Medium 1 4 2 6 3. Displacements Around Expanding C y l i n d e r i n I n f i n i t e E l a s t i c Medium 1 4 3 6 4. S t r e s s Path f o r E l a s t i c M a t e r i a l 1 4 4 6 5. S t r e s s Path f o r F a i l e d Elements 1 4 4 6 6. T r i a x i a l Model F i n i t e Element . 1 4 7 i x LIST OF TABLES Table T i t l e Page I Description of F i n i t e Element Program Subroutines 6 6 II S o i l Properties f o r Pressuremeter Analysis. 9 3 III Comparison of F r i c t i o n Angles and D i l a t i o n Angles 1 0 8 IV Comparison of Calculated I n i t i a l Shear Modulus Values; Pressuremeter Tests at Sea Island Test S i t e 1 2 0 V Comparison of I n i t i a l Horizontal Stress; Pressuremeter Tests at Sea Island Test S i t e 1 2 3 X ACKNOWLEDGEMENT The author would l i k e to express his appreciation for the advice and support extended by his research supervisor, Dr. Peter M. Byrne and by Dr. Y.P. Vaid throughout t h i s research and during the author's period of study at U.B.C. The author would also l i k e to acknowledge the f i n a n c i a l support extended by the Natural Sciences and Engineering Research Council of Canada. 1 Chapter 1  Introduction The self-boring pressuremeter i s a r e l a t i v e l y new s o i l t e sting device, o r i g i n a l l y being developed i n the early 19 70's. However, the concept of introducing a balloon-like device into the s o i l and i n f l a t i n g i t was used i n the 19 30's. The pressure-meter i n i t s present form was f i r s t developed i n 1954 by Louis Menard, a French engineer working on a model for a f i n a l year design project i n engineering. The Menard pressuremeter was further developed by a company founded by Menard and has been extensively used i n France. Few r e a l changes were made to the pressuremeter u n t i l the early 19 70's when the self-boring pressuremeter was indepen-dently developed by Hughes (19 72) at Cambridge and by Jezequel and Baguelin (19 72) i n France. The self-boring pressuremeter was a major step forward as i t s i g n i f i c a n t l y reduced the amount of s o i l disturbance on introduction of the pressuremeter into the s o i l . S o i l strength and deformation c h a r a c t e r i s t i c s can be determined from pressuremeter t e s t s . The f r i c t i o n angle or cohesion and the shear modulus can be obtained from analysis of the pressure expansion curve. In addition, the d i l a t i o n angle can be determined for f r i c t i o n a l materials. Theories for the analysis of the pressuremeter assume e l a s t i c - p l a s t i c behaviour. Linear d i l a t i o n can be added into the e l a s t i c -p l a s t i c model when analyzing f r i c t i o n a l s o i l s . 2 The e l a s t i c - p l a s t i c model with l i n e a r d i l a t i o n incor-porates some of the important features of s o i l behaviour. Both y i e l d and shear volume coupling are accounted for to some extent. However, the e l a s t i c - p l a s t i c model does not account for the nonlinearity of the s o i l s t r e s s - s t r a i n re-la t i o n s , the va r i a t i o n of the f r i c t i o n angle with stress l e v e l nor the suppression of d i l a t i o n with increasing stress. A f i n i t e element analysis incorporating nonlinear stress-s t r a i n relations and shear volume coupling i s required to determine the e f f e c t of this s o i l nonlinearity. The f i n i t e element model used i n the present analysis incorporates modi-f i e d hyperbolic s t r e s s - s t r a i n r e l a t i o n s as proposed by Kondner and Zelasko (196 3). An incremental approach i s used, with the i n i t i a l tangent modulus varying with s t r e s s . Shear volume coupling i s added to the model through use of a modi-f i e d form of Rowe's stress-dilatancy equation. The pressuremeter i s an axisymmetric, plane s t r a i n prob-lem, so a plane s t r a i n f i n i t e element i n polar coordinates can s i g n i f i c a n t l y reduce the mesh size required i n the anal-y s i s . For thi s purpose, a f i n i t e element i n polar coordinates was developed. In addition, a f i n i t e element that models t r i a x i a l conditions was developed to allow comparison of the shear-volume coupling model with t r i a x i a l t e s t data. Shear-volume coupling i s one of the predominant features of s o i l behaviour that separates i t from most other engineering materials. Vaid, Byrne and Hughes (19 80) have proposed a method of estimating l i q u e f a c t i o n resistance from s o i l 3 volumetric response. By empirically c o r r e l a t i n g d i l a t i o n angle with r e l a t i v e density, an estimate of l i q u e f a c t i o n resistance can be made. As more data becomes available, the d i l a t i o n angle w i l l be correlated d i r e c t l y with li q u e -f a c t i o n resistance. However, to use th i s method, a r e a l i s -t i c evaluation of the i n s i t u d i l a t i o n angle must be made. Hughes, Wroth, and Windle (1977) have proposed a method for determining the d i l a t i o n angle from pressuremeter r e s u l t s . This method assumes e l a s t i c - p l a s t i c behaviour and l i n e a r d i l a t i o n . The f i n i t e element analysis c a r r i e d out i n t h i s work checks the reasonableness of these assumptions f o r a nonlinear, f r i c t i o n a l material. Determination of the shear modulus of the s o i l i s also b r i e f l y investigated i n th i s analysis. Chapter 2  Pressuremeter Theories  2.1 E l a s t i c Analysis A useful base from which to develop an understanding of the pressuremeter i s the expansion of a long cylinder i n an i n f i n i t e e l a s t i c medium. The solution of t h i s problem i s well known i n the theory of e l a s t i c i t y . For a cylinder of i n f i n i t e length, the problem becomes one of plane s t r a i n , and only a plane perpendicular to the long axis of the cylinder need be analyzed. The problem i s axisymmetric and i s conveniently analyzed in.polar coordinates, r and 6, Figure 1. For the plane problem i n polar coordinates, with u and v being the displacements i n the r and 8 directions respectively, the strains are: £ = - — (2-1) r or v _ .u 1 5vx ,„ „. eQ ~ ~ ( F + r tt } { 2 _ 2 ) ,1 6u , Sv v, ,„ _> Y r e = - ( F 5 e t ^ - ? ) (2-3) These are Cauchy s t r a i n s , and are accurate f o r small strains only. At large s t r a i n s , second order terms become s i g n i f i c a n t and cannot be neglected, as i n Cauchy s t r a i n s . For large s t r a i n problems, alternate d e f i n i t i o n s of s t r a i n must be used. By the symmetry of the problem, E g and y r g must be constant with respect to 8 . Therefore, the strains become: 5 7 Also, i t may be seen that f o r the axisymmetric case, no d i s -placements can occur i n the circumferential d i r e c t i o n . There-fore , v = 0 and Y A = 0. i r0 The r a d i a l and circumferential planes are p r i n c i p a l planes for s t r a i n . From the strain-displacement r e l a t i o n s , the problem i s e s s e n t i a l l y one dimensional. The displacement f i e l d , u(r) i n the r a d i a l d i r e c t i o n uniquely defines the s t r a i n conditions i n the material. Figure 2 shows the state of stress within the material. For equilibrium the following equations must hold: 6a . 1 6xr0 . a r - a 6 . T — r + — -. + = 0 (2-4) 6x r 60 • r 16a0 . 6xr0 , oxr0 _ ,„ c. r 6 0 6 r r Using the argument of symmetry again, the stresses cannot change i n the 0 d i r e c t i o n , and there w i l l be no shear stress on r a d i a l or circumferential planes. The equilibrium equations then reduce to one equation: da cr - a f i 3 / + - ^ = 0 <2-4a) For a homogeneous, i s o t r o p i c , e l a s t i c material the stresses and strains are related by the constitutive r e l a t i o n s : r • (l+o)(l-2o) (1+u)(l-2u) ^ b ; A A = (l-^)Eee . "uEe r , 0 7, (l+u)_(l-2u) (1+u) (l-2o) Substituting equations 2-6 and 2-7 into the equilibrium equation results i n the d i f f e r e n t i a l equation for the displacement f i e l d : d r z r a r r^ which has general solutions: u = ^  + C 2 r The boundary conditions are: u = 0 at r = °°, y i e l d i n g C 2 = 0, u = u at r = a, y i e l d i n g C, = u a a x a Where u = the r a d i a l displacement at the a face of the expanding cylinder. Therefore, the displacement f i e l d i s : u = u a § (2-9) and the st r a i n s are: e r = u a f 2 (2-10) e Q = - u a | 2 (2-11) Note that for the e l a s t i c case, e -=-e„ so no volumetric r 6 s t r a i n occurs. The stresses are given by: °r = ( W U a ? 2 = 2 G u a F z < 2" 1 2 ) a G = I W U a r> =" 2 G ua P < 2 " 1 3 ) For an i n i t i a l l y unstressed medium undergoing a pressure increase, Ap, a = Ap at r = a. Therefore: a r = 2Gu a | 2 = A P u a = ( 2 _ 1 4 ) Therefore: a r = A p ~ f * (2-15) a Q =-Ap |z (2-16) £>- = Apa 2 (2-17) r 2Gr r £ 8 = -§§§t (2-18) and, u = |E|1 (2-19) 9 For an i n i t i a l l y stressed medium, with a r ^ = , a 2 a r = p ± + Ap - 2 (2-15a) a Q = p ± - Ap |z (2-16a) and the strains and displacements are as given above. Note that a r = -<jg so the mean normal stress does not change i n an e l a s t i c material. The material i s subject to a pure shear as the cylinder expands. The shear modulus may be determined from the pressure-expansion curve of the e l a s t i c material, Figure 3. By measuring the displacement of the cylinder as the pressure i s increased, a l i n e a r p l o t of pressure versus circumferential s t r a i n u /a a w i l l r e s u l t . As shown i n the figure, the shear modulus i s one hal f of the slope of the curve: G = i ^-r- (2-20) 2 u /a a 2.2 E l a s t i c - P l a s t i c Incompressible Material The domain remains e l a s t i c u n t i l the stresses at the inner face of the cylinder reach the y i e l d state defined as: a l + s i n * ( 2 . 2 1 ) — 1-smcb y ' °3 I f the pressure within the cylinder i s raised beyond th i s l e v e l , the s o i l w i l l y i e l d and a p l a s t i c annulus w i l l form around the cylinder. Within t h i s annulus, the stresses w i l l change such that they w i l l obey the y i e l d c r i t e r i o n . Beyond the p l a s t i c annulus the material w i l l remain e l a s t i c and w i l l obey the rela t i o n s developed previously. A small s t r a i n approximation w i l l be used to develop expressions for the stre s s , s t r a i n and displacements i n the p l a s t i c region. Within the p l a s t i c annulus, the equilibrium equation must 10 11 s t i l l hold. Upon substitution of equation 2-21 into the equi-librium equation 2-4a, the following equation r e s u l t s : da o l l - h ^ + r r = 0 (2-22) For an i n i t i a l l y unstressed material, the boundary condition i s : a - Ap at r = a, r and the solution i s l n T = _ ( 1 " I ) l n f ( 2 _ 2 3 ) or, within the p l a s t i c zone: a r ='AP<f> f o r r < r f (2-24) where r f = the radius of the p l a s t i c annulus i 1 (1-—) Note that the stress decays as (—) v R'in the p l a s t i c material while i t decays as ^  i n the e l a s t i c material. The circumferen-t i a l stress i s : A, a 9 = ^ ® ( f ) ( 1 " ^ (2-25) R^V v r For an i n i t i a l l y stressed material, with free f i e l d s t ress, p^: 0 r = PL + Ap(|) ( 1 _R ) (2-24a) ° 6 = pL + ^Ap(|) ^ " i * (2-25a) A comparison of the stresses i n an e l a s t i c material and an e l a s t i c - p l a s t i c material i s shown i n Figure 4. The two im-portant points are that the stress decays slower i n the p l a s t i c material and that the mean normal stress increases i n the plas-t i c material while i t does not increase i n the e l a s t i c material. The assumption of no volume change can be used to determine the displacement f i e l d and stra i n s i n the p l a s t i c material. For no volume change: du u r £ v = "ar - ^ = 0 < 2 - 2 6 ) 12 F i g u r e 4. C o m p a r i s o n o f S t r e s s D i s t r i b u t i o n i n E l a s t i c and E l a s t i c - P l a s t i c I n c o m p r e s s i b l e M a t e r i a l s 13 T h e b o u n d a r y c o n d i t i o n f o r t h i s d i f f e r e n t i a l e q u a t i o n i s : u = u a t r = a . a ' a n d t h e s o l u t i o n i s : , u , r l n — = - l n — u a a o r u = u a ( | ) ( 2 - 2 7 ) T h i s i s t h e s a m e d i s p l a c e m e n t f i e l d a s i n t h e e l a s t i c m a t e r i a l . T h e r e a s o n t h a t t h e d i s p l a c e m e n t f i e l d s a r e t h e s a m e f o r b o t h t h e e l a s t i c a n d p l a s t i c c a s e s i s t h a t n o v o l u m e c h a n g e o c c u r s i n e i t h e r t h e e l a s t i c o r p l a s t i c c a s e . A l s o , t h e s m a l l s t r a i n a p p r o x i m a t i o n w a s u s e d i n t h e d e r i v a t i o n o f t h e d i s p l a c e m e n t f i e l d . N o t u s i n g t h e s m a l l s t r a i n a p p r o x i m a t i o n w o u l d p r o d u c e a s l i g h t l y d i f f e r e n t d i s p l a c e m e n t f i e l d . F o r s t r a i n s l e s s t h a n a b o u t t e n p e r c e n t t h e e r r o r i n m a k i n g t h e s m a l l s t r a i n a p p r o x -i m a t i o n i s s m a l l . T h e p r e s s u r e - e x p a n s i o n c u r v e o f a n e l a s t i c - p l a s t i c i n c o m -p r e s s i b l e m a t e r i a l c a n b e d e r i v e d u s i n g t h e s m a l l s t r a i n a p p r o x -i m a t i o n . B e f o r e y i e l d t h e c u r v e r i s e s l i n e a r l y a t a s l o p e o f 2 G . A f t e r y i e l d , c u r v a t u r e b e g i n s . A t t h e p l a s t i c - e l a s t i c i n t e r f a c e t h e m a t e r i a l i s j u s t a t y i e l d , a n d t h e s t r a i n i s e ^ . T h e v a l u e o f d e p e n d s o n t h e i n i t i a l s t r e s s s t a t e a n d m a t e r i a l p r o p e r t i e s G a n d <j>, b u t n o t o n l o c a t i o n w i t h i n t h e m a t e r i a l . F o r a n i n i t i a l l y s t r e s s e d m a t e r i a l , y i e l d o c c u r s w h e n : a e P ± - a a r o r : °r _ P i + A a r a e P i - A r Aa r = P i ^ R + I ; i S i n t J ) ( 2 - 2 8 ) T h e s h e a r s t r e s s i s t h e n : a r - a e T = = p i s m < | ) ( 2 - 2 9 ) 14 T h e c o r r e s p o n d i n g s h e a r s t r a i n , y ^ , a t w h i c h y i e l d o c c u r s i s : p . sin<j) Y f = 1 = ~^~Q ( 2 - 3 0 ) T h e c o r r e s p o n d i n g c i r c u m f e r e n t i a l s t r a i n , e f / = - j Y f / t h e r e f o r e , • p . s i n < { ) £ f = -TG— ( 2 _ 3 1 ) F o r n o v o l u m e c h a n g e , t h i s c o r r e s p o n d s t o a s t r a i n , e , a t c l t h e c y l i n d e r f a c e o f : r p . s i n < j > r £ a = £ f ( " i > = ( 2 " 3 2 ) I n t h e p l a s t i c m a t e r i a l : A a r = A p ( | ) (1"P:) ( 2 - 2 4 ) r f ( 1 - - ) o r ; Ap = p sincj> ( - f ) ^ x R ; b u t f r o m 2 - 3 2 : r f _ 2 G e Q > „ , . " (pTIInV* ( 2 3 3 ) o p F /R—1\ T h e r e f o r e : Ap = p . s i n ( Q : ) v 2 R ; 1 p^sin<j> T h e c y l i n d e r p r e s s u r e i s t h e n : 2 G e Q ( R - i ^ p = p . (l+sin<J>) ( r — j ) ( 2 - 3 4 ) t- C - L ^ p ^ s i n c j ) 2 R v ' T h i s c u r v e i s p l o t t e d i n F i g u r e 5 . T h e e l a s t i c - p l a s t i c i n c o m -p r e s s i b l e m a t e r i a l i s m u c h s o f t e r t h a n a n e l a s t i c m a t e r i a l . 2 . 2 . 1 L a r g e S t r a i n s a n d L i m i t P r e s s u r e W i t h i n t h e p l a s t i c z o n e a s s t r a i n s b e c o m e l a r g e , t h e s m a l l s t r a i n a p p r o x i m a t i o n w i l l n o t h o l d . I n t h i s c a s e , A l m a n s i s t r a i n s c a n b e u s e d . T h e d e f i n i t i o n o f t h e A l m a n s i s t r a i n , a i s : a = 1 d l 2 - d l 2 ( 2 - 3 5 ) 2 ai w h e r e : d l Q = t h e o r i g i n a l l e n g t h d l = l e n g t h a f t e r d e f o r m a t i o n 15 Figure 5 . Pressure-Expansipn Curve f o r E l a s t i c - P l a s t i c M a t e r i a l .16 The Almansi s t r a i n and Cauchy s t r a i n are related by: ( 1 + : £ ) 2 = T ^ r <2-36> Even at high strains the equilibrium equation must hold. For large s t r a i n conditions, the equilibrium equation i s written f o r the material i n the deformed state: da a - a 6 -s-i- + — = 0 (2-37) dp p where: p = radius a f t e r deformation The stresses must remain on the y i e l d surface, therefore the equilibrium equation becomes: da c - ( 1 - i ) - J L + _£ - / 0 (2-38) dp p This d i f f e r e n t i a l equation may be solved for the stress d i s -t r i b u t i o n within the p l a s t i c material. The boundary conditions are: p = p and o = p at the face of the c y l i n -r po r ^o 2 der, and: p = p^ and o r = p^ at the l i m i t of the p l a s t i c zone where: p^ = pressure at which material y i e l d s The solution of this d i f f e r e n t i a l equation with the above boundary conditions i s : p f po (=T- = ( T 2) ( R; (2-39) po p f or i n terms of at any radius p i n the p l a s t i c zone: °r p o ( 1 - i ) — = R; (2-40) P P The circumferential stress i s then: % - ^ V 1 - * ' (2-4i> 17 T h i s i s t h e s a m e e x p r e s s i o n a s d e v e l o p e d u s i n g t h e s m a l l s t r a i n a p p r o x i m a t i o n , e x c e p t p r e p l a c e s r . W h e n t h e p l a s t i c r e g i o n e x p a n d s f r o m r ^ t o p^, t h e c o n d i t i o n o f n o v o l u m e c h a n g e i m p l i e s : T T ( P 2 - p 2) = r r ( r 2 - r 2 ) w h e r e ; r = r a d i u s o f c y l i n d e r o J p Q = d e f o r m e d r a d i u s o f c y l i n d e r o r : P f " r f = p o " r o (2-42) T h i s m a y b e t r a n s f o r m e d t o : 2 2 „ 2 2 „ ~ 2 ,p - r . 