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Non linear elastic undrained stress-strain model for anisotropicaly consolidated clay Samarasekera, Lal 1982

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NON LINEAR ELASTIC UNDRAINED STRESS-STRAIN MODEL FOR ANISOTROPICALY CONSOLIDATED CLAY by LAL SAMARASEKERA B.Sc., U n i v e r s i t y of S r i Lanka THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1982 --© • "Lal ..Samar.ase.kera , 1-982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or pu b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C iv i l Engjneerinj  The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 28* April 1982 A b s t r a c t A n o n l i n e a r e l a s t i c model f o r normally c o n s o l i d a t e d c l a y s which accounts fo r the i n f l u e n c e of i n i t i a l s t r e s s a n i s o t r o p y and s t r e s s system change d u r i n g shear, i s presented h e r e i n . The a n a l y s i s of a number of K 0 normally c o n s o l i d a t e d s t r e s s - s t r a i n r e s u l t s ( p u b l i s h e d t r i a x i a l compression and extension) of d i f f e r e n t c l a y , r e v e a l e d t h a t , a l l the s t r e s s - s t r a i n curves c o u l d be adequately represented by hyperbolas, when the d e v i a t o r s t r e s s i s measured r e l a t i v e to i t s i n i t i a l v a l u e . However, the w r i t e r b e l i e v e s ( a l t h o u g h not i n v e s t i g a t e d ) that the above f i n d i n g holds t r u e even f o r over c o n s o l i d a t e d c l a y s and t h e r e f o r e , the proposed model i s b e l i e v e d to be a p p l i c a b l e f o r over c o n s o l i d a t e d c l a y s too. The model was i n c o r p o r a t e d i n an incremental l i n e a r f i n i t e element method of a n a l y s i s programme. U t i l i z i n g the parameters from K 0 c o n s o l i d a t e d t r i a x i a l t e s t s , the programme was used to p r e d i c t other l a b o r a t o r y s t r e s s - s t r a i n behaviour such as plane s t r a i n and simple shear. The p r e d i c t e d s t r a i n s were in e x c e l l e n t agreement with the measured valu e s i n the working range of s t r a i n s . In an e f f o r t to compare the p r e d i c t i o n s of the proposed model and widely used i s o t r o p i c h y p e r b o l i c models(Duncan et a l . , 1970) a l o a d deformation a n a l y s i s of a s t r i p foundation was performed. I s o t r o p i c models always overestimated settlements and in some cases by as much as ten times that of the proposed model p r e d i c t i o n s . The i s o t r o p i c model p r e d i c t i o n s of the u l t i m a t e bearing s t r e s s e s were g e n e r a l l y higher than those of i t s c o u n t e r p a r t . The a n i s o t r o p i c model y i e l d e d shallower and wider f a i l u r e zones than those produced by the i s o t r o p i c model f o r the same l o a d i n g c o n d i t i o n s . A l a r g e number of p u b l i s h e d data on normally c o n s o l i d a t e d c l a y s were analysed and c o r r e l a t i v e trends were found between the model parameters, s o i l index p r o p e r t i e s and c o e f f i c i e n t of e a r t h pressure at r e s t , K 0. These c o r e l a t i o n s can provide v a l u a b l e t o o l s i n judging the reasonableness of the parameters obtained from l a b o r a t o r y t e s t s or they can be used to o b t a i n parameters when such t e s t r e s u l t s are u n a v a i l a b l e . i i i Table of Contents A b s t r a c t i i L i s t of Tables v i i L i s t of F i g u r e s v i i i N o t a t i o n x i i i Acknowledgement xv CHAPTER I. INTRODUCTION 1 CHAPTER I I . ISOTROPIC AND ANISOTROPIC STRESS-STRAIN MODELS 4 2. 1 BACKGROUND 4 2.2 ISOTROPIC MODEL 5 2.2.1 DETERMINATION OF MODEL PARAMETERS 7 2.2.2 LIMITATIONS OF ISOTROPIC MODEL 10 2.2.2(a) S t r e s s System Change During Shear 10 2.2.2(b) I n i t i a l C o n s o l i d a t i o n S t r e s s e s 14 2.3 ANISOTROPIC MODEL 16 2.3.1 DETERMINATION OF MODEL PARAMETERS 25 CHAPTER I I I . PROCEDURES FOR NONLINEAR STRESS ANALYSIS 32 3.1 FINITE ELEMENT TECHNIQUE 32 3.2 NONLINEAR SOLUTION PROCEDURES 35 3.2.1 INCREMENTAL APPROACH 36 3.2.2 MODIFIED INCREMENTAL METHOD 36 3.2.3 ITERATIVE APPROACH 39 i v 3.3 LINEAR ELASTICITY AND CONSTITUTIVE MATRIX 39 3.3 INCREMENTAL METHOD AND CONSTITUTIVE MATRIX 42 3.5 ISOTROPIC MODEL AND NONLINEAR PROCEDURES 43 3.5.1 TANGENT MODULUS 44 3.6 ANISOTROPIC MODEL AND NONLINEAR PROCEDURES 4 5 3.6.1 TANGENT MODULUS1 4 7 3.7 FINITE ELEMENT PROGRAMME 55 CHAPTER IV. THE ANISOTROPIC MODEL PREDICTION COMPARISONS 57 4.1 LABORATORY PREDICTIONS 57 4.1.1 PLANE STRAIN TESTS 57 4.1.2 SIMPLE SHEAR CONDITION 61 4.2 APPLICATIONS OF ISOTROPIC AND K 0 MODELS 63 4.4.1 GEOMETRY, SOIL PROPERTIES AND BOUNDARY CONDITIONS 64 4.2.2 MODEL PARAMETERS 67 4.2.3 COMPARISON OF RESULTS 68 4.2.3(A) LINEARLY VARYING UNDRAINED STRENGTH WITH DEPTH 68 (a) . D e f l e c t i o n s 69 (b) . Bearing S t r e s s e s and Surface Pressures 69 (c) . F a i l u r e Zones 74 4.2.3(B) CONSTANT UNDRAINED STRENGTH WITH DEPTH 74 (a) . D e f l e c t i o n s 75 (b) . Bearing S t r e s s e s and Surface Pressures 79 (c) . F a i l u r e Zones 79 4.3. DISCUSSION 82 V CHAPTER V. EMPIRICAL CORRELATIONS FOR MODEL PARAMETERS 84 5.1 NORMALIZED BEHAVIOUR OF NORMALLY CONSOLIDATED CLAY 85 5.1.1 NORMALIZED MODULUS 89 5.1.1(a) Normalized Modulus and K 0 89 5.1.1(b) Normalized Modulus and Index P r o p e r t i e s 91 5.3 NORMALIZED UNDRAINED STRENGTH 93 5.3.1 STRENGTH IN COMPRESSION OF ISOTROPICALLY CONSOLIDATED CLAYS 93 5.3.2 STRENGTH IN COMPRESSION OF ANISOTROPICALLY CONSOLIDATED CLAYS 95 5.3.3 COMPARISON OF STRENGTH CORRELATIONS 96 5.3.4 UNDRAINED STRENGTH IN EXTENSION 99 5.4 FAILURE RATIO 101 5.5 DISCUSSION «. 101 CHAPTER VI. SUMMARY AND CONCLUSION 104 REFERENCES 107 APPENDIX A. PUBLISHED DATA AND DERIVED TRANSFORMED STRESS-STRAIN PLOTS ON NORMALLY CONSOLIDATED CLAY 116 APPENDIX B. PUBLISHED DATA AND DERIVED MODEL PARAMETERS OF NORMALLY CONSOLIDATED CLAY 147 APPENDIX C. FAILURE ZONES UNDER THE FOUNDATION 155 APPENDIX D. v i STABILITY ANALYSIS METHOD OF SWEDISH SLICES 164 APPENDIX E. . SEMI EMPIRICAL APPROACH FOR K 0 CONSOLIDATED COMPRESSION STRENGTH • 165 E-1 THE CONCEPTS AND DERIVED FORMULAE 165 E-2 MATHEMATICAL DERIVATIONS 173 APPENDIX F. DESCRIPTION OF INPUT DATA 176 v i i L i s t of Tables 2-1. UNDRAINED ANISOTROPY OF NORMALLY CONSOLIDATED BOSTON BLUE CLAY 11 4-1. COMPARISON OF ULTIMATE BEARING.STRESSES 80 B-1. INITIAL MODULUS,FAILURE RATIO, PLASTICITY INDEX, AND CONFINING PRESSURE 147 B-2. UNDRAINED STRENGTH IN COMPRESSION; ISOTROPICALLY CONSOLIDATED TRIAXIAL TESTS . . 149 B-3. UNDRAINED STRENGTH IN COMPRESSION; K 0 CONSOLIDATED TRIAXIAL TESTS 152 B-4. COMPRESSION/EXTENSION STRENGTH RATIO AND PLASTICITY . INDEX 154 E-1. PLASTICITY INDEX AND 6/6 0 169 v i i i L i s t of F i g u r e s 2-1. H y p e r b o l i c S t r e s s - S t r a i n Curve 6 2-2. H y p e r b o l i c S t r e s s - S t r a i n Curve on the Transformed Plane 9 2-3. D i f f e r e n t S t r e s s Systems along the F a i l u r e Surface .. 13 2-4. I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d S t r e s s - S t r a i n Curves 15 2-5. A n i s o t r o p i c a l l y C o n s o l i d a t e d Compression and Exte n s i o n S t r e s s - S t r a i n Curves 19 2-6. Tangent Modulus as a F u n c t i o n of D e v i a t o r S t r e s s .... 20 2-7. Transformed P l o t s of I s o t r o p i c a l l y C o n s o l i d a t e d Extension T e s t s 26 2-8. Transformed P l o t s of A n i s o t r o p i c a l l y C o n s o l i d a t e d E x t e n s i o n T e s t s 27 2-9. Transformed P l o t s of A n i s o t r o p i c a l l y C o n s o l i d a t e d Compression Tests 28 2- 10. Transformed P l o t of the A n i s o t r o p i c a l l y C o n s o l i d a t e d H y p e r b o l i c S t r e s s - S t r a i n Curve 30 3- 1. L i n e a r Incremental Method 37 3-2. M o d i f i e d L i n e a r Incremental Approach 38 3-3. I t e r a t i v e Procedure 40 3-4. The Implied S t r e s s - S t r a i n Curve i n E x t e n s i o n of the I s o t r o p i c Model 46 3-5. A n i s o t r o p i c a l l y C o n s o l i d a t e d S t r e s s - S t r a i n Curves i n Compression and in Extension 48 3-6. A n i s o t r o p i c a l l y C o n s o l i d a t e d S t r e s s - S t r a i n Curves i n Compression and i n Extension on {(o~,-cr 3) , e} Plane ...... 49 ix 3 -7 . Case I, S u 9 0 > ( o - 1 - c r 3 ) 0 53 3 - 8. Case I I , S u 9 0^(cr,-o- 3) 54 4- 1. Comparison of Model P r e d i c t i o n s with Plane S t r a i n Laboratory R e s u l t s 58 4-2. S o i l . Parameters and Plane S t r a i n F i n i t e Element 60 4-3. S o i l Parameters and Simple Shear F i n i t e . Element 60 4-4. Simple Shear P r e d i c t i o n s 62 4-5. Geometry and S o i l P r o p e r t i e s 65 4-6. F i n i t e Element Mesh 66 4-7. Pressure D e f l e c t i o n Curves of C e n t r a l A x i s ; F l e x i b l e F o o t i n g ; V a r i a b l e Undrained Strength 70 4-8. Pressure D e f l e c t i o n Curves of C e n t r a l A x i s ; R i g i d F o o t i n g ; V a r i a b l e Undrained Strength 71 4-9. F l e x i b l e F o o t i n g Surface D e f l e c t i o n P r o f i l e s ; Proposed Model 72 4-10. F l e x i b l e F o o t i n g Surface D e f l e c t i o n P r o f i l e s ; I s o t r o p i c Model 72 4-11. R i g i d F o o t i n g Surface Pressure P r o f i l e s ; Proposed Model 73 4-12. R i g i d F o o t i n g Surface Pressure P r o f i l e s ; I s o t r o p i c Model 73 4-13. Pressure D e f l e c t i o n Curves of C e n t r a l A x i s ; F l e x i b l e F o o t i n g ; Constant Undrained Strength 76 4-14. Pressure D e f l e c t i o n Curves of C e n t r a l A x i s ; R i g i d F o o t i n g ; Constant Undrained Strength 77 4-15. F l e x i b l e F o o t i n g Surface D e f l e c t i o n P r o f i l e s ; Proposed Model 78 X 4-16. F l e x i b l e F o o t i n g Surface D e f l e c t i o n P r o f i l e s ; I s o t r o p i c Model 78 4-17. R i g i d F o o t i n g Surface Pressure P r o f i l e s ; Proposed Model 81 4- 18. R i g i d F o o t i n g Surface Pressure P r o f i l e s ; I s o t r o p i c Model 81 5- 1. Normalized Behaviour of I s o t r o p i c a l l y C o n s o l i d a t e d C l a y s 87 5-2. Normalized Behaviour of A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s 88 5-3. Normalized Modulus E j/cr^ vs K 0 90 5-4. Normalized Modulus and P l a s t i c i t y Index R e l a t i o n f o r I s t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s .... 92 5-5. Normalized Undrained Strength, P l a s t i c i t y Index C o r r e l a t i o n f o r I s o t r o p i c a l l y C o n s o l i d a t e d C l a y s 94 5-6. Normalized Strength and P l a s t i c i t y Index C o r r e l a t i o n f o r A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s 97 5-7. Comparison of the Undrained S t r e n g t h E m p i r i c a l R e l a t i o n s f o r I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d Clay .... 98 5-8. Compression/Extension Strength R a t i o and P l a s t i c i t y Index f o r I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s ... 100 5-9. V a r i a t i o n of Rf with P l a s t i c i t y Index 102 A-1. UCI on S o f t Bangkok C l a y ( U ) ; Balasubramaniam, 1976 .. 117 A-2. UCI on D i f f e r e n t C l a y s ; Ladd, 1964 118 A-3. UCK on Kawasaki C l a y ( U ) ; Ladd, 1964 119 x i A-4. UEK on Kawasaki C l a y ( U ) ; Ladd, 1964 120 A - 5 . UCK on Boston Blue C l a y ( R ) ; Ladd, 1964 121 A-6. UEK on Boston Blue C l a y ( R ) ; Ladd, 1964 122 A-7. UCI on Kinnegar C l a y ( U ) ; Crooks, 1 976 123 A-8. UCK on Kinnegar C l a y ( U ) ; Crooks, 1976 124 A-9. UCI on Clay; Pender, 1 978 125 A-10. UEI on Clay; Pender, 1978 126 A-11. UCI and UCK on Clay; Pender, 1978 127 A-12. UCI on London C l a y ( U ) ; Bishop et a l . , 1965 128 A-13. UCI on Skabo C l a y ( U ) ; Bjerrum et a l . , 1963 129 A-14. UCK on Skabo C l a y ( U ) ; Bjerrum et a l . , 1963 130 A-15. UCI on Fat C l a y ( R ) ; Richardson et a l . , 1963 131 A-16. UCI and UEI on Weald Clay(R) P a r r y , 1960 132 A-17. UEI on Weathered Bangkok C l a y ( U ) ; Balasubramaniam et a l . , 1 977 133 A-18. UEI on Weathered Bangkok C l a y ( U ) ; Balasubramaniam et a l . , 1977 134 A-19. UCI on Weald C l a y ( R ) ; Skempton, 1 964 135 A-20. UCI on Weald C l a y ( R ) ; Henkel, 1956 136 A-21. UCI on S o f t dark grey C l a y ( U ) ; Parry, 1968 137 A-22. UCK on Drammen C l a y ( U ) ; Prevost, 1978 138 A-23. UEK on Drammen C l a y ( U ) ; Prevost, 1978 139 A-24. UCI on Grundite C l a y ( R ) ; Lade et a l . , 1978 140 A-25. UCI on A r t i f i c i a l l y Prepared K a o l i n ; Subhas, 1978 .. 141 A-26. UCI on Haney C l a y ( U ) ; V a i d et a l . , 1977 142 A-27. UCI on Haney C l a y ( U ) ; V a i d et a l . , 1977 143 A-28. UCI on Osaka A l l u v i a l C l a y ( R ) ; S h i b i t a et a l . , 1965 144 x i i A-29. UCI on Clay; Ladd et a l . , 1963 145 A-30. UCK on C l a y ( U ) ; Bozozuk, 1 980-81 146 C-1. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; V a r i a b l e S t r e n g t h ; I s o t r o p i c Model 156 C-2. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; V a r i a b l e S t r e n g t h ; A n i s o t r o p i c Model 157 C-3. F a i l u r e Zones Under R i g i d F o o t i n g ; V a r i a b l e Strength; I s o t r o p i c Model 158 C-4. F a i l u r e Zones Under R i g i d F o o t i n g ; V a r i a b l e Strength; A n i s o t r o p i c Model 159 C-5. F a i l u r e Zones Under R i g i d F o o t i n g ; Constant St r e n g t h ; I s o t r o p i c Model 160 C-6. F a i l u r e Zones Under R i g i d F o o t i n g ; Constant Strength; A n i s o t r o p i c Model ' 161 C-7. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; Constant S t r e n g t h ; I s o t r o p i c Model 162 C-8. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; Constant S t r e n g t h ; A n i s o t r o p i c Model 163 D-1. C i r c u l a r F a i l u r e Surface 164 E-1. Undrained S t r e s s Paths of I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d Clays ... 166 E-2. S i m p l f i e d Undrained S t r e s s Paths of I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s 168 E-3. V a r i a t i o n of 6 / e 0 with P l a s t i c i t y Index 170 E-4. S /S vs P l a s t i c i t y Index Curves f o r UCK UCI D i f f e r e n t K 0 Values 172 E-5. S i m p l i f i e d S t r e s s Paths 174 x i i i Notat ion E Young's modulus I n i t i a l modulus E f c Tangent Young's modulus E t Q Tangent modulus corresponding to p r i n c i p a l axes r o t a t i o n 6° E e Young's modulus corresponding to p r i n c i p a l axes r o t a t i o n e ° K 0 C o e f f i c i e n t of e a r t h pressure at r e s t PI P l a s t i c i t y index q = (cr, - 0-3) - (cr, - 0-3) o on 0 ° s t r e s s - s t r a i n curve f o r (o-, -0-3 (cr, -0-3) o = (cr,-0-3) 0-(o-,-0-3) on 0 ° s t r e s s - s t r a i n curve f o r 0<(o-,-a3 )<(a,-0-3 ) 0 = (cr,-cr 3) + (cr,-o-3) 0 on 9 0 ° s t r e s s - s t r a i n curve f o r ( o - 1 - c r 3 ) > 0 q = (°yy"°xx)~ (°yy~°xx" 0 i n compression = (cr^-cc^) 0 - (cr^-cr^ i n extension (pi l t ) c T n e asymptotic value of q i n compression ( q u l t ) £ The asymptotic value of q i n e x t e n s i o n Q = {(cr -C£x)-(cr y-cr^^ 0 l/o-'i c f o r compression t e s t s = {(o^ y-cr x) 0-(c> y-cr^ }/a'1c f o r e x t e n s i o n t e s t s R Remoulded R f The a c t u a l strength/(c,-o- 3 ) s u Undrained shear s t r e n g t h (a,-o- 3) f/2 S u Undrained s t r e n g t h (o- 1-o- 3) f S u 0 Undrained s t r e n g t h corresponding to 0 ° p r i n c i p a l axes r o t a t i o n su90 Undrained s t r e n g t h corresponding to 9 0 ° p r i n c i p a l axes r o t a t i o n S U C I Undrained s t r e n g t h i n compression of i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s under t r i a x i a l c o n d i t i o n s S Undrained s t r e n g t h i n compression of K 0 c o n s o l i d a t e d c l a y s under t r i a x i a l c o n d i t i o n s x i v Undrained s t r e n g t h i n ext e n s i o n of K 0 c o n s o l i d a t e d c l a y s under t r i a x i a l c o n d i t i o n s S u Q Undrained s t r e n g t h corresponding to p r i n c i p a l axes r o t a t i o n e° U Undisturbed UCI See Appendix A UCK See Appendix A UEI See Appendix A UEK See Appendix A Y' Buoyant u n i t weight Y Shear s t r a i n xy v Poisson's r a t i o v Tangent poisson's r a t i o cr. Major p r i n c i p a l s t r e s s o-3 Minor p r i n c i p a l s t r e s s o~'1c Major e f f e c t i v e c o n s o l i d a t i o n s t r e s s o-' E f f e c t i v e c o n s o l i d a t i o n s t r e s s i n i s o t r o p i c a l l y c c o n s o l i d a t e d c l a y s a' Mean normal e f f e c t i v e c o n s o l i d a t i o n s t r e s s m cr Normal s t r e s s in X - d i r e c t i o n XX o-yy Normal s t r e s s i n Y - d i r e c t i o n (cr}-<r3)f F a i l u r e d e v i a t o r s t r e s s ( c i - c 3 ) u l t Asymptotic value of (o-^-cr3) in the h y p e r b o l i c curve o-d Deviator s t r e s s o-d0 I n i t i a l d e v i a t o r s t r e s s (°d)ult S a m e a s ( o * i _ 0 3 ) u l t X V Acknowledgement The authour would l i k e to thank Dr. P. M. Byrne and Dr. Y. P. V a i d f o r t h e i r i n v a l u a b l e guidance while s u p e r v i s i n g t h i s r e s e a r c h p r o j e c t . S p e c i a l thanks go to Dr. D. L. Anderson f o r many u s e f u l ideas r e s u l t i n g from d i s c u s s i o n s . The r e s e a r c h a s s i s t a n t s h i p awarded by the Department of C i v i l E n g i n e e r i n g i s g r a t e f u l l y acknowledged. 1 CHAPTER I_ INTRODUCTION The undrained l o a d i n g c o n s t i t u t e s only a p a r t of the l o a d i n g c o n d i t i o n s encountered i n g e o t e c h n i c a l e n g i n e e r i n g problems. N e v e r t h e l e s s , i t i s important to design f o r undrained l o a d i n g , so that the s t r u c t u r e i s not unstable due to a gross shear f a i l u r e of the s o i l . A l s o , the design must ensure that the undrained deformations are w i t h i n t o l e r a b l e l i m i t s . In a d d i t i o n , p r e d i c t i o n s of s t r e s s and displacement f i e l d s under undrained l o a d i n g are important not only from s t a b i l i t y viewpoint, but a l s o because the s t r e s s d i s t r i b u t i o n at the end of undrained l o a d i n g governs the subsequent c o n s o l i d a t i o n behaviour. In p r e d i c t i n g deformation and s t r e s s d i s t r i b u t i o n i n loaded s o i l masses, the f i n i t e element method p r o v i d e s a very powerful technique. The method r e q u i r e s s e v e r a l steps of i d e a l i z i n g the s o i l mass and modelling i t s s t r e s s - s t r a i n behaviour. T h e r e f o r e , the a c c u r a t e p r e d i c t i o n s are dependent not only on the accuracy of the d i s c r e t i z a t i o n and numerical procedures, but a l s o on the a b i l i t y to i d e a l i z e the s u b s o i l c o n d i t i o n s and formulate mathematical models s i m u l a t i n g the s t r e s s - s t r a i n behaviour of r e a l s o i l . T h i s t h e s i s c o n c e n t r a t e s on mode l l i n g the n o n l i n e a r s t r e s s - s t r a i n behaviour of r e a l s o i l . The s t r e s s - s t r a i n c h a r a c t e r i s t i c s of r e a l s o i l are complex, being n o n l i n e a r , i n e l a s t i c and s t r e s s system(and s t r e s s system change) dependent. Among s e v e r a l other methods of mode l l i n g n o n l i n e a r i t y , curve f i t t i n g methods i n v o l v i n g h y p e r b o l i c f u n c t i o n s have widely been used. The h y p e r b o l i c f u n c t i o n was 2 adapted for f i n i t e element a n a l y s i s by Duncan and h i s cowokers(1970), using Kondner's(1963) f i n d i n g that the s t r e s s -s t r a i n curve i n t r i a x i a l compression of i s o t r o p i c a l l y c o n s o l i d a t e d s o i l approximates a hyperbola. Most of the e x i s t i n g n o n l i n e a r s o i l s t r u c t u r e i n t e r a c t i o n programmes(e.g. The one m o d i f i e d h e r e i n , the N L S S I P by Byrne et a l . , 1979) use the h y p e r b o l i c s t r e s s - s t r a i n curves obtained from i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l compression t e s t s . These models assume i d e n t i c a l c o n s t i t u t i v e behaviour i n e x t e n s i o n , compression, or under any other s t r e s s system change. But t h i s i s not true f o r even i s o t r o p i c a l l y c o n s o l i d a t e d s o i l s . U s u a l l y the undrained s t r e n g t h i s lower in extension than in compression, and i s i n general dependent on the change in s t r e s s system d u r i n g shear. A l s o the s t i f f n e s s c h a r a c t e r i s t i c s are q u i t e d i f f e r e n t under d i f f e r e n t s t r e s s system changes. In most f i e l d s i t u a t i o n s however, the s o i l d e p o s i t s are not i s o t r o p i c a l l y c o n s o l i d a t e d . The s t r e s s - s t r a i n behaviour under such c o n d i t i o n s show l a r g e d i f f e r e n c e s , when compared with i s o t r o p i c a l l y c o n s o l i d a t e d behaviour. Thus, the adequacy of i s o t r o p i c models, when used in p r e d i c t i n g the deformation behaviour of a n i s o t r o p i c a l l y c o n s o l i d a t e d s o i l masses i s q u e s t i o n a b l e . A model s i m u l a t i n g the a n i s o t r o p i c a l l y c o n s o l i d a t e d s o i l behaviour i s presented h e r e i n . An e x i s t i n g s o i l s t r u c t u r e i n t e r a c t i o n programme NLSSIP (Byrne et a l . , I979)is m o d i f i e d to i n c o r p o r a t e the proposed model and the p r e d i c t i o n s of undrained deformation under f o o t i n g s are compared with the unmodified 3 programme r e s u l t s . The s t r e s s - s t r a i n r e l a t i o n s f o r use i n the a n a l y s i s can be obtained d i r e c t l y from l a b o r a t o r y t e s t s . I f such r e s u l t s are u n a v a i l a b l e design c h a r t s are presented a l l o w i n g estimates of the s t r e s s - s t r a i n r e l a t i o n s to be made from such parameters as the p l a s t i c i t y index, c o e f f i c i e n t of e a r t h pressure at r e s t K 0 and i n s i t u c o n f i n i n g p r e s s u r e s . A l s o these r e l a t i o n s can be used to judge the reasonableness of the s t r e s s - s t r a i n parameters obtained from the l a b o r a t o r y t e s t s . 