A TWO-DIMENSIONAL NON-LINEAR STATIC AND DYNAMIC RESPONSE ANALYSIS OF SOIL STRUCTURES By RAJARATNAM B.Sc. SIDDHARTHAN The U n i v e r s i t y of S r i - L a n k a , Peradeniya Campus, 1977 M.A.Sc. The U n i v e r s i t y of B r i t i s h Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF CIVIL ENGINEERING 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 MARCH 1984 (c) Rajaratnam Siddharthan, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of requirements f o r an advanced degree a t the the University o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be department o r by h i s or her granted by the head o f representatives. my It is understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed w i t h o u t my permission. Department o f C l y\/^ The U n i v e r s i t y o f B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3 Date C^ -^ \ < v ) Columbia r^Lfi-fi written ii ABSTRACT A method dynamic response has of of been presented The incrementally behaviour structures, and dynamic behaviour elastic in of was as the two assumed to and normal s t r e s s was assumed to be Slip elements contact analyses soil unloading or including represent the interface elements. The properties has static and be performed in been modelled Non-linear by be hyperbolic using Responses non-linear, elastic and incorporated in been characteristics the slip between to material with changes stress the soil i n mean dependent. g i v e n by the to structural elements were assumed to be Interface Masing analysis and an tangent The coupled reloading. have of parameters. either elastic, Mohr-Coulomb criterion. In the static analysis once f o r the completed modelled layer the predict soil-structure interaction can "elastic" p e r f e c t l y p l a s t i c , w i t h f a i l u r e at the failure to approach i n which tangent shear modulus and taken shear during two-dimensions s t r e s s mode or a combination of both modes. modulus were response in herein. e f f e c t i v e or t o t a l bulk soil static stress-strain analysis by static soil proposed h e r e , g r a v i t y may structure analysis. analysis give the The or the be construction switched on at sequence can be determined by s t r e s s - s t r a i n conditions in-situ stress condition before the dynamic analysis. In the dynamic e f f e c t i v e s t r e s s a n a l y s i s , the pressures are calculated Martin, et al. modified for the using (1975). effects of a The modification of parameters, residual porewater the G m a x r e s i d u a l porewater model proposed and pressure. _ The a x by are dynamic iii response study i n c l u d e s the p r e d i c t i o n of post earthquake The verified two p r e d i c t i v e c a p a b i l i t y of the new by comparing the r e c o r d e d porewater c e n t r i f u g e d models computed by the new This offshore Results suggest i n t e r a c t i o n may type of computing reported has island that been p r e s s u r e and a c c e l e r a t i o n s of simulated earthquakes, to the also been supporting common used a practice to compute tanker of response mounted those neglecting response At of present these one-dimensional islands. to asses the v a l i d i t y of t h i s methods Comparative procedure. of drilling an rig. soil-structure not be a p p r o p r i a t e f o r i s l a n d s which support heavy structures. the to method of a n a l y s i s has method. method drilling subjected deformations. are tanker used s t u d i e s are for also iv TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS iv LIST OF FIGURES viii LIST OF TABLES xiv NOMENCLATURE xv ACKNOWLEDGEMENTS ix CHAPTER 1 - INTRODUCTION 1 1.1 - SCOPE OF THIS THESIS 4 1.2 - ORGANIZATION OF THE THESIS 6 CHAPTER 2 - CRITICAL REVIEW OF SEED, ET AL. METHOD FOR COMPUTING DYNAMIC DEFORMATIONS CHAPTER 3 - GENERAL GUIDELINES FOR DYNAMIC ANALYSES 3.1 - ONE-DIMENSIONAL RESPONSE ANALYSIS 7 14 BY FINN, ET AL. (1977) 16 3.1.1 - Shear S t r e s s - S t r a i n R e l a t i o n s h i p 17 3.1.2 - Porewater P r e s s u r e Model 21 3.1.3 - M o d i f i c a t i o n of P r o p e r t i e s f o r Residual Porewater P r e s s u r e 22 3.1.4 - I n f l u e n c e of S t r a i n Hardening 25 3.1.5 - D i s s i p a t i o n of Porewater P r e s s u r e 26 3.2 - LABORATORY VERIFICATION OF EFFECTIVE STRESS RESPONSE ANALYSIS 3.3 - 27 FIELD VERIFICATION OF EFFECTIVE STRESS REPONSE ANALYSIS 30 V PAGE 3.4 - POREWATER PRESSURE MODEL IN PRACTICE 33 3.5 - DISCUSSION 36 CHAPTER 4 - TWO-DIMENSIONAL STATIC ANALYSIS OF SOIL STRUCTURES 38 38 4.1 - STRESS-STRAIN RELATIONSHIP 4.1.1 - Reasons f o r S e l e c t i n g G 4.1.2 - H y p e r b o l i c Shear S t r e s s - S t r a i n t and B t Relationship 39 40 4.1.2.1 - E s t i m a t i o n of G m a v 40 4.1.2.2 - E s t i m a t i o n of x m 43 4.1.2.3 - I n f l u e n c e of O v e r - C o n s o l i d a t i o n 45 4.1.2.4 - E f f e c t s of Unloading 45 4.1.3 - Tangent Bulk Modulus B 4.2 - PHYSICAL MODELLING 4.3 - SIMULATION OF CONSTRUCTION 4.3.1 - Method of A n a l y s i s 50 4.3.2 - Incremental 50 4.3.3 - Computation o f Incremental and a x 47 t 48 SEQUENCE Porewater P r e s s u r e 49 Stresses Strains 52 4.4 - SHEAR-VOLUME COUPLING 55 4.5 - INTERFACE REPRESENTATION 63 4.5.1 - Method of A n a l y s i s o f S l i p Element 68 4.6 - SELECTION OF SOIL PARAMETERS 70 4.6.1 - Obtaining Shear S t r e s s - S t r a i n Parameters 4.6.2 - Obtaining 71 Bulk Modulus Parameters 72 vi PAGE CHAPTER 5 - TWO-DIMENSIONAL 5.1 - FORMULATION OF THE PROBLEM 73 5.1.1 - Incremental 74 5.1.2 - C o r r e c t i o n Forces 76 5.2 - DYNAMIC STRESS-STRAIN RELATIONSHIP 78 5.2.1 - Volume Change Behaviour 80 5.2.2 - Dynamic Shear S t r e s s - S t r a i n Behaviour 82 5.2.2.1 - S k e l e t o n Curve f o r Dynamic Loading 82 5.2.2.2 - Unloading 83 5.2.3 M o d e l l i n g the E f f e c t s of R e s i d u a l - DYNAMIC ANALYSIS E q u i l i b r i u m Equations and R e l o a d i n g Porewater P r e s s u r e 5.2.3.1 - The Behaviour 5.2.3.2 - R e s i d u a l Porewater 73 85 of Samples w i t h T s 86 Pressure G e n e r a t i o n Model 95 99 5.3 - DAMPING MATRIX [ C j 5.4 - BOUNDARY CONDITIONS 102 5.5 - ANALYSIS OF SOIL STRUCTURE SYSTEMS 104 5.5.1 - S l i p Element i n Dynamic A n a l y s i s 104 5.6 - SOLUTION SCHEME 105 5.7 - COMPUTATION OF POST EARTHQUAKE DEFORMATION 106 CHAPTER 6 - VERIFICATION OF THE METHOD OF ANALYSIS 108 6.1 - CAMBRIDGE CENTRIFUGE TESTS 108 6.2 - COMPARATIVE STUDY I l l 6.2.1 - R e s u l t s of T e s t 1 115 6.2.2 - R e s u l t s of T e s t 2 123 vii PAGE 6.3 - A P P L I C A B I L I T Y OF THE METHOD OF A N A L Y S I S CHAPTER 7 - A P P L I C A T I O N OF THE METHOD OF A N A L Y S I S : 132 TANKER ISLAND RESPONSE 137 7.1 - INTRODUCTION 137 7.2 - A N A L Y S I S OF A T Y P I C A L TANKER ISLAND 138 7.2.1 - Results 142 7.3 - SOME PRACTICAL CONCLUSIONS 151 CHAPTER 8 - SUMMARY AND CONCLUSIONS 153 8.1 - SUMMARY 153 8.2 - CONCLUSIONS 154 8.3 - SUGGESTIONS FOR FURTHER STUDY 156 REFERENCES ' of Tanker I s l a n d Problem 157 APPENDIX I 166 APPENDIX I I 178 APPENDIX I I I 185 viii LIST OF FIGURES PAGE FIGURE 1.1 - Caisson-Retained Island 3 (De Jong and Bruce, 1978) FIGURE 2.1 - C o n v e r s i o n of Shear S t r a i n P o t e n t i a l t o E q u i v a l e n t Shear F o r c e s 10 FIGURE 3.1(a) - Initial 19 FIGURE 3.1(b) - Masing S t r e s s - S t r a i n Curves f o r U n l o a d i n g FIGURE 3.2 - FIGURE 3.3(a) - Loading Curve and R e l o a d i n g 19 Hysteretic Characteristics 20 I n i t i a l E f f e c t i v e Stress Condition i n a Simple Shear Apparatus FIGURE 3.3(b) - Intermediate E f f e c t i v e Stress Condition i n a FIGURE 3.4 - 23 Simple Shear Apparatus 23 R e l a t i o n s h i p Between V o l u m e t r i c S t r a i n s and Porewater P r e s s u r e s i n Constant S t r a i n Cyclic Simple Shear T e s t s , D_ = 45% FIGURE 3.5 - C y c l i c S t r e s s R a t i o Vs. Number of C y c l e f o r Initial FIGURE 3.6 - 29 L i q u e f a c t i o n f o r V a r i o u s Sands P r e d i c t e d and Measured 31 Porewater P r e s s u r e s i n Constant S t r e s s C y c l i c Simple Shear Tests, D r = 45% FIGURE 3.7 - Measured and Computed Ground A c c e l e r a t i o n s FIGURE 3.8 - Measured and Computed Porewater P r e s s u r e s at FIGURE 4.1 - 32 34 a Depth of 6m 34 Element S t r e s s e s 44 ix LIST OF FIGURES PAGE FIGURE 4.2 - Mohr C i r c l e Diagram 44 FIGURE 4.3 - Mohr Envelope 46 FIGURE 4.4 - Effects 46 FIGURE 4.5 - Sequential FIGURE 4.6 - Schematic of Loading and Unloading Procedure i n Dam C o n s t r u c t i o n Diagram Showing the 51 Incremental A n a l y t i c a l Procedure FIGURE 4 . 7 ( a ) - Typical Plots Loose FIGURE 4 . 