2 ( i f f ) p f = ( 1 2 o ) p n 2 1 2 U p f p o o r : a f p , — = (—) (2-43) p f w h e r e : = A l m a n s i s t r a i n t o y i e l d a = A l m a n s i s t r a i n a t c y l i n d e r o J S u b s t i t u t i n g e q u a t i o n 2-43 i n t o e q u a t i o n 2-40 r e s u l t s i n t h e e q u a t i o n : P D = P f (% = P Q ( % ( t r " ) (2-44) o T h i s e q u a t i o n a l l o w s p r e d i c t i o n o f a l i m i t p r e s s u r e . T h e s t r a i n a t t h e f a c e o f t h e c y l i n d e r , a Q , i s r e l a t e d t o t h e v o l u m e o f t h e c y l i n d e r b y : w h e r e : V = c u r r e n t v o l u m e V = i n i t i a l v o l u m e o I n t h e l i m i t , a s t h e v o l u m e b e c o m e s l a r g e , V Q b e c o m e s n e g l i g i b l e w i t h r e s p e c t t o V , s o a Q = ^ . T h e s t r a i n , a f / a t w h i c h y i e l d 18 occurs i s equal to the Cauchy s t r a i n to f a i l u r e , e^: p^sin<j) e f = 2G = a f The l i m i t pressure for a f r i c t i o n a l material then becomes: R— 1 P L = P f ( P T i W ) ( " ^ ) ( 2 " 4 6 ) where: p^ = p^(l+sin<)>) For a li n e a r e l a s t i c - p l a s t i c incompressible f r i c t i o n a l material, the f r i c t i o n angle and the shear modulus may be ob-tained from the pressure expansion curve. The shear modulus i s obtained from the i n i t i a l l i n e a r portion of the l i n e and the f r i c t i o n angle i s obtained from the curved portion of the l i n e . 2.3 Limitations of E l a s t i c - P l a s t i c Analysis The simple e l a s t i c - p l a s t i c model presented allows an analytic solution for the pressuremeter conditions and yi e l d s c h a r a c t e r i s t i c pressure-expansion curves. The l i n e a r portion occurs before y i e l d , when the material i s subject to pure shear, and the curved portion occurs at the onset of p l a s t i c behaviour. The model also shows that the l i m i t pressure i s related to the s o i l strength <j> and the shear modulus G or the s t r a i n to f a i l u r e e^ .. This type of analysis does have several major l i m i t a t i o n s . I t can only handle compressive volumetric strains due to changes i n the mean normal stress i n the e l a s t i c phase, although no volume change occurs along the stress path followed i n this case. Also, the s o i l i s incompressible i n the p l a s t i c phase, which i s v a l i d only for undrained t e s t s . The theory also neg-lec t s stress l e v e l dependency of the s o i l properties, which has 1? a major e f f e c t on the response of s o i l . Linear d i l a t i o n can be added into the closed form solution, but stress l e v e l dependency requires numerical analysis. 2.4 Linear Dilatant Analysis The e f f e c t of d i l a t i o n on the response of s o i l to the pressuremeter was examined by Baguelin, Jezequel and Shields (19 78) using the small s t r a i n case and assuming l i n e a r d i l a t i o n . For the small s t r a i n case, Cauchy strains are used. Volume changes due to d i l a t i o n are added into the analysis by: Aa A T £ = _ JB E^x (2-47) v B D K ' where: D^_ = d i l a t a n t parameter (positive for volume increase) For plane s t r a i n conditions: e = ea-+e = -- - 1^ (2-48) v 0 r r dr or: , Aa Ax - t R + ^H) = ™ - m a x (2-49) Kr dr' B D. K* The shear s t r a i n , y, i s given by: But: Therefore: Y = ? r ~ £ 0 = ~ § r + r ( 2" 5 0> A Tmax A g r - A g 0 Y = — — = ^ (2-51) •j Aa -Aa n H _ du r 0 r dr 2G u b Z ) Equations 2-49 and 2-52 allow determination of Aa andAa -Aa n r r 0 . j- u -, du i n terms of — and - s — : r dr Aa r - Aa e = 2G(H - |H) ( 2-53) A a = u( 3BG 3B _ du 3BG 3B r r l2(l+U)D 2(l + u ) + G J dr (2(l+u)D, + 2 (1+u) G ) t (2-54) 3B G The term „ , n ,— r can be s i m p l i f i e d to -r\—~—r / i n which case: 2(1+u) * (l-2u) 20 An = u f .  g 2 _ 2 U G N du, G 2 2G(l-u) , r r((l-2u)D l-2u' " d r l ( l - 2 u ) D l-2u ; t (2-55) o r : A a r = ^ - (2-55a) On s u b s t i t u t i o n o f e q u a t i o n 2-55a i n t o the e q u i l i b r i u m e q u a t i o n , the f o l l o w i n g d i f f e r e n t i a l e q uation r e s u l t s : r,alu + U ^ i ) (r|B . u ) . o ( 2 - 5 6 ) whe r e : 2G - g = 2 D t ( 1 - U ) - G n 2 D t(l-u) + G n The d i f f e r e n t i a l e q uation has s o l u t i o n s o f the form: u = Ar + B r n The boundary c o n d i t i o n of u = 0 a t r = °°, y i e l d s A = 0 and n >0. The c o n d i t i o n t h a t n >0 y i e l d s : Q D t > 2 (1-u) "^ o r v o l u m e i n c r e a s e Q D, < T T T - T r- f o r volume decrease t 2(1-u) which can be combined as: -2 (1-u) . 1 . 2(l-u) < G < D t ( 2 _ 5 7 ) The boundary c o n d i t i o n a t the face of the c y l i n d e r , r = a, i s u = u and a_ = Ap + p.. This g i v e s the s o l u t i o n f o r d i s -EL JL X placements: u = u ( f ) n (2-58) where: u = a 2G As developed p r e v i o u s l y , i n e l a s t i c m a t e r i a l w i t h no d i l a t i o n : u = u Q ( | ) (2-9) For a gi v e n value o f U q , d i l a t i o n causes the displacement t o propagate f u r t h e r from the c y l i n d e r . In the d i l a t a n t m a t e r i a l : 21* Aa r = + nn) which on s i m p l i f i c a t i o n becomes: Aa r = 2GH = 2GeQ = A p ( | ) n + 1 (2-59) Note that l i n e a r d i l a t i o n does not a f f e c t the i n i t i a l s t r a i g h t l i n e portion of the pressuremeter curve, i t s t i l l r i s e s at a slope of 2G. The stresses are given by: Aa r - Aa Q = 2x m = 2G(H)(l+n) = (l+n)Aa r a n+1 ( 2 _ 6 0 ) Aa e = -nAa r = - n A p ( | ) n + x (2-61) and: Aa = (Aa +AaQ) = Aa ( 1 ~ n ) (1+u) m r 8 ' 3 r 3 v or: Aa m = ^ffd+u)(1-n) = §£(§) ( n + 1 )(1+u)(1-n) (2-62) The strains are given by: e r = f§£(f) ( n + 1 ) (2-63) and: £ q = -||(|) ( n + 1 ) (2-64) Figures 6, 7 and 8 show a comparison of the displacements, strains and stresses, respectively, i n a d i l a t a n t material and an e l a s t i c material. In Figure 8 note that higher shear stresses e x i s t at the face of the cylinder i n the e l a s t i c material but the shear stresses propagate further into the material i n the d i l a t a n t material. Also note that d i l a t i o n increases the mean normal stress i n the material. Y i e l d of the d i l a t a n t material w i l l occur when — = R. ° e But: Aa = p. + 2G(-) (2-65) r I r and: Aa e = P ± - nAa r = p ± - 2Gn(^) (2-66) For y i e l d j u st occurring at the cylinder p . + 2 G e . i = R = p ? - 2 G e : ( 2 - 6 7 > 0a' 1 f 2 2 Figure Comparison of Displacements i n D i l a t a n t and E l a s t i c M a t e r i a l 23 M a t e r i a l 24 M a t e r i a l 2 5 w h i c h s i m p l i f i e s t o : _ p ± ( R - D e f " 2 G ( R n * l ) ( 2 - 6 8 ) F o r a n e l a s t i c m a t e r i a l w i t h n o d i l a t i o n D = ° ° s o n = 1 a n d : p ± ( R - l ) p ± * e f e = 2 G ( R + 1 ) = 2 G s i n * ( 2 _ 6 9 ) b u t i n a d i l a t a n t m a t e r i a l w i t h , f o r e x a m p l e , = a n d n = £ f d = P i - W " > £ f e ( 2 - 7 0 ) 0 D i l a t i o n m a k e s a s o i l h a v e a h i g h e r s t r a i n t o y i e l d . S i n c e t h e p r e s s u r e - e x p a n s i o n c u r v e s t i l l r i s e s a t a s l o p e o f 2 G b e f o r e y i e l d , t h e d i l a t a n t s o i l a p p e a r s s t r o n g e r b u t n o t s t i f f e r t h a n t h e n o n d i l a t a n t e l a s t i c s o i l . O n a m o d i f i e d M o h r d i a g r a m , t h e s t r e s s p a t h f o r t h e d i l a t a n t m a t e r i a l i s i n c l i n e d t o t h e r i g h t , s o t h e s o i l f a i l s a t a h i g h e r s t r e s s a n d a p p e a r s s t r o n g e r . H u g h e s , W r o t h a n d W i n d l e ( 1 9 7 7 ) h a n d l e d i l a t i o n i n b a s i -c a l l y t h e s a m e m a n n e r . B y a s s u m i n g t h a t v o l u m e t r i c s t r a i n d u e t o i n c r e a s e s i n m e a n n o r m a l s t r e s s a r e n e g l i g i b l e , H u g h e s e t a l g i v e a n e q u a t i o n f o r s t r a i n d i s t r i b u t i o n w i t h i n t h e p l a s t i c z o n e p . , n + l . ( e Q + | ) = ( e r + |) ( ^ ) U - N ; ( 2 - 7 1 ) R w h e r e : K = t h e i n t e r c e p t o f t h e e v s . y p l o t F i g u r e 9 . p . = i n i t i a l s o i l s t r e s s a R = r a d i a l s t r e s s a t e l a s t i c - p l a s t i c b o u n d a r y T h e r e a s o n i n g b e h i n d t h i s e q u a t i o n c a n b e a p p l i e d t o t h e e l a s t i c b e h a v i o u r b e f o r e a n y m a t e r i a l y i e l d s . B y a s s u m i n g K = 0 , w h i c h H u g h e s e t a l d o , t h e d i s p l a c e m e n t f i e l d i s t h e s a m e a s d e v e l o p e d b y B a g u e l i n e t a l . F o r t h i s c a s e , t h e d i l a t a n t p a r a m e t e r D ^ i s r e l a t e d t o t h e s h e a r m o d u l u s a n d d i l a t i o n a n g l e : Figure 9 . Volume Change Response used i n Pressuremeter Analysis 2 7 which, for the p r e - y i e l d behaviour, results i n : ,a, n u = u (—) a r' , 1-sinv A ^ ^ n where: n = — i , 0 < n < 1 1+sinv Going back to the case where a p l a s t i c annulus has formed, at radius r ^ , the s o i l i s just at y i e l d . The displacement f i e l d i s given by: ur 1 1 = - f r ( n + 1 ) + ( 2 _ 7 3 ) where: = constant The boundary condition at the e l a s t i c - p l a s t i c interface i s : u r f r = r f ' T J = e f which, when combined with 2-73 produces: C 1 = U f + |,r f'« + 1> The displacement f i e l d within the p l a s t i c zone i s , therefore: £ = - | + <ef + | ) ( ^ | ) ( n + 1 ) (2-74) where: a < r < r ^ As developed previously, the stress d i s t r i b u t i o n within the p l a s t i c zone i s : a r _ ., _1. = (-£•) U R} (2-24) 0 r Upon sub s t i t u t i o n into 2-74, a r e l a t i o n between stress and s t r a i n i s produced: o ,R(n+l) u = _ K j . _ L K w r, ( ? = " I + ( £ f + S> ( 0 ^ R - X (2" 7 5) R At the cylinder face, r = a, and = p, therefore: u ,R(n+l)  £ a = - | = - | + ( £ f + S><£r) R " 1 (2-76) R. Taking logarithms: l o g ( e a + |) = l o g ( e f + + R ^ ^ 1 } logg- (2-77) 28 This reduces to: R— 1 K log p = R ( n + 1 ) 1 Q g ( £ + 2") + constant (2-78) where: e = e a By neglecting K and p l o t t i n g the applied pressure (in terms of e f f e c t i v e stress) against the s t r a i n — on log-log paper a the slope of the straight l i n e portion of the curve w i l l be: = = R-l _ sincj) (1+sinv) , Q> R(n+1) l+sincj) U-/y; But, as w i l l be developed i n Chapter 3 , 1+sinv = (l+sin<h l - s i n ^ c v x t o - a n ) ^l-sind) ; ^1+sind) ; (2-80) 1-sinv vl-sind) ; vl+smd) D = f where: <}> = constant volume f r i c t i o n angle cv 3 Equations 2 - 7 9 and 2 - 8 0 may be solved f o r the d i l a t i o n angle, v, and the f r i c t i o n angle, §: s i n * = ( K - l ) s + 2 < 2 - 8 1 ' S i n v = 2 K ; ; l K - » ( 2 - 8 2 ) A pressuremeter curve for fine sand i n Sea Island i s shown i n Figure 1 0 . This data i s plotted on the log-log p l o t i n Figure 10 . The slope of the l i n e a r portion of the curve i s s = 0 . 4 2 9 . In order to solve for the f r i c t i o n angle and d i l a t i o n angle, the value of <\> i s required. Figure 1 1 shows a graphical method of solution for <j> and v for various values of 0 c v . As can be seen i n the figure, the value of <(> markedly affects the calculated values of <J> and v. Changing <j> from 3 0 ° to 3 6 ° changes the f r i c t i o n angle from 3 7 ° to 4 0 ° and changes the d i l a t i o n angle from 8 . 3 ° to 5 . 3 ° . In order to accurately determine the s o i l properties, the constant volume f r i c t i o n angle must be determined accurately. 2 9 500 J (0 ft 400 "i d) u 0 cn s 3 0 0 u ft ts <u 200 •rH rH Cu Cu <C 100 H (0 +J O En Total Applied Pressure vs. Radial Displacement Self boring pressuremeter test MacDonald's Farm Research Site - Sea Island Test Depth - 4.6m Test Date - Dec. 03, 19 81 S i t u Technology, Vancoi 1.0 2.0 TIT 5.'0 6.-0 u 7:o Radial Displacement (%) __a a 8 * 0 9'.0 Pressure Expansion Curve for Fine Sand (0 400 ft . 300 in cn CD £• 200 .w > •H $ 100 m- 90 w 80 d 7 0 3 60 (X 50 40-Radial E f f e c t i v e Stress vs. Radial Displacement i 1— l — I I I I .4 .6 .8 1 0. 429 1 1 1 1 1—I I I I 2 3 4 5 6 7 8 9 10 .2 Radial Displacement (%) u a Figu r e 10 Log-Log Pressure Expansion Curve for Fine Sand 3 0 31 Hughes et a l show that the value of K i s small and can be neglected, however, examination of the simple shear data on loose sands by Vaid et a l (19 80), Stroud (19 71) and the constant mean normal stress t r i a x i a l t e s t data by Kokusho (1978), Figures 12 and 13, shows that for loose sands the value of K may be i n the range of one to two percent. The pressuremeter data from Sea Island i s replotted i n Figure 14, assuming various values of K from 0.1% to 2.0%. The f r i c t i o n angle and d i l a t i o n angle are graphically determined for 4> c v = 32° i n Figure 15. A value of K = 0.1% changes the calculated values of <J> and v only s l i g h t l y , but a value of K = 2.0% changes the f r i c t i o n angle from 38° to 44° and changes the d i l a t i o n angle from 7.2° to 15.6°. This i s a s i g n i f i c a n t difference. Neglecting the value of K pro-duces low estimates of $ , which may or may not be conservative, depending on the problem at hand. The calculated values of the d i l a t i o n angle and f r i c t i o n angle are also sensitive to the c a l c u l a t i o n of the slope of the log-log p l o t . Minor errors i n p l o t t i n g or determining the slope of the l i n e a r portion of the curve can e a s i l y change the value of the slope by ten percent. Figure 16 shows the v a r i a - ' t i o n of the f r i c t i o n angle and d i l a t i o n angle f o r a $10% change i n the slope of the log-log p l o t of the Sea Island sand using a <j> = 32°. The 10% change i n the slope changes the f r i c -t i o n angle by about 4° and changes the d i l a t i o n angle by about 4°. The method of analyzing the pressuremeter curve developed by Hughes et a l i s t h e o r e t i c a l l y sound, however, i t i s sensitive 32 Shear Strain ,/ - 0 /© F i g u r e 12. S t r e s s - S t r a i n b e h a v i o u r o f O t t a w a S a n d i n d r a i n e d s i m p l e S h e a r . ( A f t e r V a i d , B y r n e & H u g h e s ) . 33 > w - 3 — I -3-2-1 - P o •H U H O > Volumetric Strain vs. Shear Strain Simple Shear Test on Leighton Buzzard Sand Y (%) (After Stroud, 1971) Figure 13. Volume Change Data for Loose Sand 34 F i g u r e 15. V a r i a t i o n o f C a l c u l a t e d V a l u e s o f tj, and v w i t h  Changes i n t h e V a l u e o f ' K ' 36 Figure 16. V a r i a t i o n of C a l c u l a t e d Values of (fr and v with Changes i n the Value of 's' 37 to the value of <!>cv and cal c u l a t i o n of the slope of the log-log p l o t of r a d i a l e f f e c t i v e stress and s t r a i n . Combining the three uncertainties discussed above, the f r i c t i o n angle and d i l a t i o n angle w i l l probably only be accurate to about $2°. The error may be greater i f K i s large or the estimated value of 0> c v i s grossly i n error. D i l a t i o n s i g n i f i c a n t l y affects the manner i n which the s o i l responds to the pressuremeter. Accurate strength and deformation c h a r a c t e r i s t i c s cannot be determined unless the e f f e c t of d i l a t i o n can be cor r e c t l y incorporated i n the analy-s i s . The l i n e a r d i l a t a n t model shows q u a l i t a t i v e l y the e f f e c t of d i l a t i o n , but a numerical analysis with shear-volume coup-l i n g and stress l e v e l dependent s o i l properties i s required to determine the f u l l e f f e c t of d i l a t i o n on the pressuremeter. 