4 CHAPTER II_ ISOTROPIC AND ANISOTROPIC STRESS-STRAIN MODELS 2.1 BACKGROUND The s t r e s s - s t r a i n behaviour of s o i l i s very complex, being n o n l i n e a r , s t r e s s and s t r e s s system dependent. There have been many schemes developed to model these c h a r a c t e r i s t i c s . B a s i c a l l y there are four c a t e g o r i e s of such procedures. Namely, curve f i t t i n g methods, higher order n o n l i n e a r e l a s t i c t h e o r i e s , p l a s t i c i t y models and v i s c o e l a s t i c t h e o r i e s . Out of the above methods, the h y p e r b o l i c model devoloped by Duncan and h i s collegues(1970) has been widely used in mo d e l l i n g n o n l i n e a r i t y of s o i l s . T h i s i s based on Kondner's(1963) f i n d i n g s that the s t r e s s - s t r a i n curve i n an i s o t r o p i c a l l y c o n s o l i d a t e d compression t e s t c l o s e l y approximates a hyperbola. These h y p e r b o l i c models are not only simple, but a l s o the parameters used i n the model have r e a d i l y v i s u a l i z e d p h y s i c a l meanings, ( d i s c u s s e d i n s e c t i o n 2.2) When the m a t e r i a l i s assumed to be l i n e a r and i s o t r o p i c , the s t r e s s - s t r a i n r e l a t i o n s can be completely d e f i n e d using two independent e l a s t i c parameters. U s u a l l y Young's modulus and bulk modulus are the two parameters used. However, i n the case of undrained behaviour the bulk modulus i s t h e o r e t i c a l l y i n f i n i t e , and Young's modulus i s the governing f a c t o r . T h i s t h e s i s d i s c u s s e s the undrained behaviour of s a t u r a t e d c l a y s and thus the s t r e s s - s t r a i n f o r m u l a t i o n s h e r e i n presented w i l l 5 b a s i c a l l y i n v o l v e the determination of Young's modulus. 2.2 ISOTROPIC MODEL Kondner(1963) proposed the h y p e r b o l i c f u n c t i o n given below to d e s c r i b e the no n l i n e a r s t r e s s - s t r a i n curve of i s o t r o p i c a l l y c o n s o l i d a t e d s o i l s under t r i a x i a l compression c o n d i t i o n s . (0 , -0 -3) = e a + be where, c, i s the major p r i n c i p a l s t r e s s 0*3 i s the minor p r i n c i p a l s t r e s s e i s the a x i a l s t r a i n ( i n the major p r i n c i p a l d i r e c t i o n ) a, b, are constants whose values may be determined e x p e r i m e n t a l l y . These constants have r e a d i l y v i s u a l i z e d p h y s i c a l meanings. As shown i n F i g u r e 2-1 i t can be proved that the r e c i p r o c a l of "a" i s the i n i t i a l tangent modulus and the r e c i p r o c a l of "b" i s the asymptotic value of ( o ^ - o ^ ) . As such, these constants are b e t t e r i d e n t i f i e d by a = j_, b = 1 E. T o T ^ T T u l t where, E i i s the i n i t i a l tangent modulus (cr,-cr 3 ) u l t i s the u l t i m a t e s t r e n g t h . U s u a l l y the u l t i m a t e s t r e n g t h exceeds the a c t u a l compressive s t r e n g t h due to lack of f i t i n the curve near f a i l u r e F i g u r e 2-1. Hyperbolic S t r e s s - S t r a i n Curve 7 s t r a i n s . T herefore, a t h i r d parameter, R f, i s introduced(Duncan et a l . , 1970) to r e p l a c e the t h e o r e t i c a l q u a n t i t y , the u l t i m a t e s t r e n g t h , i n terms of the a c t u a l s o i l s t r e n g t h . Where, R = The a c t u a l s t r e n g t h  f The u l t i m a t e s t r e n g t h Thus the h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n s h i p can be w r i t t e n in the f o l l o w i n g manner. 1 + R f C E i S u c i Where, i s the d e v i a t o r s t r e s s S U C I i s the i s o t r o p i c a l l y c o n s o l i d a t e d undrained compression s t r e n g t h . For number of d i f f e r e n t s o i l s , the q u a n t i t y R has been found to l i e between .75 and 1.00 and to be e s s e n t i a l l y independent of c o n f i n i n g pressure(Duncan et a l . , 1970). 2.2.1 DETERMINATION OF MODEL PARAMETERS The l a b o r a t o r y t e s t i n g programme to determine model parameters i n c l u d e a s e r i e s of t r i a x i a l compression t e s t s performed on undisturbed samples, which are i s o t r o p i c a l l y c o n s o l i d a t e d ( s e e a l s o s e c t i o n 4.2.2). In order to o b t a i n the complete s t r e s s - s t r a i n curve they are sheared under undrained con d t i o n s u n t i l f a i l u r e . To o b t a i n the model parameters the 8 l a b o r a t o r y s t r e s s - s t r a i n curves are analysed as f o l l o w s . The h y p e r b o l i c r e l a t i o n °d = — £ 1 + e E i " ^ u l t can be w r i t t e n i n the form e = e + J_ a d " r a ? u l t . E i and e / c d vs e (Figure 2-2) r e p r e s e n t s a s t r a i g h t l i n e . Thus, to determine the best f i t hyperbola f o r the s t r e s s - s t r a i n curve the v a l u e s of e/crd are computed from the t e s t r e s u l t s and p l o t t e d a g a i n s t e. The best f i t s t r a i g h t l i n e on (e/cr d,e) plane corresponds to the best f i t hyperbola on the s t r e s s -s t r a i n p l o t . Of the best f i t l i n e a r r e l a t i o n s h i p on the transformed plane ( e / o £ , e ) ; The i n i t i a l modulus i s the r e c i p r o c a l of the i n t e r c e p t The u l t i m a t e s t r e n g t h i s the r e c i p r o c a l of the g r a d i e n t R =(The i s o t r o p i c compressive s t r e n g t h ) . ( T h e g r a d i e n t ) It has been found(Duncan et a l . , 1970) that the s t r e s s - s t r a i n data d i v e r g e somewhat from a s t r a i g h t l i n e on the transformed p l a n e ( i . e . they d i v e r g e from the hyperbola on the (o" d,e) plane.) at low and high s t r a i n s . For a v a r i e t y of s o i l s , i t has been found(Duncan et a l . , 1970) that the best f i t s t r a i g h t l i n e with regard to o v e r a l l agreement passes through 9 F i g u r e 2-2. H y p e r b o l i c S t r e s s - S t r a i n Curve on the Transformed Plane 10 S=.7 and S=.95 (S= ( c r , - c r 3 )/strength m o b i l i s e d ) . Therefore i n p r a c t i c e , only the above two p o i n t s are p l o t t e d on the transformed diagram to o b t a i n the model parameters. 2.2.2 LIMITATIONS OF ISOTROPIC MODEL Undrained s t r e s s - s t r a i n and s t r e n g t h behaviour i s dependent on the i n i t i a l c o n s o l i d a t i o n s t r e s s e s , s t r e s s system change durin g shear and r a t e of l o a d i n g . To make r e l i a b l e deformation behaviour p r e d i c t i o n s , the c o n s t i t u t i v e models must be abl e to simulate a l l the c h a r a c t e r i s t i c s mentioned above. The f o l l o w i n g s e c t i o n s d i s c u s s the above t o p i c s i n d e t a i l and the treatment of rate of l o a d i n g however, i s beyond the scope of t h i s t h e s i s . 2.2.2(a) S t r e s s System Change During Shear The f o l l o w i n g d i s c u s s i o n attempts to show(more q u a l i t a t i v e l y ) the i n f l u e n c e of s t r e s s system change durin g shear on the s t r e s s - s t r a i n behaviour of c l a y which has been c o n s o l i d a t e d a n i s o t r o p i c a l l y under R 0 c o n d i t i o n s . I t i s assumed that the shape of the s t r e s s - s t r a i n curve i s very approximately determined by the f a i l u r e s t r e n g t h and s t r a i n at f a i l u r e . Table 2-1 prese n t s data from v a r i o u s t y p e s ( v a r i o u s s t r e s s system changes) of undrained s t r e n g t h tests(Ladd,1973) on a normally c o n s o l i d a t e d marine c l a y of moderate s e n s i t i v i t y . In the f i r s t two t e s t s ( p l a n e s t r a i n a c t i v e and t r i a x i a l compression) there i s no r o t a t i o n of p r i n c i p a l s t r e s s p l a n e s . The undrained s t r e n g t h i s highest 11 TABLE 2-1. UNDRAINED ANISOTROPY OF NORMALLY CONSOLIDATED BOSTON BLUE CLAY (LL« 41%, PI. - 21%) TYPE OF TEST M SITU CONDITION SHEAR STRAIN Iff . (%) S u " C " * v o * P Plane strain active 0 8 0 34 IJ Triaxial companion Circular footing 0.5 0 33 3J Direct - simple stteor 6 0 2 0 ±1 Plane strain passive 9 0 19 5J Triaxial extension Circular excavation Q 15 0155 £ j Field vone — 0.19 u •Model strip footing I 'ult — 0 26 ( N e = 5rl4) Note : S u * 0 5 (<r, - crj)f except for DSS and field vane tests 12 and s t r a i n at f a i l u r e i s lowest. During simple shear t e s t p r i n c i p a l planes r o t a t e so as to produce a h o r i z o n t a l f a i l u r e s u r f a c e . The undrained s t r e n g t h i s lower and s t r a i n at f a i l u r e i s h i g h e r . In the plane s t r a i n p a s s i v e and t r i a x i a l e x t e n s i o n t e s t s , s t r e n g t h values are the lowest and corresponding f a i l u r e s t r a i n s are the h i g h e s t . During these two t e s t s p r i n c i p a l s t r e s s planes make a f u l l 90° r o t a t i o n . These however, are some of the b a s i c modes of f a i l u r e and there can be many other d i f f e r e n t combinations. N e v e r t h e l e s s , i t c l e a r l y shows that the s t r e s s - s t r a i n and s t r e n g t h c h a r a c t e r i s t i c s of a c l a y i s dependent on the changes i n the s t r e s s system d u r i n g shear. In p r a c t i c e however, problems u s u a l l y cannot be c l a s s i f i e d under a p a r t i c u l a r mode of f a i l u r e , i n s t e a d s e v e r a l modes may occur s i m u l t a n e o u s l y . For example, i n the case of a slope f a i l u r e , a l l three modes of f a i l u r e ( S e e F i g u r e 2-3) d i s c u s s e d above are encountered. T h i s emphasizes the importance of the c o n s t i t u t i v e model's a b i l i t y t o simulate d i f f e r e n t modes of f a i l u r e . The i s o t r o p i c models used i n f i n i t e element programmes(e.g. NLSSIP by Byrne et a l . , 1970) can resonabl y simulate only the plane s t r a i n a c t i v e ( o r t r i a x i a l compression) s t a t e under i s o t r o p i c a l l y c o n s o l i d a t e d c o n d i t i o n s . For a l l other s t r e s s system changes, i t assumes that the same s t r e s s - s t r a i n curve i s a p p l i c a b l e . But t h i s does not h o l d t r u e even f o r i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s . For example, i n Direct - simple (hear Figure 2-3. D i f f e r e n t Stress Systems along the F a i l u r e Surface 14 i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s , undrained s t r e n g t h i s lower in extension than that i n compression,(as was s t a t e d before f o r K 0 c o n s o l i d a t e d c l a y s ) e s p e c i a l l y i n low p l a s t i c i t y c l a y s ( s e e s e c t i o n 5.3.4). A l s o the s t i f f n e s s c h a r a c t e r i s t i c s are d i f f e r e n t under d i f f e r e n t s t r e s s system changes in i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s as w e l l . 2.2.2(b) I n i t i a l C o n s o l i d a t i o n S t r e s s e s The i n i t i a l c o n s o l i d a t i o n s t r e s s s t a t e has a great i n f l u e n c e on the undrained s t r e s s - s t r a i n and s t r e n g t h c h a r a c t e r i s t i c s of a c l a y . The behaviour of a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y i s q u i t e d i f f e r e n t from i s o t r o p i c a l l y c o n s o l i d a t e d c l a y , from both s t r e n g t h and s t i f f n e s s view p o i n t s . T h i s i s t r u e f o r a l l kinds of s t r e s s system changes. F i g u r e 2-4 shows t y p i c a l undrained compression and e x t e n s i o n s t r e s s - s t r a i n curves(schematic) of i s o t r o p i c a l l y and a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s . The lower the K 0 value the l a r g e r such d i f f e r e n c e s are. E m p i r i c a l r e l a t i o n s c o r r e l a t i n g s t r e s s - s t r a i n parameters and i n i t i a l c o n s o l i d a t i o n c o n d i t i o n s such as K 0 and v e r t i c a l c o n f i n i n g p r e s s u r e s are presented in Chapter 5. ;Most g e o t e c h n i c a l problems encountered in p r a c t i c e u s u a l l y i n v o l v e a n i s o t r o p i c a l l y c o n s o l i d a t e d s o i l s . T h e r e f o r e , to make r e l i a b l e p r e d i c t i o n s of deformation behaviour i n such s o i l masses, the e f f e c t s of i n i t i a l s t r e s s a n i s o t r o p y must be i n c o r p o r a t e d i n the s t r e s s -15 ANISOTROPIC STRESS-STRAIN CURVE Figure 2-4. Is o t r o p i c a l l y Stress and A n i s o t r o p i c a l l y -Strain Curves Consolidated 1 6 stra i n model. The influence of stress system change i s also important. In view of the above facts, an undrained stress-st r a i n model for a n i s o t r o p i c a l l y consolidated clay is proposed. This model can simulate both, plane s t r a i n a c t i v e ( t r i a x i a l compression) and p a s s i v e ( t r i a x i a l extension) under a n i s o t r o p i c a l l y consolidated conditions. And in general i t takes into account the influence of stress system change and i n i t i a l stress anisotropy on s t r e s s - s t r a i n r e l a t i o n s . This takes the model closer to the real s o i l behaviour. The model can be used to analyse s a t i s f a c t o r i l y the undrained deformation behaviour under monotonic loading conditions. F i n a l l y , the proposed model is further ahead of the isotropic model in two v i t a l aspects. It can simulate the e f f e c t s of stress system change on deformation behaviour and also can take in-to account the influence of consolidation stress anisotropy. 2.3 ANISOTROPIC MODEL The new model discussed herein attempts to characterize the undrained behaviour of clay in a more r e a l i s t i c way. This undrained model i s intended to be employed in pseudo-linear methods of f i n i t e element stress analysis. It provides a method for evaluating Young's modulus of soiKunder any stress conditions) needed in such analysis. As discussed before the i n i t i a l stress anisotropy aff e c t s the undrained s t e s s - s t r a i n curve with respect to both 1 7 the strength and s t i f f n e s s . A l s o the strength and strain at failureCthus the modulus) depend s i g n i f i c a n t l y on the stress system change or the angle of p r i n c i p a l axes rotation. For example, the strength in t r i a x i a l extension i s lower than that in compression which represent the extreme cases of p r i n c i p a l axes rotations of 90° and 0° respectively. In a given geotechnical problem however, most of the stress 0 p o i n t s ( f i n i t e elements) undergo some rotation of p r i n c i p a l axes, which l i e between the above two extremes. The proposed model incorporates the above s t r e s s - s t r a i n behaviour and i t b a s i c a l l y involves determining a s t r e s s - s t r a i n curve for each angle of p r i n c i p a l axes rotation. The following are the basic assumptions made in developing the new model. 1. Under a l l loading conditions, the angle of rotation of p r i n c i p a l axes l i e s between 0° and 90°(including both) 2. For a given clay, Young's modulus depends only on the i n i t i a l consolidation stresses, current deviator stress and the angle of rotation of p r i n c i p a l axes. The second assumption implies that i f , the i n i t i a l consolidation stress conditions are known, for a given angle of p r i n c i p a l axes rotation, s t r e s s - s t r a i n curve can be uniquely determined. Consider the complete compression-extension stress-18 s t r a i n curve shown i n F i g u r e 2-5. Along OC and OA the angle of p r i n c i p a l axes r o t a t i o n i s 0°, whereas along AE the r o t a t i o n i s 90°. But, as d i s c u s s e d i n the p r e v i o u s paragraph, the s t r e s s - s t r a i n curves corresponding to a given angle of p r i n c i p a l axes r o t a t i o n i s u n i q u e ( f o r a given c l a y under a given i n i t i a l s t a t e ) . T h e r e f o r e , by running compression and e x t e n s i o n t e s t s on a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s , the unique s t r e s s - s t r a i n curves corresponding to 0° and 90° p r i n c i p a l axes r o t a t i o n s can e a s i l y be obtained. In the subsequent d i s c u s s i o n these two curves w i l l simply be r e f e r r e d to as 0° and 90° s t r e s s - s t r a i n curves r e s p e c t i v e l y . The two curves p l o t t e d together i n F i g u r e 2-6 give a comparison of the. two p l o t s . The corresponding tangent or secant moduli values can be d i r e c t l y determined from the above two curves. A schematic diagram of the tangent moduli v a l u e s vs the d e v i a t o r s t r e s s i s shown in F i g u r e 2-6. As d i s c u s s e d above, o b t a i n i n g the 0° and 90° s t r e s s - s t r a i n curves i s f a i r l y s t r a i g h t f o r w a r d . A l s o , these curves can be used to e v a l u a t e the moduli values d i r e c t l y . But the d e t e r m i n a t i o n of the intermediate curves; that i s the curves c o r r e s p o n d i n g to p r i n c i p a l axes r o t a t i o n s other than 0° and 90°(0°<e<90°) i s much i n v o l v e d and rather complicated. T h e r e f o r e , to o b t a i n the tangent modulus corre s p o n d i n g to any other p r i n c i p a l axes r o t a t i o n , the two moduli v a l u e s o b t a i n e d ( f o r the same d e v i a t o r s t r e s s ) from 0° and 90° s t r e s s - s t r a i n curves are i n t e r p o l a t e d . The formula proposed by Duncan et a l . , (1969) for e v a l u a t i n g what i s c a l l e d an 19 a yy 4*-F i g u r e 2-5. A n i s o t r o p i c a l l y C o n s o l i d a t e d Compression and Extension S t r e s s - S t r a i n Curves Figure 2-6. Tangent Modulus as a Function of D e v i a t o r S t r e s s 21 a p p a r e n t e l a s t i c m o d u l u s f o r a n i s o t r o p i c s o i l s i s u s e d a s t h e i n t e r p o l a t i o n f o r m u l a f o r t h e a b o v e c o m p u t a t i o n . T h u s , E = E 0 - ( E 0 - E 9 0 ) S i n 2 e 8 w h e r e , 6° i s t h e a n g l e o f p r i n c i p a l a x e s r o t a t i o n E 0 i s t h e e l a s t i c m o d u l u s when t h e p r i n c i p a l a x e s r o t a t i o n i s G ° . The u n d r a i n e d s t r e n g t h a l s o v a r i e s w i t h t h e i n i t i a l c o n s o l i d a t i o n s t a t e a s w e l l a s t h e a n g l e o f p r i n c i p a l a x e s r o t