7 ( b ) - for 53 vs f o r Dense and Sands 56 Typical Plots for y vs f o r Dense and .Loose Sands FIGURE 4.8 - 56 S t r e s s - S t r a i n Behaviour of Ottawa Sand i n Drained Simple Shear FIGURE 4.9 - (a),(b) ( A f t e r V a i d et a l . Dense Sacramento R i v e r Sand P r i n c i p a l S t r e s s R a t i o vs A x i a l b) V o l u m e t r i c S t r a i n vs A x i a l (After Lee, - 58 T y p i c a l Drained T r i a x i a l Test R e s u l t s on a) FIGURE 4.10 1981).... Strain Strain 59 59 1965) V a r i a t i o n of D i l a t i o n Angle w i t h Mean Normal S t r e s s f o r V a r i o u s Sands Robertson, (After 1982) 60 FIGURE 4.11 - Idealised Relationship FIGURE 4.12 - D e f i n i t i o n of FIGURE 4.13 - P l o t of T y p i c a l Between ^ and 62 S l i p Element Shear/Normal S t r e s s vs Normal Displacement 65 Shear/ 65 X L I S T OF FIGURES PAGE FIGURE 5.1 - Dynamic Shear Stress-Strain Relationship: S k e l e t o n Curve FIGURE 5.2 - Dynamic Shear 84 Stress-Strain Relationship: U n l o a d i n g and R e l o a d i n g FIGURE 5.3 FIGURE 5.4 - - 84 Permanent and C y c l i c Strains and C y c l i c Porewater Pressures C y c l i c and R e s i d u a l and A x i a l p' Strain Plot for or Permanent 88 Behaviour of Pore P r e s s u r e for ICT and ACT T e s t s 88 FIGURE 5.5 - q vs ICT and ACT T e s t s 91 FIGURE 5.6 - Normalized Curves f o r Pore P r e s s u r e B u i l d - u p Equation FIGURE 5.7 - Stregth 94 Curves f o r D„ = 85%; (After FIGURE 6.1 - (b) E k o f i s k Sands w i t h Sand w i t h D_ = 77% Rahman, et Submerged (a) Island al. 94 1977) Showing Transducer P o s i t i o n s ... 109 FIGURE 6 . 2 ( a ) - Input Base Motion i n Test 1 112 FIGURE 6.2(b) - Input Base Motion i n Test 2 112 FIGURE 6.3 L i q u e f a c t i o n R e s i s t a n c e Curves of Medium - Dense L e i g h t o n - B u z z a r d Sand FIGURE 6 . 4 ( a ) - Recorded A c c e l e r a t i o n of ACC1244 i n Test (With and Without S l i p FIGURE 6 . 4 ( b ) - FIGURE 6 . 5 ( a ) - 1 116 1 116 1 117 Elements) Computed A c c e l e r a t i o n of ACC1244 i n Test (With and Without S l i p 114 Elements) Recorded A c c e l e r a t i o n of ACC1225 i n Test xi LIST OF FIGURES PAGE FIGURE 6 . 5 ( b ) - Computed A c c e l e r a t i o n of ACC1225 i n Test (With and Without S l i p FIGURE 6 . 6 ( a ) - FIGURE 6 . 6 ( b ) - Recorded a c c e l e r a t i o n - 118 Computed A c c e l e r a t i o n of ACC734 i n Test 1 118 PPT68 i n Test Pressure 1 120 P r e s s u r e of 1 and Computed Porewater PPT2342 i n Test FIGURE 6 . 8 ( a ) 120 Recorded and Computed Porewater recorded P r e s s u r e of 1 Recorded and Computed Porewater PPT2338 i n Test FIGURE 6 . 7 ( d ) - Elements) Recorded and Computed Porewater of FIGURE 6 . 7 ( c ) - Elements) 1 PPT2330 i n Test FIGURE 6 . 7 ( b ) - 117 of ACC734 i n Test (With and Without S l i p FIGURE 6.7(a) 1 121 P r e s s u r e of 1 121 - Recorded A c c e l e r a t i o n of ACC1244 i n Test 2 125 FIGURE 6 . 8 ( b ) - Computed A c c e l e r a t i o n of ACC1244 i n Test 2 125 (With S l i p Elements) FIGURE 6 . 9 ( a ) - Recorded A c c e l e r a t i o n of ACC1225 i n Test 2 126 FIGURE 6 . 9 ( b ) - Computed A c c e l e r a t i o n of ACC1225 i n Test 2 126 (With S l i p Elements) FIGURE 6 . 1 0 ( a ) - Recorded A c c e l e r a t i o n of ACC734 i n Test 2 127 FIGURE 6 . 1 0 ( b ) - Computed A c c e l e r a t i o n of ACC734 i n Test 2 127 (With S l i p FIGURE 6 . 1 1 ( a ) - Elements) Recorded and Computed Porewater PPT2330 i n Test 2 Pressure of 130 xii LIST OF FIGURES PAGE FIGURE 6.11(b)- Recorded and Computed Porewater P r e s s u r e of PPT68 i n T e s t 2 FIGURE 6 . 1 1 ( c ) - 130 Recorded and Computed Porewater P r e s s u r e of PPT2338 i n T e s t 2 FIGURE 6.11(d)- 131 Recorded and Computed Porewater P r e s s u r e o f PPT2342 i n T e s t 2 131 FIGURE 6.12(a)- E f f e c t i v e S t r e s s Paths i n T e s t 1 135 FIGURE 6.12(b)- E f f e c t i v e S t r e s s Paths i n T e s t 2 136 FIGURE 7.1 - Schematic o f Tanker I s l a n d 139 FIGURE 7.2 - D i s t r i b u t i o n of Maximum Dynamic Shear R a t i o i n Sand FIGURE 7.3 - 143 D i s t r i b u t i o n o f Maximum Dynamic Shear Stress R a t i o i n Sand FIGURE 7.4 - 143 D i s t r i b u t i o n of R e s i d u a l Porewater P r e s s u r e R a t i o i n Sand FIGURE 7.5 - 145 D i s t r i b u t i o n of Maximum Dynamic Shear Strain i n Sand 145 FIGURE 7.6 - D i s t r i b u t i o n of Maximum Dynamic Displacement .... 147 FIGURE 7.7 - Post Earthquake X and Y Displacements FIGURE 7.8 - A c c e l e r a t i o n Response Spectrum f o r the M o t i o n at Berm S u r f a c e FIGURE 7.9 FIGURE A l . l - - 148 150 D i s t r i b u t i o n of R e s i d u a l Porewater P r e s s u r e R a t i o i n Sand 150 I s o - P a r a m e t r i c Element 167 xiii LIST OF FIGURES PAGE FIGURE A2.1 - S l i p Element 179 xiv LIST OF TABLES PAGE TABLE 4.1 V a r i a t i o n of Exponent r\, w i t h Plastic Index PI 42 TABLE 6.1 S o i l Properties 113 TABLE 6.2 Recorded and Computed Maximum Accelerations TABLE 6.3 119 Recorded and Computed Maximum R e s i d u a l Porewater P r e s s u r e s TABLE 6.4 124 Recorded and Computed Maximum Accelerations TABLE 6.5. 129 Recorded and Computed Maximum R e s i d u a l Porewater P r e s s u r e s 133 138 TABLE 7.1 Static TABLE 7.2 Dynamic S o i l P r o p e r t i e s TABLE 7.3 ^ ^ vo a s S o i l Properties a n d K r V a l u e s 140 1 4 1 xv NOMENCLATURE A = Variables defined i n s l i p element s t i f f n e s s matrix f o r m u l a t i o n i n Appendix I I . a = Constant used to compute damping m a t r i x , a^ - a^ = Constants d e f i n e d i n t e x t . B = V a r i a b l e d e f i n e d i n s l i p element s t i f f n e s s matrix f o r m u l a t i o n i n Appendix I I . B = Strain-displacement matrix. = Tangent b u l k modulus. b = Constant b^ - b^ = Constants d e f i n e d i n t e x t . [C] = Damping m a t r i x . C_ = I n t e r p o l a t i o n m a t r i x i n s l i p element s t i f f n e s s B t used to compute damping m a t r i x , matrix formulation. C^ - C^ = Volume change c o n s t a n t s . C = Cohesion C' = Effective I) = E l a s t i c i t y matrix. D = R e l a t i v e d e n s i t y of s o i l . = Depth of c e n t r e of g r a v i t y of an element below the top g r d is slip element. Cohesion. surface. {E} = M a t r i x d e f i n e d i n Appendix I I I . E = Young's modulus. E_ = One-dimensional e = Void t ratio. rebound modulus, NOMENCLATURE Maximum v o i d ratio. Minimum v o i d ratio. M a t r i x d e f i n e d i n Appendix I I I . Column v e c t o r of g l o b a l i n e r t i a f o r c e s a t time t Column v e c t o r o f g l o b a l s p r i n g f o r c e s a t time t . Column v e c t o r of element s p r i n g f o r c e s a t time t Shear s t r e n g t h of a s l i p element. Shear f o r c e per u n i t area i n a s l i p element. Shear f o r c e per u n i t area i n a s l i p element. Tangent shear modulus f o r l o a d i n g and u n l o a d i n g . Tangent shear modulus. Maximum shear Hardening modulus. constants. Unit vector. M a t r i x d e f i n e d i n Appendix I . Jacobian matrix. Non-dimensional shear modulus c o n s t a n t . Bulk modulus constant. Anistropic consolidation ratio. Normal j o i n t stiffness/unit area. Experimental constant used i n porewater pressure model. Shear j o i n t Slip stiffness element s t i f f n e s s G l o b a l tangent unit area. m a t r i x i n element stiffness matrix. axis. xvii NOMENCLATURE [K ] = S l i p element s t i f f n e s s m a t r i x i n xy a x i s . K = C o - e f f i c i e n t of l a t e r a l e a r t h p r e s s u r e a t r e s t . = Shear modulus c o n s t a n t used K = Constant * [K ] = Porewater p r e s s u r e m a t r i x . K = Shear modulus c o n s t a n t f o r c l a y . = P e r m i a b i l i t y of s o i l , L = Length H = D i s t a n c e of a p o i n t of s l i p x y Q (K ) 2 K m a x z to c a l c u l a t e G_ . ax which depends on the type of m a t e r i a l . J of a s l i p element. element i n the element direction. [M] = Mass m a t r i x . m = Experimental N = S c a l e f a c t o r o f the c e n t r i f u g a l [N] = Interpolation N^ = Number of c y c l e s to cause i n i t i a l N = Number o f c y c l e s to cause a porewater p r e s s u r e r a t i o o f 5 Q constant used i n porewater p r e s s u r e model. acceleration. matrix. liquefaction. 