38 Chapter 3  S o i l Model  3.1 D e f i n i t i o n of Stress and Strain The s o i l i s modelled as a homogeneous, i s o t r o p i c , incre-mental l i n e a r e l a s t i c continuum. From a microscopic view, s o i l i s f a r from being a homogeneous continuum, but from a macro-scopic view, when the geometrical dimensions defining the form of the body are large i n comparison with the dimension of i n d i -v i dual s o i l grains, the assumption of homogeneity and continuity can be made with s u f f i c i e n t accuracy. The strength and deformation of s o i l i s controlled by the e f f e c t i v e stress. This was defined by Terzaghi as: a ' = a - y (3-1) where: a = the t o t a l stress u = the pore pressure a' = the e f f e c t i v e stress Other relations have been proposed f o r the e f f e c t i v e stress that take account of the intergranular contact area, but for most engineering problems at low pressures, Terzaghi's d e f i n i -t i o n i s s u f f i c i e n t l y accurate. The s o i l model w i l l be derived i n terms of e f f e c t i v e stress parameters, but the computer pro-gram i s able to handle eith e r e f f e c t i v e stress or t o t a l stress. In further work, a l l references to stress w i l l mean e f f e c t i v e stress. The model w i l l be presented for the general case of a three dimensional rectangular Cartesian coordinate system. Within the computer program, the model i s used for two dimen-sio n a l problems i n ei t h e r rectangular or polar coordinates or three dimensional problems under t r i a x i a l conditions ( a 1 , a 9 = a ^ 39 or o = a 2 , o ^ ) . In a three dimensional space, with a rectangular coordinate system using x, y and z, the following stresses may occur i n a body: °x' V a z ' Txy' T y z ' T z x ' xz yx' zy' These stresses are shown i n Figure 17. In order to s a t i s f y equilibrium of the body: xy T — T zy yz = T zx XZ Therefore s i x quantities are required to uniquely define the state of stress of a body i n a three dimensional space. A body undergoing a change of i t s stress state w i l l also undergo deformations. In the three dimensional rectangular coordinate system the displacements u, v and w are the d i s -placements i n the x, y and z di r e c t i o n s , respectively. A com-pati b l e set of strains may be determined from the displacement f i e l d . For the three dimensional case: ex = du dx' e y = dv dy' dw Z dz' Y x y : : ,dv ~ ldx + du, dy ; Y y z = ,dw My + dv, dz ; Y z x = -du (dz + dw, dx ; Y y x = ,du ~ My + dv. dx ; ' zy = ,dv " ldz + dw, dy ; Y x z = -dw ldx + du. dz ; ( 3 - 2 ) 40 F i g u r e 17 . S t r e s s e s i n C a r t e s i a n Coordinates 41 I t may be seen that: Yxy ~ Yyx' Y y z ' Y z y ' Y z x ~ Yxz* Therefore, only s i x s t r a i n s define the state of deformation of the body. This d e f i n i t i o n may be used only when strains are small and second order e f f e c t s may be neglected. When second order e f f e c t s must be included, an alternate d e f i n i t i o n of s t r a i n must be used. Almansi strains were used i n Chapter 2 to determine the pressuremeter, l i m i t pressure. Six strains are derived from three independent displace-ments. Three equations of compatibility must e x i s t to uniquely determine the s i x strains from the three displacements. By deriving the strains from the displacement f i e l d , the compat-i b i l i t y conditions are automatically s a t i s f i e d . 3.2 Constitutive Relation In order to be of any use i n solving stress-deformation problems, stresses and strains must be related i n some fashion. The constitutive relations define the s t r e s s - s t r a i n r e l a t i o n . The constitutive relations for s o i l are highly complex. S o i l i s anisotropic, nonhomogeneous, nonlinear and i n e l a s t i c . Under these conditions 36 c o e f f i c i e n t s are required to relate stress to s t r a i n . In addition, these 36 c o e f f i c i e n t s are stress l e v e l dependent and w i l l also depend on the stress path followed during deformation. Specifying such a r e l a t i o n would be a formidable task, at best, and i n many cases, impossible. In order to reduce the complexity of the problem a number of simplifying assumptions must be made. By assuming that the material i s l i n e a r e l a s t i c , homogeneous and i s o t r o p i c , the num-ber of c o e f f i c i e n t s i s reduced from t h i r t y - s i x to two. The 42 assumption of l i n e a r i t y means that strains due to a set of stresses may be obtained through superposition of the strains due to each stress. E l a s t i c means that stress and s t r a i n are uniquely related, regardless of stress path. Isotropy means that the material constants are the same i n a l l directions. Homogeneity means that the material i s the same at a l l points i n the body and no shear volume coupling means that no volu-metric strains occur on application of a pure shear stress. For a l i n e a r e l a s t i c i s o t r o p i c medium, the s t r e s s - s t r a i n r e l a -tion i s defined by the matrix (D): 1-u u u u 1-u u u 1-0 0 0 0 0 0 (D) = (l-2u)(1+u) u 0 0 0 0 0 0 0 0 0 0 0 0 l-2u 2 u u u l-2u 2 u u u l-2u 2 where: and: ( 3 - 3 ) ia) = (D) {e} {a} = stress vector {e} = s t r a i n vector Four e l a s t i c constants are defined, with any two being independent and s u f f i c i e n t to uniquely specify the constitutive r e l a t i o n . The four constants are: Young's modulus, E Bulk modulus, B Shear modulus, G Poisson's r a t i o , u The four constants are related by: ^3 E G = 2(l+u) (3-4) B = 3(1-20) ( 3 _ 5 ) The computer f i n i t e element program has been developed u s i n g the Young's modulus and Bulk modulus as i n p u t parameters. A l i n e a r e l a s t i c i s o t r o p i c m a t e r i a l would n o t model s o i l b ehaviour v e r y w e l l . N o n l i n e a r i t y , p l a s t i c s t r a i n s and shear volume c o u p l i n g are three c h a r a c t e r i s t i c s o f s o i l b ehaviour t h a t must be i n c o r p o r a t e d i n a s o i l model. 3.3 N o n l i n e a r i t y N o n l i n e a r i t y o f the s t r e s s - s t r a i n r e l a t i o n i s accounted f o r by u s i n g s t r e s s l e v e l dependent moduli. T y p i c a l d e v i a t o r s t r e s s versus a x i a l s t r a i n behaviour f o r sand i n d r a i n e d t r i a x i a l t e s t s i s shown i n F i g u r e 18. Kondner and Zelasko (196 3) and Duncan and Chang (19 70) have shown t h a t these curves may be approx-imated by m o d i f i e d hyperbolas of the form: ( 0 1 " a 3 } = ~ ( 3 " 6 ) ( fI + <°1 " ^ u l t When an i n c r e m e n t a l e l a s t i c approach i s used, the tangent Young's modulus i s g i v e n by: E. = E. (1 - R f ( l - s i n j > ) ( o 1 - c 3 ) 2 ( 3 _ 7 ) 1 2cCos<j> + 2a3sin<(> where: E^ = i n i t i a l tangent Young's Modulus a 1 = 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 a^ = 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 <j> = f r i c t i o n angle R£ = the f a i l u r e r a t i o , the r a t i o o f maximum d e v i a t o r s t r e s s from t r i a x i a l t e s t t o the asymptotic d e v i a t o r s t r e s s d e f i n e d by the hy b e r b o l a . F i e u r e 19 shows a t y p i c a l 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 . 44 Figure 18 Stress Strain Curves for Drained Triaxial Tests on Sand. (After Varadarajan and Mishra). ^5 . < o - . - a ~ ) , . 1 3 u l t ( a 1 - a 3 ) f cu o c cu u CU m m •rH Q -co w cu u • P Stress Difference vs. A x i a l Strain ZLsymptnte Ax i a l Strain, e Figure 19. Modified Hyperbolic Stress-Strain Relation 46 The i n i t i a l tangent Young's modulus i s also stress l e v e l dependent. Janbu (196 3) presented an empirical r e l a t i o n for E. : E i = k e P a ( i r ) n ( 3 " 8 ) a where: k„ = Young's modulus number n = modulus exponent p = atmospheric pressure a T r i a x i a l tests are generally conducted with a constant confining pressure ,<T^ . For t h i s reason i s used i n the r e l a -t i o n f o r the i n i t i a l modulus. However, the s o i l s t i f f n e s s i s related to the l e v e l of the mean normal str e s s , i n which case: E i = V a ^ ^ ( 3 " 9 ) a In order to use t h i s r e l a t i o n , t r i a x i a l tests must be conducted at constant mean normal stress. The pressure control •system i s complex, so tests of this nature are not usually con-ducted. The f i n i t e element program i s set up to use eit h e r a ^ or 0 i n the c a l c u l a t i o n of E., at the user's d i s c r e t i o n . • m I The method of obtaining the hyperbolic parameters from t r i a x i a l tests i s discussed i n d e t a i l by Duncan and Chang and by Duncan et a l (19 73) and w i l l not be presented here. The f r i c t i o n angle, <{>, i s also stress l e v e l dependent. Work by Vesic and Clough (196 8) has shown that the v a r i a t i o n of <J> with the logarithm of the mean normal stress i s nearly l i n e a r , as shown i n Figure 20. The f r i c t i o n angle at any stress l e v e l i s given by: 0 $ = cj^ - A<j>log(^) (3-10) a where: <J>^  = the f r i c t i o n angle at one Atm. Atj) = the change i n f r i c t i o n angle over one log cycle increase i n pressure 47 50 e 40 7 35 30 25 _ • Oense samples Loose samples i >A A A A * '~t *~&~£> • • • • • 1 -1 1 l i i 10 10 10 crm -kNm" 2 10 F i g u r e 2 0 . 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 C h a t t a h o o c h e e R i v e r S a n d t e s t e d a t d i f f e r e n t S t r e s s l e v e l s i n t h e T r i a x i a l A p p a r a t u s . ( A f t e r V e s i c a n d C l o u g h , 1 9 6 8 ) . 48 The volumetric response of the s o i l to increases i n the mean normal stress i s also nonlinear. C h a r a c t e r i s t i c mean nor-mal stress versus volumetric s t r a i n behaviour for loose Calais sand as tested by El-Sohby and Andrawes (19 72) i s shown i n Figure 21. The tangent modulus at any stress l e v e l i s given by: B t = V a ( p - ) m ( 3 - 1 X ) *a where: k = modulus number B m = modulus exponent The modulus number and exponent may be determined from i s o t r o p i c consolidation data. The volumetric s t r a i n may be related to the mean normal stress by: E = a(a ) ( 1 _ m ) (3-12) v m D i f f e r e n t i a t i n g y i e l d s : ds = a(l-m) ( a m ) " m (3-13) m or: da (a ) m B = _ J H = (3-14) t ds a (1-m) v ' v By equating 3-11 and 3-14 the modulus number and exponent can be determined: k B = - (3-15) a (1-m) ( p ) ( 1 " m ) a The values of 'a' and *m' can be determined by p l o t t i n g E V and on log-log paper. Taking logarithms results i n : l o g s v = log(a) + (1-m)log(a^) (3-16) So, as shown i n Figure 22, the intercept on the log-log p l o t gives the value 'a' and the slope i s (1-m). This method can be used with any E versus a data to obtain 'm', however, J v m i s o t r o p i c consolidation t e s t data s t a r t i n g from the unstrained 49 Mean Normal Stress vs. Volumetric S t r a i n Oj 2 4 6 8 10 12 Volumetric S t r a i n e x l O - 3 (%) v Figure 21. I s o t r o p i c C o n s o l i d a t i o n Data (After El-Sohby&Andrawes) 5 0 , 10-rn u •P CO o ' M •P I o > Volumetric S t r a i n vs. Mean Normal Stress a m 390 P a ( a m / P a ) 0.0015 0.525 0.525 1«.0 ~i i — i i i i i i i i — r — i — ' i 1 1 , ' f t 2.0 3.0 5 6 7 8910 20 30 40 50 70 100 200 Mean Normal S t r e s s , a m (psi) Figure 242 . Log-Log Volumetric S t r a i n vs. Mean Normal Stress 51 condition i s required to c o r r e c t l y determine 'a'. 3 .4 P l a s t i c Strains On application of stress, s o i l undergoes both e l a s t i c (recoverable) and p l a s t i c (irrecoverable) s t r a i n s . Under the loading conditions found i n the pressuremeter t e s t , i n which the stresses and strains continually increase, the a b i l i t y to d i f f e r e n t i a t e between the p l a s t i c s t r a i n and the e l a s t i c s t r a i n i s not c r u c i a l . However, fo r those cases i n which the load cycles and the s t r a i n increment reverses, the separation of the e l a s t i c and p l a s t i c s t r a i n accumulated during the loading i s important. Both p l a s t i c shear s t r a i n and p l a s t i c volumetric s t r a i n occur. P l a s t i c shear s t r a i n may be taken into account with reasonable accuracy by the use of a rebound shear modulus, as shown i n Figure 2 3. The program i s not as yet set up to corre c t l y handle the p l a s t i c s t r a i n on stress r e v e r s a l . P l a s t i c volume changes occur both from changes i n the mean normal stress and changes i n the shear stress. Most of the shear induced volume change i s p l a s t i c , while a portion of the consolidation volume change i s e l a s t i c , even for v i r g i n loading. The proportion of e l a s t i c volumetric s t r a i n from consolidation increases with the number of load cycles. 3.5 Shear Volume Coupling The phenomenon of shear volume coupling i s one of the predominant differences between s o i l and other engineering materials. Upon application of a pure shear stress, materials such as s t e e l w i l l undergo no volume changes. S o i l , on the other hand, may undergo s i g n i f i c a n t volume changes depending on the l e v e l of the stress. This shear volume coupling i s 5 2 S h e a r S t r e s s v s . S h e a r S t r a i n S h e a r S t r a i n , Y F i g u r e -2-3 . D e f i n i t i o n o f R e b o u n d M o d u l u s 53 shown i n the simple shear t e s t data i n Figure 12. The simple shear test i s not one of pure shear. Changes i n the mean normal stress do occur, so some of the volumetric s t r a i n i s due to consolidation. However, as can be seen i n the figure, even the loose samples undergo d i l a t i o n a f t e r the i n i t i a l con-t r a c t i o n , so a l l sands have some shear volume coupling. The phenomenom of shear volume coupling has been studied by Rowe (1962,1971). He has proposed a stress dilatancy theory that has been used by other researchers. His theory i s for the separation of p l a s t i c and e l a s t i c volume changes and cannot be d i r e c t l y used i n the present model. A simple modification i s proposed so that the theory can be incorporated i n the model. 3.5.1 Rowe's Stress Dilatancy Theory Rowe developed his theory from a study of pa r t i c u l a t e mechanics. The stress dilatancy equation i s based upon the conditions of equilibrium, l i m i t i n g f r i c t i o n and minimization of absorbed energy f o r r i g i d p a r t i c l e s i n s l i d i n g contact. Rowe looked at both the p l a s t i c s t r a i n s and the e l a s t i c s t r a i n s . In Rowe's theory, p l a s t i c s t r a i n s are due to two components. One component i s r i g i d p a r t i c l e s l i p and i s related to the p r i n c i p a l stress r a t i o . The other component i s p a r t i c l e crushing and i s related to the l e v e l of the mean normal stress. The model derived by Rowe i s l i m i t e d to cases i n which the major p r i n -c i p a l s t r a i n d i r e c t i o n does not reverse. Only Rowe's theory developed f o r s l i p strains w i l l be di s -cussed. The s l i p s t r a i n s so calculated w i l l be used to model shear induced volume changes. The stress dilatancy equation has been discussed by Rowe and by Gunaratne (19 81), so only the 54 major points and the modification w i l l be discussed i n this work. A number of assumptions underlie the theory. These are: 1) P a r t i c l e s are r i g i d . This means that e l a s t i c deformation of the p a r t i c l e s does not occur. 2) Deformations occur as a r e s u l t of r e l a t i v e motion between groups of p a r t i c l e s . The number of r o l l i n g contacts within the s o i l mass i s ne g l i g i b l e compared to the number of s l i d i n g contacts. The stress dilatancy equation as developed by Rowe i s : !§! 2 * f N ( a 2 d e 2 s + a 3 d e 3 s ) = tan (45 + - j J (3-17) where: a,, a 2 and c_ = p r i n c i p a l e f f e c t i v e stresses de^ g, d e 2 s a n d d e 3 S = P r i n c i p a l s l i p s t r a i n increments <t>£ = a f r i c t i o n angle which l i e s between a) and cb u cv where: cb = angle of f r i c t i o n of u p a r t i c l e s i n s l i d i n g <b = constant volume f r i c t i o n c v angle Under i d e a l conditions, i f a l l s l i d i n g takes place at the c r i t i c a l angle, B , and no r o l l i n g contacts occur, <j>_ w i l l be equal to <j>u. Since not a l l s l i d i n g takes place at B c, more energy i s absorbed and cb^  approaches 4 ) c v* Rowe has exper-imentally determined that f o r t T r i a x i a l compression a 2 = and extension = o"2 Dense sand, pre-peak cb^  = $ Dense sand, large st r a i n s cb^  = a) c v Loose sand, a l l stra i n s cb,- = cb ' Y f T cv Plane s t r a i n , a l l str a i n s cb^  =-<j> Under plane s t r a i n conditions, and assuming small s t r a i n s : de 2 = 0 = d £ 2 s + d e 2 e (3-18) 55 S l i p strains can occur i n the 2 d i r e c t i o n , but are balanced by e l a s t i c strains such that the t o t a l s t r a i n i s zero. I t w i l l be assumed that de„ = de„ =0. For th i s case: 2s 2e de = de., + de _ (3-19) vs Is 3s and the stress dilatancy equation can be written as: a, de 0 5-7 = d - ^ ) t a n 2 ( 4 5 + / ) (3-20) 3 Is This i s often written as: R = DK (3-21) °1 where: R = — = p r i n c i p a l stress r a t i o 3 de D = (1 - -3 ) = dilatancy factor 2 * f K = tan z(45 + -^ -) and for plane s t r a i n conditons: di, = d> * Y f T c v Brinch Hansen (195 8) proposed the introduction of the d i l a t i o n angle, which defines the relationship between the volumetric s t r a i n increment and the shear s t r a i n increment, Figure 24: d f v sinv = (3-22) where: v = the d i l a t i o n angle In terms of the shear-induced volume change: de sinv = - 3 - ^ (3-23) ^ s where: dy g = de^ g - de^g (3-24) The dilatancy factor i s related to the d i l a t i o n angle: D = 1- = t a n 2 (45 + (3-25) s Then, as proposed by Hughes (19 77) the stress dilatancy equation for plane s t r a i n becomes: 9 § tan (45 + -A t a n ( 4 5 + = 7 2~777T~Tcv. ( 3 " 2 6 ) tan (45 + V ) 5 6 Volumetric Strain vs. Shear Strain Figure 'Z'r D e f i n i t i o n of D i l a t i o n Angle 5 7 This i s a convenient form of the stress dilatancy equation. For developed f r i c t i o n angles below cb c v equation 3-26 predicts contraction. For cb ^ = c j > c v equation 3-26 predicts v = 0, or no shear induced volume change. The maximum d i l a t i o n rate depends on the values of <t>(jmax a n d <f>cv« A s the stress l e v e l i s increased, ^ ^ a x ^ s reduced and, hence, d i l a t i o n i s suppressed. When c b ^ = 0, the i s o t r o p i c consolidation case, 3-26 predicts that the d i l a t i o n angle, v = ~^ c v* When considering shear induced volume changes, t h i s i s obviously wrong, as no shear induced volume changes occur during i s o t r o p i c consolidation. In order to use Rowe's theory to model shear volume coup-l i n g , a simple modification i s proposed. The stress dilatancy equation w i l l be used to model volume strains due to changes i n shear stress. The bulk modulus term w i l l be used to model volume changes due to changes i n the mean normal stress. Various researchers (Varadarajan and Mishra (1980), Krishnamurthy, Nagaraj and Sridrahan (19 81), Tobita and Yanagisawa (19 80), Lindenberg and Konig (19 81) and Kokusho (1978))have published data f o r constant mean normal stress t r i -a x i a l t e s t s . Tests of this nature separate volume changes due to shear stress from volume changes due to mean normal stress. This data, Figure 25, shows that the shear induced contractions are small when compared to the shear induced expansions, and that the contractions may be neglected without introducing appreci-able errors into the model. The proposed modification i s to assume no shear induced volume change for c b ^ < c b c v and to use Rowe's stress dilatancy equation to determine the shear induced volume change for c b ^ , > < ! > „ . < , • -2.0H -1.8 -1.6, -1.4. 