a t i o n . A n i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l c o m p r e s s i o n and e x t e n s i o n t e s t s c a n be u s e d t o o b t a i n t h e u n d r a i n e d s t r e n g t h c o r r e s p o n d i n g t o 0° and.90° r o t a t i o n s o f p r i n c i p a l a x e s . B u t , a s m e n t i o n e d b e f o r e , t h e d e t e r m i n a t i o n o f t h e u n d r a i n e d s t r e n g t h f o r a n g l e s o f p r i n c i p a l a x e s r o t a t i o n s o t h e r t h a n 0° a n d 90° i s r a t h e r c o m p l i c a t e d a n d d i f f c u l t . The u n d r a i n e d s t r e n g t h v a l u e s f o r any r o t a t i o n G° a r e e v a l u a t e d by i n t e r p o l a t i n g t h e s t r e n g t h v a l u e s c o r r e s p o n d i n g t o 0° and 90° a x e s r o t a t i o n s . The f o r m u l a p r o p o s e d by C a s a g r a n d e e t a l . , ( 1 9 4 4 ) t o r e p r e s e n t c o h e s i v e a n i s o t r o p y i s u s e d a s t h e i n t e r p o l a t i o n f o m u l a f o r t h e ab o v e c a s e . T h u s , ^u0 = ^ uo~ ( S u 0 _ S u 9 o ) S i n 2 9 w h e r e , 6° i s t h e a n g l e o f p r i n c i p a l a x e s r o t a t i o n S u Q i s t h e u n d r a i n e d s t r e n g t h when t h e p r i n c i p a l a x e s r o t a t i o n i s 9 ° . U s i n g t h e ab o v e f o r m u l a , t h e u n d r a i n e d s t r e n g t h c o r r e s p o n d i n g t o any a n g l e o f p r i n c i p a l a x e s r o t a t i o n c a n be 22 computed. In order to use the proposed model in n o n l i n e a r s t r e s s a n a l y s i s procedures, i t must be expressed i n a mathematical form so that the modulus values can r e a d i l y be computed. The model assumes the compression and e x t e n s i o n p a r t s of the s t r e s s - s t r a i n curve to be h y p e r b o l i c . T h i s i s e s s e n t i a l l y V a i d ' s ( l 9 8 l ) approach of mathematically r e p r e s e n t i n g K 0 c o n s o l i d a t e d compression and e x t e n s i o n s t r e s s - s t r a i n c u rves. That i s the curves OC and OE(Figure 2-5) are assumed to be p a r t s of separate hyperbolas s t a r t i n g from the i n i t i a l c o n s o l i d a t i o n s t r e s s s t a t e . For s i m p l i c i t y i t assumes the i n i t i a l moduli v a l u e s i n compression and i n e x t e n s i o n to be e q u a l . In other words i t assumes t h a t the slope of the s t r e s s - s t r a i n curve to be continuous at the p o i n t 0 ( F i g u r e 2-5). Even though t h i s i s not s t r i c t l y t r u e , the d i f f e r e n c e s are u s u a l l y small and can be c o n s i d e r e d to be i n s i g n i f i c a n t . However such d i f f e r e n c e s can be i n c o r p o r a t e d i n the model at the expense of s i m p l i c i t y . These curves are used to d e r i v e the 0° and 90° s t r e s s - s t r a i n curves which are the b a s i s of the proposed model. Once again i t must be emphasised that in t h i s model s t r a i n s are measured from the i n i t i a l c o n s o l i d a t i o n s t a t e . In other words the p o i n t 0 ( F i g u r e 2-4) i s the i n i t i a l s t a t e assumed by the proposed model as opposed to the p o i n t 0' assumed by the i s o t r o p i c models. The compression part of the s t r e s s - s t r a i n curve i s d e p i c t e d by the l i n e OC and i s expressed i n the form(Vaid,1981) 23 q = e 1 + e E i KJc where, q = (cr -cr )- (cr -cr ) o yy xx yy xx cr^ i s the a x i a l p r i n c i p a l s t r e s s i n the X d i r e c t i o n Oyy i s the a x i a l p r i n c i p a l s t r e s s i n the Y d i r e c t i o n ( c r - a ) 0 i s the i n i t i a l d e v i a t o r s t r e s s yy xx 0 (q ) i s the asymptotic value of q i n compression u l t c e i s the s t r a i n i n Y d i r e c t i o n . The e x t e n s i o n p a r t of the s t r e s s - s t r a i n curve i s represented by the l i n e OE. I t can a l s o be mathematically expressed i n the form q = e 1 +" e where, q = (cr -cr ) Q - (cr -cr ) ^ yy xx u yy xx ( Q u l 1 ) E i s the asymptotic value of q in e x t e n s i o n . The corresponding 0° and 90° s t r e s s - s t r a i n curves can be mathematically expressed using the above equations. Along AOC the angle of p r i n c i p a l axes r o t a t i o n i s 0°. T h e r e f o r e the 0° s t r e s s - s t r a i n curve(AOC) can be given i n the f o l l o w i n g form. g - - 1 f 0 r ^ y - ^ o ^ V ^ ^ u o i OC 1 + e y * E i " ^ i 7 c 24 q = e f o r (cr - c r j 0> (cr -cr x x^0 ; OA j_ + e n xx E i ( ( W E And along AE the angle of p r i n c i p a l axes r o t a t i o n i s 90° and thus the 90° s t r e s s - s t r a i n curve can be expressed in the form q = e f o r 0 > ( c r - o x x ) > - S u 9 0 ; AE - + e E i (qult^E A c c o r d i n g to the second assumption(sect ion 2.3) of the proposed model the e l a s t i c modulus of a given c l a y under a given angle of p r i n c i p a l axes r o t a t i o n depends only on the i n i t i a l c o n d i t i o n s and the c u r r e n t d e v i a t o r ' s t r e s s . T herefore i t i s worthwhile to express equations d e s c r i b i n g 0° and 90° s t r e s s - s t r a i n curves i n terms of d e v i a t o r s t r e s s as given below. The 0° s t r e s s - s t r a i n curve can be expressed i n the form q = e f o r (o-,-cr3 ) 0< (cr,-cr 3) <S u 0 ; OC 1 + e  E i (QuitJc where, q = (o-,-o-3 )-(cr,-cr 3 ) o ( q u l t ) c i s the asymptotic value of q i n compression and q = __e f o r 0<(<Ti-cr3 )<(cr1 -cr 3) 0 ; OA 1 + e  E i ( W E where, q = ( c r , - ^ ) 0-(o- 1-cr 3) 25 ( q u l 1 ) E i s the asymptotic value of q i n extension The 90° s t r e s s - s t r a i n curve can be expressed i n the form q = e f o r 0<(o-1-o-3 ) < S u 9 0 ; AE J_ + e E i ( C W E q = (o-T-o-a ) + (cr1 -o-3) o ( q u l t ) E i s the asymptotic value of q i n ext e n s i o n The proposed s t r e s s - s t r a i n model was v e r i f i e d f o r i t s h y p e r b o l i c nature using p u b l i s h e d d a t a ( d e r i v e d transformed p l o t s are given i n Appendix A) these i n c l u d e a n i s o t r o p i c a l l y as w e l l as i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s . F i g u r e s 2-7, 2-8 and 2-9 show t y p i c a l r e s u l t s having good h y p e r b o l i c f i t on the transformed planes f o r i s o t r o p i c a l l y c o n s o l i d a t e d e x t e n s i o n t e s t s and a n i s o t r o p i c a l l y c o n s o l i d a t e d s t r e s s -s t r a i n r e s u l t s . In general the p u b l i s h e d data show good agreement with the proposed model(Appendix A). The best f i t s t r a i g h t l i n e on the transformed p l a n e ( e , e / q ) i s c o n s i d e r e d to correspond to the best f i t hyperbola. 2.3.1 DETERMINATION OF MODEL PARAMETERS The a n i s o t r o p i c model parameters can be obtained from l a b o r a t o r y t e s t s as f o l l o w s ( s e e a l s o s e c t i o n 4.2.2). Undisturbed s o i l samples brought to the l a b o r a t o r y are a n i s o t r o p i c a l l y c o n s o l i d a t e d to the f i e l d K 0 value i n t r i a x i a l c e l l s . Then they are sheared under undrained c o n d i t i o n s i n extension as we l l as i n compression. S t r e s s -s t r a i n data are recorded u n t i l f a i l u r e and are analysed to WEATHERED BANGKOK CLAY(U) Balasubramaniam et a l . , 1977 kN/m: 207.0 276.0 345.0 414.0 • A -r a CD A CD A -r CO A CD A + A + 4 -f 0.0 0.02 0.04 1 0.06 STRAIN 0.08 0.1 Figure 2-7. Transformed P l o t s of I s o t r o p i c a l l y C o n s o l i d a t e d Extension T e s t s 27 L O O HANEY CLAY(U); K 0 V a i d , et a l . , 1974 = .56 o o era ' i— CO CM a o a a a 0.3 1.0 2.0 / S T R A I N 3.0 4.0 F i g u r e 2-8. Transformed P l o t s of A n i s o t r o p i c a l l y C o n s o l i d a t e d E x t e n s i o n T e s t s 28 LO o LO o •—i • era I— CO C N a a a a a HANEY CLAY(U); K 0 = .56 V a i d , et a l . , 1974 0 . 0 I 1— 0 . 2 0 . 4 ^STRAIN 0 . 6 F i g u r e 2-9. Transformed P l o t s of A n i s o t r o p i c a l l y C o n s o l i d a t e d Compression T e s t s 29 o b t a i n the model parameters. The computational procedures i n v o l v e d a re very s i m i l a r t o the methods used i n the i s o t r o p i c a l l y c o n s o l i d a t e d case. The ge n e r a l hyperbola f o r both extension and compression can be w r i t t e n i n the form q = e 1 + e E i * u l t and i t re p r e s e n t s the s t r a i g h t l i n e e = e + J_ on (e,e/q) plane. The t e s t r e s u l t s are f i t t e d to the hyperbola i n order to o b t a i n the model parameters. The best f i t i s ob t a i n e d using the transformed p l o t on (e,e/q) p l a n e ( c . f . i s o t r o p i c case; s e c t i o n 2.2.1). The best f i t s t r a i g h t l i n e on transformed plane i s co n s i d e r e d to give the best f i t hyperbola on (e,q) plane f o r the a n i s o t r o p i c case too. T h i s s t r a i g h t l i n e can be used to o b t a i n the necessary parameters(Figure 2-10) and; The i n i t i a l modulus i s the r e c i p r o c a l of the i n t e r c e p t . The u l t i m a t e q i s the r e c i p r o c a l of the g r a d i e n t . Rf i s S„ ± (oyy-cfrd o + f o r extension q u - f o r compression q = (Oyy ' S c^'^y"^ 0 i n compression q = (o- -or ) 0 - ( c r -cr ) i n ex t e n s i o n . yy xx yy xx Chapter 5 presents c o r r e l a t i o n s between these parameters 30 F i g u r e 2-10. Transformed P l o t of the A n i s o t r o p i c a l l y C o n s o l i d a t e d H y p e r b o l i c S t r e s s - S t r a i n Curve 31 and s o i l p r o p e r t i e s such as p l a s t i c i t y index, i n s i t u c o n f i n i n g p r e s s u r e s and c o e f f i c i e n t of e a r t h pressure at r e s t K 0. These c o r r e l a t i o n s can be used as v a l u a b l e t o o l s i n judging the r e l i a b i l i t y of the parameters obtained from l a b o r a t o r y t e s t r e s u l t s . A l s o these r e l a t i o n s can be u s e f u l i n the absence of s u f f i c i e n t data or i n cases where expensive l a b o r a t o r y t e s t i n g procedures are not j u s t i f i e d . 32 CHAPTER I I I  PROCEDURES FOR NONLINEAR  STRESS ANALYSIS 3.1 FINITE ELEMENT TECHNIQUE For many g e o t e c h n i c a l problems i t i s not p o s s i b l e to obt a i n c l o s e d form s o l u t i o n s . Such s o l u t i o n s are a v a i l a b l e only f o r c e r t a i n s i m p l i f i e d s i t u a t i o n s . A method which employs Boussinesq's a n a l y t i c a l e l a s t i c s t r e s s s o l u t i o n s i s one commonly used c o n v e n t i o n a l technique f o r computing i n i t i a l s e t t l e m e n t s . Although the pre v i o u s method i s simple i t has s e v e r a l p r a c t i c a l drawbacks. The assumptions that the s o i l i s homogeneous and l i n e a r l y e l a s t i c d r a s t i c a l l y o v e r s i m p l i f i e s many f i e l d s i t u a t i o n s encountered i n p r a c t i c e . The other commonly used c o n v e n t i o n a l method i s the s t r e s s path method put forward by Lambe(1964,1967). I t assumes t h a t , i f a l a b o r a t o r y sample i s sub j e c t e d to the same i n i t i a l s t r e s s e s and the same s t r e s s changes due to l o a d i n g as a corresponding element i n the ground, then the l a b o r a t o r y sample w i l l undergo the same s t r a i n s as the f i e l d element. A major disadvantage of t h i s method i s that the i n s i t u s t r e s s e s and s t r e s s changes must be known before s t r e s s path t e s t s can be performed. Except f o r few cases such as, the change i n l e v e l of ground water t a b l e , i t i s not p o s s i b l e to get a c c u r a t e values f o r the i n s i t u s t r e s s changes. Boussinesq's e l a s t i c s o l u t i o n s have been used f o r t h i s purpose with the d i f f i c u l t i e s a l r e a d y mentioned. 33 In view of the above d i f f i c u l t i e s , f i n i t e element method p r o v i d e s a very powerful technique i n s o l v i n g complex g e o t e c h n i c a l e n g i n e e r i n g problems. T h i s technique has been used to s o l v e problems having m a t e r i a l s with l i n e a r , n o n l i n e a r and many other forms of c o n s t i t u t i v e r e l a t i o n s , under plane s t r a i n , a x i s y m e t r i c and general three dimensional c o n d i t i o n s . The method r e p l a c e s a continuum with an e q u i v a l e n t f i n i t e assemblage of d i s c r e t e s m a l l e r continua c a l l e d f i n i t e elements. These elements are connected at a f i n i t e number of nodes. A l l of the m a t e r i a l p r o p e r t i e s can be r e t a i n e d i n the i n d i v i d u l elemen'ts. The a n a l y s i s uses the averages(or weighted averages) of these p r o p e r t i e s as element p r o p e r t i e s . The governing equations of the f i n i t e element mesh can g e n e r a l l y be solved- without r e s o r t i n g to approximate mathematical procedures. Thus, the accuracy of the s o l u t i o n i s dependent s o l e l y on the d e s c r e t i z a t i o n procedure and i d e a l i z a t i o n of the p h y s i c a l and c o n s t i t u t i v e p r o p e r t i e s . The method assumes a d i s t r i b u t i o n p a t t e r n ( f u n c t i o n ) f o r the unknown q u a n t i t y over the domains of each element. U s u a l l y t h i s unknown i s the displacement or s t r e s s or both. If the unknown i s displacement, s t r e s s or both the f o r m u l a t i o n i s c a l l e d the displacement method, s t r e s s method or mixed method r e s p e c t i v e l y . These d i s t r i b u t i o n f u n c t i o n s are u s u a l l y p o l y n o m i a l s . G e n e r a l l y , higher order polynomials g i v e s o l u t i o n s t h a t are c l o s e r to the exact answer. However, t h i s i s not the only c r i t e r i o n i n s e l e c t i n g the d i s t r i b u t i o n 34 f u n c t i o n . These f u n c t i o n s must be c o n t i n u o u s and must have c o n t i n u o u s d e r i v a t i v e s , a c o n d i t i o n r e f e r r e d t o as a d m i s s i b i l i t y . T h i s c o n d i t i o n i s u s u a l l y t aken f o r g r a n t e d and sometimes not mentioned a t a l l . The d i s p l a c e m e n t must be c o m p a t i b l e a l o n g the edges of the elements and such elements a r e c a l l e d c o n f o r m i n g elements. The d i s p l a c e m e n t f u n c t i o n must be a b l e t o s i m u l a t e b o t h the r i g i d body motion of the element and a c o n s t a n t s t r a i n s t a t e over the e n t i r e element. The l a t t e r two c o n d i t i o n s a r e r e f e r r e d t o as completeness of the d i s t r i b u t i o n f u n c t i o n . These above c o n d i t i o n s are s u f f i c i e n t f o r the convergence of the f i n i t e element s o l u t i o n t o the c o r r e c t answer. However elements which s a t i s f y , o n l y the c o n s t a n t s t r a i n c o n d i t i o n seem t o converge a c c e p t a b l y . A l s o elements which a r e complete but n o n c o n f o r m i n g ( n o n c o m p a t i b l e ) have been w i d e l y and s u c c e s s f u l l y used. The g r e a t d i s a d v a n t a g e of nonconforming elements i s t h a t i t i s not known whether the s t i f f n e s s i s an upperbound or n o t . A l s o nonconforming elements are more f l e x i b l e and thus the r a t e of convergence i s l o w e r . The next s t e p i s the d e r i v a t i o n of element e q u a t i o n s . Two w i d e l y used methods f o r t h i s purpose a r e v a r i a t i o n a l and r e s i d u a l methods. These methods l e a d t o e q u a t i o n s t h a t can be w r i t t e n i n the form { Q } = [ k ] { q } where, [k] i s the element p r o p e r t y m a t r i x { Q j i s the n o d a l f o r c i n g v e c t o r 35 {q} i s the unknown q u a n t i t y v e c t o r . In the displacement f o r m u l a t i o n [ k ] , {Q} and {q} are s t i f f n e s s matrix, nodal f o r c e v e c t o r and the unknown displacement v e c t o r r e s p e c t i v e l y . Then the equations f o r each element i s combined to o b t a i n a s i m i l a r r e l a t i o n f o r the e n t i r e f i n i t e element mesh. T h i s i s c a l l e d the g l o b a l r e l a t i o n and i s a set of simultaneous equations. The s o l u t i o n f o r the primary unknowns i n these equations are used to compute the necessary secondary q u a n t i t i e s . In the case of displacement method, the primary unknowns are displacements and these displacements are used to compute the secondary q u a n t i t i e s such as s t r a i n s and thus, s t r e s s e s . The great advantage of the f i n i t e element method, compared to other s o l u t i o n techniques i s t h a t , gross s i m p l i f i c a t i o n s of s o i l p r o f i l e s , boundary c o n d i t i o n s and s o i l p r o p e r t i e s that p o o r l y resemble the r e a l s o i l c o n d i t i o n s i s not r e q u i r e d i n the s o l u t i o n p r o c e s s . 3.2 NONLINEAR SOLUTION PROCEDURES F i n i t e element method uses two b a s i c procedures to so l v e n o n l i n e a r problems, namely incremental l i n e a r and i t e r a t i v e l i n e a r approaches. 36 3.2.1 INCREMENTAL APPROACH T h i s method i s s c h e m a t i c a l l y shown i n F i g u r e 3-1. In t h i s method the load i s s u b d i v i d e d i n to smaller increments not n e c e s s a r i l y e q u a l . Then the loa d i s a p p l i e d increment by increment. During each increment the m a t e r i a l i s assumed to be l i n e a r . For a given l o a d increment, corresponding l i n e a r e l a s t i c moduli ,for each element are computed on the b a s i s of t h e i r s t a t e of s t r e s s and s t r a i n at the end of the pre v i o u s lo a d increment. T h i s i s done f o r each and every element i n the f i n i t e element mesh. Then the corresponding increments i n s t r e s s e s s t r a i n s and displacements are computed f o r every element. In order to o b t a i n the t o t a l values of s t r e s s e s s t r a i n s or displacements, each increment up to the r e q u i r e d load increment i s added to t h e i r i n i t i a l v a l u e s . 3.2.2 MODIFIED INCREMENTAL METHOD Fi g u r e 3-2 shows a schematic diagram of the procedure i n v o l v e d . In t h i s method, f o r every load step, s t r e s s , s t r a i n and displacement increments corresponding to h a l f the f u l l l o a d step increment are computed as d e s c r i b e d above. Then the t o t a l s t r e s s e s ( n o t s t r e s s increments) corresponding to t h i s l o a d i n g s t a g e ( h a l f f u l l l o a d step) are computed. These values are used to r e - e v a l u a t e the new tangent moduli values cor r e s p o n d i n g to the c u r r e n t l o a d i n g s t a g e ( h a l f f u l l l oad s t e p ) . Then, these new tangent moduli values are used to estimate the s t r e s s , s t r a i n and displacement increments corresponding to the f u l l l o a d increment. The same procedure Figure 3-1. L i n e a r Incremental Method ure 3-2. M o d i f i e d L i n e a r Incremental Approach 39 i s repeated f o r each l o a d step up to the r e q u i r e d t o t a l l o a d . The accuracy can f u r t h e r be improved by i n c r e a s i n g the number of r e p e t i t i o n s at each h a l f l o a d s t e p . 3.2.3 ITERATIVE APPROACH Fi g u r e 3-3 i s a schematic diagram of the i t e r a t i v e l i n e a r approach. The procedure i n v o l v e s , the body being loaded f u l l y before s t a r t i n g the i t e r a t i o n . For the f i r s t i t e r a r t i o n some estimates of secant moduli(normally the i n i t i a l moduli) are used. In the subsequent i t e r a t i o n s the secant moduli corresponding to the s t r a i n s i n the previous i t e r a t i o n s are used. I t e r a t i o n s are repeated u n t i l a l l c o n s e c u t i v e sets of moduli v a l u e s agree to a r e q u i r e d accuracy. The incremental method can be used to a l l types of n o n l i n e a r i t i e s and i t simulates c o n s t r u c t i o n sequences. However the method i s not a p p l i c a b l e f o r work s o f t e n i n g m a t e r i a l s . P r i n c i p a l advantage of the i t e r a t i v e procedure i s , i t can simulate the s t r a i n s o f t e n i n g behaviour. But the method does not assure that the procedure w i l l always converge to the exact answer. 