50%. n = Experimental n = Bulk modulus exponent. OCR = Over c o n s o l i d a t i o n r a t i o . {P} = G l o b a l column v e c t o r of a p p l i e d l o a d s . P = Atomospheric = Correction force vector. Q = Transformation matrix i n s l i p p' = Average e f f e c t i v e p r i n c i p l e fl {P 1 P } corr-" constant used i n porewater p r e s s u r e model. pressure. element, stress. xviii NOMENCLATURE 0^,0^ = Constant m a t r i c e s d e f i n e d i n the q = Principle stress R^ ,R_2 = Constant m a t r i c e s d e f i n e d i n the T_ = Transformation matrix. t = time. U = R e s i d u a l porewater = Maximum r e s i d u a l porewater = Column v e c t o r of i n c r e m e n t a l porewater U max {U} text. difference. text. pressure. r pressure, pressures i n elements. {TJ} = Displacement u^ = T a n g e n t i a l displacement {u } = Element porewater V = Volume of an v^ = Normal displacement Wg = E x t e r n a l work done. Wjjy = I n t e r n a l work done. w = Normal displacement = Shear dispalcement = Column v e c t o r s whose components are Q X n w s {X},{X},{X} vector. of node i . pressure v e c t o r . element. of node i . in a slip in a slip element. element. relative d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n of nodes. X^ = Base a c c e l e r a t i o n , x = Horizontal distance, z = Vertical distance. ix X ACKNOWLEDGEMENTS I would research like to extend my supervisor, interest and for Professor making p r e s e n t a t i o n of t h i s I R.G. am being many indebted and A.T. constructive suggestions. for W.D. Liam F i n n , valuable a p p r e c i a t i o n to f o r h i s guidance and suggestions to my keen improve the thesis. greatly Campanella, s i n c e r e g r a t i t u d e and to Bui Professors, for reading the Thanks are a l s o due always a v a i l a b l e to d i s c u s s any P.M. Byrne, thesis V.P. Vaid, for making and to P r o f e s s o r D.L. problem and Anderson, for his useful and stimulating discussions. Special data and test data British to thanks Mr. to a are Scott form also due to Mr. F.H. Mr. Richard Steedman and compatible with the Lee, f o r use Dean f o r computer at the of his test converting University the of Columbia. The Research financial Council of Government of and Production Exxon support of the Natural Canada, Department of Sciences Energy Canada, E a r t h Technology C o r p o r a t i o n , Research and Mines Engineering and Resources, Long Beach California Texas, is gratefully w i f e V i s h a , f o r her p a t i e n c e , sacrifices Company, Houstan, acknowledged. Finally, and for the I wish to thank my excellent job have seen t h i s day without I continued also remain this thesis. her whole-hearted indebted moral s u p p o r t . that I have completed i n typing to I t was t h i s work. my due parents to t h e i r This t h e s i s would not good and support. for their b l e s s i n g s and wishes encouragement 1 CHAPTER 1 INTRODUCTION In performance evaluated safety. feet engineering of i n terms structures on the middle the f u n c t i o n a l two-dimensional The vary and into two broad and that aspects the should be of f a c t o r s of a few i n c h e s to many of the s t r u c t u r e s categories: methods only. e t a l . (1966), considered. These methods one-dimensional assume that methods and properties deformations are e i t h e r The methods proposed assume that failure the method proposed Often one-dimensional soil develops of zoned be along (1965) well- b l o c k s of s o i l s . o f the s o i l d e p o s i t . deposits must c o n s t a n t or by I a i and F i n n (1982) f o r l o n g s l o p e s cannot be characterized d e p o s i t s , and the v a r i a b i l i t y three-dimensions occur i n by Newmark f a i l u r e planes and compute d i s p l a c e m e n t s of r i g i d accounts f o r the f l e x i b i l i t y considered. For adequately as of p r o p e r t i e s i n two or even example, in the response dams, two or t h r e e - d i m e n s i o n a l a n a l y s e s a r e e s s e n t i a l . When the t h i r d dimension and i n terms may v a r y from the m a t e r i a l to the planes More r e c e n t l y analyses than loading s t r u c t u r e s have been proposed. one-dimensional Goodman, defined rather that methods. planes normal to s e i s m i c agreed s i x t i e s many a n a l y t i c a l methods f o r a s s e s s i n g earthquake be c l a s s i f i e d parallel subjected of d e f o r m a t i o n s induced deformations i n s o i l may i t i s generally The a l l o w a b l e d i s p l a c e m e n t s depending Since soil practice, the p r o p e r t i e s do i s v e r y much l a r g e r than the o t h e r two-dimensions not vary significantly in this direction, a 2 two-dimensional the modes of analysis. response analysis deformations may i s usually also adequate. dictate two The or geometry three-dimensional For example i n the a n a l y s i s of embedded s t r u c t u r e s r o c k i n g may be an important d e f o r m a t i o n mode i n a d d i t i o n to t r a n s l a t i o n and at of l e a s t a two-dimensional There are a a n a l y s i s i s necessary. number of compute s e i s m i c d e f o r m a t i o n s . plastic soil behaviour 1979). These methods validation. The method two-dimensional methods available Some of these methods are based on (Finn, are therefore, et a l . 1973; complicated proposed Mroz, to use by Seed, and et et elastic- a l . 1979; have had to Prevost, very limited to compute a l . (1973), s e i s m i c deformations of e a r t h dams has found wide a p p l i c a t i o n i n p r a c t i c e . This i s a s e m i - a n a l y t i c a l method and data from displacements using cyclic with the computed In it is for gas drilling are and the tests Non-linearity elastic approach are of computing design to soil estimate i s taken analysis potential into properties account compatible to determine deformations man-made sand islands on e i t h e r which A earthquake support Beaufort sand-filled consideration. under Sea. drilling These loading. platforms islands c a i s s o n s or t a n k e r s ( F i g . 1.1). i s l a n d s d u r i n g earthquakes general satisfactory carry a r e an method for d e f o r m a t i o n s of these s t r u c t u r e s i s not a v a i l a b l e at p r e s e n t . Indeed recently the of s t r e s s types of s t r u c t u r e s have emerged, f o r which The deformations of these s t r u c t u r e s and important used to a c h i e v e s o i l o i l e x p l o r a t i o n s i n the equipment the r e s u l t s strains. r e c e n t years new important Examples triaxial i n dams. an i t e r a t i v e i n which assessed the in a state-of-the-art report on for earthquake analysing deformations e n g i n e e r i n g r e s e a r c h by was the F i g . 1.1. C a i s s o n - R e t a i n e d I s l a n d ( D e J o n g a n d Bruce, 1978). 4 National Research C o u n c i l i n the f o l l o w i n g of the United States (USNRC, 1982; Finn, 1983) terms: "Many problems i n s o i l mechanics, such as s a f e t y s t u d i e s of e a r t h dams, r e q u i r e that the p o s s i b l e permanent d e f o r m a t i o n s t h a t would be produced by earthquake shaking of prescribed i n t e n s i t y and d u r a t i o n be e v a l u a t e d . Where f a i l u r e develops along well - defined failure planes, relatively simple e l a s t o p l a s t i c models may s u f f i c e to c a l c u l a t e d i s p l a c e m e n t s . However, i f the permanent deformations are distributed throughout the s o i l , the problem i s much more complex, and practical, r e l i a b l e methods of a n a l y s i s are not available. Future progress w i l l depend on development of suitable p l a s t i c i t y models f o r s o i l undergoing r e p e t i t i v e l o a d i n g . This i s c u r r e n t l y an important area of r e s e a r c h " . For realistic structures the predictions of stresses s t r e s s - s t r a i n behaviour c l o s e l y as p o s s i b l e . This strain soils behaviour of of and soils i s of course a d i f f i c u l t is extremely displacements should task complex. be soil modelled since Using in the stress- simple stressv s t r a i n r e l a t i o n s h i p s , a two-dimensional method f o r computing t r a n s i e n t permanent d e f o r m a t i o n s i n s o i l 1.1 and i s presented i n t h i s t h e s i s . SCOPE OF THIS THESIS This seismic thesis presents for two-dimensional static s t i f f n e s s which c o n t r o l the effective stresses in-situ analysis i n the static which the S o i l properties response soil into to earthquake structure. stresses. takes and structures. have been e s t a b l i s h e d . and evaluate method earthquake l o a d i n g o c c u r s a f t e r s t a t i c e q u i l i b r i u m or state conditions static a response a n a l y s i s of s o i l The to structures as loading Therefore, These s t r e s s e s account such as are non-linear steady strength depend on i t i s important determined by a stress-dependent 5 response path, of s o i l the construction modelled. available to l o a d . (Kulhawy, method m a t e r i a l parameters the soil structure is loading carefully Duncan, et a l . 