5-1.2 c-0. 8. •H U u i - 0 . 2 . o > 4-o A o + V X -o-Varadarajan Dense Sand, Krishnamurthy Loose Sand, a Loose Sand, Loose S a n d , Loose Sand, Dense Sand, Dense Sand, Dense Sand, Dense Sand, Volumetric S t r a i n vs. Developed F r i c t i o n Angle and Mishra 2(1980) a = 4kg/cm m ^' a * -a m = a m = a m = a m _ a m = a m = m Nagaraj, 2and Sridharan (1981) = 2kg/cm 2 3kg/cm 2 4kg/cm 2 4.8kg/cm 2kg/cm 2 3kg/cm 2 4kg/cm _ 4.8kg/cm T o b i t a and Yanagisawa 11980) Dense Sand, o = lkg/ciru Dense Sand, a m = 3kg/cm m ^' Lindenberg and Konig (1981) Loose Sand, a = 5kg/cnu Medium S a n d , a m — Dense Sand, a m m Kokusho (1978) Dense Sand, a m • • 5kg/cm 2 5kg/cm = 2kg/cm' 2 4 ^ 6 8 10*8*1? 1 4 ^ 6 48* 2^' 22* 2 4^2* *2*% ^ 5 ^ 3 7 * 4 F ^ 0.2-0.4 4 '8   v  * <6 *2*%Developed F r i c t i o n Angle (°) 34 36 — r " 42 Fi g u r e 2S • Volume Change i n Constant Mean Normal S t r e s s T r i a x i a l T e s t s 59 3.5.2 V a r i a t i o n of D i l a t i o n Angle with Stress Level Vaid e t a l (19 80) use simple shear t e s t data from Cole (196 7) to determine a l i n e a r v a r i a t i o n of the d i l a t i o n angle with v e r t i c a l confining s t r e s s , Figure 2<o. Rowe's stress dilatancy equation can also be used to determine a rel a t i o n s h i p between d i l a t i o n angle and st r e s s . The maximum d i l a t i o n angle f o r a given stress occurs when the peak f r i c t i o n angle i s mobilized. The peak f r i c t i o n angle, $, varies with stress as: o 4> = 4>, - A<Hog(-2) (3-10) *a Substituting 3-10 int o the stress dilatancy equation 3-26 y i e l d s : tan(45 + - ± 5 ^ ) tan (45 + 7) = T ~ T (3-34) 2 tan(45 + *§X) This equation i s pl o t t e d i n Figure 27 f o r various values of <f>_.# $ and As can be seen i n the f i g u r e , f o r the stress range of i n t e r e s t , a nearly l i n e a r v a r i a t i o n of the d i l a t i o n angle with the stress occurs. The greater the value of A<J> , the f a s t e r the decrease i n the d i l a t i o n angle, and the more curvature i n the r e l a t i o n . For a given value of A<j> and <}> , the value of <}> s h i f t s the curve v e r t i c a l l y , but does not cv change the slope of the r e l a t i o n . 6 0 F i g u r e D i l a t i o n A n g l e v s . " C o n f i n i n g P r e s s u r e 2 T 0 2 8( ( k P a ) 26. D e p e n d e n c e o f D i l a t i o n A n g l e o n R e l a t i v e D e n s i t y a n d C o n f i n i n g P r e s s u r e ( D a t a f r o m C o l e , 1 9 6 7 ) a ' , v o ^ 1 0 H o -H -P (0 rH •iH Q 0 H - 1 0 -D i l a t i o n A n g l e v s . M e a n N o r m a l S t r e s s }0 3 0 - Acb Y c v O 3 9 ° 4 ° 3 3 3 9 4 3 6 0 2 0 - A 3 9 4 3 0 •> • 3 9 2 3 3 cu rH & tn • 3 9 3 3 1 0 0 2 0 0 M e a n N o r m a l io~o" S t r e s s , o m 4 0 0 ( k P a ) 5 0 0 ido* F i g u r e ;2;7_ D e p e n d e n c e o f D i l a t i o n A n g l e o n M e a n N o r m a l S t r e s s  B a s e d o n R o w e ' s S t r e s s - D i l a t a n c y E q u a t i o n 61 Chapter 4 F i n i t e Element Program  4.1 Incremental Programs The numerical analysis of the pressuremeter i s based on the f i n i t e element computer Nonlinear S o i l Structure Inter-action Program (NLSSIP) developed by Byrne and Duncan (19 79) to analyze long span f l e x i b l e culverts. NLSSIP i s a revised version of the Berkeley computer program ISBILD. The program NLSSIP was revised by Gunaratne (1981) to allow the determination of shear induced volume changes and stress r e d i s t r i b u t i o n from elements that v i o l a t e the Mohr-Coulomb f a i l u r e c r i t e r i a . The thrust of.the present work was to refine the changes made by Gunaratne so that the program could handle general plane s t r a i n problems involving s o i l dilatancy, and more s p e c i f i c a l l y , to modify the program so that i t could e f f i c i e n t l y handle plane s t r a i n axisymmetric problems. The NLSSIP has been well documented by Byrne and Duncan and by Gunaratne. In order to minimize r e p e t i t i o n of t h i s previous work, the program w i l l only be discussed b r i e f l y , however the new changes w i l l be discussed i n more d e t a i l . A derivation of the f i n i t e element formulation of the new t r i a x i a l element i s i n Appendix A and a derivation of the new polar coordinate element i s i n Appendix B. The program uses an incremental d i l a t a n t l i n e a r e l a s t i c approach. An increment of load may be the addition of a construction s o i l layer, the addition of a structure, the app-l i c a t i o n of an external load, or an increment of pore pressure. The basic l o g i c of the program i s shown i n the s i m p l i f i e d flow-chart, Figure 28. The functions of the various subroutines are o 62 Figure 28 S i m p l i f i e d Flowchart of Mainline Program C o n t r o l Data 7 INPTNP yes Al CALBAN no BEAM LAYOUT CALBAN IG > 0 1 0 yes DERIVE FORMST (=> LN-1 LN«-LN+1 LN&NUMLD IG=0 "DERIVE FORMST ELAW - 0 CALBLK FVECT 63 NMI> 0 y e s IPF=LN o r sIDR=0 and ILR=(K POREF NUMEL=0 ISQUAD ADDSTF A l SYMBAN 6,4 MMM=0 and IDR=1 and no yes DILAT IDR=1 IT=2 IDR=0 NMI=0 ITR=0 MMM=0 IT=1 ^ STOP ^ 65 given i n Table I. In an incremental program, s o i l properties (moduli, f r i c t i o n angle, and Poisson's r a t i o ) , stresses and strains are evaluated before the load i s applied. The load i s applied and the element deformations and stress changes are determined using the s o i l properties at the beginning of the stress i n t e r v a l . The s o i l properties are reevaluated f o r the stress conditions at the midpoint of the predicted stress i n t e r v a l . The global s t i f f n e s s matrix i s evaluated using these s o i l properties and the load i s applied again. The new deformations, stresses and strains are then computed. For programs without shear volume coupling or stress r e d i s t r i b u t i o n usually only two i t e r a t i o n s are made. For a load step, i n NLSSIP, s o i l properties are printed for the mid-point of the stress i n t e r v a l , while displacements, strains and stresses are printed f o r the end of the i n t e r v a l . When shear volume coupling and stress r e d i s t r i b u t i o n are added to an incremental program, additional i t e r a t i o n s within each load step must be made to achieve the correct shear induced volume change and the stress r e d i s t r i b u t i o n . Within the new program, stress r e d i s t r i b u t i o n from the elements that v i o l a t e the Mohr-Coulomb f a i l u r e c r i t e r i o n i s performed f i r s t . The program w i l l i t e r a t e to a maximum of four times to achieve stress r e d i s t r i b u t i o n . S o i l properties at the end of the stress i n t e r v a l are used i n stress r e d i s t r i b u t i o n . After elements are brought back onto the f a i l u r e surface, shear volume coupling i s introduced. The program w i l l i t e r a t e twice, or to a maximum number s p e c i f i e d by the user, to achieve the correct shear induced volume change. I f convergence of 6'6 SUBROUTINE INPTNP TABLE I DESCRIPTION Reads and prints nodal point data, estab-lish e s a relationship between each nodal point degree of freedom and the corres-ponding equation number and sets the equa-tion number to zero for constrained bound-ary conditions. BEAM Reads and print s beam element data and calculates the i n t e r n a l force-displacement matrix and the element s t i f f n e s s matrix. CALBAN Calculates the band width of a group of elements LAYOUT Reads and print s the s o i l input data and computes and prints the i n i t i a l stresses and the i n i t i a l moduli values for the s o i l elements. ELAW Calculates moduli values for the s o i l elements i n accordance with the magnitudes of the stresses. FORMST Cal l s subroutine DERIVE to est a b l i s h strain-displacement matrices for the three types of s o i l elements. DERIVE Forms the s o i l strain-displacement matrix for the three element types. CALBLK Determines the number of elements and nodal points for the entire mesh, the number of elements and nodal points i n the pre-existing part and the newly added 6:7 TABLE I (continued) layers, the number of equations, the number of equations i n each block, and the number of blocks for each construction layer incre-ment or load increment. FVECT Calculates nodal point forces due to weights of added elements, reads concen-trated load data and/or boundary pressure data, pri n t s nodal point forces , and.isets up the force vector-POREF Reads element incremental porewater pres-sures and computes equivalent nodal forces, ISQUAD Formulates the constitutive equations, forms the element s t i f f n e s s matrix for each element, and forms the s t r a i n - d i s -placement matrix used i n stress c a l c u l a i tions. ADDSTF The t o t a l s t i f f n e s s matrix for each incre-ment i s formed two blocks at a time by making a pass through the element s t i f f n e s s matrices and adding the appropriate coef-f i c i e n t s . SYMBAN Solves the simultaneous equations repre-senting the s t i f f n e s s matrix and the load vector for nodal point displacements using the Gaussian :.elemination technique. ISRSLT Calculates stress increments and average stresses and evaluates the modulus of each s o i l element after the f i r s t i t e r a t i o n . [J For the subsequent i t e r a t i o n s ISRSLT c a l -culates the incremental and cumulative 6:8 TABLE I (continued) stresses and strains for each s o i l element to be used i n the next i t e r a t i o n , and i n -ternal forces i n s t r u c t u r a l elements. A l Calculates the element s t i f f n e s s matrix and generalized force-displacement matrix for beam elements. MODD Calculates the f l e x u r a l s t i f f n e s s of beam elements. TAPER Calculates the generalized force-displace-ment matrix for a section with l i n e a r l y varying f l e x u r a l s t i f f n e s s . LSHED Calculates the element force vector to red i s t r i b u t e stress from a..\soil element that v i o l a t e s the Mohr-Coulomb f a i l u r e c r i t e r i a . DILAT Calculates the element force vector to produce shear-volume coupling in s o i l elements. the volume change i s achieved within the s p e c i f i e d number of i t e r a t i o n s , the program w i l l p r i n t out data as i n NLSSIP. If convergence i s not achieved, the convergence of those elements v i o l a t i n g the convergence c r i t e r i o n for the volume change w i l l be printed along with the other r e s u l t s . The program w i l l then go on to the next load case. The number of i t e r a t i o n s to achieve stress r e d i s t r i b u t i o n and shear volume coupling can be minimized by using small load steps, e s p e c i a l l y near f a i l u r e when dense sand dilates rapidly. I t should be noted that since shear volume coupling follows stress r e d i s t r i b u t i o n , some elements may be forced to v i o l a t e the f a i l u r e c r i t e r i o n when the shear volume coupling i s per-formed. These elements w i l l be brought back onto the f a i l u r e surface on the next load step. Since shear volume coupling i s the primary feature of the program, i t was thought better to achieve the correct volume change and to possibly have a few elements marginally v i o l a t i n g the f a i l u r e c r i t e r i o n than to have a l l elements within the f a i l u r e surface, but to have the wrong volume change. The stress r e d i s t r i b u t i o n technique i s not required i n the present work, and since i t i s covered by Gunaratne, i t w i l l not be discussed i n t h i s work. However, care should be taken when using stress r e d i s t r i b u t i o n as i t can cause elements to v i o l a t e stress boundary conditions. 4.2 Shear Induced Volume Change Shear induced volume changes are handled i n an analogous manner to thermal volume changes i n thermoelasticity. The sand i s considered to be i s o t r o p i c with i t s incremental l i n e a r e las-70 t i c response modelled by two stress l e v e l dependent e l a s t i c parameters, the Youngs 1s modulus and the bulk modulus. The shear induced volume change i s modelled by an additional d i l a -tant parameter, forming a three parameter model. The volumetric response of the s o i l i s divided into two components, one due to changes i n the mean normal stress and the other due to changes i n the shear s t r e s s . Volume change due to changes i n the mean normal stress are modelled by the bulk modulus term. This volumetric s t r a i n w i l l have both elas-t i c and p l a s t i c components, the p l a s t i c component being mainly due to grain crushing and c l i p . C y c l i c hydrostatic loading w i l l s i g n i f i c a n t l y reduce the amount of p l a s t i c volume change occur-ring on loading. Within the program i t does not matter i f the problem i s for v i r g i n loading or for loading a f t e r c y c l i c loading. Provided that the bulk modulus term has been deter-.:..: mined from tests that reasonably model the f i e l d conditions, the program w i l l give reasonable r e s u l t s . Shear induced volume changes are handled by a d i l a t a n t parameter, D^. The exact form of Dfc does not a f f e c t the program' l o g i c . The d i l a t a n t parameter may be input i n any fashion that f i t s the t e s t data or theory used to model the shear volume coupling by making some minor changes to subroutine DILAT i n the program. The incremental shear induced volume change, Ae , i s vs obtained from: Ae = D Ay (4-1) vs t ' Rowe developed his theory using the p l a s t i c shear s t r a i n increment, however, as discussed below, using the t o t a l shear s t r a i n i n c r e m e n t i n t h e d e t e r m i n a t i o n o f t h e s h e a r i n d u c e d v o l u m e c h a n g e i n t r o d u c e s l i t t l e e r r o r . A c o n v e n i e n t m e t h o d o f s p e c i f y i n g t h e d i l a t a n t p a r a m e t e r i s w i t h t h e d i l a t i o n a n g l e : D = - s i n v ( 4 - 2 ) W i t h i n t h e p r o g r a m , t w o o p t i o n s a r e a v a i l a b l e f o r c a l c u l a t i o n o f t h e d i l a t a n t p a r a m e t e r , F i g u r e 2 9 . O p t i o n 1 u s e s a c o n s t a n t d i l a t i o n a n g l e a f t e r a c e r t a i n s h e a r s t r a i n . T h i s s p e c i f i c a t i o n o f t h e s h e a r i n d u c e d v o l u m e c h a n g e h a s b e e n u s e d b y H u g h e s e t a l ( 1 9 7 7 ) a n d b y B y r n e e t a l ( 1 9 8 0 ) . H u g h e s s p e c i f i e d t h e c o n s t a n t d i l a t i o n a n g l e a s t h e m a x i m u m d i l a t i o n a n g l e t h a t o c c u r s a t t h e p e a k s t r e s s r a t i o , a n d t h e s t a r t i n g s t r a i n , Y , a s t h e s t r a i n a t w h i c h y i e l d - f i r s t o c c u r s . B y r n e s p e c i f i e d t h e s t a r t i n g s t r a i n a s t w i c e t h e s t r a i n a t w h i c h t h e c o n s t a n t v o l u m e f r i c t i o n a n g l e i s f i r s t r e a c h e d . T h e s e c o n d o p t i o n o f s p e c i f y i n g D u s e s R o w e ' s s t r e s s d i l a t a n c y t h e o r y a s m o d i f i e d i n C h a p t e r 3 . T h e p r o g r a m u s e s R o w e ' s t h e o r y a b o v e a d e v e l o p e d f r i c t i o n a n g l e e q u a l t o t h e c o n s t a n t v o l u m e f r i c t i o n a n g l e . A t d e v e l o p e d f r i c t i o n a n g l e s b e l o w <J> , n o s h e a r i n d u c e d v o l u m e c h a n g e i s a s s u m e d t o o c c u r . r c v ^ T h i s a s s u m p t i o n m o d e l s t h e v o l u m e c h a n g e b e h a v i o u r w e l l , a s o s h o w n i n F i g u r e 3 6 . T h e f i g u r e c o m p a r e s t h e v o l u m e c h a n g e p r e d i c t e d b y O p t i o n 1 , b y O p t i o n 2 u s i n g e i t h e r t h e t o t a l s h e a f u s t r a i n i n c r e m e n t o r t h e p l a s t i c s h e a r s t r a i n i n c r e m e n t a n d R o w e ' s t h e o r y f r o m <J)^  = 0 . I t m a y b e s e e n t h a t O p t i o n 1 a n d O p t i o n 2 m o d e l t h e t e s t d a t a e q u a l l y w e l l . R o w e ' s t h e o r y f r o m ct)^ = 0 o v e r p r e d i c t s v o l u m e t r i c c o m p r e s s i o n , a s e x p e c t e d . T h e f i g u r e a l s o s h o w s t h a t u s i n g t h e t o t a l s h e a r s t r a i n i n c r e -m e n t i n s t e a d o f t h e p l a s t i c s h e a r s t r a i n i n c r e m e n t c h a n g e s t h e 7 2 S t r e s s R a t i o V e r s u s S h e a r S t r a i n O p t i o n 1 V o l u m e t r i c S t r a i n v s . S h e a r S t r a i n •P CO w cu u • p CO R - K C O / S t r e s s R a t i o v e r s u s S h e a r ^ S t r a i n <R s i h v = 2 ( a r c t a n ( —) - 4 5 ) L K Y Q S h e a r S t r a i n , y O p t i o n 2 V o l u m e t r i c S t r a i n v s . S h e a r S t r a i n F i g u r e 2 * 3 : . S p e c i f i c a t i o n o f S h e a r I n d u c e d V o l u m e C h a n g e 7 3 C N e u \ CD X m to I rH to 0) o c <D U CD m m • H Q cn cn cu U • P CO o\o c •iH c« M - P CO O • H M - P CU g r H O > S t r e s s D i f f e r e n c e v s . A x i a l S t r a i n o_ D a t a i o n 1 o n 2 , t o t a l s t r a i n o n 2 , p l a s t i c s t r a i n s S t r e s s - D i l a t a n c y T h e o r y , t o t a l s h e a r s t r a i n c r e m e n t —i—I—'—H 9 1 0 1 1 1 2 1 S t r a i n , e 1 (%) F i g u r e 3D . C o m p a r i s o n o f S h e a r I n d u c e d V o l u m e C h a n g e M o d e l s A n a l y s i s o f a , C o n s t a n t T r i a x i a l T e s t 74. prediction of volume changes very l i t t l e . Once d i l a t i o n begins the e l a s t i c strains are small compared to the p l a s t i c s t r a i n s , so l i t t l e e rror i s introduced i n Option 2 by using the t o t a l shear s t r a i n increment. The advantage of Option 2 over Option 1 i s that Option 2 suppresses d i l a t i o n as the stress l e v e l increases and the f r i c t i o n angle decreases. 