3.3 LINEAR ELASTICITY AND CONSTITUTIVE MATRIX In the p r e v i o u s s e c t i o n , two procedures f o r s o l v i n g n o n l i n e a r problems have been d i s c u s s e d . In i t s s o l u t i o n technique, incremental method assumes the m a t e r i a l to be l i n e a r w i t h i n a small l o a d increment, whereas the i t e r a t i v e method assumes the m a t e r i a l to be l i n e a r and the n o n l i n e a r i t y i s accounted Figure 3-3. I t e r a t i v e Procedure 41 for by using the secant modulus. In other words, both methods use p s e u d o - l i n e a r approaches. And the f o l l o w i n g i s a d i s c u s s i o n on the l i n e a r e l a s t i c s t r e s s - s t r a i n m a t r i x . The ge n e r a l l i n e a r e l a s t i c s t r e s s - s t r a i n r e l a t i o n s , f o r most numerical purposes, are given i n the form {cr} = [D]{e} where, {&} = { C£ x Oyy cr z z c x y c r y z cgx} {e} = { e e e e e e } -xx yy zz xy xz zx [D] i s the c o n s t i t u t i v e m atrix. The c o n s t i t u t i v e matrix [D] has 36 c o e f f i c i e n t s which r e l a t e s s t r e s s e s and s t r a i n s i n the l i n e a r case. But on energy c o n s i d e r a t i o n s i t can be shown t h a t [ D ] i s symmetric, thus having only 21 independent e l a s t i c parameters. If the medium i s c r o s s - a n i s o t r o p i c the number of independent e l a s t i c parameters reduces to f i v e . But i f the continuum i s i s o t r o p i c only two parameters are needed to d e f i n e [D]. In most s i t u a t i o n s Young's modulus and Poisson's r a t i o are used as the fundamental e l a s t i c parameters, and thus [D] becomes (1+v)(1-2v) 1 -v V V 0 0 0 V 1 -v V 0 0 0 V V 1 -v • 0 0 0 0 0 0 1-2v 0 0 0 0 0 0 1-2v 0 0 0 0 0 0 1-2v In p r a c t i c e , most g e o t e c h n i c a l f i e l d s i t u a t i o n s can a c c u r a t e l y be modelled as plane s t r a i n problems. In other words, e„ = 0 e = 0 and e = 0 zz zx zy and s t r e s s - s t r a i n r e l a t i o n s can be w r i t t e n i n the form °kx ' l - v v 0 exx °xy E ( 1+v)(1-2v) v 1 -v 0 eyy 0 0 1-2v . exy and cr = 0 zx cr and = v (o-.+ or. zy " z z "xx "y S i m i l a r c o n s t i t u t i v e matrix can be d e r i v e d f o r the incremental l i n e a r .approach. 3 INCREMENTAL METHOD AND CONSTITUTIVE MATRIX The incremental l i n e a r method assumes l i n e a r e l a s t i c behaviour d u r i n g each step of l o a d increment. The moduli values are d e f i n e d by U c r } = [D t ]{Ae} under a given s t r e s s and s t r a i n c o n d i t i o n . Where, [D t] = E< (1+v t)(1-2v t ) 1 ~ v t v t v t v t 1 " v t v t v t v t 1 ~ v t 0 0 0 0 0 0 0 0 0 0 0 0 1-2v. 0 0 0 0 0 0 1-2v t 0 0 0 0 0 0 1-2v t ) 43 Under plane s t r a i n c o n d i t i o n s [D t] can be w r i t t e n i n the form Jx-( 1+v. ) ( 1 - 2 v J 1-v t v t 0 v t 1-v t 0 0 1-2v. OR V G t V G t 0 where, 2 U + v t ) ( 1-2v t ) and 2(1+v t) A l l the above e l a s t i c parameters are d e f i n e d with respect to incremental s t r e s s e s and s t r a i n s and a c c o r d i n g l y they are c a l l e d tangent moduli. 3.5 ISOTROPIC MODEL AND NONLINEAR PROCEDURES The f o l l o w i n g i s a very b r e i f d e s c r i p t i o n of the procedure of o b t a i n i n g tangent moduli i n a n o n l i n e a r a n a l y s i s using i s o t r o p i c models. 44 3.5.1 TANGENT MODULUS Fi g u r e 2-1 shows a best f i t hyperbola(schematic) obtained from an i s o t r o p i c a l l y c o n s o l i d a t e d undrained compression t e s t . T h i s can be used to determine the tangent modulus at any p o i n t i n the s t r e s s - s t r a i n curve. By d e f i n i t i o n of E t Ae = 1 {ACT -V ( A C +AO* ) } yy F yy t xx z.± t For the undrained t r i a x i a l s i t u a t i o n V t = ' 5 A C r x x = A C r:z E = AOyy-ACryx = ^(Cyy-cr^ Ac,.,. dc yy yy But, (cr -ty ) = e,r,7 ' v yy xx 1 + R f £ E i SUCI T h e r e f o r e , E t = E ±{1 -R f (O y y - c y y ) } 2 ^uci S U C I i s the undrained s t r e n g t h i n i s o t r o p i c a l l y c o n s o l i d a t e d compression t e s t . It i s assumed that the tangent modulus i s dependent on (cry-c r 3 ) and the i n i t i a l c o n s o l i d a t i o n s t a t e only. T h e r e f o r e , the tangent modulus E f c can be expressed uniquely i n terms of the d e v i a t o r s t r e s s ( o ' 1 - c r 3 ) and the i n i t i a l c o n s o l i d a t i o n c o n d i t i o n s , as given below. E. = E. {1-Rf (Q-,-0-3 ) } s s UCI 45 T h i s formula is used in isotropic models to analyse the undrained deformation behaviour. The influence of the stress system change has completely been disregarded in t h i s approach. In other words, the analysis implies that the s t r e s s - s t r a i n behaviour under any stress system change is i d e n t i c a l l y the same. For example the stress st r a i n curves under compression and extension are i m p l i c i t l y assumed to be i d e n t i c a l ( F i g u r e 3 - 4 ) . A l s o , the influence of the i n i t i a l stress anisotropy can not be incorporated in t h i s model. 3 .6 ANISOTROPIC MODEL AND NONLINEAR PROCEDURES The proposed s t r e s s - s t r a i n model can be used either in the linear incremental technique or in the i t e r a t i v e technique. In the following discussion attention i s focused exclusively on the linear incremental technique. In t h i s method, at every stage of loading the tangent modulus E f c must be known. Since the deformation behaviour under consideration is undrained, the Young's modulus E of an isotropic materiaKthe proposed model assumes clay to be isotropic) completely defines(when a large value for the bulk modulus i s assumed, to simulate undrained behaviour) a l l the con s t i t u t i v e r e l a t i o n s . 46 ( a r a 3 ) IMPLIED t EXTENSION CURVE / / / / / / ° C ISOTROPIC COMPRESSION CURVE OA AND OC ARE SKEW SYMMETRIC F i g u r e 3 - 4 . The Implied S t r e s s - S t r a i n Curve i n Extension of the I s o t r o p i c Model 47 3.6.1 TANGENT MODULUS 1 F i g u r e 3-5 shows the s t r e s s - s t r a i n c u r v e s ( p r o p o s e d model) under u n d r a i n e d t r i a x i a l c o n d i t i o n s . As proved b e f o r e , the tangent modulus E t = ^ ( O y y - o y de The 0° s t r e s s - s t r a i n c u r v e c o n s i s t s of two p a r t s which r e p r e s e n t two d i f f e r e n t m a t h e m a t i c a l f u n c t i o n s . The p a r t OC i s g i v e n by q = —s. f o r ( o y y - s ; x ) o ^ ( o - y y - ^ < s u o E i < q u l 1 J c and thus the tangent modulus E t = E ± [ 1 "Rf { ( Cyy-Oxx) - ( C y y - Oxx) O } ] 2 S u 0 ~ (°yy ° X X 0 i n the above (cyy-cr^ range. The o t h e r p a r t of the 0° s t r e s s - s t r a i n c u r v e ( O A ) i s g i v e n by q = e f o r 0< (Qfa-aQ < ( o j y - o ^ 0 — + e and thus the tangent modulus i n the above d e v i a t o r s t r e s s range i s E t = Ei [ 1 -R f { ( Oyy-0- x^ p - ( OyyCfo) } ] 2 S u 9 0 + ( c r y - c r J 0 1 R e f e r F i g u r e s 3-5 and 3-6 f o r t h i s s e c t i o n . 48 F i g u r e 3-5. A n i s o t r o p i c a l l y C o n s o l i d a t e d S t r e s s - S t r a i n Curves i n Compression and i n Extension ( a r o 3 , 50 The f o l l o w i n g equation d e s c r i b e s the complete 90° s t r e s s -s t r a i n curve (AE) q = e f o r - S u 9 0 < ( o - y - ^ < 0 1 + e y y E i ( C W E and i t s tangent modulus i s given by E t = E ± M ~Rf {(Qyy-o^x) o - (<ryy«nj } ] 2 S U 9 o + ( a y y - c r < ^ 0 i n the same ( c r - c r ) range. yy x x A l l the above exp r e s s i o n s d e s c r i b e tangent modulus values i n terms of (cyy-o^. But i t i s more u s e f u l to express them i n terms of d e v i a t o r s t r e s s e s . Because the d e v i a t o r s t r e s s , angle of p r i n c i p a l axes r o t a t i o n and i n i t i a l c o n d i t i o n s uniquely determines the tangent modulus(according to the assumptions; s e c t i o n 2.3). F i g u r e 3-6 shows the 0° and 90° s t r e s s - s t r a i n curves on {(cr,-o-3 ) , e } plane. The f o l l o w i n g are e x p r e s s i o n s f o r the tangent moduli i n terms of d e v i a t o r s t r e s s . The tangent modulus of the 0° s t r e s s - s t r a i n curve can be expressed i n the form E t = Ej . [ 1 -R f {(or, -cr 3) - (cr, -<y3) 0 } ] 2 S u 0 - (o- 1-cr 3) o f o r S uo >(cr,-cr 3 )>(cr,-cr3 ) 0 on OC and E t = E j f 1~Rf{ ( C i - c 3 ) o - ( o - i - o v ) } ] 2 f o r 0<(cr,-cr3 )<(a,-cr 3) „ S U 9 0 +(cri"O -3 ) 0 51 on OA. The tangent modulus on the 9 0 ° s t r e s s - s t r a i n curve(AE) can be expressed i n the f o l l o w i n g form. E t = E s [ 1 -Rf {(cr 1-o' 3) + (o'1-cr, ) 0 } ] 2 f o r 0< (cr,-cr 3) <S U 9 0 S ug 0 + ( © " I 0*3 5 0 Above e x p r e s s i o n s d e f i n e the tangent moduli v a l u e s i n terms of the d e v i a t o r s t r e s s e s f o r the complete 0 ° and 9 0 ° s t r e s s -s t r a i n c u r v e s . The moduli values c o r r e s p o n d i n g to intermediate angles( 0 ° < e < 9 0 °) of p r i n c i p a l axes r o t a t i o n are determined using the i n t e r p o l a t i o n formula E t e = E t 0 + ( E t 0 - E t 9 O ) S i n 2 e sugested i n the s e c t i o n 2.3. E t Q i s the tangent modulus, when the p r i n c i p a l axes r o t a t i o n i s G °. The 0 ° s t r e s s - s t r a i n curve c o n s i s t s of two parts(OC,OA) r e p r e s e n t i n g two d i f f e r n t mathematical e x p r e s s i o n s . T h e r e f o r e , when i n t e r p o l a t i n g f o r the tangent modulus, a p p r o p r i a t e f u n c t i o n f o r the c u r r e n t d e v i a t o r s t r e s s must be chosen. In some cases the f a i l u r e d e v i a t o r s t r e s s under 9 0 ° p r i n c i p a l axes r o t a t i o n i s lower than the i n i t i a l d e v i a t o r s t r e s s (c r , - c r 3 ) 0 so that when i n t e r p o l a t i n g f o r the tangent modulus two b a s i c cases must be c o n s i d e r e d . Case I S u 9 0 > ( a 1 - a 3 ) 0 (Figure 3 -7) (a) . For (o-,-o-3 ) 0 >(o- 1-o- 3 )>0 Eto = Ej [ 1 -Rf {(cr, - 0*3) o - (cr, -cr 3)} 3 2 S u 9 o + (cr,-cr 3) 0 E t 9 o = E -; [ 1 -Rf {(o-, -0*3) + (o"i-a*,) 0 } 32 Su9 0 + (o- 1-cr 3 ) 0 (b) . For S u 9 0 > {0^-0-3 )> (o-,-a 3) 0 E t 0 = E j [ 1 -Rf {(<Ti - c r 3 ) - (cr, - c r 3) o } 32 S u 0 ~ (cr, -Cr3 ) o E t s o = Ej [ 1-Rf{ (o-T-cTa ) + ( c r 1 - c r , ) 0 } ] 2 S U 9 o + (cr 1-o- 3) 0 (c) . For Su 0>(o-,-cr3 ) > S U 9 0 E to = E j [ l - R f {(o- 1-cr 3)-(cr 1-cr3) n}3 2 S uo " (cr, -C 3 ) o E t9 0 = 0 ( d ) . For S u 0^(o-,-o- 3) Eto = 0 Etso = 0 Case II S u90<(er,-cr3 ) 0 (Figure 3 -8) ( a ) . For S u 9o >(o-,-cr 3 )£0 E E t 0 = Es [ 1-Rf { (Q-,-0-3 ) o-(cr,-g- 3 ) } 3 2 Su9 0 + vO-,-o-3) o t 9 0 = E x [ 1-R£ { ( c r x - c r 3 ) + i c r , - c r 3 ) Q] 32 S u 9 o + (0"-i~°"3)o (b) . For (o-,-o-3 )0>(a-,-o-3 ) ^ S u 9 0 E t 0 = Es [ 1-Rf { (cr,-Q-3 ) o~ (0*1-03 ) } 3 2S U 9 0 + ( c r , -c r 3 ) 0 E t 9 o = 0 (c) . For S u o>(o - , -0 '3 )>(cr,-0-3 ) 0 Eto = Ej [ 1-Rf{(cr 1-o - 3)-(cri-cr 3) 0}] 2 S u 0 ~ (cr, -cr 3 ) 0 F i g u r e 3-7. Case I, S U 9 O > ( o " i -( V a 3 ) 55 E t 9 0 = 0 (d ) . For (cr,-0-3 ) ^ S U 0 E t 0 = 0 Et9 0 = 0 In a l l the above cases E f c can be computed using the i n t e r p o l a t i o n formula Efcg = E to _ (E t 0 - E t 9 o ) Sin 2 6 3.7 FINITE ELEMENT PROGRAMME The f i n i t e element programme NLSSIP(Byrne et a l . , 1979) was mo d i f i e d to i n c o r p o r a t e the proposed s t r e s s - s t r a i n model. T h i s programme employs n o n l i n e a r incremental s o l u t i o n t e c h n i q u e ( s e c t ion 3.2.2) with i s o p a r a m e t r i c elements. T h e o r e t i c a l l y , the bulk modulus of a s o i l under undrained c o n d i t i o n s i s i n f i n i t e . In p r a c t i c e however, i n f i n i t e bulk modulus can not be accomodated i n the s t i f f n e s s m a trix. T h e r e f o r e , i n s t e a d of an i n f i n i t e v a l u e , a s u f i c i e n t l y l a r g e bulk modulus value i s assumed to simulate the undrained c o n d i t i o n s . A value of E /3(1-2V); v=.495, i s assumed f o r the bulk modulus i n the programme and i s kept constant throughout the a n a l y s i s . As a pa r t of the computational procedure of moduli v a l u e s , S the undrained s t r e n g t h corresponding to the c u r r e n t angle of r o t a t i o n of p r i n c i p a l axes i s computed f o r every element using ^uG = ~ ( S u 0 - S U 9 0 ) S i n 2 8 If the c u r r e n t d e v i a t o r s t r e s s i s g r e a t e r than S u 0 , that i s i f the element has f a i l e d , the tangent modulus i s set to 56 .1E ( 1 - R f ) 2 . T h i s i s a 90% r e d u c t i o n of the modulus value given by the h y p e r b o l i c r e l a t i o n c orresponding to the f a i l u r e d e v i a t o r s t r e s s . T h i s procedure s i m u l a t e s , f a i r l y reasonably the great l o s s of s t i f f n e s s i n the f a i l i n g s o i l elements in a c t u a l p r a c t i c e . 57 CHAPTER IV THE ANISOTROPIC MODEL PREDICTION COMPARISONS 1 LABORATORY PREDICTIONS The proposed s t r e s s - s t r a i n model can be completely d e f i n e d by using the model parameters obtained from a n i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l compression and extension t e s t s . The f o l l o w i n g s e c t i o n i n c l u d e s some s t r e s s - s t r a i n p r e d i c t i o n s made by the a n i s o t r o p i c model f o r Haney c l a y ( o f moderate s e n s i t i v i t y found i n Canada). These p r e d i c t i o n s were made for d i f f e r e n t s t r e s s s y s t e m s ( i . e . other than the K 0 c o n s o l i d a t e d t r i a x i a l c o n d i t i o n ) using the a n i s o t r o p i c a l l y c o n s o l i d a t e d  t r i a x i a l model parameters. As a t e s t of the r e l i a b i l i t y of the model, the above p r e d i c t i o n s are compared(for cases where l a b o r a t o r y data are a v a i l a b l e ) with the l a b o r a t o r y s t r e s s -s t r a i n curves obtained under such d i f f e r e n t s t r e s s systems. 1.1 PLANE STRAIN TESTS In t h i s case the s t r e s s system under c o n s i d e r a t i o n i s the plane s t r a i n c o n d i t i o n . The model p r e d i c t i o n s are compared with the l a b o r a t o r y r e s u l t s . F i g u r e 4-1 presents the comparison of plane s t r a i n s t r e s s - s t r a i n curves f o r Haney c l a y obtained from the l a b o r a t o r y t e s t s and those p r e d i c t e d by the f i n i t e element programme. Curve-1 i s the experimental s t r e s s - s t r a i n r e l a t i o n o b t a i n e d ( V a i d et a l . , 1974) under plane s t r a i n c o n d i t i o n s and curve-2 d e p i c t s the p r e d i c t e d c o n s t i t u t i v e behaviour. The 58 F i g u r e 4-1. Comparison of Model P r e d i c t i o n s with Plane S t r a i n Laboratory R e s u l t s 59 l a b o r a t o r y specimens t e s t e d have been normally c o n s o l i d a t e d under K 0(=.56) c o n d i t i o n s to a v e r t i c a l e f f e c t i v e s t r e s s of 6kg/cm 2. In p r e d i c t i n g the plane s t r a i n behaviour the specimen was co n s i d e r e d to have a normally c o n s o l i d a t e d i n i t i a l s t r e s s c o n d i t i o n i d e n t i c a l to the l a b o r a t o r y c o n s o l i d a t i o n s t r e s s s t a t e . That i s , the v e r t i c a l c o n s o l i d a t i o n s t r e s s was 6kg/cm 2 and K 0 was .56. The model parameters used, which are given i n F i g u r e 4-2 were obtained from a n i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l t e s t r e s u l t s ( V a i d et a l . , 1974). A l s o , F i g u r e 4-2 shows the plane s t r a i n element u t i l i z e d i n the programme f o r the s t r e s s - s t r a i n p r e d i c t i o n s . In the simulated compression t e s t the v e r t i c a l s t r e s s was i n c r e a s e d u n t i l f a i l u r e , i n steps of .02kg/cm 2, keeping the h o r i z o n t a l s t r e s s constant, whereas i n the exte n s i o n t e s t v e r t i c a l s t r e s s r e d u c t i o n was .20kg/cm 2 i n each step with constant h o r i z o n t a l s t r e s s . Load steps smaller than the above d i d not make a s i g n i f i c a n t d i f f e r e n c e i n the p r e d i c t e d s t r e s s - s t r a i n curve. The l a b o r a t o r y s t r e s s - s t r a i n r e s u l t s and the programme p r e d i c t i o n s compare remarkably w e l l f o r small s t r a i n s . For l a r g e s t r a i n s ( n e a r f a i l u r e ) however, the two curves d e v i a t e somewhat from each other. T h i s i s q u i t e pronounced(when measured from the i n i t i a l d e v i a t o r s t r e s s ) i n the compression branch and the p r e d i c t e d plane s t r a i n compression s t r e n g t h i s lower than the experimental v a l u e . T h i s r e f l e c t s the inherent d i f f e r e n c e between the undrained s t r e n g t h under t r i a x i a l and plane s t r a i n c o n d i t i o n s . Whether to use plane s t r a i n c r \ c = 6kg/cm2 KQ = .56 R f = .9 E i / o - i , 'UCK UEK 200 .54 .34 F i g u r e 4-2. S o i l Parameters and Plane S t r a i n F i n i t e Element xy K 0 E ./cr'. i 1 c 'UCK 'UEK /cr\, 200 .54 .34 F i g u r e 4-3. S o i l Parameters and Simple Shear F i n i t e Element 61 parameters or t r i a x i a l parameters f o r a given model under given f i e l d c o n d i t i o n s must be i n v e s t i g a t e d s e p a r a t e l y f o r each model. In t h i s chapter, i n comparing the two models, only t r i a x i a l parameters have been u t i l i z e d so as to make the comparisons f a i r . ( i . e . not to allow the inherent d i f f e r e n c e s i n experimental r e s u l t s i n t e r f e r e with the p r e d i c t i o n s made by the two models.) 4.1.2 SIMPLE SHEAR CONDITION The programme p r e d i c t i o n s of simple shear behaviour of Haney c l a y i s shown in F i g u r e 4-4. U n f o r t u n a t e l y , there are no l a b o r a t o r y r e s u l t s on simple s h e a r ( f o r Haney c l a y ) f o r making any comparisons. An element modelled as shown i n F i g u r e 4-3 was used f o r the computer programme i n p r e d i c t i n g the simple shear behaviour. In the simulated simple shear t e s t the i n i t i a l v e r t i c a l c o n s o l i d a t i o n s t r e s s was 6kg/cm 2 and K 0 was .56(same as f o r plane s t r a i n l a b o r a t o r y t e s t s ) . In s i m u l a t i n g the simple shear t e s t the a p p l i e d h o r i z o n t a l shear s t r e s s was i n c r e a s e d i n steps of ,0500kg/cm 2 i n i t i a l l y and the increment was g r a d u a l l y reduced to .0005kg/cm 2, c l o s e to f a i l u r e c o n d i t i o n s . S o l u t i o n s with l o a d increments s m a l l e r than the above d i d not make a s i g n i f i c a n t d i f f e r e n c e i n the p r e d i c t e d s t r e s s - s t r a i n curve. The same model parameters u t i l i z e d f o r p r e d i c t i n g p l a n e - s t r a i n s t r e s s - s t r a i n behaviour were used f o r simple shear p r e d i c t i o n s too. These parameters were obtained from a n i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l t e s t s ( V a i d et a l . , 1974) and are shown on the r i g h t hand*side of F i g u r e 4-3. I t 63 i s the n o r m a l i z e d T v s y curve(under simple shear xy xy r c o n d i t i o n s ) that i s shown in F i g u r e 4-4. 4.2 APPLICATIONS OF ISOTROPIC AND K 0 MODELS The f o l l o w i n g i s a parametric study of the undrained behaviour of f o o t i n g s analysed using both, the i s o t r o p i c and a n i s o t r o p i c s t r e s s - s t r a i n models. The load settlement curves, u l t i m a t e b e a r i n g p r e s s u r e s , f a i l u r e zones, s u r f a c e d e f l e c t i o n p r o f i l e s and contact pressure p r o f i l e s p r e d i c t e d by the two models are compared in order to i n v e s t i g a t e the importance of t a k i n g i n t o c o n s i d e r a t i o n the i n f l u e n c e of i n i t i a l s t r e s s a n i s o t r o p y and the s t r e s s system change d u r i n g shear on the undrained s t r e s s - s t r a i n behaviour of c l a y . In s e l e c t i n g the problem, two other t h i n g s were a l s o of concern. F i r s t l y , the p r a c t i c a l i t y of the problems; secondly and most importantly t h e i r a b i l i t y to i l l u s t r a t e extreme cases with respect to the d i f f e r e n c e between the ( s e t t l e m e n t ) p r e d i c t ions of the two models. A l s o , using these two examples, i t was intended to v e r i f y whether, the q u a n t i t y OHo/Su i s a measure of the d i f f e r e n c e between the p r e d i c t i o n s ( s e t t l e m e n t ) made by the two models. The constant s t r e n g t h c a s e ( s e c t i o n 4.2.2(B)) was expected to represent the l e a s t d i f f e r e n c e whereas the v a r i a b l e s t r e n g t h ( s e c t i o n 4.2.2(A)) case was expected to represent the l a r g e s t d i f f e r e n c e as f a r as the percentage d i f f e r e n c e i n settlement p r e d i c t i o n s are concerned(see s e c t i o n 4.3 f o r a d i s c u s s i o n ) . 