1978). is here presented i n both s t a t i c cost and effective which uses a N e v e r t h e l e s s an consistent dynamic a n a l y s e s . solution set of This results In to the problem of dynamic analysis. The method hysteretic carried of behaviour depends on the et a l . 1969; a much more e f f i c i e n t , linear sequence soil A number of s a t i s f a c t o r y methods of s t a t i c a n a l y s e s are a l r e a d y independent response Because out for stress-strain i n either appropriate dynamic analysis behaviour an e f f e c t i v e stress-strain analysis, residual porewater porewater pressure g e n e r a t i o n model seismically induced porewater or total For pressures pressures. account The be been The the analysis non- may be s t r e s s mode, u s i n g an effective must has into of s o i l s . stress relation. takes stress known. developed porewater response Therefore, for a predicting p r e s s u r e model i s a g e n e r a l i z a t i o n of the one-dimensional model of M a r t i n , et a l . (1975). The p r e d i c t i v e c a p a b i l i t y of the new method f o r dynamic a n a l y s i s has been verified by comparing the recorded porewater pressures and a c c e l e r a t i o n s of a c e n t r i f u g e d model s u b j e c t e d to s i m u l a t e d earthquakes those computed by the new This offshore porewater method drilling pressures, method. has island also computing conducted the been used supporting a stresses, i s l a n d have been determined. for response of compute accelerations and response drilling of rig. displacements in an The the At p r e s e n t one-dimensional methods are used of a to tanker-mounted these islands. to a s s e s s the v a l i d i t y of t h i s Development to non-linear Comparative studies were procedure. method of analysis is very 6 difficult. A number practical useful of program. l e n g t h i n t h i s t h e s i s and 1.2 computing The developed The These have to be approximations made have to been achieve examined a at s u g g e s t i o n s have been made f o r f u t u r e r e s e a r c h . ORGANIZATION OF THESIS A critical for approximations review of the method proposed by s e i s m i c deformations i s presented i n Chapter f o r m u l a t i o n s , b a s i c assumptions f o r the a n a l y s i s complete Seed, treatment of s t a t i c of et a l . (1973) 2. and l i m i t a t i o n s of the model response are p r e s e n t e d i n Chapter soil-structure interaction has also 3. been included. The proposed two-dimensional dynamic response analysis is an e x t e n s i o n of the one-dimensional response a n a l y s i s of F i n n , et a l . (1977). Therefore, a detailed description of their model, p r a c t i c e and i n the l a b o r a t o r y i s g i v e n i n Chapter The in Chapter Chapter proposed 5. method Details in 4. f o r dynamic response of the v e r i f i c a t i o n i t s application analysis i s presented of the method i s p r e s e n t e d i n 6. The tanker i s l a n d method i s used to compute response s u b j e c t e d to s e i s m i c l o a d i n g . The of a results typical drilling of the a n a l y s i s i n c l u d i n g i m p l i c a t i o n s f o r e n g i n e e r i n g d e s i g n are d i s c u s s e d i n Chapter A brief g i v e n i n Chapter 7. summary, s u g g e s t i o n s f o r f u t u r e work and c o n c l u s i o n s are 8. 7 CHAPTER 2 CRITICAL REVIEW OF SEED, ET METHOD FOR AL. COMPUTING DYNAMIC DEFORMATIONS The state-of-the deformations i n e a r t h simple art structures report (USNRC, 1982), e l a s t o p l a s t i c models f o r computing deformations develop a l o n g w e l l d e f i n e d slip surface does not the soil The most w i d e l y used by Seed, structure, et occur and an analysis 1979). analysis of suggests that deformations may slip planes. permanent the use When a w e l l at l e a s t i n two-dimensions method Detailed i s the one defined out i s necessary. that was d e s c r i p t i o n and of be adequate i f d e f o r m a t i o n s are d i s t r i b u t e d through two-dimensional a l . (1973, on proposed l i m i t a t i o n s of t h e i r method are p r e s e n t e d below. 2.1 SEED, ET AL. METHOD (1973, The b a s i c steps i n the Seed, a) Determine in b) et a l . method can be summarized as pre-earthquake the s o i l Determine Compute the soil or steady s t r u c t u r e b e f o r e the the the state condition time that exists a c c e l e r a t i o n f o r the structure i s situated. history structure follows: earthquake. d e s i g n time h i s t o r y of base s i t e where the e a r t h c) 1979) using of a dynamic shear stresses two-dimensional dynamic throughout response 8 analysis. A p p r o p r i a t e dynamic stress-strain relationship and damping s h o u l d be used. d) Apply these s t r e s s e s to u n d i s t u r b e d samples to the initial static stresses in of s o i l the soil determine the s t r a i n s and r e s i d u a l porewater e) Based on factor the porewater of s a f e t y methods after against reducing developed s i g n i f i c a n t f) pressure data, total the failure strength structure pressures. the minimum by l i m i t i n g equilibrium elements which the o v e r a l l strains induced by d e f o r m a t i o n of the s o i l the combined have pressures. I f the s o i l s t r u c t u r e i s found to be s a f e a g a i n s t t o t a l assess to determine of s e i s m i c porewater consolidated effects structure of s t a t i c failure, from the and dynamic l o a d s as determined from the l a b o r a t o r y t e s t d a t a . Seed, et a l . (1973) proposed an e q u i v a l e n t l i n e a r e l a s t i c method to model the dynamic fundamental response assumption of a satisfactorily are non-linear, i n this non-linear hysteretic type of hysteretic behaviour approach material of soils. i s that may the be The dynamic approximated by a damped, e l a s t i c model i f the p r o p e r t i e s of t h a t model chosen a p p r o p r i a t e l y . The a p p r o p r i a t e properties a r e o b t a i n e d by an i t e r a t i v e process. In the dynamic finite element analysis, p r o p e r t i e s o f the s o i l a r e d e f i n e d i n each f i n i t e ratio, v, and shear damping r a t i o s . strain dependent the stress-strain element by the P o i s s o n ' s shear moduli and e q u i v a l e n t An average or e f f e c t i v e shear s t r a i n viscous ( u s u a l l y assumed to be 65% of the maximum shear s t r a i n ) i s computed i n each f i n i t e element and shear moduli and damping r a t i o s a r e s e l e c t e d compatible w i t h these average 9 strains until The and no are used i n the significant response next changes determined iteration. i n moduli d u r i n g the or last The procedure damping ratios iteration i s repeated are necessary. i s c o n s i d e r e d to be a r e a s o n a b l e approximation of the n o n - l i n e a r response. Since is elastic computed the f i n a l the by a n a l y s i s with s t r a i n permanent deformation method. Deformations this dynamic s t r e s s e s w i t h a i d of is assumed element are triaxial finite shear strain not as closely as dam. potential by strain specified structure and i n the i s the the static axial In from dynamic possible strain develops loading. Since indication strains of that its excitation. will potential i n an the cyclic stresses on a sample factor p o t e n t i a l of (1+v). The soil elements in structure the stresses i s determined are then determined induced to shear ( F i g . 2.1b). by permanent a the shear strain a a strain element soil procedure but given potential in from the s t r e s s - s t r a i n curve ( F i g . 2.1a). converted f i n i t e element to proposed to the the above the are an seismic The corresponding have cyclic i s converted under It finite in a unconstrained soil and tests. i n a given strain be the s t a t i c triaxial o b t a i n e d by straining (1976), from cannot c o n v e r t i n g the s t r a i n p o t e n t i a l s to a set of compatible d e f o r m a t i o n s . element al. structures finite i n the properties for stress et for a soil procedure shear Serff, develop this by are i n t e r c o n n e c t e d , the s t r a i n s the soil i s the s t r a i n practice multiplying that the are e s t i m a t e d data the the r e s u l t i n g element under when simulated test, potential are that, strain in compatible static force and applied to the a finite The shear nodes of the The deformations under these n o d a l f o r c e s are analysis deformations. This and are assumed technique of to be computing seismically compatible F i g . 2.1. Conversion of Shear Strain Potential to Equivalent Shear Forces. 