4.3 F i n i t e Element Formulation of the Shear Induced Volume Change In thermoelasticity, the force system to prevent deforma-tions due to temperature change i s calculated for each element. The domain i s then subjected to a force system equal but op-posite to the system just calculated, allowing the deformation and stresses i n the domain due to the temperature change to be calculated. Within the present program, the force required to produce the required volume change i s calculated for each element. The forces are applied to the domain and the stresses and deforma-tions are determined. An i t e r a t i v e procedure i s required because application of forces to a l l elements w i l l cause redis-t r i b u t i o n of the stress and s t r a i n i n the f i n i t e element domain, i Figure 31 gives a s i m p l i f i e d flowchart f o r subroutine DILAT, which calculates the forces for shear volume coupling. The method of c a l c u l a t i n g the forces for an element w i l l be presented f o r the three element types. The plane s t r a i n polar coordinate element w i l l be discussed i n d e t a i l , while the other two elements w i l l be discussed b r i e f l y . Assuming that Option 2 i s used, the d i l a t i o n angle i s determined by: v = 2 (arctan ( (|F ' - J ) ) , R > K (4-3) 7 5 . Figure 31. Flowchart f o r Subroutine DILAT Control Data Rewind 9 Rewind 11 / Read(9) DISP(N,M) STRAIN (N,M)/ MMM=0 CHEKS(N)=0 I FX(N)=0 FY(N)=0 FZZ(N)=0 N — 1 N~-N+l N *NUMEL READ(11) STR(I,J) DILFAC(N)=0 y e s DELGAM=PRS3(N)-EPS(3) EE TA= 2(ARCTAN(/D)-45) DIL=DELGAM(DSIN(EETA)) I DELVOL= -DIL/2 .ye^l DELVOL= -DIL/3 yes DELEPS(N)=DELVOL 73 SIGM= - D S I G ( N ) ( 1 + v ) SIGM=- • D S I G ( N ) E P E = S I G M / B I E P P = E P T - E P E 1 R 1 = - E P P / D I L C H E K S ( N ) = D A B S ( 1 - R 1 ) MMM=MMM+1 D I F E V P = D E L E P S ( N ) E P P K - l K--K+1 K-^- 8 D I F E V P = D E L E P S ( N ) E P P K«-l K < 8 K«-K+l F ( K ) = - 2 * A R E A ( N ) * B U L K ( N ) * D I F E V P ( N ) * ( S T R ( 1 , K ) + S T R ( 2 , K ) ) K*-K+2 K £ 7 F ( K ) = B U L K ( N ) * D I F E V P * R R ( K ) * ( Z Z ( 2 ) - Z Z ( 1 ) ) F ( K ) = - 2 A R E A ( N ) * B U L K ( N ) * ( 1 + P O I S ( N ) ) * D I F E V P ( N ) * ( 2 S T R ( 1 , K ) + S T R ( 2 , K ) ) 7 8 DILFAC(N)=0 I i 4 M=INP(N,I) FX(M)+F(2*I-1)+FX(M) FY(M)=F(2*I)+FY(M) PRINT CHEKS(N) RETURN 7 9 o r v = 0 f o r R < K 2 * d w h e r e : R = t a n ( 4 5 + ^=-) K = t a n ( 4 5 + - ^ - ) T h e s h e a r i n d u c e d v o l u m e c h a n g e i s t h e n c a l c u l a t e d b y : d e = - d y s i n v v s ' U s i n g t h e a s s u m p t i o n o f i s o t r o p i c v o l u m e c h a n g e a n d p l a n e s t r a i n d e v s = d £ l s + d e 3 s = 2 d £ l s ( 4 ~ 4 ) I n t h e f i n i t e e l e m e n t f o r m u l a t i o n , t h e i n t e r n a l v i r t u a l w o r k d o n e b y a s e t o f v i r t u a l d i s p l a c e m e n t s i s : d W i = v / { e " } T { a } d V ( 4 _ 5 ) w h e r e : { e } = v i r t u a l s t r a i n v e c t o r { a } = s t r e s s v e c t o r b u t : { e } = ( B j { 6 } ( 4 - 6 ) w h e r e : ( B ) = s t r a i n d i s p l a c e m e n t m a t r i x { 6 } = v i r t u a l d i s p l a c e m e n t v e c t o r T h e i n t e r n a l v i r t u a l w o r k i s t h e r e f o r e : d W i = v - r { ^ } T ( B ) T { a } d v ( 4 ~ 7 ) T h e s t r e s s i s d e t e r m i n e d f r o m t h e s t r a i n : { 0 } = (D) { e } ( 4 - 8 ) w h e r e : (D) = s t r e s s - s t r a i n m a t r i x U p o n s u b s t i t u t i o n o f 4 - 8 i n t o 4 - 7 , t h e i n t e r n a l v i r t u a l w o r k i s : d W ± = v / { 6 } T ( B ) T ( D ) { e } d V ( 4 - 9 ) T h e e x t e r n a l v i r t u a l w o r k d o n e b y t h e e x t e r n a l f o r c e s u n d e r g o i n g t h e s e d i s p l a c e m e n t s i s : dW = { 6 } T { f } ( 4 - 1 0 ) e s w h e r e : { f } = n o d a l f o r c e v e c t o r s 8.0 {6 } T { f } = /{6} T(B) T(D) { e } d V O V E q u a t i n g t h e i n t e r n a l v i r t u a l w o r k a n d t h e e x t e r n a l w o r k : (4-11) T h e 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 , {6} x , i s c o n s t a n t o v e r t h e v o l u m e o f t h e e l e m e n t . A l s o , f o r a n e l e m e n t o f u n i t t h i c k n e s s , t : d V = t d A (4-12) C a n c e l l i n g t h e 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 i n 4-11 a n d s u b s t i t u -t i n g 4-12 i n t o 4-11 g i v e s a n e x p r e s s i o n f o r t h e n o d a l f o r c e s : (4-13) T h e s t r a i n v e c t o r f o r t h e p o l a r c o o r d i n a t e e l e m e n t i s : { f s } = A / ( B ) ( D ) { e } t d A d e ] < ( d e ( d e v s d e v s d e v s a n d t h e f o r c e v e c t o r w i l l b e : T 1 d e . { f s } = A / ( B ) (D) {J } ~2 1 1 0 v s t d A (4-14) T h e s t r e s s e s a n d s t r a i n s a r e e v a l u a t e d a t t h e c e n t r e o f t h e e l e m e n t . T h i s i s e s s e n t i a l l y o n e p o i n t G a u s s q u a d r a -t u r e . T h e f o r c e r e q u i r e d t o g i v e t h e c o r r e c t v o l u m e t r i c s t r a i n t h e n b e c o m e s : , .. m 1 d e „ • * ' - V S (4-15) { f } = W(B) (D){1}-^ S 0 Z T h e s t r a i n d i s p l a c e m e n t m a t r i x (B) a n d t h e s t r e s s - s t r a i n m a t r i x (D) a r e e v a l u a t e d a t t h e m i d p o i n t o f t h e e l e m e n t . T h e c o e f f i -c i e n t W i s a w e i g h t i n g f u n c t i o n f r o m t h e G a u s s q u a d r a t u r e . I t s h o u l d b e n o t e d h e r e t h a t f o r t h e a x i s y m m e t r i c c a s e , i f t h e f o r m o f t h e s h e a r i n d u c e d v o l u m e c h a n g e w a s n o t a s s u m e d t o b e i s o t r o p i c , o n l y t h e f o r m o f t h e r e q u i r e d s t r a i n v e c t o r m u s t b e c h a n g e d . F o r t h e p o l a r c o o r d i n a t e e l e m e n t i n F i g u r e 32, d u e t o t h e b o u n d a r y c o n d i t i o n o f n o d i s p l a c e m e n t s i n t h e c i r c u m f e r e n -t i a l d i r e c t i o n , f o r c e s a r e a p p l i e d o n l y i n t h e r a d i a l d i r e c t i o n . 81 2 F i g u r e 32 . Four Node P o l a r Coordinate F i n i t e Element 8 2 Only four forces need be calculated, forces 1, 3, 5 and 7 i n the figure. These forces are given by equation 4-15. The above formulation i s used on the f i r s t i t e r ation i n a load step. On the second and subsequent i t e r a t i o n s , i f the volume change i s not correct, a force vector i s calculated on the basis of the additional volume change required to achieve the correct shear volume coupling i n the element. This force vector i s added into the t o t a l load vector for the element. When ca l c u l a t i n g the stresses i n the element, the addi-t i o n a l strains introduced i n the element due to shear volume coupling must be taken into account. B a s i c a l l y : {Aa} = (D){Ae - Ae } (4-16) s where: {Aa} = stresses due to applied loads {Ae} = strains calculated from solution of f i n i t e element problem {Ae }= s t r a i n due to the forces c a l -s culated for shear volume coupling i f the element were completely free to move 1 Ae_ {Ae }= {1} vs S 0 Z Equation 4-16 i s used i n the form: 1 Ae {Aa} = (D)( B ) i { 6 } - (D){1}^ S- (4-17) 0 where: { 6 } = nodal displacements from solution of f i n i t e element problem with external loads and forces for shear volume coupling. 83 The two other element types are handled i n exactly the same manner. The only differences are that the s t r e s s - s t r a i n matrix, (D), i s d i f f e r e n t and the stresses and strains are for the x, y coordinate system. The forces must be applied i n both the x and y directions at each node, since for the general case the element w i l l not be f i x e d i n one of these directions. In the t r i a x i a l element, the s l i p s t r a i n i n any one d i r e c t i o n w i l l be one-third the t o t a l volumetric s l i p s t r a i n , while i n the plane s t r a i n element, the s l i p s t r a i n i n the x or y d i r e c t i o n w i l l be one-half of the t o t a l volumetric s l i p s t r a i n . Figure 33' shows the results of a = constant t r i a x i a l t e s t on medium dense sand. The t o t a l volume change has been divided into the components due to hydrostatic stress change and due to shear volume coupling. The curve fo r the shear induced volume change was used fo r the d i l a t a n t parameter, D . Figure 3^ shows the comparison between the program results and t e s t r e s u l t s . This comparison does not validate the theory of separating the volume changes into the hydrostatic and shear induced components, but i t does show that t h i s numerical tech-nique can be used to r e p l i c a t e t e s t r e s u l t s . This comparison i s a check on the l o g i c of the solution technique. The shear volume coupling technique was also checked against the closed form l i n e a r d i l a t a n t model developed i n Chapter 2. Figures 35, 3fc and 37 show the comparison of the closed form solution and the f i n i t e element c a l c u l a t i o n . Agree-ment i s excellent. .84 E u \ I I u c a> cn tn (V 16 14 12 10 8 6 4 2 Stress Difference vs. Strain Difference 1 <T3 = 4kg/cm* 1 ± 0 2 4 6 8 10 12 Strain Difference, e-e_ — % I 9 14 - 1.4 - 1 . 2 ° - 1 . 0 > 0 . 8 .E - 0 . 6 o ~ - 0 . 4 CO o - 0 . 2 E o > 0 0 . 2 0 . 4 0 . 6 0 . 8 Total Volumetric Strain Volumetric Strain due to Consolidation / Volumetric Strain due to . / Shear - Volume Coupling / / ' / ' / ' / / / Figure33: Constant CT3 Triaxial Test on Medium Dense Sand. 3 (After Varadarajan and Mishra, 1980). 8 5 L Stress Difference vs. Axial Straii 7 8 9 10 II 12 Axial Strain , € , - % Figure 34 ' 4 Finite Element Analysis of tx3 constant Tr iax ia l Test. 86 Figure 35. Comparison of Displacements i n Linear Dilatant Material from Closed Form Solution and F i n i t e Element Calculation 87 8 8 F i g u r e Comparison of Stresses i n L i n e a r D i l a t a n t M a t e r i a l from Closed Form S o l u t i o n and F i n i t e Element Program C a l c u l a t i o n . 8 9 4 . 4 S t r e s s R e d i s t r i b u t i o n F o r l o a d i n g t y p e p r o b l e m s , s t r e s s r e d i s t r i b u t i o n p r e s e n t s n o d i f f i c u l t y w h e n w o r k i n g w i t h p r i n c i p a l p l a n e s , a s i n t h e a x i s y m m e t r i c c a s e i n p o l a r c o o r d i n a t e s . I n l o a d i n g p r o b l e m s , t h e n e e d f o r s t r e s s r e d i s t r i b u t i o n o c c u r s b e c a u s e t h e s h e a r m o d u l u s c a n n o t u s u a l l y b e s e t t o z e r o w i t h o u t c a u s i n g n u m e r i c a l i n s t a b i l i t i e s . T h i s m e a n s t h a t e l e m e n t s t h a t h a v e f a i l e d , a n d s h o u l d n o t p i c k u p a n y m o r e s h e a r s t r e s s , w i l l b e f o r c e d t o p i c k u p s h e a r s t r e s s a n d h e n c e , w i l l v i o l a t e t h e f a i l u r e c r i -t e r i o n . P r i n c i p a l p l a n e s c a r r y n o s h e a r s t r e s s , s o w i t h i n t h e s o l u t i o n t e c h n i q u e , n o e q u a t i o n s m u s t b e s o l v e d t h a t w o u l d h a v e a z e r o d i a g o n a l i f t h e s h e a r m o d u l u s w e r e s e t t o z e r o . T h e r e -f o r e , f o r t h i s s p e c i a l c a s e , t h e s h e a r m o d u l u s m a y b e s e t t o z e r o . E l e m e n t s t h a t f a i l w i l l n o t p i c k u p s h e a r s t r e s s . E l e -m e n t s c a n b e f o r c e d u p t h e f a i l u r e s u r f a c e , f o l l o w i n g a z i g - z a g p a t h , a s s h o w n i n t h e p - q p l o t o f F i g u r e 3 8 . T h i s p a t h c a n b e m a d e t o c l o s e l y a p p r o a c h t h e f a i l u r e s u r f a c e b y u s i n g s m a l l l o a d i n c r e m e n t s . 9.0 Chapter 5  Results of F i n i t e Element Analysis 5.1 F i n i t e Element Mesh The response of sand to the expansion of a c y l i n d r i c a l cavity was investigated using the f i n i t e element mesh i n Figure 39. The mesh was extended to f i f t y cylinder r a d i i to ensure that the e f f e c t of the e l a s t i c springs, which extend the domain to i n f i n i t y , was minimized. S o i l properties used i n the analy-sis are shown i n Table I I . S o i l s t i f f n e s s properties were chosen for a s o i l of about 70% r e l a t i v e density. However, by changing the value of <J> C V/ the volumetric response of the s o i l can be changed from that of a dense sand to that of a loose sand. Four cases were looked at: no d i l a t i o n (the standard incremental l i n e a r e l a s t i c model) and three d i f f e r e n t values of 2 4 > c v « A11 i n i t i a l horizontal stress of 1.0kg/cm was used to investigate the e f f e c t s of changing dilatancy. This corresponds to a t e s t depth of about ten meters and a k G = 1.0. The e f f e c t of confining stress was investigated using i n i t i a l h orizontal stresses of 1.0, 1.5 and 2.0kg/cm2 and a 4> c v = 30°. 5.2 Pressure Expansion Curves Pressure expansion curves f o r the various s o i l conditions o are shown i n Figure 4p. The curves have the c h a r a c t e r i s t i c shape of f i e l d r e s u l t s . Figure 41; compares the f i n i t e element results with closed form e l a s t i c p l a s t i c r e s u l t s . The figure shows that the nonlinear e l a s t i c material i s much softer than the e l a s t i c p l a s t i c material. In assessing the s o i l model, for the s t r e s s - s t r a i n r e l a t i o n used, the shape of the r e l a t i o n has a major e f f e c t on the s o i l response, while the addition-of d i l a t i o n has a smaller e f f e c t . 92 Figure 33. F i n i t e Element Mesh 93 TABEE II  SOIL PROPERTIES k^ = 800. n = 0.5 k D = 800. m =0.5 r f = 0.9 c 0.0 Volume Change Charact e r i s t i c I n i t i a l Horizontal 2 "Stress (kg/cm ) No d i l a t i o n 1.0 1.0, 1.5, 2.0 1.0 1.0 94 95 9 6 5.3 Volumetric Response Volumetric response of the s o i l i s shown i n Figure 4 2. The nonlinear s o i l with no d i l a t i o n has only volume contraction. This i s the response that a l l f i n i t e element programs without shear volume coupling produce. As shown i n Figure 40, t h i s produces the s o f t e s t s o i l response and cannot possibly be used to estimate volume changes i n d i l a t a n t materials. Volumetric response varied from what would be termed a loose sand, with expansion beginning only a f t e r large s t r a i n s , to a medium dense sand with expansion beginning at low s t r a i n s . The three tests on sand with the same constant volume f r i c t i o n angle but varying i n i t i a l stress show the suppression of d i l a t i o n with increasing stress l e v e l . D i l a t i o n angles, determined from the slopes of the near l i n e a r portion of the volumetric s t r a i n versus shear s t r a i n curves, vary from v = -0.3° for the case with no d i l a t i o n to v = 6.7° f or' the s o i l with cb = 30° and an i n i t i a l cv 2 stress of l.Okg/cm . 