64 4.4.1 GEOMETRY, SOIL PROPERTIES AND BOUNDARY CONDITIONS The geometry of the foundation problem c o n s i d e r e d i s shown i n F i g u r e 4-5. The s t r i p s u r f a c e f o o t i n g has a width B of 20FT. and the s o i l below i s Haney c l a y ( o f moderate s e n s i t i v i t y found i n Canada) to a depth of 30FT., o v e r l y i n g bed rock. The water t a b l e i s assumed to be at the top of the s o i l s u r f a c e with, at r e s t s t r e s s e s i n the s o i l . The d i s c r e t i z a t i o n p a t t e r n u t i l i z e d f o r the f i n i t e element a n a l y s i s i s shown i n Fi g u r e 4-6. C o n s i d e r i n g symmetry o n l y " h a l f of the continuum was analys e d . Therefore the l e f t v e r t i c a l boundary has no h o r i z o n t a l displacements. The bottom h o r i z o n t a l boundary between s o i l and rock i s c o n s i d e r e d to be f i x e d . That i s , there i s no displacement at the s o i l rock i n t e r f a c e . The r i g h t v e r t i c a l boundary i s placed f a r enough away from the loaded area that i t s pressure does not i n f l u e n c e the s o l u t i o n s i g n i f i c a n t l y A f i x e d l a t e r a l boundary a c t s to produce a c o n f i n i n g e f f e c t whereas a free boundary has an op p o s i t e e f f e c t ( H o e g et a l . , 1968). Ther e f o r e the c o n d i t i o n of zero h o r i z o n t a l displacements and fr e e v e r t i c a l displacements i s maintained at a d i s t a n c e 4B from the ce n t r e l i n e of the foundation, so that the two e f f e c t s are compensated. The top h o r i z o n t a l boundary i s f r e e except under the loaded area, where f l e x i b l e or r i g i d boundary c o n d i t i o n s e x i s t . A r i g i d boundary c o n d i t i o n i s s t i p u l a t e d to have equal v e r t i c a l displacements whereas under a f l e x i b l e boundary the nodes are f r e e . 20FT. A y' =60 K 0 = .56 E i / o " i c = 2 0 0 R f = . 9 HANEY CLAY 30FT. / V / III /// Ml /// Ml /// Ml /// 777" BED ROCK Figure 4-5. Geometry and S o i l P r o p e r t i e s as ui k b h k k k h k k k k k k & k h h t t - K - * ? - Jf -a? -85 1 Figure 4 - 6 . F i n i t e Element Mesh 67 4.2.2 MODEL PARAMETERS G e n e r a l l y , to o b t a i n the model parameters, l a b o r a t o r y t e s t s must be performed on undisturbed specimens obtained at d i f f e r e n t depths of the s o i l mass. Because, samples at the same depths are s u b j e c t e d to the same c o n s o l i d a t i o n pressure and have the same s t r e s s h i s t o r y and consequently the undrained s t r e s s - s t r a i n behaviour i s almost e x a c t l y the same. To o b t a i n the i s o t r o p i c parameters, i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l t e s t s are performed, where the l a b o r a t o r y specimens are u s u a l l y c o n s o l i d a t e d to a pressure equal to the i n s i t u v e r t i c a l e f f e c t i v e s t r e s s , and then sheared undrained. On the other hand, a n i s o t r o p i c c o n s o l i d a t i o n i s done to simulate the exact i n s i t u s t r e s s c o n d i t i o n i n the K 0 c o n s o l i d a t e d t e s t s . These r e s u l t s are used to compute the a n i s o t r o p i c model parameters. However, the normally c o n s o l i d a t e d c l a y s show normalised behaviour and thus higher c o n s o l i d a t i o n s t r e s s e s may be used to e l i m i n a t e the e f f e c t s of sample d i s t u r b a n c e . But, f o r the o v e r c o n s o l i d a t e d c l a y s under i s o t r o p i c a l l y c o n s o l i d a t e d c o n d i t i o n s the SHANSEP concept(Ladd et a l . , 1973) i s a p p l i c a b l e . Hence i n the l a b o r a t o r y experiments the specimens can be c o n s o l i d a t e d to h i g h e r ( t h a n i n s i t u ) s t r e s s e s to e l i m i n a t e the e f f e c t s of sample d i s t u r b a n c e . I n v e s t i g a t i o n of such p o s s i b i l i t i e s f o r a n i s o t r o p i c a l l y c o n s o l i d a t e d t e s t s ( o n OC c l a y ) can be very u s e f u l . A l s o , i f normalized behaviour (or SHANSEP c o n c e p t ) i s a p p l i c a b l e , the l a b o r a t o r y s t r e s s -s t r a i n parameters of a s i n g l e sample can be e x t r a p o l a t e d to 6 8 get the corresponding parameters of the same c l a y under d i f f e r e n t c o n f i n i n g p r e s s u r e s ( s e e a l s o Chapter 5). In the absence of s u f f i c i e n t data or i n case, expensive l a b o r a t o r y s t r e s s - s t r a i n t e s t s are not j u s t i f i e d , the parameters f o r normally c o n s o l i d a t e d c l a y s may be estimated from e m p i r i c a l c o r r e l a t i o n s ( s e e Chapter 5). The model parameters u t i l i z e d f o r t h i s a n a l y s i s were obtained from the r e s u l t s of t r i a x i a l t e s t s run on Haney c l a y by V a i d et a l . , 1974. 4.2.3 COMPARISON OF RESULTS 4.2.3(A) LINEARLY VARYING UNDRAINED STRENGTH WITH DEPTH A s i t u a t i o n r e p r e s e n t i n g a normally c o n s o l i d a t e d ( e x c e p t w i t h i n the c r u s t ) clay(Haney) of undrained s t r e n g t h and i n i t i a l modulus v a r y i n g l i n e a r l y with the c o n f i n i n g pressure has been s e l e c t e d . A d e s i c a t e d c r u s t of 2.5FT., deep having an undrained s t r e n g t h of 225PSF., i n i t i a l modulus of 10 5PSF., and K 0 and R f v a l u e s of .9 were a l s o assumed(these values correspond to an OCR of about 5). A l s o , i n a d i t i o n to the common parameters shown in F i g u r e 4-5 u t i l i z e d i n the a n a l y s i s , the i s o t r o p i c and a n i s o t r o p i c parameters given below were used. S U C K / o ' 1 c and S U C I /o' were .54 and S ^ K / O I C was .34. The above are normally c o n s o l i d a t e d parameters of Haney c l a y . T h i s problem was expected to show a l a r g e d i f f e r e n c e ( % d i f f e r e n c e ) between the load settlement curves p r e d i c t e d by the" i s o t r o p i c and a n i s o t r o p i c models. 69 (a) . D e f l e c t i o n s Load d e f l e c t i o n curves of the c e n t r a l a x i s of the f l e x i b l e and r i g i d foundations are shown i n F i g u r e s 4-7 and 4-8 r e s p e c t i v e l y . In both cases the c e n t r a l d e f l e c t i o n s p r e d i c t e d by the i s o t r o p i c model are much l a r g e r than that p r e d i c t e d by the a n i s o t r o p i c model. The i s o t r o p i c c e n t r a l d e f l e c t i o n s ( e x c e p t f o r near f a i l u r e loads) are as l a r g e as ten times that of a n i s o t r o p i c d e f l e c t i o n s . However, f i g u r e s 4-9 and 4-10 show the s i m i l a r i t y between the s u r f a c e d e f l e c t i o n p a t t e r n s p r e d i c t e d by the two models under the f l e x i b l e f o u n d a t i o n . (b) . Bearing S t r e s s e s and Surface Pressures The u l t i m a t e bearing s t r e s s p r e d i c t e d by the i s o t r o p i c model i s about 15% higher than that p r e d i c t e d by the a n i s o t r o p i c model. Both models p r e d i c t u l t i m a t e b e a r i n g p r e s s u r e s that are s u b s t a n t i a l l y lower than that of l i m i t e q u i l i b r i u m v a l u e s ( T a b l e 4-1). In t h i s context, the u l t i m a t e b e a r i n g s t r e s s i s d e f i n e d as the e x t e r n a l f o o t i n g p r e s s u r e corresponding to the p o i n t ( o n the p r e s s u r e d e f l e c t i o n curve) where the pressure d e f l e c t i o n curve becomes h o r i z o n t a l . In a simpler, but l e s s p r e c i s e way, a f o o t i n g p r essure which y i e l d s l a r g e s ettlements i s d e f i n e d as the u l t i m a t e b e a r i n g s t r e s s . The c o n t a c t pressure p r o f i l e s ( F i g u r e s 4-11 and 4-12) of the r i g i d foundation p r e d i c t e d by the two models are approximately the same. G e n e r a l l y , the a n i s o t r o p i c model F i g u r e 4 - 7 . Pressure D e f l e c t i o n Curves of C e n t r a l Axis F l e x i b l e F o o t i n g ; V a r i a b l e Undrained Strength o in A PROPOSED MODEL Q ISOTROPIC MODEL 0.0 ~ i 1 r r -0- + O.S 1.2 1 6 CENTRAL DEFLECT ION I FT) F i g u r e 4-8. Pressure D e f l e c t i o n Curves of C e n t r a l A x i s R i g i d F o o t i n g ; V a r i a b l e Undrained S t r e n g t h 72 I—o bin'" O UJ CJ d o U-Ln -_i cm i FOOTING PRESSURE(PSF) 1 , 2, 3, 4, 100 200 300 370 10.0 DISTANCE T " —I , 15.0 20.0 FROM C-LJNE(FT) 25.0 30.0 F i g u r e 4-9. F l e x i b l e F o o t i n g Surface D e f l e c t i o n P r o f i l e s ; Proposed Model FOOTING PRESSURE(PSF) I I 10.0 ]5.0 20.0 DISTANCE FROM C-LINE(FT) 25.0 30.0 F i g u r e 4-10. F l e x i b l e F o o t i n g Surface D e f l e c t i o n P r o f i l e s ; I s o t r o p i c Model ~ to o «—1 X £° on ZD CO co° L U Q _ | Q_ LU C_) CE U_ 0.0 FOOTING PRESSURE(PSF) 1 . 100 2. 200 3. 300 4. 380 ' T 1 r — 4 - ° 8.0 12.0 16.0 DISTANCE FROM C - L I N E ( F T ) 20.0 Figure 4-11. Rigid Footing Surface Pressure P r o f i l e s ; Proposed Model o CO O « 1 X cogH LU on ZD co co ° L U d _ | Q_ (_) CE era =D Q-CO 0.0 FOOTING PRESSURE(PSF) 1 . 100 2. 200 3. 300 4. 380 ' " " 1 r— * ° 8.0 12.0 IB 0 DISTANCE FROM C - L I N E ( F T ) 20.0 Figure 4-12. Rigid Footing Surface Pressure P r o f i l e s ; Isotropic Model 74 p r e d i c t e d s l i g h t l y l a r g e r p r e s s u r e s towards the c e n t r e and s m a l l e r p r e s s u r e s near the edge than that p r e d i c t e d by the i s o t r o p i c model. ( c ) . F a i l u r e Zones The spreading of f a i l u r e zones under the f l e x i b l e f o o t i n g i s shown i n F i g u r e s C-1 and C-2 f o r the i s o t r o p i c and a n i s o t r o p i c models r e s p e c t i v e l y . F i g u r e s c-3 and c-4 shows s i m i l a r diagrams f o r the r i g i d f o undation. The a n i s o t r o p i c model y i e l d s shallower and wider f a i l u r e zones than the i s o t r o p i c model f o r the corresponding l o a d i n g c o n d i t i o n s . In t h i s context, the f a i l u r e i s d e f i n e d as a c o n d i t i o n , where the maximum shear s t r e s s e s i n an element reaching the corresponding undrained s t r e n g t h of that p a r t i c u l a r element. 4.2.3(B) CONSTANT UNDRAINED STRENGTH WITH DEPTH In t h i s case, the s o i l mass was assumed to have a constant undrained s t r e n g t h with depth. At a depth of 30FT.(at the bedrock) the c l a y was assumed to be j u s t normally c o n s o l i d a t e d . Therefore the constant s t r e n g t h s i n compression and i n e x t e n s i o n were 972PSF. and 612PSF. r e s p e c t i v e l y ( V a i d et a l . , 1974) and the i s o t r o p i c undrained s t r e n g t h was 972PSF. Because for normally c o n s o l i d a t e d Haney c l a y S U C K/o"' 1 c and S /cr' are .54 and S /cr; i s .34. A v a r i a t i o n of UCI C UEK C undrained modulus s i m i l a r t o the p r e v i o u s case was assumed so t h a t , the e f f e c t of o d 0 / S u o n t n e % d i f f e r e n c e of settlements 75 p r e d i c t e d by the two models c o u l d be compared i n a meaningful way. The other s o i l parameters are shown i n F i g u r e 4-5. T h i s s i t u a t i o n r e p r e s e n t s s m a l l e r ( t h a n i n the p r e v i o u s case) o'do/Su v a l u e s along the depth and thus, was expected to show a smaller(compared with the pr e v i o u s case) d i f f e r e n c e ( % d i f f e r e n c e ) between the load settlement curves p r e d i c t e d by the two models. The f i e l d s i t u a t i o n c o n s i d e r e d above i s s i m i l a r to an over c o n s o l i d a t e d c o n d i t i o n , except f o r the low Compression/Extension s t r e n g t h r a t i o . The l i n e a r v a r i a t i o n of the modulus with depth assumed i n the a n a l y s i s r e p r e s e n t s a lower bound i n modulus f o r an over c o n s o l i d a t e d c l a y . ( a ) . D e f l e c t i o n s The l o a d d e f l e c t i o n curves of the c e n t r a l a x i s of the s t r i p are shown i n F i g u r e s 4-13 and 4-14 f o r f l e x i b l e and r i g i d boundary c o n d i t i o n s r e s p e c t i v e l y . As expected, (compared to the p r e v i o u s c a s e ) t h e d e f l e c t i o n s do not show a very l a r g e d i f f e r e n c e ( b u t l a r g e enough f o r concern, i n a g e o t e c h n i c a l e n g i n e e r i n g problem) between the two models. The c e n t r a l d e f l e c t i o n s p r e d i c t e d by the i s o t r o p i c model are as much as 100% higher than that p r e d i c t e d by, the a n i s o t r o p i c model, d u r i n g the i n i t i a l stages of l o a d i n g . N e v e r t h e l e s s , i t i s i n t e r e s t i n g to note the very c l o s e s i m i l a r i t y between the s u r f a c e d e f l e c t i o n p a t t e r n s ( F i g u r e s 4-15 and 4-16) of both models, i n the case of the f l e x i b l e f oundation c o n d i t i o n . T h i s s i m i l a r i t y was observed f o r the normally c o n s o l i d a t e d case too. D PROPOSED MODEL + ISOTROPIC MODEL Fi g u r e 4-13. Pressure D e f l e c t i o n Curves of C e n t r a l A x i s ; F l e x i b l e F o o t i n g ; Constant Undrained S t r e n g t h 0 PROPOSED MODEL + ISOTROPIC MODEL Figure 4-14. Pressure D e f l e c t i o n Curves of C e n t r a l A x i s ; R i g i d Footing; Constant Undrained Strength 78 FOOTING PRESSURE(PSF) o r— UJ 0 a UJ CJ C C a ^ - L O _l C£ i cn 1. 400 2. 800 3. 1200 4. 1600 5. 1800 1 -5-J3 10.0 15.0 20.0 25.0 30.0 DISTANCE FROM C - L I N E ( F T ) F i g u r e 4-15. F l e x i b l e F o o t i n g S u r f a c e D e f l e c t i o n P r o f i l e s ; P r o p o s e d M o d e l FOOTING PRESSURE(PSF) I— o O 1 . 2. 3. 4. 5. 400 800 1200 1600 1800 CJ UJ, U_ca UJ 01 Q UJ CJ C C a o r i i i 10.0 15.0 20.0 DISTANCE FROM C - L I N E ( F T ) 25.0 30.0 CO F i g u r e 4-16. F l e x i b l e F o o t i n g S u r f a c e D e f l e c t i o n P r o f i l e s ; I s o t r o p i c M o d e l 79 (b) . Bearing S t r e s s e s and Surface Pressures As i n the pre v i o u s case, the u l t i m a t e bearing pressure p r e d i c t e d by the i s o t r o p i c model i s higher than that p r e d i c t e d by the a n i s o t r o p i c model(Figures 4-13 and 4-14). It i s higher by approximately 10% and 15% under the f l e x i b l e and r i g i d foundation c o n d i t i o n s r e s p e c t i v e l y . Both models p r e d i c t e d bearing p r e s s u r e s that are lower than that of l i m i t e q u l i b r i u m v a l u e s ( T a b l e 4-1). The s u r f a c e pressure p r o f i l e s ( F i g u r e s 4-17 and 4-18) under the r i g i d foundations p r e d i c t e d by the two models are approximately the same. The s u r f a c e p r e s s u r e s p r e d i c t e d by the a n i s o t r o p i c model i s s l i g h t l y l a r g e r around the c e n t r e l i n e and s l i g h t l y s maller towards the edge of the s t r i p foundation, than that p r e d i c t e d by the i s o t r o p i c model. S i m i l a r behaviour was observed f o r the normally c o n s o l i d a t e d case too. (c) . F a i l u r e Zones The zones of s o i l undergoing f a i l u r e at two load steps f o r the i s o t r o p i c and a n i s o t r o p i c models are shown i n F i g u r e s C-5, C-6 and C-7, C-8, under r i g i d and f l e x i b l e boundary c o n d i t i o n s r e s p e c t i v e l y . As i n the pre v i o u s case the f a i l u r e zones p r e d i c t e d by the a n i s o t r o p i c model are shallower and wider than that of t h e i r c o u n t e r p a r t s f o r the corresponding l o a d s . T h i s i s apparent under both the f l e x i b l e and r i g i d foundation c o n d i t i o n s . 80 TABLE 4-1. COMPARISON OF ULTIMATE BEARING STRESSES BOUNDARY CONDITIONS TYPE OF ANALYSIS BEARING AND STRENGTH VARIATION STRESS(PSF) FLEXIBLE FOOTING STABILITY SWEDISH1 2680 AND CONSTANT ISOTROPIC MODEL 2150 UNDRAINED STRENGTH ANISOTROPIC MODEL 1 975 FLEXIBLE FOOTING STABILITY SWEDISH1 500 AND VARIABLE ISOTROPIC MODEL 425 UNDRAINED STRENGTH ANISOTROPIC MODEL 375 1See Appendix-D f o r the method fM o X U_d , CO«o_| Q _ ~ L U CH ZD CO C Q O L L J O _ | Crrco •_ L U O C E U_ C O 0.0 FOOTING PRESSURE(PSF) 1 . 400 2. 800 3. 1200 4. 1600 5. 2000 - I 1 1 ' 1 4-0 8.0 J2.0 16.0 DISTANCE FROM C-LINE(FT) 20.0 F i g u r e 4-17. R i g i d F o o t i n g S u r f a c e P r e s s u r e P r o f i l e s ; Proposed Model 3 ^ CM o f — I X COCO _t Q_ " LU or ZD C O CO ca a. 'Ld C J cr ZDd CO 0.0 FOOTING PRESSURE(PSF) 1 . 400 2. 800 3. 1200 4. 1600 5 . 2000 ' 1 " 1 — r — 4.0 8 0 12 D IR n DISTANCE FROM C-UNEIFT) 20.0 F i g u r e 4-18. R i g i d F o o t i n g S u r f a c e P r e s s u r e P r o f i l e s ; I s o t r o p i c Model 82 4.3. DISCUSSION The comparison of model p r e d i c t i o n s with the l a b o t a t o r y t e s t data show some encouraging r e s u l t s f o r small s t r a i n s but show some d i s c r e p a n c i e s between the f a i l u r e s t r e n g t h s . Hence, using K 0 t r i a x i a l parameters, one can expect the proposed, model to p r e d i c t f a i r l y a ccurate s t r e s s deformation behaviour fo r a plane s t r a i n s i t u a t i o n i n the working range of s t r a i n s . Whether to use K 0 t r i a x i a l or plane s t r a i n parameters, under a given f i e l d s i t u a t i o n , f o r a given model; and t h e i r r e l a t i v e m e r i t s must be i n v e s t i g a t e d s e p a r a t e l y f o r each model. The c e n t r a l d e f l e c t i o n s p r e d i c t e d by the i s o t r o p i c model are always l a r g e r ( e x c e p t f o r near f a i l u r e c o n d i t i o n s ) than that p r e d i c t e d by the a n i s o t r o p i c model. T h i s i s because, g e n e r a l l y the tangent modulus corresponding to a given s t r e s s l e v e l i s l a r g e r in the proposed model than that i n the i s o t r o p i c modeKsee F i g u r e 2-4). The percentage difference(compared to a n i s o t r o p i c d e f l e c t i o n s ) in the p r e d i c t e d c e n t r a l d e f l e c t i o n s of the two models are very much l a r g e r i n the case of l i n e a r l y v a r y i n g undrained s t r e n g t h than i n the case of constant undrained s t r e n g t h . T h i s i s due to the f a c t that the r a t i o , i n i t i a l d e v i a t o r s t r e s s to undrained strength(o-d 0/Su) i s sm a l l e r f o r the constant undrained s t r e n g t h case than that f o r the l i n e a r l y v a r y i n g s t r e n g t h case. In the case of constant s t r e n g t h , the r a t i o i n c r e a s e s with the depth and at the r o c k - s o i l i n t e r f a c e i t becomes equal to the r a t i o c orresponding to the l i n e a r l y 83 v a r y i n g case, which i s a constant with the depth. When t h i s r a t i o i s l a r g e r the d i f f e r e n c e between the two s t r e s s - s t r a i n models are l a r g e r and thus the percentage d i f f e r e n c e between the settlement p r e d i c t i o n s of the two models are l a r g e r too. The u l t i m a t e bearing s t r e s s e s p r e d i c t e d by the a n i s o t r o p i c model are g e n e r a l l y lower than that of the i s o t r o p i c model p r e d i c t i o n s . T h i s i s because, i n the a n i s o t r o p i c model the s t r e n g t h decreases with the r o t a t i o n of p r i n c i p a l axes whereas i n the i s o t r o p i c model the s t r e n g t h s t a y s constant i r r e s p e c t i v e of the p r i n c i p a l plane r o t a t i o n s . However, the d i f f e r e n c e i n u l t i m a t e bearing s t r e s s e s p r e d i c t e d by the two models should depend on the r e l a t i v e magnitudes(K 0 compression s t r e n g t h and i s o t r o p i c compression s t r e n g t h are equal f o r Haney c l a y ) of the undrained s t r e n g t h v a l u e s ( K 0 compression, K 0 e x t e n s i o n and i s o t r o p i c compression). L i m i t e q u i l i b r i u m p r e d i c t i o n s are the highest i n both cases.' I t can be a t t r i b u t e d to the upperbound nature of the r e s u l t s of l i m i t e q u i l i b r i u m a n a l y s i s . The c o n t a c t p r essure p r o f i l e s do not show a s i g n i f i c a n t d i f f e r e n c e between the two models. However, the f a i l u r e zones produced show a s u b s t a n t i a l d i f f e r e n c e between the two models. The a n i s o t r o p i c model y i e l d e d f a i l u r e zones that are much wider and shallower than those produced by the i s o t r o p i c model. 