11 d e f o r m a t i o n i s sometimes r e f e r r e d to as a s t r a i n harmonizing One of the s e r i o u s used to model method may solutions non-linear not be Equivalent The occurs corresponds in is the s e t of compatible time of depend analysis methods period properties Since method, given by the i s because the on the assumed may occurs period overestimate non-linear materials soil because of of the resonance. input motion of the d e p o s i t as d e f i n e d by the i n the i t e r a t i v e equivalent linear the a n a l y s i s i s c a r r i e d out w i t h response the ( F i n n , e t a l . 1978a). the f i n a l s e t s t i f f n e s s e s f o r the e n t i r e d u r a t i o n of the i n p u t motion, f o r resonant response may fundamental to the fundamental of constant iteration This hysteretic materials linear when method of a n a l y s i s . the s o l u t i o n s et a l . 1977). method response of n o n - l i n e a r Resonance i s that elastic iteration. linear overestimation final (Desai, i n the l a s t p r o p e r t i e s of the f i r s t seismic behaviour, unique obtained l i m i t a t i o n s of any i t e r a t i v e technique. to b u i l d up. The s t i f f n e s s change c o n s t a n t l y f o r every time s t e p . there properties i n When resonant i s a f u n c t i o n p r i m a r i l y of the method of a n a l y s i s , i t i s c a l l e d pseudo-resonance. into The Seed account the stiffness. the et Since validity method effects response of a t o t a l a l . (1978) compared stress methods. (Schnabel, i s a total They of s t r e s s method and i t does not take increasing of s o i l s porewater pressure i s c o n t r o l l e d by e f f e c t i v e on stresses, s t r e s s response a n a l y s i s i s q u e s t i o n a b l e . responses p r e d i c t e d used by t o t a l two one-dimensional soil Finn, s t r e s s and e f f e c t i v e computer programs, et a l . 1972) and DESRA1 (Lee, e t a l . 1975) f o r t h i s SHAKE purpose. 12 SHAKE i s a total linear elastic models soil stress program which models s o i l material as a and non-linear concluded that the response porewater when DESRA1 i s an hysteretic a damped effective stress material. total stress analysis pressures as equivalent program Finn, et al, tends to o v e r e s t i m a t e the exceeded about practice with 30% of which the (1978) dynamic effective overburden p r e s s u r e . The method is major that possible. a The difficulty direct concept permanent deformations cies this in iteration not decision the correct, and one ( S e r f f , et First, has a l . 1976). the stresses to one from the i n the relationship the a be used to two final ignored, assumed stresses loading, not estimate the are are is inconsistenin properties given linear deformations strains Knowing that for equivalent There are stresses and the to as be strains arbitrary computed s t r a i n s i s somewhat i n c o n s i s t e n t . Second, last strains i t e r a t i o n are ignored, i n i n t e r m e d i a t e i t e r a t i o n s were used damping r a t i o s . to computed computed ground. the permanent strain potentials whereas to i g n o r e the strains computed of of o b t a i n e d w i t h s t r a i n compatible s o i l being a computation procedure. representative have in This type of whereas to o b t a i n inconsistent the compatible moduli assumptions make the final e s t i m a t e d deformations somewhat a r b i t r a r y . When subjected to porewater cyclic deformation i s not Noting use for about undrained are tests, allowed deformations l i m i t a t i o n s with years, the the structures the for of dissipate occur. active research the in samples This plastic Seed. Seed method, which has state-of-the-art permanent deformations i n e a r t h subject to accounted f o r i n the approach proposed by the 10 pressures report on been analysis recommends t h i s t o p i c s h o u l d next ten years (USNRC, 1982). in of be In 13 this and thesis, an attempt permanent presented. behaviour which deformations will of s o i l allow direct structures Procedures have been developed of soil, g e n e r a t i o n on s o i l taking into properties. account the computation of i n a consistent transient manner, i s to model n o n - l i n e a r h y s t e r e t i c effect of porewater pressure 14 CHAPTER 3 GENERAL GUIDELINES FOR DYNAMIC ANALYSES The motion Is an important accelerations throughout during If performance of an e a r t h induced by the dam. i n seismically active the earthquake These an earthquake these concern dam under earthquake induced forces induce which can cause reverse and a s s o c i a t e d slumping and slope instability eventual f a i l u r e of the dam. displacements may result, Newmark (1965), i n h i s p i o n e e r i n g on dams, seismic recommended loading that should surface. The a l l o w a b l e functional role deposit. and inches; deposit for critical platforms, important factors analysis that enough, t o over soil topping during and of d i s p l a c e m e n t failure" along during and not i n an assumed failure structures of t h i s displacement thesis of s o i l influence such as n u c l e a r may the behaviour be on the reactors only and a f t e r the earthquake. a few acceptable. i s to present structures, founded taking a two-dimensional into of s o i l account a l l deposits. a n a l y s i s p r e d i c t s d i s p l a c e m e n t s , s t r e s s e s , s t r a i n s and a c c e l e r a t i o n etc, large structure or the s t r u c t u r e the a l l o w a b l e main o b j e c t response are large of a however f o r e a r t h dams many f e e t may be The times or s a t i s f a c t o r y d i s p l a c e m e n t s depend m a i n l y on the the s o i l F o r example, gravity dynamic of forces work on e f f e c t s of earthquakes i n terms terms of the " f a c t o r of s a f e t y a g a i n s t inertia and s t r a i n s i n the dam. leading the performance be a s s e s s e d large The ground i n d i r e c t i o n many alternating stresses strains areas. ground The fields 15 The d e t e r m i n a t i o n of s t r e s s e s , s t r a i n s and d i s p l a c e m e n t s induced i n a dam by an earthquake i s a complex a n a l y t i c a l s i m p l i f y i n g assumptions must be made. performance structure, of can an earth be dam which interpreted subjected condition i s assumed to p r e v a i l The main reasons to the for this same the dynamic i s the assumption essentially the performance seismic t h a t the a three-dimensional of transverse loading. i n the proposed assumption s t o r a g e requirements needed In is from sections Foremost problem and a number of The plane two-dimensional crossstrain analysis. are the h i g h c o s t and h i g h computer f o r a three-dimensional a n a l y s i s . response a n a l y s i s of continuous systems such as e a r t h dams, non-uniform mass and s t i f f n e s s d i s t r i b u t i o n s are p r e s e n t . The finite and element approach, mass w i t h extreme In stress subjected pressures, and to if or e f f e c t i v e repetitive sufficient stresses will effective stresses and functions of e f f e c t i v e preferable analysis for i s more generation and loading drainage result. soil those to dissipation of on excitation the response showed that of level effective occur, and moduli reduction in controlled by and Effective It requires additional saturated stress are porewater strength are s t r e s s response a n a l y s i s i s soils. perform. models not as decision cohesionless residual deformations such a s h a l l be c a r r i e d out i n generate an e f f e c t i v e type soils, Saturated loose does Since properties stresses, difficult stress. n e c e s s a r y to e s t i m a t e c u r r e n t e f f e c t i v e s t r e s s e s . (1978a) in stiffness of s a t u r a t e d as to whether the a n a l y s i s effective always the v a r i a t i o n the dynamic response a n a l y s i s of t o t a l soils can model ease, has been adopted. must be made i n i t i a l l y terms which stress response porewater pressure computations are S t u d i e s by F i n n , et a l . sandy analyses sites are not to seismic generally 16 required unless the e f f e c t i v e the porewater overburden pressures excitation is response insitu of soil effective stress the porewater p r e s s u r e develops material dynamic a n a l y s i s has assumed on long to computation is been the analysis soil this thesis is a review of linear, of The loading, current for laboratory and 3.1 of the account in by conditions. which c o n t r o l on initial effective stress effective stresses a static both s t a t i c and as analysis, subsequent all a o n e - d i m e n s i o n a l dynamic important response factors analysis that analysis, of has their field. including specifically been modelled method Some has details