5.4 Stress Paths The computed stress paths followed by the s o i l around the 5 expanding cavity are shown i n Figure 43. For a given i n i t i a l s t r ess, a l l s o i l s follow the same, stress path, regardless of volume change c h a r a c t e r i s t i c s . Also, a l l elements within a s o i l follow the same stress path, although each element i s at a d i f f e r e n t point on the stress path at any given instant. The e l a s t i c - p l a s t i c stress path i s also shown. At low stress l e v e l s the e l a s t i c - p l a s t i c path varies considerably from the nonlinear s o i l stress path, but once y i e l d occurs, a l l s o i l s on the same f a i l u r e surface follow the same stress path. Once y i e l d occurs, the stresses must obey the y i e l d c r i -98 R a d i a l Stress vs. C i r c u m f e r e n t i a l S t r e s s C i r c u m f e r e n t i a l S t r e s s , a (kg/cm ) Figure 43. 3 S t r e s s Paths Followed by S o i l Around Expanding  C y l i n d e r 99 t e r i o n . The equilibrium equation a f t e r y i e l d produces a single d i f f e r e n t i a l equation that may be solved with the stress bound-ary condition. S o i l properties, aside from the y i e l d condition, are not required to determine the stress f i e l d . A l l s o i l s with the same f r i c t i o n angle w i l l follow the same stress path a f t e r f a i l u r e , but the displacements and s t r a i n i n the s o i l s w i l l depend on the s o i l s t i f f n e s s properties. 5.5 Displacement F i e l d , Strains and Stresses Figure 44- shows a comparison of the displacement f i e l d s f o r 2 a t o t a l cylinder pressure of 6.25kg/cm and an i n i t i a l stress of 2 l.Okg/cm . The nonlinear material with no d i l a t i o n has the l a r -gest displacement at the cylinder face, but the displacements i n the d i l a t a n t material propagate further into the material. The difference that d i l a t i o n makes i s marginal beyond about 3 c y l i n -der r a d i i . A purely e l a s t i c material has much smaller displace-ments, while the e l a s t i c - p l a s t i c material has displacements that are smaller but of the same order of magnitude as the nonlinear material displacements. Displacements decay approximately as ^ in a l l cases. The nonlinear material with no d i l a t i o n has d i s -,1.03 placements decaying as — , while the d i l a t a n t material has ^.96 displacements decaying as — Figure 45 shows a comparison of the stra i n s f o r the same cylinder pressure. As with the displacements, the nondilatant s o i l has the highest shear s t r a i n at the cylinder face, but the d i l a t a n t sail has higher shear strains further away from the c y l i n -der. Again, beyond about 3 cylinder r a d i i , d i l a t i o n makes l i t t l e difference. S o i l n o n l i n e a r i t y i s the major factor changing the s o i l response. 1 0 0 6 Strain versus Radial Distance Figure 4 5 . Comparison of Strains in S o i l Around Expanding Cylinder 10.2. The s t r e s s f i e l d f o r t h i s c y l i n d e r p r e s s u r e i s shown i n F i g u r e 4 6 . As p r e v i o u s l y d i s c u s s e d , volume change makes no d i f f e r e n c e t o the n o n l i n e a r s o i l s t r e s s f i e l d once y i e l d has been reached. S t r e s s e s decay slower i n the n o n l i n e a r s o i l than i n the e l a s t i c m a t e r i a l , so h i g h e r shear s t r e s s e s e x i s t f u r t h e r from the c y l i n d e r i n the n o n l i n e a r s o i l . The s t r e s s e s i n the e l a s t i c - p l a s t i c m a t e r i a l are s l i g h t l y d i f f e r e n t from the non-l i n e a r m a t e r i a l because the f r i c t i o n angle does not change i n the e l a s t i c - p l a s t i c m a t e r i a l as i t does i n the n o n l i n e a r s o i l . The mean normal s t r e s s i n c r e a s e s s i g n i f i c a n t l y i n the non-l i n e a r s o i l s and i n the e l a s t i c - p l a s t i c m a t e r i a l . The i n c r e a s e i n mean normal s t r e s s s i g n i f i c a n t l y a f f e c t s the s o i l response, as both the bul k modulus and the i n i t i a l tangent shear modulus i n c r e a s e w i t h s t r e s s l e v e l i n the n o n l i n e a r m a t e r i a l . As the t e s t p r o g r e s s e s the s o i l becomes s t i f f e r f o r a g i v e n s t r e s s d i f f e r e n c e . 5.6 D i s t r i b u t i o n o f Shear Modulus i n F i n i t e Element Domain The d i s t r i b u t i o n o f shear modulus w i t h i n the s o i l i s shown i n F i g u r e 4 7 . In the n o n l i n e a r s o i l s volume change c h a r a c t e r i s t i c s have l i t t l e e f f e c t on the v a r i a t i o n o f shear modulus. T h i s i s as expected, as the s t r e s s d i s t r i b u t i o n w i t h i n the s o i l s are the same. The e l a s t i c m a t e r i a l has a c o n s t a n t shear modulus w h i l e the e l a s t i c - p l a s t i c m a t e r i a l has zero shear modulus i n the p l a s t i c zone and a con s t a n t shear modulus i n . t h e e l a s t i c r e g i o n . The n o n l i n e a r m a t e r i a l shear modulus i n c r e a s e s s l o w l y w i t h r a d i a l d i s t a n c e towards the e l a s -t i c v a l u e . Figure 46. Comparison of Stresses in S o i l Around Expanding Cylinder 1 0 5 5 . 7 Determination of F r i c t i o n Angle and D i l a t i o n Angle Computed log-log pressure expansion curves are shown i n & Figure 46. A summary of the values of f r i c t i o n angle and d i l a -t ion angle calculated by Hughes' method i s shown i n Table I I I . The method gives good agreement with the results of the f i n i t e element analysis. The d i l a t i o n determined from the f i n i t e element analysis was taken from the slope of the volumetric Z s t r a i n versus shear s t r a i n curves shown i n Figure 42. The f r i c t i o n angles were determined from an average stress r a t i o near the end of the test. The s o i l does not have a unique f r i c t i o n angle, as the f r i c t i o n angle changes with stress l e v e l . The f r i c t i o n angle continually decreases as the t e s t progresses, but the decrease i n the angle i s small at higher s t r a i n s , when the mean normal stress does not change rapidly with s t r a i n . The e l a s t i c - p l a s t i c material plots as a s t r a i g h t l i n e at a l l strains on a log-log pressure expansion p l o t . 5 . 8 Stress-Strain Curves St r e s s - s t r a i n curves can be developed from the pressure ex-pansion curves. An example of the method i s shown i n Figure 4-9. The value of f r i c t i o n angle i s determined by Hughes' method. Over most of the test the s o i l i s at f a i l u r e with a „ = ho , 0 R r ' where R i s calculated from the f r i c t i o n angle. The value of R changes as strains increase, but the error i n using a constant value of the stress r a t i o i s small. At low s t r a i n the s o i l may be approximated as an e l a s t i c material, i n which case : A a Q = Aa o r This e l a s t i c response i s shown i n the figure. The s o i l i s assumed to y i e l d at the s t r a i n at which the e l a s t i c a Q curve intersects the a A curve determined from the r a d i a l stress and 1,06 Radial E f f e c t i v e Stress vs. Circumferential Strain 5.0 0. 398 <j> = 33 r c v P i = l.Okg/cm^ 1-rftl 0.05 0.1 — I — ' I ' M — 0.5 l'. 0 10.0, 5 . 0 Circumferential Strain, e (%) " i — l i i • i I 5.0 l'0. 0 Radial E f f e c t i v e Stress vs. Circumferential Strain <f> = 36* cv P i = l.Okg/cm 10. On 5.0J ' ' ' d.i ' ' ' I ' . O R ~ Circumferential Strain, e (%) s:o' ' '10.0 l-r& Radial E f f e c t i v e Stress vs. Circumferential Strain 0. 371 No d i l a t i o n 1.Okg/cm^ ' ' i i I 0.05 0.1 ' ' — " » — I ' ' ' 1 | ' " 1 | I I 1 1 | °- 5 1-0 5.0 10.0 Circumferential Strain, e (%) Figure 48. Log-Log Pressure Expansion Curves from F.E. Analysis IG:7. lO.O-i 5.0-1 l . - e -Radial E f f e c t i v e Stress vs. Circumferential Strain *cv = 3 0 1.0kg/cm i i i • i 0.05 O i l lO.O-i 5.0-1 1 « ' 1—i i i t t r— 0'. 5 l'. 0 Circumferential Strain, e (%) 5'.0 10.0 Radial E f f e c t i v e Stress vs. Circumferential *cv = 3 0 ° 1.5kg/cm' 5.0 1 : ° I ' 1 1 ' I " 1 1 « 1 i i i i I i r ° - 0 5 0.1 0.5 1.0 1 0 0 Circumferential Strain, e (%) Radial E f f e c t i v e Stress vs. Circuj&ftSrential Strain 5.0-^ - u 10.0 l . - a E l a s t i c - P l a s t i c Incompressible i i i i i 0.05 0.1 1 I * ' ' 1 1 0.5 1.0 i i 1—i—•—r—r 1 5.0 10.0 Circumferential Strain, e (%) Figure 46 Log-Log Pressure Expansion Curves from F; E . ^Anal Y s i s -1 6 8 TABLE: H I Comparison of F r i c t i o n Angles and D i l a t i o n Angles Volume Hughes et a l F i n i t e Element Results Change Method cv <\> cv No D i l a t i o n 36.4 -0.18 36.5 •0.3 cb = 36. cv 36. 7 0.90 36.5 0.8 cb = 33. cv 36. 3 4.0 36. 3 3.5 cb = 30. cv 35. 1 6.0 36.1 6.7 109: S t r e s s v e r s u s C i r c u m f e r e n t i a l S t r a i n 4 5 6 S t r a i n , e . (%) F i g u r e 4 9 . C a l c u l a t i o n o f S h e a r S t r e s s F r o m P r e s s u r e - E x p a n s C u r v e 1M) a s s u m i n g t h e s o i l w a s a t a c o n s t a n t s t r e s s r a t i o . T h e v a r i a t i o n o f Og w i t h s t r a i n f o l l o w s t h e e l a s t i c c u r v e t o t h e i n t e r s e c t i o n , t h e n f o l l o w s t h e t o t h e e n d o f t h e t e s t . T h e v a r i a t i o n o f R r a Q w i t h s t r a i n a s d e t e r m i n e d f r o m t h e f i n i t e e l e m e n t a n a l y s i s i s s h o w n i n t h e f i g u r e f o r c o m p a r i s o n . T h e r e i s s o m e e r r o r n e a r t h e y i e l d s t r a i n , b u t o v e r a l l a g r e e m e n t i s g o o d . T h e s h e a r s t r e s s d e t e r m i n e d a s x = (a - o „ ) / 2 i s a l s o s h o w n i n t h e m a x r 9 ' f i g u r e . T h e c u r v e s o f a r d e v e l o p e d h a s T v e r s u s e „ . T h e s t r e s s e m a x G s t r a i n c u r v e o f x v e r s u s y i s d e s i r e d . A n a p p r o x i m a t i o n o f t h e s h e a r s t r a i n c a n b e m a d e f r o m t h e c i r c u m f e r e n t i a l s t r a i n b y a s s u m i n g t h a t a l l t h e v o l u m e c h a n g e i s s h e a r i n d u c e d , i n w h i c h c a s e : d e = d e + d e „ v r 0 a n d , d y = d e r - deQ b u t , d e v = - d y s i n v T h e r e f o r e : d y = - 2 e Q / ( l + s i n v ) ( 5 - 1 ) T h e r e l a t i o n b e t w e e n y a n d 2 e Q / ( l + s i n v ) d e t e r m i n e d f r o m t h e 9 o f i n i t e e l e m e n t a n a l y s i s i s s h o w n i n F i g u r e 5 0 . A s c a n b e s e e n , a s m a l l e r r o r i s i n t r o d u c e d b y n e g l e c t i n g t h e v o l u m e t r i c s t r a i n d u e t o i n c r e a s i n g m e a n n o r m a l s t r e s s . A l s o p l o t t e d o n t h e f i g u r e i s a c o m p a r i s o n o f y a n d 2 e 2 - G r e a t e r e r r o r w o u l d b e i n t r o d u c e d b y a s s u m i n g y = 2 e Q , a s i n t h e e l a s t i c c a s e . S h e a r s t r e s s - s h e a r s t r a i n r e l a t i o n s c a l c u l a t e d b y t h e a b o v e m e t h o d a r e s h o w n i n F i g u r e 51 . T h e c u r v e s d e t e r m i n e d f r o m t h e f i n i t e e l e m e n t a n a l y s i s a r e s h o w n f o r c o m p a r i s o n . A g r e e -m e n t i s v e r y g o o d . T h e e r r o r w o u l d n o t b e s i g n i f i c a n t i n p r a c -t i c e . i l l : S train 112 113 5.9 H y p e r b o l i c S t r e s s - S t r a i n P a r a m e t e r s H y p e r b o l i c p a r a m e t e r s a r e u s u a l l y d e t e r m i n e d u s i n g a t r a n s f o r m e d * p l o t o f y / x v e r s u s y . A n a p p r o x i m a t e l y l i n e a r c u r v e r e s u l t s , a n d t h e i n i t i a l t a n g e n t s h e a r m o d u l u s a n d m a x i -mum s h e a r s t r e s s a r e d e t e r m i n e d f r o m t h e i n t e r c e p t a n d s l o p e o f t h e c u r v e . T h i s t r a n s f o r m e d p l o t w a s m a d e f o r t h e t h r e e z h i g h l y d i l a t a n t s a n d s , F i g u r e 5 2 . T h e r e s u l t i n g c u r v e s a r e h i g h l y n o n l i n e a r . A s t r a i g h t l i n e f i t t o t h e u p p e r p o r t i o n o f t h e c u r v e s p r o d u c e s u n r e a l i s t i c a l l y l o w e s t i m a t e s o f t h e i n i t i a l t a n g e n t s h e a r m o d u l u s . I t i s t h o u g h t t h a t t h e r e a s o n f o r t h e n o n l i n e a r b e h a v i o u r i s t h e i n c r e a s e i n m e a n n o r m a l s t r e s s d u r i n g t h e t e s t . T h e v a r i a t i o n o f t h e m e a n n o r m a l s t r e s s w i t h s t r a i n i s s h o w n i n •3 F i g u r e 5 3 . F o r a g i v e n t e s t a l l s o i l e l e m e n t s l i e o n t h e c u r v e , t h o u g h e a c h i s a t a d i f f e r e n t p o i n t a t a n y t i m e . T h e i n c r e a s e i n m e a n n o r m a l s t r e s s i n c r e a s e s b o t h t h e b u l k m o d u l u s a n d t h e i n i t i a l t a n g e n t s h e a r m o d u l u s , F i g u r e 5 4 a n d 5.5. T h e s o i l r e s p o n d s t o a p r e s s u r e i n c r e a s e b y m o v i n g a c r o s s a s e t o f s h e a r s t r e s s v e r s u s s h e a r s t r a i n c u r v e s , e a c h w i t h a h i g h e r i n i t i a l m o d u l u s . D e t e r m i n i n g t h e v a r i a t i o n o f t h e i n i t i a l m o d u l u s w i t h m e a n n o r m a l s t r e s s o n l y f r o m t h e p r e s s u r e e x p a n s i o n c u r v e i s n o t p o s s i b l e f o r t h i s c o m p l i c a t e d s t r e s s p a t h . 5 . 1 0 D e t e r m i n i n g I n i t i a l S t r e s s a n d S h e a r M o d u l u s A n a l t e r n a t e m e t h o d o f d e t e r m i n i n g t h e i n i t i a l t a n g e n t s h e a r m o d u l u s , G ^ , w a s d e v e l o p e d . I n a n e l a s t i c m a t e r i a l t h e s l o p e o f t h e p r e s s u r e e x p a n s i o n c u r v e i s 2 G . A t l o w s t r a i n s s o i l m a y b e a p p r o x i m a t e d a s a n e l a s t i c m a t e r i a l . B y p l o t t i n g P - P i - ~ — a g a i n s t e , t h e i n t e r c e p t i s G ^ . T h e c u r v e i s n o n l i n e a r a t H -cQ Hi CD Ui i-3 cn Hi o g H O r t r t O a CD r t CD r i 3 H -(D ffi >•< t 3 (D r< tr o H -O CU 1-i fU 3 CD r t CD 1-i cn o tn 7 4 6 -4 CN u - 5 4 >-3 4 2 H 1 1 cv = 30 A o = 1.0 kg/cm p_^  = 1.5 kg/cm"2 p. = 2.0 kg/cm^ O A « O y/x versus y "T» 3 T -5 r 8 10 11 12 13 IT ll 18 15 2u~ Shear Strain, y (%) 4 Figure 54. Variation of Bulk Modulus with Strain Level 117; c •H -fti U -P CO rH rd •H -P CD SH CD m g u SH •H u cn rs cn 5-1 CD > cn rH o a u CO CD X! CO o o rH o o < O CN CN CN o £ E g ro O O O \ \ \ II in tn tn ^ ^ * > o in o o • -S- rH rH CN rH II || II •H 0 -H -H -rH CO Cu Cu Cu • <3 0 o 00 o KO O M O « « o o CN 2 o - • o 00 o o O m o o ro O CN o\o C --H cd i n +J CO rH Figure 5 5 . Variation of Shear Modulus with Strain Level 118 h i g h s t r a i n s , b u t a t s t r a i n s l e s s t h a n 1%, i t i s a p p r o x i m a t e l y (c, l i n e a r . T h r e e c a s e s a r e p l o t t e d i n F i g u r e 5 6 . T h e c a l c u l a t e d v a l u e s o f a r e i n g o o d a g r e e m e n t w i t h t h e i n p u t v a l u e s . P l o t t i n g ^2e* a 9 " a i n s t £ Q 1 S n o t a n a t t e m p t t o p r o d u c e 8 h y p e r b o l i c p a r a m e t e r s . I n s t e a d i t i s a m e t h o d t o d e t e r m i n e t h e i n i t i a l t a n g e n t s h e a r m o d u l u s . T h e v a l u e ^ 9 ~ ^ i i s a n e q u i v a l e n t e l a s t i c s h e a r m o d u l u s t h a t c o n t i n u a l l y d e c r e a s e s w i t h s t r a i n . T o d e t e r m i n e t h e s h e a r m o d u l u s , t h e c u r v e s p e c -i f y i n g t h e v a r i a t i o n o f t h e e q u i v a l e n t e l a s t i c m o d u l u s w i t h s t r a i n i s p r o j e c t e d t o t h e i n t e r c e p t w i t h t h e z e r o s t r a i n a x i s . T h i s m e t h o d c o u l d b e u s e d w i t h o t h e r v a r i a b l e s b e i n g s u b s t i t u t e d f o r t h e c i r c u m f e r e n t i a l s t r a i n , f o r e x a m p l e , t h e e q u i v a l e n t e l a s t i c s h e a r m o d u l u s c o u l d b e p l o t t e d a g a i n s t c y l i n d e r p r e s s u r e s i n c e t h e m o d u l u s d e c r e a s e s a s t h e p r e s s u r e i n c r e a s e s . T h i s i s a d i f f i c u l t m e t h o d t o u s e w i t h f i e l d d a t a . U s u a l l y v e r y f e w r e a d i n g s a r e t a k e n a t l o w s t r a i n s . M i s s i n g t h e l o w s t r a i n r e a d i n g s c a n c a u s e t h e i n i t i a l m o d u l u s t o b e u n d e r -e s t i m a t e d . A c o m p a r i s o n o f s h e a r m o d u l i f r o m f i v e t e s t s a t M a c D o n a l d ' s t e s t s i t e o n S e a I s l a n d i s s h o w n i n T a b l e I V . A l s o s h o w n a r e t h e m o d u l i c a l c u l a t e d f r o m t h e i n i t i a l s l o p e o f t h e p r e s s u r e e x p a n s i o n c u r v e s a n d t h e m o d u l u s c a l c u l a t e d f r o m t h e r e b o u n d p o r t i o n o f t h e c u r v e . T h e p r e d i c t e d m o d u l u s s h o u l d a g r e e w i t h t h e m o d u l u s f r o m t h e i n i t i a l s l o p e o f t h e p r e s s u r e e x p a n s i o n c u r v e , b u t a g r e e m e n t i s v a r i a b l e . A n o t h e r p r o b l e m i s t h e d e t e r m i n a t i o n o f t h e i n i t i a l h o r -i z o n a t a l s t r e s s , p . . T h e c a l c u l a t i o n o f G . i s s e n s i t i v e t o t h e c h o s e n v a l u e o f p . . C h o o s i n g a v a l u e o f p . t o o s m a l l w i l l l x 7 c a u s e G.. t o b e o v e r p r e d i c t e d , a s s h o w n i n F i g u r e 5 7 . C h o o s i n g 1.1.9 700 600 -\ CM E o \ Cn C D CM 500 1 400 •H 300 J i a, 200 H 100 J ( p - P i ) / 2 e Q vs. cQ S o i l = 30 cv 4 ? i = l'0kg/cm* • P i = l-5kg/cm^ o = 2.0kg/cm' G = 410kg/cm G = 365kg/cm" G = 2 85kg/cm' O T r i i i 1 1 1 1 1 1 1— 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 S t r a i n , e Q (%) F i g u r e 5fe. C a l c u l a t i o n of I n i t i a l Shear Modulus From P r e s s u r e -meter Data 1-2-0 TABLE IV Comparison of Calculated I n i t i a l Shear Modulus Values Pressuremeter Tests at Sea Island Test Site Test Depth I n i t i a l Shear Modulus, Number (m) I n i t i a l Unload New Slope of Portion Method p-e Curve. of Curve 2 (kg/cm ) 2 (kg/cm ) (kg/cm2) 1 3.0 110 130 110 2 3.8 75 260 130 3 4.6 400 170 60 4 5.3 70 280 130 5 6.3 50 300 280 121 700 600 _n ( p - P i ) / 2 e 0 vs. e ( G = 660kg/cm' e ^ 500 Cn 400 HN CD CJ CM •H 300 Cu i Cu G = 410kg/cm S o i l <|) =30 u cv • = 2 . Okg/cm' + p. = 1.8kg/cm' xp^ = 2.2kg/cm' 200 H + loo A — i 1 1 1 1 1 1 1 1 1 1 1 — ' 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 S t r a i n , £ Q (%) F i g u r e 5.7 S e n s i t i v i t y of C a l c u l a t e d Shear Modulus t o I n i t i a l E stimate o f H o r i z o n t a l S o i l S t r e s s 122 a value of too large causes the equivalent e l a s t i c shear modulus to f i r s t increase then decrease. This behaviour can be used to estimate p^, since p^ w i l l be no larger than the • value that causes the equivalent e l a s t i c shear modulus to i n -crease with s t r a i n . A comparison of p. from the above method with the i n i t i a l stress predicted by Hughes i s shown i n Table V. Agreement i s f a i r l y good. A better estimate of both the shear modulus and the i n -i t i a l horizontal.stress could be made i f more data was taken at low st r a i n s . However, th i s would be pressing the l i m i t of s e n s i t i v i t y of measurement of the cylinder displacement and possibly would not produce r e l i a b l e and reproducable r e s u l t s . 