84 CHAPTER V EMPIRICAL CORRELATIONS FOR  MODEL PARAMETERS The undrained model parameters f o r both i s o t r o p i c and a n i s o t r o p i c c o n d i t i o n s can be obtained from l a b o r a t o r y t e s t s on undisturbed samples. Depending on the s o i l type, t e s t i n g c o n d i t i o n s such as K 0, c o n f i n i n g p r e s s u r e s and type of t e s t s ( e x t e n s i o n or compression), some of these parameters vary c o n s i d e r a b l y . T h e r e f o r e , i t i s very u s e f u l to develop e m p i r i c a l r e l a t i o n s h i p s between the model parameters and the v a r i a b l e s mentioned above. These c o r r e l a t i o n s can provide v a l u a b l e t o o l s for judging the reasonableness of parameter values determined from l a b o r a t o r y t e s t data. In a d d i t i o n , they can pr o v i d e estimates of the parameters, i f s o p h i s t i c a t e d and expensive l a b o r a t o r y t e s t i n g procedures are not warranted, or when i n s u f f c i e n t data i s a v a i l a b l e f o r these e s t i m a t e s . In the f o l l o w i n g s e c t i o n s an attempt to o b t a i n such c o r r e l a t i o n s f o r normally c o n s o l i d a t e d c l a y s i s made. Proceedings of g e o t e c h n i c a l conferences, symposia and v a r i o u s j o u r n a l s of s o i l mechanics were scanned, i n order to o b t a i n a data base f o r the c o r r e l a t i o n s . v 85 5.1 NORMALIZED BEHAVIOUR OF NORMALLY CONSOLIDATED CLAY In some s o i l s the undrained s t r e s s - s t r a i n curve p l o t t e d in the form o-d/o'1c vs e y i e l d s a (almost)unique s t r e s s - s t r a i n r e l a t i o n i r r e s p e c t i v e of the value of c o n s o l i d a t i o n s t r e s s cr' 1 c(or f o r a c e r t a i n range of cr'1c ). T h i s i s r e f e r r e d to as the normalized behaviour. The normalized behaviour of s o i l s i s important from two p o i n t s of veiw. F i r s t l y , i t i n d i r e c t l y p r o v i d e s s o l u t i o n s f o r the problems of sample d i s t u r b a n c e i n l a b o r a t o r y t e s t s . The d i s t u r b a n c e s caused by sampling procedures and specimen p r e p a r a t i o n s can be e f f e c t i v e l y e l i m i n a t e d by c o n s o l i d a t i n g the s p e c i m e n ( i s o t r o p i c a l l y or a n i s o t r o p i c a l l y ) to a higher pressure than i t was under in s i t u c o n d i t i o n s . Then, using the normalized behaviour, these r e s u l t s can be back c a l c u l a t e d to o b t a i n the parameters co r r e s p o n d i n g to the i n s i t u s t r e s s c o n d i t i o n s . Secondly and most i m p o r t a n t l y , ( r e l a t i v e to the t o p i c under d i s c u s s i o n ) i t reduces the number of v a r i a b l e s that has to be co n s i d e r e d ( e .g. i n s t e a d of E.^  and cr, c ; only E^/cr\c can be d e a l t with) i n t r y i n g to to develop e m p i r i c a l c o r r e l a t i o n s . The i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l t e s t s on normally c o n s o l i d a t e d c l a y s ( H e n k e l , 1960 and Parry, 1960) and on a wide range of normally c o n s o l i d a t e d c l a y s t e s t e d at MIT e x h i b i t the normalized behaviour i n undrained s t r e n g t h and i n undrained modulus. That i s , the q u a n t i t i e s , S U C I / c r ' c and E±/c£ do not vary with the i n i t i a l c o n s o l i d a t i o n p ressure i n the normally c o n s o l i d a t e d range of s t r e s s e s . Also,, the normalized behaviour c o n s t i t u t e s the b a s i s of what i s c a l l e d the s t a t e 86 boundary s u r f a c e , conceived i n developing the c r i t i c a l s t a t e m o d e K r e f e r Atkinson and Bransby) moreover, the normalized behaviour i s a s p e c i a l case of the SHANSEP concept proposed by Ladd et a l . , ( 1 9 7 4 ) . T y p i c a l r e s u l t s d i s p l a y i n g the normalized behaviour i n undraied s t r e n g t h and i n i n i t i a l modulus are shown in F i g u r e 5-1. In the subsequent d i s c u s s i o n i t i s assumed that f o r a given K 0, a n i s o t r o p i c a l l y c o n s o l i d a t e d s t r e s s - s t r a i n curves can be normalized with respect to t h e i r i n i t i a l major p r i n c i p a l c o n s o l i d a t i o n s t r e s s cr\c . T h i s has been used by s e v e r a l r e s e a r c h e s ( V a i d et a l . , 1974; Ladd et a l . , 1973) and thus t h e i r s t r e s s - s t r a i n curves f o r normally c o n s o l i d a t e d c l a y s ( a n i s o t r o p i c a l l y c o n s o l i d a t e d ) h a v e been u s u a l l y p u b l i s h e d i n the form of a s i n g l e normalized c u r v e ( f o r each type of c l a y ) . T y p i c a l r e s u l t s demonstrating t h i s behaviour i n s t r e n g t h and i n i n i t i a l modulus f o r two d i f f e r e n t o' 1 c are shown in F i g u r e 5-2. The normalized parameters do not depend on the c o n s o l i d a t i o n s t r e s s e s , but they depend on the s o i l type and i n some cases depend on K 0 too. T h e r e f o r e , any normalized p r o p e r t y of a normally c o n s o l i d a t e d c l a y X,(e.g. Ei/o*'1c) can be expressed i n the form; X = f ( K 0 , S o i l type) For the i s o t r o p i c case; X = f ( S o i l type) In normally c o n s o l i d a t e d undisturbed and remoulded c l a y s , the s o i l type i s w e l l i d e n t i f i e d by t h e i r index p r o p e r t i e s . In WEATHERED BANGKOK CLAY Balasubramaniam et a l . , 1977 i 1 T 1 0-0 20.0 40.0 60 0 80 CONSOLIDATION PRESSURE ( KN /M 1 ) ' WEATHERED BANGKOK CLAY Balasubramaniam et a l . , 1977 l 1 1 1 1 0.0 20.0 40.0 60.0 80.0 CONSOLIDATION PRESSURE ( KN /M 1 ) F i g u r e 5-1. Normalized Behaviour of I s o t r o p i c a l l y C o n s o l i d a t e d C l a y s o , L O . O X o * — N CM . CJ CD CO •si EABPL CLAY K 0 = .67 Donaghe et a l . , 1978 88 0.0 2.0 4.0 C O N S O L I D A T I O N P R E S S U R E ~ i i 6.0 8.0 ( K G / C M 2 ) CJ LDrsi CO CD • J IxJ or i— CO S * cr or Q EABPL CLAY K 0 = .67 Donaghe et a l . , 1978 I 1 1 0.0 2.0 4.0 6.0 C O N S O L I D A T I O N P R E S S U R E ( K G / C M * ) F i g u r e 5-2. Normalized Behaviour of A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s B.O 89 other words, the index p r o p e r t i e s are f u n c t i o n s of s o i l type and v i c e v e r s a . Hence, the normalized p r o p e r t i e s can be d e s c r i b e d by the equation given below. X = g(K 0,Index p r o p e r t i e s ) In the subsequent s e c t i o n s a l a r g e number of p u b l i s h e d l a b o r a t o r y t e s t data are analysed i n hopes of d i s c e r n i n g any c o r r e l a t i v e trends of the above form. 5.1.1 NORMALIZED MODULUS 5.1.1(a) Normalized Modulus and K 0 As d i s c u s s e d i n the p r e v i o u s s e c t i o n , E±/cr\c can be expressed i n the form; E±/cr\c = g(K 0,Index p r o p e r t i e s ) However, i t i s l o g i c a l to c o n s i d e r the p o s s i b i l i t y ; E i being dependent only on the mean normal c o n s o l i d a t i o n s t r e s s cr^, under d i f f e r e n t K 0 c o n d i t i o n s . T h i s i s r e a d i l y demonstrated(Figure 5-3) by a n i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l t e s t s ( l l ) performed with d i f f e r e n t K 0 v a l u e s , on s o f t grey c l a y . In other words, E^/cr^ appears to be t o t a l l y independent of cr^ and K 0. The i n i t i a l moduli values computed on the b a s i s of h y p e r b o l i c f i t show some d i f f e r e n c e between extension and compression. U s u a l l y these d i f f e r e n c e s are not very s i g n i f i c a n t and may be d i s r e g a r d e d . T h e r e f o r e , E ^ c r ^ of undisturbed and remoulded c l a y s can be co n s i d e r e d to depend only on t h e i r s o i l index p r o p e r t i e s and t h i s w i l l be d i s c u s s e d i n the next s e c t i o n . N O R M A L I Z E D I N I T I A L M O D U L U S E i / V 0.0 40.0 80.0 m 1 I (XlO1 ) c ro i z o n 3 N CD C L o a c t— 1 c in w < in o O O m n 0 i — i . Z D c n 7 0 T J m o CO X 3 o 06 91 5.1.1(b) Normalized Modulus and Index P r o p e r t i e s As p o i n t e d out i n the p r e v i o u s s e c t i o n , the normalized modulus E^/cr^ of normally c o n s o l i d a t e d c l a y (undisturbed or remoulded) depends only on t h e i r s o i l index p r o p e r t i e s . Hence the normalized modulus can be expressed i n the form Ei/o-^=g (Index p r o p e r t i e s ) . However, E^/o^ of undisturbed c l a y s are c o n s i d e r a b l y l a r g e r than that of remoulded ones, e s p e c i a l l y i n the low p l a s t i c i t y range(PI<60%). P u b l i s h e d s t r e s s - s t r a i n d a t a ( T a b l e B-1 g i v e s d e r i v e d model parameters) on l a b o r a t o r y t r i a x i a l t e s t s of undisturbed and remoulded c l a y s have been analysed i n order to formulate c o r r e l a t i o n s of the form d e s c r i b e d above for Ej^/cr^. A c o r r e l a t i v e t rend between the normalized modulus and p l a s t i c i t y index was found. The normalized undrained modulus E±/cr^ p l o t t e d a g a i n s t p l a s t i c i t y index i n F i g u r e 5-4 shows l i n e a r c o r r e l a t i o n s between the two parameters fo r the undisturbed and remoulded c l a y s . Both, the i s o t r o p i c a l l y and a n i s o t r o p i c a l l y c o n s o l i d a t e d undisturbed c l a y s f o l l o w the same c o r r e l a t i o n and can be w r i t t e n in . the form; E ^ C T ; = 400 -10PI/3 For remoulded c l a y s and appears to be t o t a l l y independent of the p l a s t i c i t y index. Hence, in undisturbed c l a y s the normalized undrained modulus decreases l i n e a r l y with the p l a s t i c i t y index whereas for remoulded c l a y i t i s a constant and i s a _ ID a '3 O X gr\i LU a a 0.0 A- UNDISTURBED D REMOULDED I I 1 1 20.0 40.0 60.0 80 0 PLASTICITY INDEX (J) 100.0 Figure 5-4. Normalized Modulus and P l a s t i c i t y Index R e l a t i o n f o r I s t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s 93 independent of the p l a s t i c i t y index. 5.3 NORMALIZED UNDRAINED STRENGTH 5.3.1 STRENGTH IN COMPRESSION OF ISOTROPICALLY CONSOLIDATED CLAYS The f o l l o w i n g i s an attempt to f i n d a c o r r e l a t i o n of the form, S U C I /o-J, = f (Index p r o p e r t i e s ) d e s c r i b e d i n s e c t i o n 5.1. S U C I i s . t h e undrained s t r e n g t h i n compression of i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s under t r i a x i a l c o n d i t i o n s . A l a r g e number of p u b l i s h e d t r i a x i a l t e s t r e s u l t s have been analysed in order to o b t a i n such a c o r r e l a t i o n between S^j/cr^, the normalized undrained s t r e n g t h and p l a s t i c i t y index. The l e a s t square r e g r e s s i o n a n a l y s i s on the data base(Table B-2) y i e l d s the c o r r e l a t i o n given below. S /2o-' = .38-.0015PI UCI c F i g u r e 5-5 shows the data p o i n t s and the above c o r r e l a t i o n . According to the above r e l a t i o n s h i p , the normalized undrained s t r e n g t h decreases with i n c r e a s i n g p l a s t i c i t y index. However, ac c o r d i n g to the Skempton's equation, s u / p = .11+.0037PI ; the undrained s t r e n g t h i n c r e a s e s with the p l a s t i c i t y index. When compared with the data base under c o n s i d e r a t i o n , Skempton's equation underestimates s t r e n g t h f o r low p l a s t i c i t y ( P I < 4 0 % ) c l a y s and over estimates f o r high p l a s t i c i t y ( P I > 6 0 % ) c l a y s ( s e e F i g u r e 5-5). However, i t i s important to note the f a c t t h a t , v e r i f i c a t i o n s on Skempton's SKEMPTON EQUATION PROPOSED CORRELATION a a a a CD a CD CD ~I 1 1 1— 2 0 . 0 4 0 . 0 6 0 . 0 BO.O PLASTICITY INDEX(J) 0.0 1 0 0 . 0 F i g u r e 5-5. Normalized Undrained Strength, P l a s t i c i t y Index C o r r e l a t i o n f o r I s o t r o p i c a l l y C o n s o l i d a t e d C l a y s 95 equation have been mainly done(Bjerrum 1954, Karlso n et a l . , 1967) based on the r e s u l t s of vane shear and unconfined compression t e s t s . The vane shear t e s t g i v e s shear s t r e n g t h under in s i t u c o n d i t i o n s (K 0) . But the shear s t r e n g t h S u/a' 1 c, under K 0 c o n d i t i o n s i s u s u a l l y lower than the shear s t r e n g t h under i s o t r o p i c c o n d i t i o n s , e s p e c i a l l y i n low p l a s t i c i t y c l a y s ( s e e F i g u r e 5-7). Due to the sample d i s t u r b a n c e , the unconfined compression t e s t s are c o n s i d e r e d to under estimate the s t r e n g t h . The new r e l a t i o n s h i p between the undrained s t r e n g t h and p l a s t i c i t y index presented i n t h i s s e c t i o n i s e n t i r e l y based on the i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l t e s t r e s u l t s . T h e r e f o r e , a d i r e c t comparison of the new c o r r e l a t i o n and Skempton's equation, without g i v i n g due regard to the above d i f f e r e n c e s may not be very meaningful. 5.3.2 STRENGTH IN COMPRESSION OF ANISOTROPICALLY CONSOLIDATED CLAYS In the pr e v i o u s s e c t i o n the r e l a t i o n s h i p between the undrained s t r e n g t h of i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s and t h e i r p l a s t i c i t y was d i s c u s s e d . But i n p r a c t i c e more of t e n a n i s o t r o p i c a l l y c o n s o l i d a t e d s o i l s are encountered. As was d i s c u s s e d b e f o r e , under given K 0 c o n d i t i o n s a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s show normalized behaviour i n s t r e n g t h with r e s p e c t to the i n i t i a l major c o n s o l i d a t i o n s t r e s s . The a v a i l a b l e data have been analysed i n order to o b t a i n a c o r r e l a t i o n between the normalized s t r e n g t h of a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s and t h e i r index 96 p r o p e r t i e s . S U C K / o " i c (Table B-3) i s p l o t t e d a g a i n s t p l a s t i c i t y index and i s shown in F i g u r e 5-6. S U C K i s the undrained s t r e n g t h i n compression of a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s under t r i a x i a l c o n d i t i o n s . The l e a s t square r e g g r e s s i o n a n a l y s i s on the above data base suggests the c o r r e l a t i o n between S U C K / c r ' l c and p l a s t i c i t y index given below. SUCK / 2 o " i c = • 30-.001 PI the data base c o n s i s t e d of c l a y s having K 0 v a l u e s ranging from .43 to . 6 4 ( n e g l e c t i n g two extreme cases) with a mean value of .56. I t must be noted t h a t , no e f f o r t has been made in t h i s a n a l y s i s to i n c l u d e K 0 as a v a r i a b l e i n the f u n c t i o n g d n d ex p r o p e r t i e s ) , s o l e l y due to the l i m i t e d s i z e of the data base with respect to the K 0 v a r i a t i o n . However, in Appendix E a semi e m p i r i c a l approach i s u t i l i z e d to f i n d a c o r r e l a t i o n between S T J C K / ^ C an& other s o i l parameters i n c l u d i n g K 0. 5.3.3 COMPARISON OF STRENGTH CORRELATIONS In the p r e v i o u s s e c t i o n an e m p i r i c a l r e l a t i o n s h i p between the undrained s t r e n g t h and p l a s t i c i t y index was obtained f o r a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s . The data base used i n t h i s a n a l y s i s had a mean K 0 value of .56. However, i t i s i n t e r e s t i n g to note the s i m i l a r i t i e s between the above r e l a t i o n s h i p and the one obtained f o r i s o t r o p i c a l l y c o n s o l i d a t e d ( K 0 = 1 ) c l a y s ( F i g u r e 5-7). Both show decreasing s t r e n g t h with i n c r e a s i n g p l a s t i c i t y . A n i s o t r o p i c s t r e n g t h i s to o F i g u r e 5-6. Normalized Strength and P l a s t i c i t y Index C o r r e l a t i o n f o r A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s LD o " a K U D CO a a a 0.0 1 ISOTROPICALLY CONSOLIDATED 2 ANISOTROPICALLY CONSOLIDATED (AVERAGE K 0 I 1 20.0 40.0 PLASTICITY 1 60.0 INDEX(J) 80.0 100. F i g u r e 5 - 7 . Comparison of the Undrained Strength E m p i r i c a l R e l a t i o n s f o r I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d Clay 99 small e r than the i s o t r o p i c s t r e n g t h . However, the a n i s o t r o p i c s t r e n g t h to i s o t r o p i c s t r e n g t h r a t i o S U C K / S u c i ' i n c r e a s e s ( a c c o r d i n g to the two equations) from .7 f o r a p l a s t i c i t y index of 10% to .9 f o r a p l a s t i c i t y index of 90%. 5.3.4 UNDRAINED STRENGTH IN EXTENSION T h i s s e c t i o n p r o v i d e s an e m p i r i c a l r e l a t i o n f o r the r a t i o of the undrained s t r e n g t h in compression to that i n e x t e n s i o n . The undrained deformation behaviour i n ext e n s i o n i s d i f f e r e n t from that i n compression and i n g e n e r a l , i s dependent on s t r e s s system change during shear. The s t r e n g t h in compression i s u s u a l l y higher than that i n e x t e n s i o n . The exte n s i o n st-rength i s one of the parameters which d e f i n e s the s t r e s s - s t r a i n r e l a t i o n s i n the proposed model. And thus i t i s as important as the compression s t r e n g t h . The f o l l o w i n g s e c t i o n attempts to develop e m p i r i c a l r e l a t i o n s f o r the undrained s t r e n g t h i n e x t e n s i o n . The undrained s t r e n g t h i n extension under a n i s o t r o p i c a l l y c o n s o l i d a t e d c o n d i t i o n s i s lower than that under the i s o t r o p i c a l l y c o n s o l i d a t e d c o n d i t i o n s ( f o r the same o-; c). However l a b o r a t o r y t e s t r e s u l t s (Table B-4) show that the r a t i o of the undrained s t r e n g t h i n compression t o the undrained s t r e n g t h i n ex t e n s i o n S U C K /S UEK' & o e s n o t s i g n i f i c a n t l y vary with K 0. I t seems to vary with the p l a s t i c i t y index. The r a t i o decreases with the p l a s t i c i t y index and approaches 1 f o r h i g h l y p l a s t i c c l a y s . F i g u r e 5-8 shows the v a r i a t i o n of t h i s r a t i o with the p l a s t i c i t y index. m F i g u r e 5 - 8 . Compression/Extension Strength R a t i o and P l a s t i c i t y Index f o r I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s o o 101 These f i n d i n g s emphasizes the importance of t a k i n g i n t o account the d i f f e r e n c e s of s t r e n g t h i n compression and in e x t e n s i o n , e s p e c i a l l y i n low p l a s t i c i t y c l a y s . A l s o i t i n d i c a t e s t h at the undrained s t r e n g t h i n e x t e n s i o n i s lower than the s t r e n g t h i n compression. 5.4 FAILURE RATIO The f a i l u r e r a t i o s computed(Table B-1) from p u b l i s h e d s t r e s s -s t r a i n data are p l o t t e d a g a i n s t the p l a s t i c i t y index i n F i g u r e 5-9. These s t r e s s - s t r a i n data i n c l u d e i s o t r o p i c a l l y and a n i s o t r o p i c a l l y c o n s o l i d a t e d t e s t s i n compression as w e l l as i n e x t e n s i o n . The f a i l u r e r a t i o R f does not show any c o r r e l a t i o n with the p l a s t i c i t y index of the m a t e r i a l . A l s o K 0 does not seem to have any s i g n i f i c a n t e f f e c t on R f. G e n e r a l l y R f l i e s between .8 and 1.0, but the lowest value of R f i n the data set under c o n s i d e r a t i o n i s .7. So, i n the absence of l a b o r a t o r y s t r e s s - s t r a i n data, a value f o r R f i s chosen using the e n g i n e e r i n g judgement and past experience, otherwise .9 i s a reasonably good c h o i c e . 