is been on affect proposed t h e i r one-dimensional a n a l y s i s . behaviour capability in imposed depend To do t h i s , two-dimensional t h e i r method of predictive procedures The soil loading equilibrium Furthermore applicable into an e x t e n s i o n hysteretic 30% - 40% of developed. taking behaviour. the and s t i f f n e s s , seismic important. parameters term F i n n , Lee and M a r t i n ( 1 9 7 7 ) , presented response to exceed that strength condition. analysis, uses is such as structures response which it superimposed The dynamic s o i l p r o p e r t i e s , the likely pressure. In dynamic a n a l y s e s , seismic are Therefore, how the non- presented verified these in below. in the verification are a l s o i n c l u d e d . ONE-DIMENSIONAL RESPONSE ANALYSIS BY FINN, ET A L . (1977) In waves horizontally propagate pattern shear i s in the layered vertically deposit. r e q u i r e d i n the leads Then, analysis. deposits to only a the shear the assumption that beam of type stress-strain the shear deformation relationship in 17 The important f a c t o r s that must be c o n s i d e r e d dynamic response of s o i l s when computing the are; a) The nonlinear s t r e s s dependent s t r e s s - s t r a i n b) The modelling of c) Contemporaneous g e n e r a t i o n behaviour. unloading-reloading. and d i s s i p a t i o n of porewater pressures. d) H y s t e r e t i c and e) S t r a i n hardening. All these viscous f a c t o r s have s t r a i n r e l a t i o n s presented 3.1.1. damping. been taken i n t o account in the stress- by F i n n , et a l . (1977). Shear S t r e s s - S t r a i n R e l a t i o n s h i p The seismic loading c o n s i s t s of l o a d i n g , u n l o a d i n g imposes and irregular reloading. The loading pulses which soils exhibit different behaviour i n each of the above phases. The for the r e l a t i o n s h i p between shear initial loading phase under s t r e s s , x, e i t h e r drained c o n d i t i o n s i s assumed to be h y p e r b o l i c and given and or shear strain, undrained y, loading by, G x = f(y) = max y fl + G max Tmax IYI) (3.1) 18 in which, strength. The = max This maximum initial unloading (Masing, if G - loading reloading 1926). This u n l o a d i n g occurs shear or s k e l e t o n has implies been that from ( x r , Y r ) , G max modulus is curve i s modelled the and shear shown i n F i g . using equation = max for Masing 3.1a. behaviour the u n l o a d i n g curve, g i v e n by, " Vr) 1 + 2 G max T v-Y I' ' r I (3-2) max which i s simply, - T The shape of should (1975) loading. f o l l o w the maximum beyond shear B, direction, becomes i.e, also follow along is the BC. by the path that exceeded. In extension the If of initial general the c u r r e n t loading curve. continued, the concept and r e l o a d i n g the of loading the in the previous curve, negative history, further l o a d i n g curve i n t e r s e c t s the to curves unloading loading curve, Two t y p i c a l examples are BC i s 3.2a, 3.1b. Masing the magnitude Fig. loading the unloading if the case previous previous of shown i n F i g . extending the curve In the for loading to be made. to u n l o a d i n g . loading rules He suggested strain described should (3.3) proposed skeleton assumptions have curve Yr) the u n l o a d i n g - r e l o a d i n g curve i s Lee irregular f(y - Tr the stress strain curve The above r u l e s should apply provided i n F i g . 3.2b;(a) loading path is assumed to if be 3.1(a). Initial L o a d i n g C u r v e . pig. 3.1(b). Masing Stress Strain C u r v e s f o r Unloading and Reloading. I (a) " first unloading (b) F i g . 3.2. Hysteretic general reloading Characteristics. O 21 BCAM, will (b) i f u n l o a d i n g be ABP' . along Newmark path CPB i s c o n t i n u e d , and Rosenblueth (1971) have the u n l o a d i n g suggested a path similar procedure. 3.1.2 Porewater P r e s s u r e Model Consider a sample of s a t u r a t e d sand under a v e r t i c a l effective i stress, o . During y shear strain, Ae j, due t o g r a i n y < the y» E„ r stress system, Ae = increment During simple shear i n volumetric an undrained the c y c l i c pressure, A U . s a t u r a t e d sands AU - E which an cyclic slip. i n c r e a s e i n porewater in drained causes same e f f e c t i v e that f o r f u l l y a shear shear test, a cycle compaction test strain, of strain, starting with y , causes an I t was shown by M a r t i n , e t a l . (1975) and assuming water to be i n c o m p r e s s i b l e , (3.4) v d one-dimensional rebound modulus of sand at an t effective stress a « v Martin, conditions total e t a l . (1975), the v o l u m e t r i c s t r a i n accumulated volumetric shear s t r a i n c y c l e , y, Ae also showed increment, strain, £ v c p that Ae j, * t simple i s a function V ( a n c under n e amplitude shear of the of the and i s g i v e n by, 3 vd = CAy-C e ) + — — vd 1 2 vd + C, e , ' 4 vd C Y e (3.5) 22 i n which , C 2 , Cg and sand type and r e l a t i v e modulus Martin, E , r at et a l . are density. any (1975), do' E = ~ — r de vr volume change c o n s t a n t s t h a t depend on the An a n a l y t i c a l e x p r e s s i o n effective stress for i level av the rebound is given by as, . = (a*) v' /{m K 1 r v (a ) vo' ' } value of (3.6) ' v i in which, aVQ Kr, m and n are e x p e r i m e n t a l c o n s t a n t s The cycle with is the initial increment i n porewater maximum shear and strain, (3.4), (3.5) 3.1.3 M o d i f i c a t i o n of P r o p e r t i e s The T max* develops. of effective stress sand. pressure, AU, d u r i n g a g i v e n l o a d i n g may now be computed u s i n g y, and equations (3.6). residual These for the values Hardin, f o r R e s i d u a l Porewater porewater p r r e s s u r e ,> r should et be al. updated (1972) as TJ reduces residual assumed that Pressure Gmax „„„ porewater Gmax is and pressure independent s t r e s s h i s t o r y and s u g g e s t e d , * G = K (a') max m i n which K is The 1/2 ' a constant, initial shear apparatus 3.3. Here i t with is (3.7) depends on s o i l type and r e l a t i v e and c u r r e n t e f f e c t i v e zero assumed initial that the porewater ratio stress density. conditions i n a pressure are simple shown i n Fig. between h o r i z o n t a l and v e r t i c a l m F i g . 3.3(a). Initial E f f e c t i v e S t r e s s C o n d i t i o n i n a Simple S h e a r A p p a r a t u s . F i g . 3.3(b). i«2K»«r- -u) 3 Intermediate Effective S t r e s s Condition i n a Simple S h e a r A p p a r a t u s . Co 24 effective lateral ^max^ stresses pressure l s 8 (G For i v e n max is a constant at r e s t . b v ) F o r the i n i t i a l K i s the c o e f f i c i e n t Q effective stress of condition, 1/2 1 +9 V ' fV>l = K* 3 stress (Ko) 1/2 v ; conditions, 1/2 (G ) = K (if*) * max where Q > o the c u r r e n t K , 1+2K 3 v n ; r; D - u ) i / 2 <3-9> ; on d i v i d i n g the e q u a t i o n (3.9) by (3.8) one o b t a i n s , ( G max ) n _ r L (G .)) max o Therefore, modulus - U . vo il/2 J o' a' vo - knowing at the c u r r e n t (3.10) J (^ m a x^o' a 'vo a n c * ^> t n e maximum shear e f f e c t i v e s t r e s s c o n d i t i o n can be c a l c u l a t e d using e q u a t i o n 3.10. The shear strength stress condition i s given ^ f ( W o 9 = {(—-—°) 1 1K + K 2 J ( T m a x ^o ^ o r t n Initial e effective by ( F i n n , et a l . 1977). i o sin 2 0 1 - K - 1/2 ( ^ - ^ ) K 2 ' 2 } 0' ' vo = C a' vo (3.11) 25 * in which 0' i s the angle depends on s o i l ( W ) of i n t e r n a l friction p r o p e r t i e s . F o r the c u r r e n t = C*(o-' - U) n vo and C i s a constant stress condition, which (3.12) D i v i d i n g e q u a t i o n (3.12) by (3.11) one o b t a i n s , (t (T max max ) * ) ^ o ^ max^o' "vo T strength at current equation (3.13). 