123 TABLE V Comparison of I n i t i a l Horizontal Stresses Test Number I n i t i a l Horizontal Stress Intercept of p-c curve (kg/cm^) New Method-(kg/cm2) 1 0.45 J 0.62 2 0.87 ' 0. 76 3 0.40 0.05 4 2.22 2.16 5 2.13 1.96 124 5.11 Determination of Hyperbolic S o i l Model Parameters Shear modulus parameters, k G and n, can be determined i f te s t data i s available from a minimum of three depths within a s o i l horizon. Parameters are determined by p l o t t i n g the non-dimensional i n i t i a l shear modulus values versus the i n - s i t u stress on a log-log p l o t . The parameter k^ i s the intercept of the l i n e at P i / D = 1.0, and the parameter n i s the slope of the l i n e . Figure 5$. shows the p l o t for the results from the f i n i t e element analysis. The calculated values of k„ and n agree with the input values. Estimates of the Young's modulus parameter k„ can be made hi from k_,. The Poisson's r a t i o for s o i l usually i s i n the range 0.0 to 0.5, which implies: 2 k G " k E " 3 k G The bulk modulus parameter, k_,, w i l l be approximately equal to k„, and the parameter 'm' i s usually assumed to be equal to 'n'. These estimates of the s o i l properties would be adequate for preliminary analyses and parametric studies of s o i l response. However, c r i t i c a l structures should have additional testing of the s o i l s to determine the properties more accurately. 125 1 0 0 0 - I 0 Figure 58 . Determination of S o i l S t i f f n e s s Parameters For Hyperbolic Stress-Strain Relation o 126 C h a p t e r 6  S u m m a r y a n d C o n c l u s i o n A f i n i t e e l e m e n t a n a l y s i s o f t h e r e s p o n s e o f s o i l t o t h e e x p a n s i o n o f a c y l i n d r i c a l c a v i t y w a s p e r f o r m e d . A p l a n e s t r a i n f i n i t e e l e m e n t i n p o l a r c o o r d i n a t e s w a s d e v e l o p e d t o a l l o w e f f i c i e n t s o l u t i o n o f t h e a x i s y m m e t r i c p r o b l e m . T h e p o l a r c o o r d i n a t e e l e m e n t g i v e s e x c e l l e n t a g r e e m e n t w i t h c l o s e d f o r m e l a s t i c s o l u t i o n s . T h e v o l u m e t r i c r e s p o n s e o f a s o i l s u b j e c t e d t o a s h e a r s t r e s s i s o n e o f t h e p r e d o m i n a n t d i f f e r e n c e s b e t w e e n s o i l a n d o t h e r m a t e r i a l s . T h e v o l u m e t r i c r e s p o n s e c a n b e a d e q u a t e l y m o d e l l e d w i t h a m o d i f i e d f o r m o f R o w e ' s s t r e s s - d i l a t a n c y e q u a -t i o n . T h e s h e a r - v o l u m e c o u p l i n g m o d e l u s e s R o w e ' s t h e o r y a b o v e a d e v e l o p e d f r i c t i o n a n g l e e q u a l t o $ . A t d e v e l o p e d f r i c t i o n a n g l e s b e l o w $ c v n o s h e a r i n d u c e d v o l u m e c h a n g e i s a s s u m e d t o o c c u r . A n e w t r i a x i a l f i n i t e e l e m e n t i n c o r p o r a t i n g t h i s s h e a r - v o l u m e c o u p l i n g m o d e l g i v e s g o o d a g r e e m e n t w i t h t r i a x i a l t e s t d a t a . T h e f i n i t e e l e m e n t a n a l y s i s s h o w s t h a t t h e n o n l i n e a r i t y o f t h e s o i l s t r e s s - s t r a i n r e l a t i o n s i g n i f i c a n t l y a f f e c t s t h e s o i l r e s p o n s e t o t h e p r e s s u r e m e t e r . A n o n l i n e a r i n c r e m e n t a l e l a s t i c s o i l i s m u c h s o f t e r t h a n a n e q u i v a l e n t e l a s t i c - p l a s t i c m a t e r i a l . D i l a t i o n t e n d s t o s t i f f e n t h e s o i l , b u t e v e n a h i g h l y d i l a t a n t s o i l i s n o t a s s t i f f a s t h e e l a s t i c - p l a s t i c m a t e r i a l . T h e v o l u m e t r i c r e s p o n s e o f t h e s o i l d o e s n o t a f f e c t t h e s t r e s s e s o n c e y i e l d h a s o c c u r r e d . A l l s o i l s w i t h t h e s a m e f r i c t i o n a n g l e w i l l h a v e t h e s a m e s t r e s s d i s t r i b u t i o n . H o w e v e r , t h e v o l u m e t r i c r e s p o n s e d o e s a f f e c t t h e s t r a i n d i s t r i b u t i o n a r o u n d t h e p r e s s u r e m e t e r . 127 T h e m e t h o d o f d e t e r m i n i n g f r i c t i o n a n g l e s a n d d i l a t i o n a n g l e s p r o p o s e d b y H u g h e s e t a l g i v e s g o o d a g r e e m e n t w i t h f i n i t e e l e m e n t r e s u l t s . H o w e v e r , i n p r a c t i c e , u n l e s s c b c v i s d e t e r m i n e d i n l a b o r a t o r y t e s t s , t h e a c c u r a c y o f t h e c a l -c u l a t e d v a l u e s o f cb a n d v w i l l d e p e n d o n t h e a c c u r a c y o f t h e e s t i m a t e o f cb Y c v T h e i n i t i a l s h e a r m o d u l u s c a n b e e s t i m a t e d f r o m p r e s s u r e -m e t e r r e s u l t s . A m e t h o d o f d e t e r m i n i n g t h e s h e a r m o d u l u s w a s p r o p o s e d . T h i s m e t h o d o f p l o t t i n g a n e q u i v a l e n t e l a s t i c s h e a r m o d u l u s a g a i n s t t h e s t r a i n g i v e s g o o d a g r e e m e n t w i t h f i n i t e e l e m e n t r e s u l t s . T h e m e t h o d i s d i f f i c u l t t o u s e i n p r a c t i c e . I t r e q u i r e s a n e s t i m a t e o f t h e i n i t i a l h o r i z o n t a l s t r e s s a n d s e v e r a l p r e s s u r e r e a d i n g s a t s t r a i n s l e s s t h a n o n e p e r c e n t . T h e p r o p o s e d m e t h o d c a n b e u s e d t o e s t i m a t e t h e i n i t i a l h o r i z o n t a l s o i l s t r e s s . I n m o s t c a s e s e s t i m a t e s o f t h e i n i t i a l s t r e s s m a d e b y t h i s m e t h o d a r e i n f a i r a g r e e m e n t w i t h t h e l i f t o f f p r e s s u r e c o r r e c t e d f o r t h e s t a t i c w a t e r p r e s s u r e . U s i n g t h e e s t i m a t e o f t h e i n i t i a l s t r e s s , a n i n i t i a l s h e a r m o d u l u s w a s c a l c u l a t e d . C o m p a r i n g t h i s v a l u e o f s h e a r m o d u l u s w i t h t h e s h e a r m o d u l u s c a l c u l a t e d f r o m t h e i n i t i a l s l o p e o f t h e p r e s s u r e e x p a n s i o n c u r v e t o w i t h t h e r e b o u n d p o r t i o n o f t h e p r e s s u r e e x p a n s i o n c u r v e d o e s n o t s h o w g o o d a g r e e m e n t . H o w e v e r , a g r e e -m e n t i s b e t t e r w i t h t h e r e b o u n d m o d u l u s t h a n w i t h t h e i n i t i a l m o d u l u s . I n t h e o r y t h e p r e s s u r e m e t e r i s a n a t t r a c t i v e m e t h o d o f d e t e r m i n i n g s t r e n g t h a n d d e f o r m a t i o n c h a r a c t e r i s t i c s o f s o i l . 128^ However, s o i l i s extremely complex and present analysis of the pressuremeter cannot account e n t i r e l y for t h i s complexity. Even i f s o i l response could be completely quantified, the pressuremeter could not give a l l the s o i l properties. There are too many s o i l variables to uniquely determine from the lim i t e d amount of data obtained i n the pressuremeter t e s t . E i ther other tests must be performed or some empirical cor-r e l a t i o n must be used to analyze pressuremeter data. R E F E R E N C E S Baguelin, F., Jezequel, J.F., and Shields, D.H., "The Pressuremeter and Foundation Engineering", Series on Rock and S o i l Mechanics, Trans Tech Publications, Ger-many, 19 78 Barden, L., and Khayatt, A.J., "Incremental Strain Rate Ratios and Strength of Sand i n the T r i a x i a l Test", Geotechnique, Vol. 19, No 4, 1966, pp. 338-357 Byrne, P.M., and Grigg, R.F., " O i l s t r e s s : A Computer Program for Nonlinear Analysis of Stresses and Deformations i n Oilsands", S o i l Mechanics Series No. 42, Dept. of C i v i l Engineering, U.B.C., July, 1980 Byrne, P.M., and Duncan, J.M., "NLSSIP: A Computer Program for Nonlinear Analysis of Soil-Structure Interaction Problems", S o i l Mechanics Series No. 41, Dept. of C i v i l Engineering, U.B.C, 1979 Boyce, W.E., and DiPrima, R.C, "Elementary D i f f e r e n t i a l Equations and Boundary Value Problems", John Wiley and Sons, New York, Third E d i t i o n , 19 77 Cook, R.D., "Concepts and Applications of F i n i t e Element Analysis", John Wiley and Sons, New York, F i r s t E d i t i o n , 1974 Cole, E.R.L., "The Behaviour of So i l s i n Simple Shear Apparatus" Ph.D. Thesis, Cambridge University, England, 1967 Duncan, J.M., Byrne, P.M., Wong, K.S., and Mabry, P., "Strength, Stress-Strain and Bulk Modulus Parameters for F i n i t e Element Analyses of Stresses and Movements in S o i l Masses", Report No. WCB/GT/78-02 to National Science Foundation, A p r i l , 1978 Duncan, J.M., and Chang, C-Y, "Nonlinear Analysis of Stress and Strain i n S o i l s " , Journal of the S o i l Mechanics and Foundations Divi s i o n , ASCE, Wol. 96, No SM5, Sept., 1970 pp. 1629-1651. El-Sohby, M.A., and Andrawes, K.Z., "Deformation Charac-t e r i s t i c s of Granular Materials Under Hydrostatic Compression", Canadian Geotechnical Journal, Vol. 9, 1972, pp. 338-350 Frydman, S., and Ze i t l e n , J.G., "Some Pseudo-Elastic Propertie of Granular S o i l " , Proceedings, 7th International Con-ference on S o i l Mechanics and Foundation Engineering, 130 Mexico, Vol 1, 1969, pp/ 135-141 Gibson, R.E., and Anderson, W.F., "In-situ Measurement of S o i l Properties with the Pressuremeter", C i v i l Engineering, Vol. 56, May, 1961, pp. 615-620 Gunaratne, M, "A Nonlinear Incremental F i n i t e Element Program for the Analysis of Shafts and Tunnels i n Oilsands", M.A.Sc. Thesis, U.B.C., 1981 Hansen, B., "Line Ruptures Regarded as Narrow Rupture Zones, Basic Equation Based on Kinematic Considerations", Proceedings, Brussels Conference on Earth Pressure Prob-lems, Brussels, Vol. 1, 1958, pp. 39-48 Home, M.R. , "The Behaviour of An Assembly of Rotund, Rigid, Cohesionless P a r t i c l e s , I&II, Proceedings of the Royal Society, London, Vol. 286, A, 1964, pp.62-96 Hughes, J.M.O., Wroth, CP., and Windle, D., "Pressuremeter Tests i n Sand", Geotechnique, Wol. 27, No. 4, Dec. 19 77, pp. 455-477 Hughes, J.M.O., "Pressuremeter Testing", Class Notes for C.E. 577, U.B.C., Nov., 1979 Kitamura, R., "Mechanical Properties of Par t i c u l a t e Material Under General Stress Conditions", S o i l s and Foundations, Vol. 21, No. 3, Sept., 1981, pp. 67-82 Kokusho, T., "Nonlinear Analysis of a S o i l with Arbit r a r y Dilatancy by F i n i t e Element Method", S o i l s and Foundations Vol. 18, No. 1, Mar., 1978, pp. 73-89 Ko. , H-Y. , and Scott, R.F., 11 Deformation of Sand at F a i l u r e " , Journal of the S o i l Mechanics and Foundations D i v i s i o n , ASCE, Vol. 94, No. SM4, July, 1968, pp. 883-898 Kondner, R.L., "Hyperbolic Stress-Strain Response: Cohesive S o i l s " , Journal of the S o i l Mechanics and Foundations D i v i s i o n , ASCE, Vol. 89, No. SMI, 1963, pp. 115-143 Kondner, R.L., and Zelasko, J.S., "A Hyperbolic Stress-Strain Formulation for Sands", Proceedings, 2nd Pan-American Conference on S o i l Mechanics and Foundations Engineering, B r a z i l , Vol. 1, 1963, pp289-324 Krishnamurthy, M., Nagaraj, T.S., and Sridharan, A., "Stress Path E f f e c t s on the Dilatant Behaviour of Sand", Geo-technical Engineering, Vol. 12, 1981, pp. 41-51 Lade, P.V., and Duncan, J.M., " E l a s t o p l a s t i c Stress-Strain Theory for Cohesionless S o i l " , Journal of the Geotech-n i c a l Engineering D i v i s i o n , ASCE, Vol. 101, No. GT10, Oct., 1975, pp. 1037-1053 1'31 Lindenberg, J., and Koning, H.L., " C r i t i c a l Density of Sand", Geotechnique, Vol. 31, No. 2, 1981, pp. 231-245 Marsland, A., and Randolph, M.F., "Comparison of the Results From Pressuremeter Tests and Large In-Situ Plate Tests i n London Clay", Geotechnique, Vol. 27, No. 2, June, 1977, pp. 217-243 Oda, M., "The Mechanism of Fabric Changes During Compressional Deformation of Sand", S o i l s and Foundations, Vol. 12, No, 2, June, 1972, pp. 1-18 Palmer, A.C., "Undrained Plane-Strain Expansion of a C y l i n -d r i c a l Cavity i n Clay: A Simple Interpretation of the Pressuremeter Test", Geotechnique, Vol. 22, No. 4, Sept. 1972, pp. 451-457 Rowe, P.W., "The Stress-Dilatancy Relation for S t a t i c Equi-librium of an Assembly of P a r t i c l e s i n Contact", Pro-ceedings of the Royal Society, London, Vol. 269, A, 1962, pp. 500-529 Rowe, P.W., "Theoretical Meaning and Observed Values of Deform-ation Parameters i n S o i l " , Stress-Strain Behaviour of S o i l , Roscoe Memorial Symposium, Cambridge, G.T. F o u l i s , Henley-on-Thames, 1971, pp. 143-194 Tatsuoka, F., "Stress-Strain Behaviour of an Idealized Anisotropic Granular Material", S o i l s and Foundations, Vol. 20, No. 3, Sept., 1980, pp. 75-90 Stroud, M.A., "The Behaviour of Sand at Low Stress Levels i n the Simple Shear Apparatus", Ph.D. thesis, Cambridge, England, 196 7 Timoshenko, S.P., and Goodier, J.N., "Theory of E l a s t i c i t y " , McGraw-Hill Book Company, New York, Third E d i t i o n , 19 70 Tobita, Y., and Yanagisawa, E., "Stress-Strain Relationship of Sand and i t s Application to FEM Analysis", Inter-national Symposium of S o i l s under C y c l i c and Transient Loading, Swansea, January, 1980, pp. 653-664 Vaid, Y.P., Byrne, P.M., and Hughes, J.M.O., "D i l a t i o n Rate as a Measure of Liquefaction Resistance of Saturated Granular Materials", S o i l Mechanics Series, No. 43, Dept. of C i v i l Engineering, U.B.C, Nov., 1980 Varadarajan, A., and Mishra, S.S., "Stress-path Dependent Stress Strain-Volume Change Behaviour", International Symposium on S o i l s under C y c l i c and Transient Loading, Swansea, Jan., 1980, pp. 109-119 Vesic, A.S., "Expansion of Cavities i n I n f i n i t e S o i l Mass", Journal of the S o i l Mechanics and Foundations D i v i s i o n , 132 ASCE, Vol. 98, No. SM3, March, 1972, pp. 265-290 Vesic. A.S., and Clough, G.W., "Behaviour of Granular Material under High Stresses", Journal of the S o i l Mechanics and Foundations D i v i s i o n , ASCE, Vol. 94, No. SM3. May, 1968, pp. 661-688 Wroth, CP., and Hughes, J.M.O., "An Instrument for the In-S i t u Measurement of the Properties of Soft Clay," 8th International Conference on S o i l Mechanics and Founda-ti o n Engineering, Moscow, 1973, Vol. 1.2, pp. 487-494 Wroth, CP., and Windle, D., "Analysis of the Pressuremeter Test Allowing for Volume Change", Geotechnique, Vol. 25, No. 3, 1975, pp. 598-604 133 APPENDIX A Four Node Polar Coordinate F i n i t e Element 134. • . A p p e n d i x A F o r m u l a t i o n o f F o u r Node P o l a r C o o r d i n a t e F i n i t e E l e m e n t The f o u r node i s o p a r a m e t r i c e l e m e n t c a n n o t m o d e l c u r v e d b o u n d a r i