5.5 DISCUSSION The e m p i r i c a l r e l a t i o n s obtained i n the above s e c t i o n s are a p p l i c a b l e only f o r normally c o n s o l i d a t e d c l a y s . Using the above r e l a t i o n s the parameters needed to d e f i n e the proposed s t r e s s - s t r a i n model can be obtained. In other words, i f the p l a s t i c i t y index i s known, E±/cr'm can be estimated from F i g u r e 5-4. F i g u r e 5-7 p r o v i d e s estimates of S U C K /cr', c and 00 o LO a <4-t a a ' o a 4 o + CD « Q fn v D • S o » (2) °-° 2 0'° - 60.0 B o ^ ^ PLRSTICITT INDEX (J) F i g u r e 5-9. V a r i a t i o n of R f w i t h P l a s t i c i t y Index 103 ^uci /°c • 0 r ' S U C K / S U C I can be estimated using the semi e m p i r i c a l curves given i n F i g u r e E-4 and thus S U C K . F i g u r e 5-8 gives the s t r e n g t h r a t i o , compression to e x t e n s i o n . Hence, the undrained s t r e n g t h i n e x t e n s i o n S r r o „ , can be e v a l u a t e d . An R f value of .9 i s a reasonably good c h o i c e , unless the e n g i n e e r i n g judgement and and past experience i s a g a i n s t i t . The parameters d i s c u s s e d above completely d e f i n e both, extension and compression p a r t s of the proposed s t r e s s - s t r a i n model. The c o r r e l a t i o n s obtained i n the above s e c t i o n can be extremely u s e f u l i n the absence of l a b o r a t o r y s t r e s s - s t r a i n data and where great accuracy i n the a n a l y s i s i s not warranted. Even when t h e . l a b o r a t o r y s t r e s s - s t r a i n r e s u l t s are a v a i l a b l e these r e l a t i o n s can be used i n judging the r e l i a b i l i t y of l a b data. However, i t must be emphasized t h a t , they can never be a s u b s t i t u t e f o r the s t r e s s - s t r a i n parameters obtained from l a b o r a t o r y t e s t s on undisturbed samples. T h i s i s e s p e c i a l l y true when the accuracy of the r e s u l t s i s a major concern. I t must a l s o be noted t h a t , the data set being l i m i t e d , no attempt has been made to c a t e g o r i z e them on the b a s i s of t h e i r g e o l o g i c a l o r i g i n , ( i . e . marine, f r e s h water, g l a c i a l , r e s i d u a l e tc.) i n t r y i n g to o b t a i n any c o r r e l a t i o n s . A l s o , the d i f f e r e n c e s i n the r a t e of s t r a i n and the c o n s o l i d a t i o n time(before shearing) has not been c o n s i d e r e d . 1 04 CHAPTER VI SUMMARY AND CONCLUSION The undrained s t r e s s - s t r a i n behaviour i s complex, being non l i n e a r i n e l a s t i c and s t r e s s system dependent. I t i s a l s o i n f l u e n c e d by the i n i t i a l s t r e s s a n i s o t r o p y . The widely used i s o t r o p i c modeKDuncan et a l . , 1970) does not take i n t o account e i t h e r the i n f l u e n c e of i n i t i a l s t r e s s a n i s o t r o p y or the s t r e s s system change duri n g shear. The l a b o r a t o r y t e s t r e s u l t s ( p u b l i s h e d ) showed t h a t , the r o t a t i o n of p r i n c i p a l planes by 90°(extension) duri n g shear, of some clays(PI<40%) reduced the s t r e n g t h (compared to compression s t r e n g t h S /a', ) as much as by 100%. A c c o r d i n g l y , s t r e s s - s t r a i n curves i n ext e n s i o n and i n compression are q u i t e d i f f e r e n t . T h i s d i f f e r e n c e depends on the angle of r o t a t i o n of p r i n c i p a l axes, g i v i n g a maximum f o r a 90° r o t a t i o n . The value of o-d 0/S u i s a measure of the i n f l u e n c e of i n i t i a l s t r e s s a n i s o t r o p y on s t r e s s - s t r a i n c u r v e s . Depending on the f a c t o r a d 0 / S u , the a n i s o t r o p i c a l l y c o n s o l i d a t e d s t r e s s -s t r a i n c u r v e s ( e . g . compression or extension) can d e v i a t e s u b s t a n t i a l l y from the i s o t r o p i c c u r v e s . Hence, the i s o t r o p i c model can simulate a c c u r a t e l y , the i s o t r o p i c a l l y c o n s o l i d a t e d compression t e s t c o n d i t i o n o n l y . In a d d i t i o n to the s t r e s s - s t r a i n c h a r a c t e r i s t i c s modelled by the i s o t r o p i c model, the proposed non l i n e a r e l a s t i c model can take i n to account the i n f l u e n c e of s t r e s s a n i s o t r o p y and the s t r e s s system change d u r i n g shear. The v a r i a t i o n of s t r e n g t h and e l a s t i c modulus with the angle of r o t a t i o n of p r i n c i p a l planes i s i n c o r p o r a t e d i n the model. These q u a n t i t i e s are assumed to 105 vary s i n u s o i d a l l y with the angle of r o t a t i o n of p r i n c i p a l axes. The a n a l y s i s of a l a r g e number of K 0 c o n s o l i d a t e d t r i a x i a l s t r e s s - s t r a i n r e s u l t s of normally c o n s o l i d a t e d c l a y r e v e a l e d that a l l these s t r e s s - s t r a i n curves can be adequately modelled by using the h y p e r b o l i c f u n c t i o n s p r o v i d e d the d e v i a t o r s t r e s s i s measured r e l a t i v e to i t s i n i t i a l v a l u e . However the w r i t e r b e l i e v e s ( n e e d s i n v e s t i g a t i o n ) that t h i s i s true even f o r over c o n s o l i d a t e d c l a y s and t h e r e f o r e the proposed model i s b e l i e v e d to be a p p l i c a b l e f o r over c o n s o l i d a t e d c l a y s too. T h i s model can be used with g r e a t e r c o n f i d e n c e to analyse the undrained deformation behaviour of K 0 c o n s o l i d a t e d s o i l masses under monotonic l o a d i n g c o n d i t i o n s . K 0 c o n s o l i d a t e d t r i a x i a l compression and extension t e s t r e s u l t s can be used to o b t a i n the parameters which completely d e f i n e the proposed a n i s o t r o p i c model. U t l i z i n g the parameters obtained from K 0 c o n s o l i d a t e d t r i a x i a l t e s t s ( V a i d et a l . , 1974) on Haney c l a y , K 0 c o n s o l i d a t e d plane s t r a i n s t r e s s - s t r a i n behaviour was p r e d i c t e d . P r e d i c t e d s t r a i n s were i n e x c e l l e n t agreement with the measured v a l u e s ( V a i d et a l . , 1974) i n the working range of s t r a i n s . T h i s proves the model's a b i l i t y to p r e d i c t a c c u r a t e s t r a i n s i n the working range of s t r a i n s under plane s t r a i n c o n d i t i o n s , using K 0 t r i a x i a l parameters. Whether to use K 0 t r i a x i a l parameters or plane s t r a i n parameters under a given f i e l d s i t u a t i o n f o r a given model and the d i s c r e p a n c i e s i n r e s u l t s such d i f f e r e n t parameters produce(when used i n a given model) needs f u r t h e r i n v e s t i g a t i o n . As expected, the comparison of p r e d i c t e d settlements of the 106 two models on a foundation a n a l y s i s showed t h a t , when o^o/Su i s l a r g e r , the d i f f e r e n c e between the two models are l a r g e r too. The i s o t r o p i c model always overestimated the sett l e m e n t s ; i n some cases as much as by ten times those p r e d i c t e d by the a n i s o t r o p i c model. The f a c t t h at the a n i s o t r o p i c model u s u a l l y g i v e s l a r g e r tangent moduli val u e s than the i s o t r o p i c model f o r the same s t r e s s l e v e l , e x p l a i n s such behaviour. In g e n e r a l , these r e s u l t s r e v e a l e d the gross inadequacy of the i s o t r o p i c model as f a r as i t s settlement p r e d i c t i o n s on K 0 c o n s o l i d a t e d s o i l masses are concerned. The u l t i m a t e b e a r i n g s t r e s s e s p r e d i c t e d by the i s o t r o p i c model were g e n e r a l l y higher(by about 10%) than those p r e d i c t e d by i t s c o u n t e r p a r t . T h i s can be expected to be the general trend, because, both S /cr'. and UCK STTT,„/crV are u s u a l l y smaller than S I T^ T /cr' . The f a i l u r e zones UEK ' c J UCI c p r e d i c t e d by the i s o t r o p i c model were g e n e r a l l y deeper and narrower than those p r e d i c t e d by the a n i s o t r o p i c model. 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P., "Foundation F a i l u r e of New L i s k e a r d Embankment," Highway Research Board B u l l e t i n , No. 463, 1973, pp. 1-17. 68. Raymond, G. P., "Kars Leda C l a y , " Performance of E a r t h and E a r t h Supported S t r u c t u r e s , V o l . 1, Part 1, ASCE, New York, N.Y., 1972, pp. 319-340. 69. Richardson, M. A., and Whitman, R. V., " E f f e c t of S t r a i n Rate upon Undrained Shear R e s i s t a n c e of a Saturated C l a y , " Geotechnique, London, England, V o l . 13, No., 1963, pp. 310-324. 70. Sangrey, D. A., Henkel, D. J . , and E s r i g , M. I., " E f f e c t i v e S t r e s s Response of Sa t u r a t e d Clay S o i l to Repeated Loading," Canadian G e o t e c h n i c a l J o u r n a l V o l . 6, No. 3, Aug., 1969, pp. 241-252. 71. Saxena, S., Hedberg, J . , and Ladd, C. C , "Geo t e c h n i c a l P r o p e r t i e s of Hackensack V a l l e y Varved Clay of New J e r s e y , " G e o t e c h n i c a l T e s t i n g J o u r n a l , GTJODJ, V o l . 1, No. 3, Sept., 1978, pp. 148-161. 72. S h e r i f , M. A., Wu, M. J . , and Bostrum, R. C , "Reduction i n S o i l Strength due To Dynamic Loading" M i c r o z o n a t i o n Conference, V o l . I I , N a t i o n a l Science Foundation and ASCE, Nov., 1972, pp. 439-454. 73. S h i b i t a , T. and Karube, D., " I n f l u e n c e of the V a r i a t i o n of the Intermediate P r i n c i p a l S t r e s s on the Mechanical P r o p e r t i e s of Normally C o n s o l i d a t e d C l a y s , " Proceedings of the 6 th I n t e r n a t i o n a l Conference on S o i l Mechanics and Foundation E n g i n e e r i n g , V o l . 1, U n i v e r s i t y of Toronto Press, 1965, pp. 359-363. 74. S h i b i t a , T., and Karube, D., "Creep Rate and Creep St r e n g t h of C l a y s , " Proceedings of the 7 th I n t e r n a t i o n a l Conference on S o i l Mechanics and Foundation E n g i n e e r i n g , V o l . 1, Sociedad Mexicana de Mecanica de Suelos, Mexico, 1969, pp. 361-368. 75. Simon, R. M. , C h r i s t i a n , J . T. , and Ladd, C. C , " A n a l y s i s of Undrained Behaviour of Loads on C l a y s , " A n a l y s i s and Design i n G e o t e c h n i c a l E n g i n e e r i n g , V o l . I, ASCE, New York, N.Y., June, 1974. 76. Simons, N. E., "The E f f e c t s of O v e r c o n s o l i d a t i o n on the Shear Strength C h a r a c t e r i s t i c s on an Undisturbed Oslo C l a y , " Shear S t r e n g t h of Cohesive S o i l s , ASCE, 1 1 4 U n i v e r s i t y of Colorado, 1960. . " 77. Singh, R., and Gardner, W., " C h a r a c t e r i s t i c s of Dynamic P r o p e r t i e s of Gulf of Alaska C l a y s , " presented at the A p r i l 2-6, 1979, ASCE, Convention and E x p o s i t i o n , h e l d at Boston, M a s s . ( P r e p r i n t 3606). 78. Skempton, A. W., " H o r i z o n t a l S t r e s s e s i n an O v e r c o n s o l i d a t e d Eocene C l a y , " Proceedings of the 5 th I n t e r n a t i o n a l Conference on S o i l Mechanics and Foundation E n g i n e e r i n g , V o l . 1, Dunod Press, P a r i s , France, 1961, pp. 351-357. 79. Skempton, A, W., "The Pore Pressure C o e f f i c i e n t s A and B," Geotechnique, London, England, V o l . 4, No., 1954, pp. 143-147. 80. Sketchley, C. J . , and Bransby, P. L., "The Behaviour of an O v e r c o n s o l i d a t e d Clay i n Plane S t r a i n , " Proceedings of the 8 th I n t e r n a t i o n a l Conference on S o i l Mechanics and Foundation E n g i n e e r i n g , V o l . 1, Moscow, USSR, 1973, pp. 377-384. 8 1 . Sparrow, R. W., Swanson, P. G., and Brown, R. E., "Report of Laboratory T e s t i n g of Gulf of Alaska Cores," Open F i l e Report to the United S t a t e s G e o l o g i c a l Survey, Law E n g i n e e r i n g T e s t i n g Company, Mar., 1979. 82. S r i d h a r a n , A., Rao, S., and Rao, G., "Shear Strength C h a r a c t e r i s t i c s of Saturated M o n t m o r i l l o n i t e and K a o l i n C l a y s , " S o i l Mechanics and Foundations, V o l . 11, No. 3, Sept., 1971, pp. 1-22. 83. Subhas, C. D., and Gangopadhyay, C.R., "Undrained S t r e s s e s and Deformations Under F o o t i n g s on C l a y , " J o u r n a l of S o i l Mechanics and Foundations D i v i s i o n , ASCE, V o l . 104, No. SM1, Proc. Paper 13463, Jan., 1978, pp. 11-24 8 4 . Swanson, P. G., and Brown, R. E., " T r i a x i a l and C o n s o l i d a t i o n T e s t i n g of Cores from the 1976 A t l a n t i c Margin Co r i n g P r o j e c t , " Open F i l e Report No. 78-124 to the United S t a t e s G e o l o g i c a l Survey, Law E n g i n e e r i n g T e s t i n g Company, Washington, D.C, Nov., 1977. 85. Tavenas, F., Blanchet, R. , Garneau, R., and L e r o u e i l , S., "The S t a b i l i t y of Stage Co n s t r u c t e d Embankments on S o f t C l a y s , " Canadian G e o t e c h n i c a l J o u r n a l V o l . 15, No. 2, May, 1978, pp. 283-305. 86. V a i d , Y. P., " E f f e c t of C o n s o l i d a t i o n H i s t o r y and S t r e s s Path on H y p e r b o l i c S t r e s s - S t r a i n R e l a t i o n s , " S o i l Mechanics S e r i e s , No. 54, U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada, Dec., 1981. 1 15 87. V a i d , Y. P., and Campanella, R. G., " T r i a x i a l and Plane S t r a i n Behaviour of N a t u r a l C l a y s , " J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 100, No. GT3, Proc. Paper 10421, March, 1974, pp. 207-224. 88. V a i d , Y. P., and Campanella, R. G., "Time Dependent Behaviour of Undisturbed C l a y , " J o u r n a l of S o i l Mechanics and Foundations D i v i s i o n , ASCE, V o l . 103, No. SM7 , Proc. Paper 13065, J u l y , 1977, pp. 693-709. 89. Whitman, R. V., "Some C o n s i d e r a t i o n s and Data Regarding the Shear Strength of C l a y s , " Shear Strength of Cohesive S o i l s , ASCE, U n i v e r s i t y of Colorado, 1960, pp. 581-614. 90. Widger, R. A., and Fredlund, D. G., " S t a b i l i t y of S w e l l i n g Clay Embankments," Canadian G e o t e c h n i c a l J o u r n a l V o l . 16, No. 1, Feb., 1979, pp. 140-151. 91. Wissa, A., Ladd, C. C , and Lambe, T. W., " E f f e c t i v e S t r e s s Parameters of S t a b i l i z e d S o i l s , " Proceedings of the 6 th I n t e r n a t i o n a l Conference on S o i l Mechanics and Foundation E n g i n e e r i n g , V o l . 1, Montreal, Canada, 1965, pp. 140-151. 92. Wu, T. H., Chang, N. and A l i , E. M., " C o n s o l i d a t i o n and Strength P r o p e r t i e s of a C l a y , " J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 104, No. GT7, Proc. Paper, J u l y , 1978, pp. 889-905. 93. Wu, T., Douglas, A., and Goughnour, R., " F r i c t i o n and Cohesion of Saturated C l a y s , " J o u r n a l of S o i l Mechanics and Foundations D i v i s i o n , ASCE, V o l . 90, No. SM3, Proc. Paper, June, 1962, pp. 1-32. 94. Yudhbir, and Varadarajan, A., "Undrained Behaviour of O v e r c o n s o l i d a t e d Saturated C l a y s d u r i n g Shear, " S o i l Mechanics and Foundations, V o l . 14, No. 4, D e c , 1 974, pp. 1-12. 1 16 APPENDIX A PUBLISHED DATA AND DERIVED TRANSFORMED STRESS-STRAIN PLOTS ON NORMALLY CONSOLIDATED CLAY ALL RESULTS ARE FROM TRIAXIAL TESTS For compression t e s t s ; cr, c i n c r e a s e d , unless otherwise s t a t e d . Q = {(cr -cr ) - (cr -cr ) 0 } /cr', „ yy xx yy xx u 1 c e = A x i a l compressive s t r a i n UCI I s o t r o p i c a l l y c o n s o l i d a t e d undrained t e s t in compression UCK A n i s o t r o p i c a l l y c o n s o l i d a t e d undrained t e s t i n compression For extension t e s t s ; cr, c decreased, unless otherwise s t a t e d . Q = { (cr -cr ) 0 - ( c r -cr )}/cr', yy yy yy xx. <= e = A x i a l t e n s i l e s t r a i n UEI I s o t r o p i c a l l y c o n s o l i d a t e d undrained t e s t in extension UEK A n i s o t r o p i c a l l y c o n s o l i d a t e d undrained t e s t i n extension R = Remoulded U = Undisturbed in o o i m o°. X a <—IO £° I CO o o a a a' l b / i n 2 c 20 30 40 50 60 -f CD X CD • CD 4 4 K 1 1 1 1 1 : 1 0 Q 0 01 0 .02 0 .03 0 .04 0 .05 STRAIN F i g u r e A-1. UCI on Soft Bangkok C l a y ( U ) ; Balasubramaniam, 1976 118 L O Q Si * a' kg/cm 2 c 1.0 L a g u n i l l a s Clay(U) 3.0 Kawasaki Clay(U) 6.0 Boston Blue Clay(R) * 6.2 V i c k s b u r g Buckshot C 1 A 8.0 Amuay Clay(U) + CD <J> $ CD CD CD I 1 1 1 1 0.0 0.05 0.1 0.15 0.2 0.25 STRAIN (X10+1 ) F i g u r e A-2. UCI on D i f f e r e n t C l a y s ; Ladd, 1964 CO cr\c = 3.00 kg/cm 2 o-3c = 1 .53 kg/cm 2 CD CD co 0.0 0.04 0.08 0 . 1 2 S T R R J N ( X 1 0 + 1 ) F i g u r e A - 3 . UCK on Kawasaki C l a y ( U ) ; Ladd, 1964 OQ (NI rsi •—> Q + o .—I X '—'LO OH oo o o CD cr\c = 3.00 kg/cm 2 cric = 1 .50 kg/cm 2 CD CO CD I 1 1 1 1 0.0 0.04 0.0B 0.12 0.16 0. STRAIN ( X 1 0 + 1 ) F i g u r e A-4. UEK on Kawasaki C l a y ( U ) ; Ladd, 1964 ro o 03 o + O ' ' CD O a Q - | a cc or ( N J a a a CO a' l c = 4.00 kg/cm 2 o-3'c = 2.16 kg/cm 2 CD CO -I 1 1 1 -I ~l I 0.0 0.0B 0.16 0.24 0.32 0.4 0.48 0. STRRIN ( X 1 0 + 2 ) Figure A-5. UCK on Boston Blue C l a y ( R ) ; Ladd, 1 9 6 4 cr'1c = 4.00 kg/cm 2 cr'3c = 2.16 kg/cm 2 CD 1 1 1 0.0 0.05 0.1 0.15 09 STRRIN (X10 + 1 ) F i g u r e A-6. UEK on Boston Blue C l a y ( R ) ; Ladd, 1964 C M o r» - 43.2 kN/m2 CD CO fNJ CC cr I—oo cr> | CD CD m ra ° CD a a I 1 J CD CD CD I T | • 1 — I °-° ° - D ) 0.D2 0.03 0.04 0.05 0 06 0 07 STRRIN F i g u r e A-7. UCI on Kinnegar C l a y ( U ) ; Crooks, 1976 124 CM o-; c = 44.5 kN/m2 K 0 = .47 cr\c = 104.7 kN/m2 K 0 = .42 c 4 CD or I— CO o CD CD CD o CD ° 1 I I 1 1 1 °-° 0.0B 0 J 2 0 16 n 2 STRAIN ( X 1 0 + ! ) F i g u r e A-8. UCK on Kinnegar C l a y ( U ) ; Crooks, 1976 Cr' c a' c 529.2 kN/m2 335.4 kN/m2 a CD 0.0 ~l 1 1 0.04 0.08 0.12 STRAIN ( X 1 0 + 1 ) F i g u r e A-9. UCI on Clay ; Pender, 1978 LD CM a CD a cr' = 529.2 kN/m2 <r'c = 335.4 kN/m2 CD CD CD 0.0 0 .01 I 0 .02 0 .03 T 0 .04 STRRJN I 0 .05 Figure A-10. UEI on Clay; Pender, 1978 CD o i c = 650.0 kN/m2 cr'3C = 383.8 kN/m cr'1c = 650.0 kN/m2 o-3C = 650.0 k N / m 4 a • i —1 1 1 ~~i 1— 0.0 0.05 0.1 0.15 0.2 0 25 0 3 STRRIN (XI0 + 1 ) Figure A-11. UCI and UCK on Cl a y ; Pender, 1978 128 CN cr' p s i c 145 372 860 CD CE on CD CD CM O CD CD o CO + o LO a CD CD CD CD a a 0.0 ~T 0.01 0.02 0.03 S T R A I N 0 .04 0 .05 F i g u r e A-12. UCI on London C l a y ( U ) ; Bishop et a l . , 1965 oo LO a + '—1 a X ^-00 1 — i a LO a CT 129 4 «=> I CONSOLIDATION TIME 3 DAYS A 2 WEEKS 4 MONTHS a CT ~j = 2.5 kg/cm 2 4 a o r a | a t — co 4 CD a a ' - ! 1 1 1 1 r 0.0 0.02 0.04 0.06 0.08 0.1 0 12 STRAIN ( X 1 0 + 1 ) F i g u r e A-13. UCI on Skabo C l a y ( U ) ; Bjerrum et a l . , 1963 130 o co a a CONSOLIDATION TIME g 3 DAYS 2 WEEKS 4 MONTHS cr', c = 2.5 kg/cm 2 c r 3 c = 1.5 kg/cm 2 ex I + cc |—CM | CD c o ° . a CT | CO ° I I I 1 1 0.0 0.04 0.08 0.12 0 16 STRAIN ( X 1 0 + 1 ) F i g u r e A-14. UCK on Skabo C l a y ( U ) ; Bjerrum et a l . , 1963 SLOW TEST(1% s t r a i n i n 500min.) FAST TEST(1% s t r a i n i n 1min.) = 60 l b / i n 2 A A a a A 1 1 1 1 0-01 0.02 0.03 0 04 STRAIN F i g u r e A - 1 5 . UCI on Fat C l a y ( R ) ; Richardson et a l . , 1963 in o cr^ = 30 l b / i n 2 COMPRESSION EXTENSION CD Q s a CO CD o - r 0.0 •'. 02 1 0 .04 STRRIN 0 .06 0 .08 F i g u r e A-16. UCI and UEI on Weald Clay(R) Parry, 1960 cV kN/m 2 69.0 103.5 172.5 CD co CD CD a CO + + A CD + A ID A -t A + A CD A + A 4 I 1 1 1 1— 0.0 0.02 0.04 0.06 0.0B 0.1 . STRRIN F i g u r e A-17. UEI on Weathered Bangkok C l a y ( U ) ; Balasubramaniam et a l . , 1977 CD co CT a f M + to " — i n cr'c kN/nv 207.0 276.0 345.0 414.0 ru O " — i CT i— CO C N CT CJ a co + CD A 4 CD + CD A 4 CD A 4 CD A 4 A 4 I 0 . 0 B a . 0 . 0 0 . 0 2 I 0 . 0 4 0 . 0 6 STRRIN 0.1 Figu r e A-18. UEI on Weathered Bangkok C l a y ( U ) ; Balasubramaniam et a l . , 1977 135 o CM o - f CO CO o CM a ra o o cr' = 1 5 l b / i n 2 c CO • a ID a a a CD CD a I I I I I 0.0 0.02 0.04 0.06 0.08 0.1 S T R A I N F i g u r e A-19. UCI on Weald C l a y ( R ) ; Skempton, 1964 cr' = 30 l b / i n 2 c CD CD CD CD CD CD CD CD CD .0 . 0 . 0 4 0 . 0 8 STRAIN — I — 0 .12 0 . F i g u r e A-20. UCI on Weald C l a y ( R ) ; H e n k e l , 1956 A CD & or* l b / i n 2 c 6 . 4 1 1 . 0 2 8 . 0 CD CD CD CD CD CD ~~1 1 0 . 0 2 0 . 0 3 STRRIN 0 . 0 0 .01 0 . 04 0 . 