3.1.4 Influence et T max • and (3.13) U effective are stress known, the maximum condition can be shear calculated from o f S t r a i n Hardening During drained a* vo seismic conditions, a l . (1977), loading the sand used of d r y sand or saturated s t r u c t u r e hardens due to g r a i n following equations to modify sand slip. Finn, G m a under x and • vd — ~ ~ } H, + H„E , 1 2 vd e (G ) = (G ) max max nn n 1 {1 + J (3.14a) and (3.14b) 26 in which, ( G m a maximum shear modulus and are hardening ) x n and shear defined a x strength are n n the modified i n the n t h c y c l e and , » H3 constants. The s t r e s s - s t r a i n completely (T_ ) and n behaviour by e q u a t i o n s , for 3.1, one-dimensional a n a l y s i s 3.2, 3.4, 3.5, 3.6, 3.10, is now 3.13 and 3.14. In laboratory cyclic changes i n dry sands and the saturated sands Therefore, curve to occur Finn, take et account increases during al. the (1977), of If will be The rate the saturated simultaneous of increase tests most i n porewater pressure portion modifications hardening and of of to the sand deposit and can drain dissipation pressure will during of be only shaking there less bility of drainage the sand, The d i s t r i b u t i o n of porewater j _ <_z_ _J-\ 5z W 5 z J 'w pressure porewater drainage depends _ . _ E ot cycle. stress-strain porewater The amount of shaking. load loading. porewater compressibility volume i n undrained the c o m p l e t e l y undrained s a n d . and the Pressure generation of used the D i s s i p a t i o n o f Porewater shear unloading strain d u r i n g the u n l o a d i n g phases of 3.1.5 simple E 8 £ vd ot p r e s s u r e at path pressure. than on the and time t i s that in permea- duration of g i v e n by, (3.15) 27 i n which, k z The term is the p e r m e a b i l i t y and y w containing ey(j porewater p r e s s u r e ( F i n n , t n u n i t weight of e represents et a l . the water. internal generation 1977). The s t r e s s - s t r a i n r e l a t i o n s h i p o u t l i n e d above can be v e r y extended to non-uniform l o a d i n g u s i n g an i n c r e m e n t a l l y e l a s t i c time domain. The dynamic response numerically solving e x p l a i n e d by F i n n , 3.2 equation et a l . of can be computed f o r (3.15) and the easily analysis in each time step by equation of motion, as (1977). LABORATORY VERIFICATION OF EFFECTIVE STRESS RESPONSE ANALYSIS The strain basic assumptions relationship groups: Those presented made in the made above in the can be formulation formulation broadly of of the categorized porewater pressure stressinto two model and those made i n m o d e l l i n g l o a d i n g , u n l o a d i n g and r e l o a d i n g . The the porewater fundamental assumption p r e s s u r e model, was that that was made i n the the porewater f o r m u l a t i o n of pressures i n an u n - d r a i n e d t e s t can be o b t a i n e d from v o l u m e t r i c s t r a i n s measured test strain loading. on a s i m i l a r means strains that sample w i t h same h i s t o r y of there should i n drained tests be a unique and porewater g i v e n sand at c o r r e s p o n d i n g s t r a i n Finn to i n v e s t i g a t e drained Ottawa subjected Porewater to (1981) this sand constant pressures were at strain also between This volumetric i n undrained t e s t s for a histories. assumption. samples relationship pressures reported results basic shear i n a drained of an e x t e n s i v e Volumetric strains relative cycles measured laboratory densities in a program were measured in D r = 45% and 60% when simple shear i n undrained c y c l i c apparatus. tests at the 28 same relative Volumetric ratios strains U/o^. represents of densities in a J equal are initial shown 3.4 of there i s a s l i g h t small V ( Fig. set c y c l e s with these E and for values cyclic = pressure will suggested E„ under from values verify E_ modulus computed not important. the rebound be static slope from in of the the the from data with shown curve number n o t i c e d that indicate (1981) higher the curve that the the al. shown in (1975) curve showed than unique to c o n f i n i n g et i n F i g . 3.4. of a in that the He an the modulus used F i g . 3.4 the to the porewater p r e s s u r e model. (equation. (1981) 3.14) when p r e d i c t i n g because hardening maintained strain hardening should not be i n c l u d e d i n the s t r e s s - s t r a i n the net conditions. behaviour volumetric I f drainage of sands under undrained strains i s allowed do not to o c c u r , occur then effect relationship conditions. during the e f f e c t s This undrained of strain should be i n c l u d e d . The porewater pressure model coupled with the stress-strain r e l a t i o n s h i p can be employed to p r e d i c t l i q u e f a c t i o n strength curves. strength shear the But ratios. respect unloading Finn is the given be Martin, the oedometer slope a pressure s t r a i n amplitudes. E_. But curve on porewater p r e s s u r e modulus evaluated for I t can The normalized conditions. measured computed curve pressures. porewater point v o i n the a p p l i e d shear this can the Finn is of give that oedometer, rebound slope U/a amplitudes. r e l a t i o n s h i p between v o l u m e t r i c s t r a i n and The Each and y ( confining against 45%. e j strain difference d e v i a t i o n s are plotted D„ of effective curve number consolidated of plots cycles of to (0CR=1) and the cyclic cause over initial stress ratio liquefaction, c o n s o l i d a t e d sands, obtained t/o for v o The versus normally analytically o b D T V- >nfin •> o» c T T T Sand type : Ottawa sand (C-109) a ' s 200 kN/m . Relative density * 4 5 % vo 2 •> M IO 0) T 1.00 - 0.75 Legend u Shear strain amplitudes o |c 0 5 0 - Drained Undrained o 0056% 0.056 % • 0.100% 0.100 % CL * 0.200% 0.210 % • A 0314% 0300 % 0 ^_ 3 to • 025 - fc. O 0. 0 L ) 02 0.4 0.6 08 1.0 12 1.4 1.6 18 Volumetric strain in percent, € j % y ( Fig. 3 . 4 . Relationship Between Volumetric Strains and Porewater Pressures in Constant Strain Cyclic Simple Shear Tests, D_ = 4 5 % . ho 30 and experimentally, are shown i n o b t a i n e d from undrained constant The strength comparison curves is 3.5. the computed T h i s means f o r m u l a t i o n of the n o n - l i n e a r h y s t e r e t i c ship are extreme valid. case. porewater shear results agreement good, stress stress from indicating that undrained tests shear the cyclic tests the assumptions relation- to an predict This strain to constant h i s t o r y as (i shear strain tests. are shown i n Fig. pressures made i n the is when s u b j e c t e d model parameters and computed porewater all model apparatus. i n an i r r e g u l a r Further made i n the prediction during model and s t r e s s - s t r a i n r e l a t i o n s h i p are 3.3 curve was liquefaction stress-strain stress undrained between measured measured effective constant two resistance curve tests. the assumptions effective i n a simple develop. from that and this loading results obtained obtained used development pressures were liquefaction (1981) c y c l i c shear porewater used Finn pressure constant cyclic But, The e x p e r i m e n t a l volume c y c l i c simple shear between very good. Fig. the = 1,4) Typical 3.6. is The remarkably porewater pressure reasonable. FIELD VERIFICATION OF EFFECTIVE STRESS RESPONSE ANALYSIS A unique opportunity to investigate the capability was p r o v i d e d r e c e n t l y when dimensional effective stress response a n a l y s i s data available on the dynamic response of became Tokyo Bay artificial to the island Mid-Chiba located constructed with materials A test record porewater site at earthquake on the of west 1980. side pressures south and end of ground the an a r t i f i c i a l Owi I s l a n d of dredged from the nearby the of Tokyo the island No.1 Bay. one- is It in an was sea. island accelerations is instrumented during to earthquakes. o - > 0.40 Sand type Ottawa sand (C-109) or ' « 2 0 0 kN/m , Relative density » 4 5 - 4 7 % ; 2 b v 0 0.30 o o w S 0.20 w •> O *•= 0.10 o O , « , A , A Analytical curve Experimental data 3 6 Number of cycles 10 30 60 for initial liquefaction, N F i g . 3 . 5 . Cyclic Stress Ratio v s . Number of C y c l e s for Initial Liquefaction for Various Sands. 100 L Sand type Ottawa sand (C-109) o- ' « 200 k N / m , Relative density = 4 5 % : 2 v 0 T / c r ' =0.074 T/crV*a065 vo / / o 6 10 30 Experimental curve Analytical curve 60 100 Number of cycles , N Fig. 3.6. Predicted and Measured Porewater Pressures in Constant Stress Cyclic Simple Shear Tests, D r = 45%. 200 33 Porewater p r e s s u r e s are r e c o r d e d by piezometers installed at depths of 6m and 14m. The Tokyo Bay area response of earthquake, on September Owi I s l a n d dimensional the Mid-Chiba effective 25, No.1 stress 4 sees, Finn, the Mid-Chiba response analysis. to are v e r y low a c c e l e r a t i o n s developed between 4 and 6 s e e s . , was recorded. are shown i n F i g . 10 sees, The ground range al. (1982) recording the computed the 10 3.7(a). Significant by F i n n , similar to oneof During the accelerations excitation et some minor d i f f e r e n c e s very a sees. only low l e v e l computed was shook using The f i r s t occurred. for 6.1, earthquake and t h e r e a f t e r Except computed et M = shown i n F i g . accelerations 3.7(b). the magnitude 1980. recorded ground a c c e l e r a t i o n s first with al. (1982) between 8 the recorded motions. The porewater No. 1 are pressures shown i n F i g . 3 . 