e s . I n o r d e r t o a c c u r a t e l y m o d e l t h e c u r v e d b o u n d a r y o f t h e p r e s s u r e m e t e r : , a f i n e mesh w o u l d be r e q u i r e d . T h i s w o u l d r e q u i r e l a r g e amounts o f c o m p u t e r t i m e t o s o l v e . An e i g h t node i s o p a r a m e t r i c e l e m e n t c o u l d be u s e d , b u t t h i s m o d e l s q u a d -r a t i c b o u n d a r i e s . To r e d u c e t h e mesh s i z e a nd t o a c c u r a t e l y m o d e l t h e c i r c u l a r b o u n d a r y o f t h e p r e s s u r e m e t e r a f o u r node p l a n e s t r a i n p o l a r c o o r d i n a t e e l e m e n t was d e v e l o p e d . T h i s e l e m e n t h a s two d e g r e e s o f f r e e d o m p e r n o d e . A h i g h e r o r d e r e l e m e n t c a n be d e v e l o p e d by a l s o s p e c i f y i n g d e r i v a t i v e s a t t h e n o d e s . The f o u r node e l e m e n t h a s t h e n u m b e r i n g scheme shown i n F i g u r e 5 9 . The d i s p l a c e m e n t s a t t h e n o d e s a r e u a n d v , t h e d i s p l a c e m e n t i n t h e r a d i a l a n d c i r c u m f e r e n t i a l d i r e c t i o n s r e s -p e c t i v e l y . The e l e m e n t h a s a l i n e a r d i s p l a c e m e n t f i e l d : u - a,+a„r+a_8+a„r6 1 2 3 4 v = b 1 + b 2 r + b 3 6 + b 4 r 0 The n o d a l d i s p l a c e m e n t s a r e t h e n g i v e n b y : ( u l v l 1 r l 6 2 \ 0 0 0 0 /a A U 2 1 r l 6 1 r l c \ 0 0 0 0 a 2 u 3 1 r 2 °1 3 1 0 0 0 0 a 3 u 4 \ _ 1 r 2 6 2 0 0 0 0 a 4 v l /- 0 0 0 0 1 r l 9 2 r l 9 2 \ b l v 2 0 0 0 0 1 r l 9 1 r l 6 l b 2 V 3 0 0 0 0 1 r 2 6 1 r 2 9 l b 3 0 0 0 0 1 r 2 6 2 r 2 6 2 W h i c h i n m a t r i x n o t a t i o n i s : {6} = (A){a} 1.3-5^  5°) F i g u r e 59 ' .. Four Node P o l a r C o o r d i n a t e F i n i t e Element 136' This i s inverted to give: {a} = ( A ) Where ( A r = < r r r 2 , ( 8 L - e 2 ) 2 91 r 2 e 2 - r l 6 2 r l e l 0 0 0 0 9 1 " e2 92 - e l 0 0 0 0 2 " r2 r l - r l 0 0 0 0 - i 1 -1 1 0 0 0 0 6 0 0 0 - r 2 e 1 r 2 e 2 - r l 9 2 r l e l 0 0 0 0 e l - Q2 92 -°1 0 0 0 0 r2 " r2 r l - r l 0 0 0 0 -1 1 1 1 The strains are obtained from the displacement f i e l d : e r 8r ,u 13v. e0 ~ ^ r r99 j Y = -r±*R 3v _ v> rr0 vr30 8r V Compressive strains are po s i t i v e . For the present case, from the symmetry of the problem, there i s no var i a t i o n of u or v with theta, and there i s no displacement i n the.theta d i r e c t i o n . This would simplify the derivation of the element s t i f f n e s s matrix, but i n order to keep the element as general as possible, the complete expressions for s t r a i n w i l l be used. The displacement f i e l d may be s p e c i f i e d as: {u} = v" l r 0 . r 0 O O O O 0 0 0 0 1 r 0 rC {a} The vector {a} i s the vector of the coordinates of the element nodes and contains only constants. This vector i s given by: {a} = ( A ) _ 1 { 6 } On d i f f e r e n t i a t i o n of the s t r a i n f i e l d , the s t r a i n vector i s determined: 137 {£} = -0 0 0 0 0' 1 ± r 0 0 - 1 -r r 0 0 1 i 1 r 0 - ^ 0 r ( A ) 1 { 6 } o r : { e } = - ( C ) ( A ) _ 1 { 6 } T h e r e f o r e t h e s t r a i n d i s p l a c e m e n t m a t r i x ( B ) i s ( B ) = - ( C ) ( A ) " 1 U p o n p e r f o r m i n g t h e m a t r i x m u l t i p l i c a t i o n : ( B ) e^-e (|2-1) ( 0 - 0 1> •^2-1 r - 0 2 + 0 ( f 2 - l ) ( 0 2 - 0 ) -§2 + 1 0 2 - 0 ( f i - i ) ( 0 - e 2> ^1-1 r -1 - 0 ^ 0 ( | i - D ( 0 X -6) -^1+1 r r 2 ) ( 9 l - 0 2 ) 6 ^2-1 r ( |2) (e^e j -e 0 --2+1 r ( |2) ( 0 - e 2 ) + 0 0 ^1-1 r ( f l ) ( 0 2 - 0 ) - 0 0 -^1+1 r ( pL) (e-e1)+e T h e e l e m e n t s t i f f n e s s m a t r i x i s t h e n g i v e n b y o r ( K ) = y / ( B ) ( D ) ( B ) d V ( K ) = A / ( B ) T ( D ) ( B ) d A f o r a u n i t t h i c k n e s s T h e s t r e s s - s t r a i n m a t r i x (D) i s t h e s t a n d a r d m a t r i x f o r p l a n e s t r a i n p r o b l e m s : 1-y u 0 TP (D) = (1+U)(l-2u) 0 1-u 0 l-2u 0 w h e r e : u = P o i s s o n ' s r a t i o E = Y o u n g ' s m o d u l u s T h i s i n t e g r a l c o u l d b e e v a l u a t e d , h o w e v e r s i n c e t h e m a t r i x ( B ) c o n t a i n s b o t h r a n d 0 t e r m s , i t w o u l d b e l o n g a n d t e d i o u s . 13&- y Instead, the i n t e g r a l i s evaluated numerically. Four point Gauss quadrature i s used. Within the program, the boundary conditions for the axisymmetric problem have been incorporated. These are: v = 0 at a l l points and -. 0 In addition, forces can only be applied i n the r a d i a l direction, E l a s t i c springs have been incorporated i n the polar coordinate mesh. The springs are attached at the outer most nodes. They model the e f f e c t of an i n f i n i t e e l a s t i c medium. Spring s t i f f n e s s i s determined by looking at the expansion of a cylinder i n an i n f i n i t e e l a s t i c medium. For a cylinder of radius, a, the pressure required to give a unit displacement of the cylinder boundary i s : 2G P = i ~ where: G = material shear> modulus The t o t a l force applied to a unit length of the cylinder i s : f = pA = p(2TTa) = —(2fra) a or: f = 4TTG independent of radius Therefore, by analogy to F = K A the spring s t i f f n e s s i s : : k = 4TTG This i s the stiffness for the complete cylinder. When analyzing a sector, with two springs, the i n d i v i d u a l spring s t i f f n e s s becomes: k = JS(4TTG) (JL) = G6 2TT where: 9 = sector angle, i n radians h accounts for two springs for the sector 139 T h e p o l a r c o o r d i n a t e e l e m e n t w a s c h e c k e d a g a i n s t c l o s e d o ' f o r m e l a s t i c s o l u t i o n s u s i n g t h e m e s h s h o w n i n F i g u r e • . 6 , 9 . . R e s u l t s a r e i n v e r y c l o s e a g r e e m e n t w h e n t h e m e s h s i z e i s s m a l l i n r e g i o n s o f l a r g e s t r e s s a n d s t r a i n g r a d i e n t s . S t r a i n s v e r s u s r a d i a l d i s t a n c e a r e s h o w n i n F i g u r e 6 1., s t r e s s e s v e r s u s r a d i a l d i s t a n c e a r e s h o w n i n F i g u r e 6 2 a n d d i s p l a c e m e n t v e r s u s r a d i a l d i s t a n c e i s s h o w n i n F i g u r e 63 • T n e s t r e s s p a t h f o l l o w e d b y •'' • A e l e m e n t 1 i s s h o w n i n F i g u r e 64 • O f s p e c i a l i n t e r e s t i n t h e p o l a r c o o r d i n a t e e l e m e n t i s t h e l i m i t o n P o i s s o n ' s r a t i o . I n m o s t n u m e r i c a l w o r k P o i s s o n ' s r a t i o m u s t b e l i m i t e d t o a v a l u e l e s s t h a n 0 . 5 t o p r e v e n t n u m e r i c a l i n s t a b i l i t i e s . H o w e v e r , i n t h i s c a s e t h e r a d i a l a n d c i r c u m f e r e n t i a l d i r e c t i o n s d e f i n e p r i n c i p a l p l a n e s . N o s h e a r s t r e s s o r s h e a r s t r a i n o c c u r s o n t h e s e p l a n e s , s o b y i n t r o d u c i n g t h e b o u n d a r y c o n d i t i o n s t h a t v = 0 a n d 6 u / 6 8 = 0 i n t h e s o l u t i o n r o u t i n e , e q u a t i o n s w i t h a p o s s i b l e d i v i d e b y z e r o e r r o r a r e e l i m i n a t e d . T h i s m e a n s t h a t P o i s s o n ' s r a t i o m a y b e s e t t o 0 . 5 w i t h o u t n u m e r i c a l i n s t a b i l i t y . T h i s a l s o a l l o w s s t r e s s p a t h s a l o n g t h e f a i l u r e s u r f a c e t o b e f o l l o w e d w i t h o u t u s i n g a s t r e s s r e d i s t r i b u t i o n r o u t i n e . T h o s e e l e m e n t s o n o r a b o v e t h e f a i l u r e s u r f a c e h a v e t h e s h e a r m o d u l u s , G = 0 . T h e s e e l e m e n t s p i c k u p n o a d d i t i o n a l s h e a r s t r e s s o n c o n t i n u e d l o a d i n g , s o w i l l m o v e b a c k o n t o t h e f a i l u r e s u r f a c e a s t h e m e a n n o r m a l s t r e s s i n c r e a s e s . . . . < F i g u r e . 6 5 s h o w s t h e s t r e s s p a t h o f a f a i l e d e l e m e n t o n a M o d i f i e d M o h r d i a g r a m . T h e u s e o f s m a l l l o a d s t e p s a l l o w s a c c u r a t e m o d e l l i n g o f s t r e s s p a t h s a l o n g t h e f a i l u r e e n v e l o p e . 1.40 F i g u r e 6 Q / » F i n i t e E l e m e n t M e s h 141 0.020 S t r a i n v e r s u s R a d i a l D i s t a n c e 0 . 0 1 5 H 0 . 0 1 0 - H 0.005H OV3 C •H u -p CO - 0 . 0 0 5 - 1 -0.010H - 0 . 0 1 5 H -0.020H C l o s e d F o r m E l a s t i c S o l u t i o n P r o g r a m C a l c u l a t i o n G = 5 0 0 k g / c m 2 2 a - l k g / c m ° 2 p = . 2 k g / c m a = 1 c m ' — 1 — 4 6 7 8 9 R a d i a l D i s t a n c e , r a 1 0 1 1 12 \ F i g u r e 6 1 . E x p a n s i o n o f C y l i n d e r i n I n f i n i t e E l a s t i c M e d i u m 1 4 2 1 . 2 0 -J S t r e s s v e r s u s R a d i a l D i s t a n c e 1.16 H 1.12 A 1.08 1 1.04 CN e u tn cn cn QJ - P C/3 0 . 9 6 0 . 9 2 0 . 8 8 H 0 . 8 4 J 0 . 8 0 H C l o s e d F o r m E l a s t i c S o l u t i o n P r o g r a m C a l c u l a t i o n G = 5 0 0 k g / c m 2 a = 1 k g / c m 2 o ^ -p = 0 . 2 k g / c m ' a = 1 . 0 c m " F i g u r e 6 2 . . E x p a n s i o n o f C y l i n d e r i n I n f i n i t e E l a s t i c M e d i u m 143 D i s p l a c e m e n t v e r s u s R a d i a l D i s t a n c e F i g u r e 63 . E x p a n s i o n o f C y l i n d e r i n I n f i n i t e E l a s t i c Medium 144 1 4 5 APPENDIX B Stress-Strain Matrix for Triaxial Finite Element 146 A p p e n d i x B S t r e s s - S t r a i n M a t r i x f o r T r i a x i a l F i n i t e E l e m e n t T h e p l a n e s t r a i n f o u r n o d e i s o p a r a m e t r i c e l e m e n t u s e d i n t h e N L S S I P c a n b e e a s i l y m o d i f i e d s o t h a t i t c a n m o d e l t h e t r i a x i a l t e s t . T h e i s o p a r a m e t r i c f o r m u l a t i o n i s r e t a i n e d . O n l y t h e s t r e s s - s t r a i n m a t r i x m u s t b e c h a n g e d f r o m t h e p l a n e s t r a i n e l e m e n t . T h e n e w e l e m e n t i s n o t c y l i n d r i c a l . I t i s a b r i c k e l e m e n t o f u n i t t h i c k n e s s w i t h f o u r n o d e s s p e c i f y i n g t h e e l e m e n t . I n t h e i s o p a r a m e t r i c e l e m e n t f o r m u l a t i o n , t h e s t i f f n e s s m a t r i x i s g i v e n b y a n i n t e g r a l o v e r t h e v o l u m e o f t h e e l e m e n t : ( K ) = V / ( B ) T ( D ) ( B ) d V w h e r e : ( K ) . = e l e m e n t s t i f f n e s s . m a t r i x ( B ) = e l e m e n t 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) = e l e m e n t s t r e s s - s t r a i n m a t r i x F o r t h e p l a n e s t r a i n e l e m e n t (D) i s : 1 - u u 0 u 1 - u 0 0 0 1 = 2 0 U n d e r t r i a x i a l c o n d i t i o n s t h e r e l a t i o n s b e t w e e n s t r a i n a n d s t r e s s a r e : e = X 1 , E ( V -u (a +a ) ) y z e -y -u (a +o )) X z e = z E ( ° z - -u (a +a ) ) x y Y -x y 1 G T x y Y = y z 1 G T y z Y = 1 z x 1 G T z x T h e t r i a x i a l e l e m e n t i s o r i e n t e d s u c h t h a t t h e y a x i s i s a l o n g t h e l o n g a x i s o f t h e e l e m e n t , F i g u r e 6 6 . . (D) = ( 1 + u ) ( l - 2 u ) 1 4 7 Y F i g u r e 66 . T r i a x i a l M o d e l F i n i t e E l e m e n t 148 F o r t h i s g e o m e t r y t h e c o o r d i n a t e p l a n e s d e f i n e p r i n c i p a l p l a n e s : e = e X z x y z x Y =0 z y T = T = T =0 x y z x z y a n d : e x Y x y \ / 1-u -u E E -2u 1 E E 0 0 / \ a x I n v e r t i n g t h i s m a t r i x g i v e s t h e s t r e s s - s t r a i n m a t r i x : 1 u 0 I (D) (1+u)(l-2u) 2u 1-u 0 l-2u 0 T h i s m a y b e s i m p l i f i e d b y u s i n g t h e p a r a m e t e r s B 1 a n d G, 3 B E w h e r e : R e s u l t i n g i n B' = 2(1+u) G = a n d B = E 2(1+u) 3 ( l-2u) 2B1 B'-G 0 (D) = 2(B'-G) B'+G 0 0 0 G T h e v a r i a b l e B i s a m o d i f i e d b u l k m o d u l u s u s e d o n l y f o r p r o g r a m -m i n g c o n v e n i e n c e . T h e s t r e s s - s t r a i n m a t r i x d e v e l o p e d a b o v e i s c o r r e c t w h e n f o r c e s a r e a p p l i e d i n t h e y - d i r e c t i o n o r t h e x - d i r e c t i o n , b u t i s n o t c o r r e c t f o r t h o s e c a s e s w h e n f o r c e s a r e a p p l i e d i n b o t h t h e x - d i r e c t i o n a n d t h e z - d i r e c t i o n , a s i s t h e u s u a l c a s e i n t h e t r i a x i a l t e s t . T h e p r i n c i p a l o f v i r t u a l w o r k c a n b e u s e d t o p r o d u c e t h e c o r r e c t s t r e s s - s t r a i n m a t r i x . I n t h e p r i n c i p a l 149-o f v i r t u a l w o r k , t h e w o r k d o n e b y t h e i n t e r n a l s t r e s s e s u n d e r -g o i n g a v i r t u a l d i s p l a c e m e n t i s e q u a t e d t o t h e w o r k d o n e b y t h e e x t e r n a l f o r c e s u n d e r g o i n g t h e v i r t u a l d i s p l a c e m e n t . T h e i n t e r -n a l v i r t u a l w o r k i s g i v e n b y : W. L i l t V _ rp v i r t u a l s t r a i n s i n t h e e l e m e n t { a } = i n t e r n a l s t r e s s e s i n e l e m e n t E x p a n d i n g t h e v e c t o r s g i v e s : W . J _ = t T / { e e e Y Y Y } m t V x y z - x y ' y z ' z x .  . = T 7 / { E } t { a } d V i n t V — T w h e r e : { e } U s i n g t h e s t r a i n a n d s t r e s s b o u n d a r y c o n d i t i o n s o u t l i n e d a b o v e g i v e s : W. i n t = T T / { e e Y ) 2 a V x y ' x y ) a x -Y d V b u t , T h e r e f o r e { a } = (D) { e } — / \ 4 B " 2 ( B ' - G ) 0 e X --- 2 ( B ' - G ) B ' + G 0 < £ v 0 0 G ^ x y = ( D ) { e ) T h e s t r a i n s a r e d e t e r m i n e d f r o m t h e n o d a l d i s p l a c e m e n t s : { e } = (B){6\> w h e r e : {6} = 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 T h e r e f o r e t h e i n t e r n a l v i r t u a l w o r k b e c o m e s : W. , i n t V / ^ 6 } T ( B ) T ( D ) ( B ) { 6 } d V 15.0 The external v i r t u a l work i s W ext w , u „ v „ W_ U-, v _ w_ u. V . w.} < / f i x \ ;iy : l z > L4x where u^ = v i r t u a l displacement of node 1 in x di r e c t i o n v^ = v i r t u a l displacement of node 1 in y di r e c t i o n w^  = v i r t u a l displacement of node 1 in z dir e c t i o n f, = force at node 1 i n x d i r e c t i o n lx f„ = force at node 1 i n y di r e c t i o n 2x 2 f_ = force at node 1 i n z d i r e c t i o n 3x For the t r i a x i a l case u w. and: 1 "1' f, =f-u Wv 2 "2' f • =f, u 3=w 3, f, =f, , 3x 3z lx ' l z , 2x 2z' Which gives the external v i r t u a l work: W.._L = { u ~ ~ ~ ~ ~ ext 1 V l U2 V2 U3 V3 U4 u . = w . 4 4 4x 4z V 2f 2f lx "iy 2x N 2 f -3x > 2f "3y 4x W . = {<5}x{f} ext The external and i n t e r n a l v i r t u a l work are equated, giving: {6} T{f} = V/{6} T(B) T(D)(B){<5}dV -.151 O r : { f } = V / ( B ) T ( D ) ( B ) d V { 6 } T h e s t i f f n e s s m a t r i x i s t h e n : ( K ) = V / ( B ) T ( D ) ( B ) d V T h e s t r e s s - s t r a i n m a t r i x (D) i s u s e d w h e n f o r m i n g t h e s t i f f n e s s m a t r i x . I t h a s b e e n m o d i f i e d t o a c c o u n t f o r f o r c e s b e i n g a p p l i e d i n b o t h t h e x a n d z d i r e c t i o n s . T h i s m a t r i x i s u s e d i n t h e p r o g r a m s u b r o u t i n e I S Q U A D , w h i c h a s s e m b l e s t h e e l e m e n t s t i f f n e s s m a t r i x . O n c e t h e n o d a l d i s p l a c e m e n t s h a v e b e e n d e t e r m i n e d , t h e s t r a i n s a r e d e t e r m i n e d b y : { e } = ( B ) { 6 } T h e s t r e s s e s a r e d e t e r m i n e d b y : { a } = ( D ) ( B ) { 5 } W h e n c a l c u l a t i n g t h e s t r e s s e s , t h e u n m o d i f i e d m a t r i x (D) i s u s I n t h e a b o v e f o r m u l a t i o n t h e f o r c e v e c t o r i s o f t h e f o r m { f } = l x l z " i y > f - + f . 4 x 4 z f . \ 4 y / T h i s r e q u i r e d m o d i f i c a t i o n o f t h e s t r e s s - s t r a i n m a t r i x . A n a l t e r n a t e a p p r o a c h c o u l d b e u s e d , i n w h i c h c a s e t h e f o r c e v e c t o r w o u l d b e o f t h e f o r m : f l x «>-<b 4 x gThe s t r e s s - s t r a i n m a t r i x r e q u i r e s n o m o d i f i c a t i o n i n t h i s c a s e W h i l e g i v i n g t h e c o r r e c t s t r e s s e s a n d s t r a i n s , t h i s a p p r o a c h 152 does not give the correct i n t e r n a l and external v i r t u a l work done during a load step. 153 APPENDIX C Pressuremeter Test Data, MacDonald's Farm, Sea Island RADTAL DISPLACEMENT (%) S.O 6.0 7.0 RADIAL DISPLACEMENT (%) 10.0 11.0/ 12.0 A R R W I T SCALE:IOOKPR- 2CH J10B SCALE; J X TO a c n SOKE MOLl No. sapTi OCPTH T t » T DATC EL 17.5 FT 03 DEC 1981 TfST No. MATERIAL TltTID FINE HEDIUI1 SAND SITU TECHNOLOGY INC B.C. UMIVERS1TT OF B.C. H.S.B PRESSUREMETER PLOT OF APPLIED PRESSURE V.S. RADIAL DISPLACEMENT U.B.C. DEHO JOB HO. 40-B1H0S3 1100 1000 900 800 V I v> S 8 5! T;i—I.< ill. J 6 700 •00 500 400 300 200 A 100 A 10.0 11.0 12.0 RADIAL DISPLACEMENT (%) A R W T JCM.€:l00KrB- 2CH jum son.*: IX TO 2CH •one Move N*. SBPTI O C P T H EL 20.S FT ARM •7HT T f I T DA TV 03 DEC 19SI oo M A T M I A I . T C S T E O FINE SPND SITU TECHNOLOGY INC. Vmcowvw, B.C. 84fltft#, Wfc UNIVERS1TT OF 8.C. H.S.B PRESSUREMETER PLOT OF APPLIED PRESSURE V.S. RADIAL DISPLACEMENT U.B.C. 0EH0 JOB HO. 40-B1H0ES 

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