05 Figure A-21. UCI on Soft dark grey C lay(U); Parry, 1968 in a • CD CD CD CD CD — | | 1 1 1 0.0 0.02 0.04 0.06 0.08 0 S T R A I N (X10+1 ) Figure A-22. UCK on Drammen Clay(U); Prevost, 1978 a • D • CO co CD CD CU 3 CD 0.0 ~~1 1 1 — 0.01 0.02 0.03 STRRIN Figure A-23. UEK on Drammen Clay(U); Prevost, 1978 1 40 cr' kg/cm 2 c 1.0 1.5 2.0 LD 4 A CD 4 A 4 A a 4 A CD 4 A 4 A CD t CD A 4 A 1 0 .06 S T R R I N 0 .1 —I 0.12 0.0 0 .02 0 .04 0 .08 F i g u r e A-24. UCI on Grundite C l a y ( R ) ; Lade et a l . , 1978 00 a r— cn a LO a Kn = .5 a CD •—i . C E ° CH I— CO CD CD n a CD C M a a" CD CD CD CD 0.0 0.0] ~I 1— 0.02 0.03 S T R R I N - 1 — 0 .04 F i g u r e A-25. UCI on A r t i f i c i a l l y Prepared K a o l i n ; Subhas, 1978 142 03 a a CO r-a I A 4 0 X CO a LOADING RATE kg/h .066 .350 1 .590 2.500 12.000 a g I 110.000 $ - a era or I— CO CM a CD O A X X s J • X X a X K O o" 0.0 0.01 0.02 0.03 0.04 STRRIN F i g u r e A-26. UCI on Haney C l a y ( U ) ; V a i d et a l . , 1977 143 IT) O STRAIN RATE l 0 3 % / m i n . .94 ^ 2.80 14.00 150.00 1100.00 g + 4 X I X a 4 D A CD H 4 1 1 1 1 1 1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 STRAIN ( X 1 0 + 1 ) F i g u r e A-27. UCI on Haney C l a y ( U ) ; V a i d et a l . , 1977 LT) cn cn o to X o i—i.—i l— CO a • 4 DD A 4 a CD A 1 9 a I 1 1 1 1 0.0 0.01 0.02 0.03 0.04 STRRIN F i g u r e A-28. UCI on Osaka A l l u v i a l C l a y ( R ) ; S h i b i t a et a l . , 1965 OC a a a cr* kN/m2 c 200 400 CD CD CD CD CD CD 0.0 0 .01 0 .02 STRAIN ~ ~ I — 0 .03 F i g u r e A-29. UCI on C l a y ; Ladd et a l . , 1963 o i c = .73 kg/cm 2 cr'3c = cr\c = .77 kg/cm 2 c r 3 c = o-*1c = .80 kg/cm 2 c r 3 c = cr',c = .83 kg/cm 2 c r 3 C = <j\c = .84 kg/cm 2 o-3c = 50 kg/cm 41 kg/cm 37 kg/cm 36 kg/cm 35 kg/cm CD CD CD CD © A 0 A X A * A * CO X CD A A • X » » A X CD A X CD A X CD A X CD CD CD 0.0 0.2 0 . 4 1 0.6 STRRIN/ 0 .B Figure A-30. UCK on C l a y ( U ) ; Bozozuk, 1980-81 APPENDIX B PUBLISHED DATA AND DERIVED MODEL PARAMETERS OF NORMALLY CONSOLIDATED CLAY TABLE B-1. INITIAL MODULUS,FAILURE RATIO, PLASTICITY INDEX, AND CONFINING PRESSURE SOIL 1 %PI E./o-i m R f TEST ! REFERENCE S o f t Bangkok C l a y ( U ) 75.4±1.3 1 19.2 .95 1 .Oc Ba1asubraman1 urn , 1976 Kawasaki C l a y ( U ) 34(20-40) 302 .0 .93 1 .Oc Ladd, 1964 Kawasaki C1ay(U) 34(20-45) 371 .0 . 75 .51c Ladd, 1964 Kawasaki C l a y ( U ) 34(20-45) 318.5 .50e Ladd, 1964 B o s t o n B l u e C l a y ( R ) 15±2 347 . 2 .92 1 .Oc Ladd, 1964 B o s t o n B l u e C l a y ( R ) 15 + 2 824 . 2 .70 ,54c Ladd,1964 Bo s t o n B l u e C l a y ( R ) 15±2 917.8 . 54e Ladd, 1964 L a g u n l l l a s C l a y ( U ) 37(29-49) 307 . 1 1 .00 1 .Oc Ladd, 1964 V l c k s b u r g Buck Shot Clay(R) 39±1.5 269.0 .98 1 .Oc Ladd, 1964 Amuay C l a y ( U ) 25+10 306.5 1 .07 1 .Oc Ladd, 1964 London C l a y ( U ) 43 68.5 .82 1 .Oc B1 shop et a l . , 1963 Skabo C l a y ( U ) 28 412.8 .83 1 .Oc Bjerrum et a l . , 1963 'R = Remoulded 'U = U n d i s t u r b e d ! K o c = K ° c o n s o l i d a t e d t r i a x i a l compression 'Koe = Ko c o n s o l i d a t e d t r i a x i a l e x t e n s i o n 148 Skabo C l a y ( U ) 28 402 .0 .6c Bjerrum et a l . . 1963 Fat C1ay(R) 38 128 .6 .92 1 ,0c R i c h a r d s o n e t a l . . 1963 Weald C l a y ( R ) 25 93.3 .93 1 .Oc P a r r y , 1960 Weathered Bangkok C1ay(U) 82±4 115.5 1 .00 1 .Oe Balasubramaniam et a l . , 1977 Haney C l a y ( U ) 18 180.0 .94 . 56e Va1d et a l . , 1974 Weald C l a y ( R ) 25 76.9 .95 1 .Oc P a r r y , 1960 Drammen C1ay(U) 29 298.3 1 .Oe P r e v o s t , 1978 G r u n d l t e C l a y ( R ) 30. 1 82.9 .89 1 .Oc Lade et a l . , 1978 A r t i f i c i a l K aol1n(R) 20 117.4 .98 . 5c Subhas et a l . , 1978 Haney C l a y ( U ) 18 294 . 2 .94 1 .Oc V a i d et a l . , 1977 Osaka A l l u v i a l C l a y ( R ) 49 116.8 .91 1 .Oc Ladd et a l . , 1974 Haney C l a y ( U ) 18 285.7 .94 1 .Oc Va1d et a l . , 1974 Haney C l a y ( U ) 18 321 .6 .90 .56c Va1d et a l . , 1974 S o f t Grey C l a y ( U ) 20 450.0 .85 Koc Bozozuk, 1980-1981 Haney C l a y ( U ) 18 252.7 .94 .56e Va1d et a l . . 1974 4^ 00 1 49 TABLE B-2. UNDRAINED STRENGTH IN COMPRESSION; ISOTROPICALLY CONSOLIDATED TRIAXIAL TESTS SOIL %PI S /cr' UCI c REFERENCE Weald 25 .279 Henkel,1960 K a o l i n 31 .313 Amerasinghe et a l . , 1975 San Fran. Bay Mud 45 .432 M i t c h e l , 1976 London Clay 52 .250 Henkel, 1960 Keuper Marl 1 4 .256 Brown et a l . , 1975 I l l i t e 31 .341 France et a l . , 1977 Oslo 18 .380 Simons, 1960 Bras w e l l 65 .300 Skempton, 1961 Shellhaven 80 .205 Skempton et a l . , Newf i e l d 10 .475 Sangrey et a l . , 1969 V i r g i n i a C o a s t a l 27 .409 Swanson et a l . , 1977 Spestone K a o l i n 32 .210 Parry et a l . , 1973 Japanese 37 .400 S h i b i t a et a l . , 1969 Soft Bangkok 49 .255 Moh et a l . , 1969 Mi l a z z o 33 .333 Croce et a l . , 1969 Calcium I l l i t e 48 .250 Olson, 1962 Drammen 15 .310 Simons, 1960 150 Backswamp 40 .280 Whitman, 1960 Kars Leda 21 .265 Raymond, 1972 P l a s t i c Holocene 38 .335 Ko u t s o f t a s et a l . , 1 976 Ghana 29 .380 DeGraft et a l . , 1969 S e a t t l e 26 .372 S h e r r i f et a l . , Rang de Fleuve 46 .378 Tavanes et a l . , 1978 Sodium I l l i t e 44 .340 Olson et a l . , 1963 Grundite 29 .356 P e r l o f f et a l . , Tera Roxa 22 .313 Dagrus, 1963 Amuay 42 .342 Lambe, 1963 Scott 12 .231 Ladanyi et a l . , 1965 Toleda 23 .200 Wu et a l . , 1978 Kawasak i 31 .370 Ladd et a l . , 1963 Concord Blue 1 0 .355 Egan, 1977 L a g u n i l l a s 37 .395 Ladd et a l . , 1963 L i skeard 33 .298 Raymond, 1973 Vi c k s b u r g 39 .287 Ladd, 1962 Massachusetts 6 .416 Wissa et a l . , 1965 Ottawa E s t u r i n e 26 .300 Kenny et a l . , 1961 L i l l a Edet 32 .290 Bjerrum et a l . , 1960 Khor A l Zubair 35 .360 Hanzawa, 1977 Hokkido S i l t B 21 .362 M i t a c h i et a l . , 1976 Hokkaido Clay 32 .410 M i t a c h i et a l . , 1976 Fao 29 .530 Hanzawa, 1977 151 Sa i n t Alban 22 .290 Tavenas et a l . , 1 978 Kanpur Clay 18 .295 Yudhbir et a l . , 1974 Rann of Kutch 49 .326 Yudhbir et a l . , 1974 K a o l i n i t e 25 .492 Broms et a l . , 1965 New England 20 .325 Ladd, 1976 Drammen Clay 26 .325 Ladd, 1976 L a n s i s a l m i 46 .215 Korhonen, 1977 Sa u l t Ste Maru 32 .327 Wu et a l . , 1962 Bath K a o l i n i t e 15 .425 Broms et a l . , 1963 Mondal 25 .254 Karlso n et a l . , 1967 Banglore K a o l i n i e 20 .422 Sridharan et a l . , 1971 Kinnegar 58 .338 Crooks et a l . , 1976 Regina 54 .308 Widger et a l . , 1979 Long l i e . C o a s t a l 34 .271 Swanson et a l . , 1977 Hackensack Varved 35 .158 Saxena et a l . , 1978 Alaskan Gulf 1 4 .408 Singh et a l . , 1979 East A t c h a f a l a y a 53 .282 Donaghe et a l . , 1978 Buckshot Clay 36 .335 Donaghe et a l . , 1978 Koyoto 57 .530 Akai et a l . , 1965 Gunabara Bay 90 .253 Costa et a l . , 1977 Kodiak I s l a n d 1 4 .510 Sparrow et a l . , 1978 Winnipeg Clay 60 .211 Crawford, 1964 Weald Clay 22 .323 Henkel et a l . , 1964 Beaumont Clay 41 .267 Mahar et a l . , 1979 Drammen Clay 27 .280 Anderson et a l . , 1980 152 TABLE B-3. UNDRAINED STRENGTH IN COMPRESSION; KQ CONSOLIDATED TRIAXIAL TESTS %PI s /cr', UCK c REFERENCE 18 .268 Vaid et a l . , 1974 32 .205 Parry et a l . , 1973 20 .320 Kitago et a l . , 1976 49 .301 Kita g o et a l . , 1976 18 .320 Simons, 1960 15 .330 Ladd, 1965 24 .270 Skempton et a l . , 1963 39 .280 Ladd, 1965 31 .308 Amerasinghe And Parry, 1975 26 .300 Ladd and Foot, 1974 21 .200 Ladd and Foot, 1974 41 .270 Ladd and Foot, 1974 75 .240 Ladd and Foot, 1974' 1 5 .230 Simon et a l . , 1 974 39 . 163 Ladd et a l . , 1974 24 .308 Eagn,l977 21 .400 M i t a c h i , 1976 1 53 21 .361 M i t a c h i et a l . , 1 976 32 .360 M i t a c h i et a l . , 1976 37 . 184 Sketchley et a l ., 1976 . 58 .338 Crooks et a l . , . 1974 20 .342 Gango'padhyay et a l . , 1974 53 .258 Donaghe et a l . , 1978 53 .281 Donaghe et a l . , 1978 36 .305 Donaghe et a l . , 1978 36 .320 Donaghe et a l . , 1978 1 6 .410 Poulos, 1978 22 .256 Henkel et a l . , 1 964 27 .210 Anderson et a l . , 1980 1 54 TABLE B-4. COMPRESSION/EXTENSION STRENGTH RATIO AND PLASTICITY INDEX" %PI W ° i c SUEK/°ric K 0 SUCl/ SUEK REFERENCE 25 .280 .230 1.0 1 .22 H e n k e l , 1960 25 .290 .240 1 .0 1.21 P a r r y , 1960 32 .215 .205 1.0 1 .05 P a r r y e t a l . , 1973 32 .205 .175 K 0 1.17 P a r r y e t a l . , 1973 1 5 .330 .165 K 0 2.00 L a d d , 1965 18 .268 .168 K 0 1 .59 V a i d e t a l . , 1974 290 .510 .510 1 .0 1 .00 Leon e t a l . , 1977 30 .490 .450 1.0 1 .09 Lade e t a l . , 1 976 49 .420 .370 1 .0 1.14 S h i b i t a e t a l . , 1965 24 .500 .330 1 .0 1 .52 Wu e t a l . , 1963 25 .430 .340 1 .0 1 .26 Broms e t a l . , 1965 49 .383 .382 1 .0 1 .00 K i t a g o e t a l . , 1976 49 .301 .197 K 0 1 .57 K i t a g o e t a l . , 1976 49 .400 .380 1.0 1 .05 K i t a g o e t a l . , 1976 4 9 .320 .210 K 0 1 .52 K i t a g o e t a l . , 1976 APPENDIX C FAILURE ZONES UNDER THE FOUNDATION EXTERNAL FOUNDATION PRESSURE(PSF) 1 . 250 2. 350 F i g u r e C-1. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; V a r i a b l e Strength; I s o t r o p i c Model EXTERNAL FOUNDATION PRESSURE(PSF) 1 . 250 2. 350 Figure C-2. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; V a r i a b l e Strength; A n i s o t r o p i c Model EXTERNAL FOUNDATION PRESSURE(PSF) 1 . 350 2. 380 Figure C-3. F a i l u r e Zones Under R i g i d F o o t i n g ; V a r i a b l e Strength; I s o t r o p i c Model co EXTERNAL 1 . 350 2. 380 FOUNDATION PRESSURE(PSF) Figu r e C-4. F a i l u r e Zones Under R i g i d F o o t i n g ; V a r i a b l e Strength; A n i s o t r o p i c Model EXTERNAL FOUNDATION PRESSURE(PSF) 1. 1800 2. 2000 F i g u r e C-5. F a i l u r e Zones Under R i g i d F o o t i n g ; Constant Strength; I s o t r o p i c Model CM O EXTERNAL FOUNDATION PRESSURE(PSF) 1. 1800 2 . 2000 F i g u r e C-6. F a i l u r e Zones Under R i g i d F o o t i n g ; Constant Strength; A n i s o t r o p i c Model EXTERNAL FOUNDATION PRESSURE(PSF) 1. 1800 2. 1950 Fi g u r e C-7. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; Constant Strength; I s o t r o p i c Model EXTERNAL FOUNDATION PRESSURE(PSF) T. 1800 2. 1950 Figu r e C-8. F a i l u r e Zones Under F l e x i b l e F o o t i n g ; Constant Strength; A n i s o t r o p i c Model APPENDIX D STABILITY ANALYSIS METHOD OF SWEDISH SLICES I b I * * F i g u r e D - 1. C i r c u l a r F a i l u r e Surface For e q u i l i b r i u m , W ^ o s a i = Nj + U ± N. = W .Cosa. - U. 1 1 i i F i s d e f i n e d as, F = Moment of shear s t r e n g t h along the f a i l u r e s u r f a c e Moment of the weight of the f a i l u r e s o i l mass n I R(c i+o- iTan0)Al i = i= j n I RW. Sine. i-1 1 n n E c . A l . + E (W.Coso.-u.)Tan0Al. . , 1 1 . , i i i i n E W.Sine. . i i 165 APPENDIX E SEMI EMPIRICAL APPROACH FOR CONSOLIDATED COMPRESSION STRENGTH E-1 THE CONCEPTS AND DERIVED FORMULAE In t h i s s e c t i o n a more conceptual approach i s u t i l i z e d in f i n d i n g a c o r r e l a t i o n of the form g(K 0,Index p r o p e r t i e s ) ; fo r S^/o-; . The concept which i s d e s c r i b e d below i s based UCK C on the normalized behaviour of i s o t r o p i c a l l y and a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y s . Along with t h i s concept and some e x i s t i n g e m p i r i c a l equations, S U C K/o-, c i s expressed as a f u n c t i o n of p l a s t i c i t y index and K 0, as opposed to the c o r r e l a t i o n developed i n s e c t i o n 5.3.2 where, K 0 was not c o n s i d e r e d as a v a r i a b l e . Consider the undrained s t r e s s p a t h s ( F i g u r e E-1) on q-p plane. Let OP be the f a i l u r e l i n e . ABC i s an undrained s t r e s s path s t a r t i n g from an i s o t r o p i c a l l y c o n s o l i d a t e d s t a t e A. The po i n t B i n d i c a t e s an a n i s o t r o p i c a l l y c o n s o l i d a t e d ( K 0 ) s t a t e . Now, c o n s i d e r an a n i s o t r o p i c a l l y c o n s o l i d a t e d undrained t e s t s t a r t i n g from the i n i t i a l c o n d i t i o n B. U s u a l l y t h i s s t r e s s path does not f o l l o w the s t r e s s path obtained f o r the i s o t r o p i c a l l y c o n s o l i d a t e d t e s t . Instead, i t f o l l o w s a s t r e s s path s i m i l a r to BD as shown i n F i g u r e E-1. However, the a c t u a l s t r e s s path i s not important i n t h i s d e r i v a t i o n , only the f a i l u r e s t r e s s s t a t e i s important. The normally c o n s o l i d a t e d c l a y s show normalized behaviour with respect to s t r e n g t h f o r a given K 0 i n i t i a l c o n d i t i o n . T h e r e f o r e , s t r e s s paths s t a r t i n g from A, and B, F i g u r e E-1. Undrained S t r e s s Paths of I s o t r o p i c a l l y and A n i s o t r o p i c a l l y C o n s o l i d a t e d C l a y s CM CM 1 6 7 must be s i m i l a r to the s t r e s s paths ABC and BD r e s p e c t i v e l y . T h e r e f o r e , the s t r a i g h t l i n e s BD and B,D, are p a r a l l e l , AC and A,C! are p a r a l l e l too. Hence f o r t h i s s o i l , f o r a given i n i t i a l s t r e s s s t a t e X , ( F i g u r e E-2) under K 0 c o n s o l i d a t e d c o n d i t i o n s ( a s B or B,) the f a i l u r e s t a t e can be obtained by drawing a l i n e ' p a r a l l e l to BD(or B T D , ) a cross X to meet the f a i l u r e l i n e at Y ( F i g u r e E-2). Then, Y represents the f a i l u r e s t r e s s s t a t e . Or i f the angle 6 ( F i g u r e E-2) i s known f o r a given K 0, one can determine the f a i l u r e s t r e s s s t a t e , p r o v i d e d the i n i t i a l s t r e s s s t a t e i s known. That i s , the p o i n t Y can be l o c a t e d by drawing a l i n e i n c l i n e d at an angle 6 to BD(or B T D , ) a cross X to meet the f a i l u r e l i n e . The angle 6 i s a measure of the d e v i a t i o n of a n i s o t r o p i c s t r e s s path from the i s o t r o p i c s t r e s s path. T h e r e f o r e , 6 can be a f u n c t i o n of K 0. But K 0 i s a f u n c t i o n of e 0 ( T a n e 0 = ( 1 -K 0 ) / ( 1 + K 0 ) ) and thus, 6 can be a f u n c t i o n of 6 0 . A set of t r i a x i a l compression data has been a n a l y s e d ( T a b l e E - 1 ) i n order to o b t a i n a r e l a t i o n s h i p between 6 and 9 0 . The values 6 /e 0 are p l o t t e d a g a i n s t p l a s t i c i t y index i n F i g u r e E-3. The value of 6/e 0 f o r d i f f e r e n t c l a y s i s approximately 2 and appears to be independent of the p l a s t i c i t y index. Using the above f i n d i n g s the e x p r e s s i o n given below can be d e r i v e d ( s e c t i o n E-2) f o r the normalized undrained s t r e n g t h SUCK-A^C of a K 0 c o n s o l i d a t e d c l a y , interms of R 0, S and p l a s t i c i t y index. q 169 TABLE E-1. PLASTICITY INDEX AND 6/0 o %PI UCI c K 0 6 / G 0 REFERENCE 18 .262 .56 .268 1.11 V a i d et a l . , 1 974 32 .215 .54 .205 1 .57 Parry et a l . , 1973 20 .400 .49 .320 1 .73 Kitago et a l . , 1976 49 .383 .58 .301 1 .74 Kita g o et a l . , 1976 18 .350 .47 .320 1.12 Simons, 1960 1 5 .440 .54 .330 2.10 Ladd, 1965 24 .320 .61 .270 1 .92 Skempton And Sowa, 1963 39 .280 .54 .280 1 .03 Ladd, 1965 a rsi I 1 3 D O <r> o | a a a CD CD 1 1 — T 1 I I 0 0 10 0 20.0 30.0 40.0 50.0 60.0 PLASTICITY INDEX U ) F i g u r e E - 3 . V a r i a t i o n of 6/© 0 with P l a s t i c i t y Index o 171 1-Kp + Tan(e-6) SjjcjL = 1 +K n (1+K 0) o-;c 1 + Tan(6-6) 2 Sin0 Where, 6 = Cot" 1 (o-^/S U C I-Cosec0) 6 = 2Tan" 1{(1"K 0)/(1+K 0)} Sin0 = ,8064-.2286Log 1 0(PI) S i s the undrained s t r e n g t h i n compression under i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l c o n d i t i o n s SUCK i s the undrained s t r e n g t h i n compression under a n i s o t r o p i c a l l y c o n s o l i d a t e d t r i a x i a l c o n d i t i o n s cr\ c i s the major e f f e c t i v e c o n s o l i d a t i o n p ressure K 0 i s the e f f e c t i v e c o n s o l i d a t i o n s t r e s s r a t i o o-'3r/a,\c Using the above formula, to estimate the normalized undrained s t r e n g t h i n compression of a n i s o t r o p i c a l l y c o n s o l i d a t e d c l a y , R 0, S U C I and p l a s t i c i t y index must be known. But f o r a given p l a s t i c i t y index, S can be estimated from F i g u r e 5-5. T h e r e f o r e f o r given p l a s t i c i t y index and K 0, S U C K / o - ' l c can be estimated. The curves i n F i g u r e E-4 show these estimates i n a g r a p h i c a l form. These curves can be u t i l i z e d to estimate the undrained s t r e n g t h i n compression, of normally c o n s o l i d a t e d c l a y s with given p l a s t i c i t y index and K 0 c o n d i t i o n s . 1 7 2 co i f f rs/ o a 4 n 1 1 1 1 0.0 20.0 40.0 60.0 80.0 100.0 P L A S T I C I T Y INDEXm F i g u r e E-4. S U C K / S u c i v s P l a s t i c i t y Index Curves f o r D i f f e r e n t K 0 V a l u e s 173 E-2 MATHEMATICAL DERIVATIONS Refer F i g u r e . OA i s the f a i l u r e l i n e P i s the i n i t i a l s t a t e PQ i s the s t r e s s path PQ' i s the s t r e s s path It should be noted t h a t , i n t h i s case a l l the s t r e s s paths have been approximated by s t r a i g h t l i n e s . T h i s i s reasonable because, i n i t i a l s t r e s s s t a t e and the f i n a l f a i l u r e s t a t e are of only concern i n t h i s a n a l y s i s . S U C I i s S U C K i s 0"lC i s K 0 i s SUCK " ( 1 KQ ) 2 = Tana 2 m where, m = Sin0 2 S U C K - (1~K 0) Tana = o-'1r(1+K0)~ (1+K Q) 1 " , 2ST 1 (-K m ( 1 +K.o ) o '1C 2SUCK 1 [ 1 + T a n g ] = Tana + (1-K 0) cr'ic ( 1 +K 0) m (1+K 0) Tana + (1-K 0) 2SUCK = (1+Kp) (1+K 0) cr'ic 1 + Tana 2 m But, from experimental r e s u l t s 6 = 29 0 F i g u r e E-5. S i m p l i f i e d S t r e s s Paths 175 where, 9 0 = T a n " 1 ( 1 - K 0 ) ( 1 + K 0 ) But, Tano = S T 1 P T °"c " S u c I m Tana = 1 _JTc " 1 S U C I m 1-K 0 + Tan(G-6) Snrv = 1+ K N ( 1 + K 0 ) cr'1c 1 + Tan (e-6) 2 Sin0 where, 9 = C o f 1 [ 1 - Cosec0] Suci 6 = 2 T a n " 1 ( 1 - K p ) (1+KQ) Sin0 = .8064 - .2286Log 1 0(PI) 176 APPENDIX F DESCRIPTION OF INPUT DATA The f o l l o w i n g cards d e s c r i b e d i n 1(next page) must be changed as g i v e n below. 4. M a t e r i a l p r o p e r t y c a r d s . ( r e f e r 5. f o r f a c i l i t y to have o r d i n a r y NLSSIP) (b ) . (I5,6F10.0/4F10.0) F i r s t c a r d 1-5 M = M a t e r i a l type number 6-15 EMPR(M,1) 16-25 EMPR(M,2) Un i t weight that g i v e s e f f e c t i v e s t r e s s Ei/° r' 1c» i f % i s computed i n the prog. E±; i f E± i s read d i r e c t l y 26-35 EMPR(M,3) = F a i l u r e d e v i a t o r s t r e s s i n comp./cr'1c i f f a i l u r e s t r e n g t h i s computed i n the prog. = F a i l u r e d e v i a t o r s t r e s s i n comp. i f f a i l u r e s t r e n g t h i s read d i r e c t l y 36-45 EMPR(M,4) 46-55 EMPR(M,5) = F a i l u r e r a t i o R f = F a i l u r e d e v i a t o r s t r e s s i n ext./o-' 1 c i f f a i l u r e s t r e n g t h i s computed i n the prog. = F a i l u r e d e v i a t o r s t r e s s i n e x t . i f f a i l u r e s t r e n g t h i s read d i r e c t l y 56-65 EMPR(M,6) = Second c a r d 1-10 EMPR(M,7) = 1 i f Ei i s computed i n the prog. 0 i f E± i s read d i r e c t l y 1 i f the f a i l u r e strengths(comp. and ext.) are computed i n the prog. 0 i f the f a i l u r e strengths(comp. and ext.) are read d i r e c t l y 11-20 EMPR(M, 8) and 21-30 EMPR(M,9) 11-20 EMPR(M,8) and 21-30 EMPR(M,9) = 1 i f the i n i t i a l d e v i a t o r s t r e s s = 1 i s computed from the p r e - e x i s t i n g s t r e s s e s = I n i t i a l d e v i a t o r s t r e s s i f i n i t i a l d e v i a t o r = 0 s t r e s s i s read d i r e c t l y 31-40 EMPR(M,10)= E a r t h p r e s s u r e c o e f f i c i e n t i n the foundation or i n s o i l l a y e r s to be p l a c e d , K 0 177 5. S o i l element cards(6I5) One card f o r each element 1- 5 N = Element number 6- 10 INP(N,2) = Number of nodal p o i n t I 1 1 -15 INP(N,3) = Number of nodal p o i n t J 16 -20 INP(N,4) = Number of nodal p o i n t K 21 -25 INP(N,5) = Number of nodal p o i n t L 26 -30 INP(N,6) = Material- number 31 -35 INP(N,6) = 0 f o r m o d i f i e d NLSSIP = 1 f o r o r d i n a r y NLSSIP; i n t h i s case, s e c t i o n 4.(b) must be input a c c o r d i n g t o ( l ) 1 Byrne, P. M., and Duncan, J . M., "NLSSIP; A Computer Programme fo r N onlinear A n a l y s i s of S o i l - S t r u c t u r e I n t e r a c t i o n Problems" S o i l Mechanics S e r i e s No. 41, U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada, J u l y , 1979. 

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