8 ( a ) . components: transient instantaneous and r e s i d u a l and r e s i d u a l . response porewater of the 6m depth on Owi I s l a n d The t r a n s i e n t porewater to changes used by F i n n , pressure component between recorded and and in total computed two pressures applied are stresses volume changes. et a l . is p r e s s u r e has porewater p r e s s u r e s occur due to p l a s t i c porewater Comparison at The r e c o r d e d porewater o n e - d i m e n s i o n a l response a n a l y s i s residual recorded (1982) computes shown porewater The in Fig. pressures the 3.8(b). is very good. 3.4. POREWATER PRESSURE MODEL IN PRACTICE To analyses, 7 apply the constants porewater must be pressure known; four model C. (i in = dynamic 1,4) effective constants to (a) (b) CJ a.: s o.o Fig. 3.7. i.o i 4.0 TIME i— 1.0 0.0 10.0 i— (.0 i 4.0 TIME 0.0 —I 10.0 Measured (a) and Computed (b) Ground Accelerations (Acc. in ft/sec . Time in Sees). 2 8- ;HWv(M*w r- o.o TIME ' • • Fig. 3.8. 10.0 Measured (a) and Computed (b) Porewater Pressure at a Depth of 6m. (Porewater Pressures in lb/ft , Time in Sees). 2 LO 35 compute incremental represent be used rebound volumetric characteristics. to o b t a i n these practical can be avoided strength curve that and liquefaction cyclic by (Finn, such development or years experiences parameters The the behaviour a) study give trial of the from the r a t e of porewater curve and the of The a experimental right t e s t s on f i e l d the of do not number of rate trial especially K r pressure of analyses typical cyclic procedure is pressure triaxial tests the undrained following: i s sensitive to C^. shifts the liquefaction potential up or the shape a p p r e c i a b l y . the values liquefaction procedure generation. porewater to p r e d i c t the shape of the l i q u e f a c t i o n and liquefaction samples. liquefaction changing i s similar error to with variation of has i t will of reality to the model match measurement r e s i s t a n c e curve shape constants. direct evolved The sands and has r e v e a l e d the In n T h i s i s done by m o d i f y i n g the the of a number of down without number apparatus of samples i n simple shear has The and et a l . 1982). the c o n s t a n t s C^, b) shear e x p e r i m e n t a l l y o b t a i n e d by doing shape m r constants strength simple shear A simple K, tests. procedure which constants A number of l a b o r a t o r i e s s t i l l to do these a 3 these give can be and Cyclic constants. have simple shear apparatus Over strain adopted i s outlined of C^ potential to below: get r e s i s t a n c e curve given i n the curves. values for a literature In p r a c t i c e for the a model 36 1) Performing a number of value K_ that for such analyses the by varying K„, computed and select liquefaction resistance curve matches the e x p e r i m e n t a l l i q u e f a c t i o n r e s i s t a n c e 2) For 3) this selected porewater pressure K_ value, with number cycles pressure curve. If pressure these porewater repeat only the parameter volumetric C^, can analysis. that strain be This type model development of constants and are should i n the first the development and not be compare similar, noted with the alter is of incremental Therefore, estimates residual of that calculation cycle. from the curve. porewater the of pressure cycle. and error such that liquefaction the ones observed It i n the interpreted trial curves i s used recorded i n the f i r s t 3.5 of l a b o r a t o r y porewater and relevant calculate the procedure the can be corresponding resistance curves are employed to o b t a i n porewater sufficiently pressure close to i n the l a b o r a t o r y . DISCUSSION In subjected pattern to the response horizontal i s assumed in relationship i n shear the parameter elastic proposed by Finn, two-dimensions, two analysis accelerations, the deposit. i s required. in et of the al. elastic horizontally a shear beam type Therefore, The tangent only parameters To are the of deposits deformation stress-strain shear modulus i s used incrementally e l a s t i c (1977). layered extend response their required. analysis model A as to detailed 37 description of relationship to two-dimensions It static shear the has extension been stress cyclic observed affects subjected to porewater p r e s s u r e model of one-dimensional in extending loading deposits, for. one-dimensional discussed in the (Finn, Finn, the the i n Chapter laboratory porewater et et al. al. where s t a t i c t h e i r model to must be accounted is of 1978; two-dimensions, stress the presence response Vaid, (1977) i s shear 4. that pressure stress-strain et al. strictly is of samples 1979). The applicable zero. the i n f l u e n c e of of to Therefore, static shear 38 CHAPTER 4 TWO DIMENSIONAL STATIC ANALYSIS OF SOIL STRUCTURES A static is necessary response a n a l y s i s to compute i n - s i t u e f f e c t i v e because the s t i f f n e s s , depend on incremental soil dynamic in-situ soil effective Byrne, e t a l , 1982). based mainly proposed both on i n this the streses. such as the methods thesis static The and uses and satisfactory sequence of (Ozawa, et a l , 1973; the Duncan, et a l , s t a t i c a n a l y s i s presented i n t h i s t h e i s i s proposed a strength A number of e l a s t i c methods, which model the c o n s t r u c t i o n s t r u c t u r e s , are a l r e a d y a v a i l a b l e 1978, properties, stresses by consistent dynamic analyses; these authors. set of m a t e r i a l procedures The method parameters also have in been i n c o r p o r a t e d to a p p l y c o r r e c t i o n f o r c e s d u r i n g the a p p l i c a t i o n of the l o a d increments. 4.1 STRESS-STRAIN RELATIONSHIP A computation divided of of models. monotonic schemes. stress-strain in-situ broadly into non-linear simple number static linear, Some of types of relations stresses bilinear, in have been soil deposits. elasto-plastic, these models are v e r y loading are expensive proposed can visco-plastic complex to use They i n the in and be and even f o r computational 39 Two isotropic, modulus G elastic and t constants. given be elastic or are required incrementally tangent bulk elastic modulus B for were t extended 5, that to model that the the s t r e s s - s t r a i n dynamic loading s e l e c t i o n of any Tangent selected as the shear elastic c o n s i d e r a t i o n s were a l s o formulation proposed here has conditions. these two-dimensional analysis. In s e l e c t i n g these e l a s t i c c o n s t a n t s , to the f a c t Chapter constants It w i l l parameters greatly be to shown i n reduces the amount of computation time i n the dynamic a n a l y s i s . The static under stress-strain fully parameters on stress-strain test models drained selected results or which stability test results. of relevant of proposed for soils, undrained to model the c o n d i t i o n s that e x i s t term model earth model stress-strain as field. dams, can like conditions, represent i n the here, one only and as for the soils should possible The be the based loading example, i n the a n a l y s i s of should model other soils. chose parameters A d e s c r i p t i o n of the s t r e s s - s t r a i n model, and parameters all saturated dry behaviour closely For almost are discussed in from long drained the s e l e c t i o n detail, in this chapter. 4.1.1 Reasons f o r S e l e c t i n g In medium can deviatoric through the stress can be general, be strain divided strain. The the evaluated t and i n an into two volumetric bulk modulus. through G The B _t isotropic, homogeneous, l i n e a r components: independently strain and s t r a i n i s r e l a t e d to mean normal s t r e s s deviatoric strain shear modulus. volumetric elastic These by two i s r e l a t e d to d e v i a t o r i c independent m a t e r i a l applying uniform moduli changes in AO corresponding tangent stresses. shear T h e r e f o r e , by s e l e c t i n g modulus as two independent tangent elastic b u l k modulus and constants, better c o n t r o l s on s t r e s s e s and s t r a i n s can be imposed. 4.1.2 H y p e r b o l i c Shear S t r e s s - S t r a i n R e l a t i o n s h i p A number relationship 1963). G max a n of r e s e a r c h e r s have to p r e d i c t the behaviour The h y p e r b o l i c r e l a t i o n s h i p d x max a s used a hyperbolic of a s o i l stress-strain d e p o s i t (Konder, et a l . between i and y i s g i v e n i n terms o f > G T = y max ' ( l + Wmax G M m l) max in which, T,y G_ T 4.1.2.1 = u l t l m a t e G data have shown that f o r sands and s i l t s conditions, G in shear s t r e n g t h E s t i m a t i o n o f max Experimental drained strain = tangent shear modulus as y+0 a x max a r e the shear s t r e s s and shear = which, max " f K » e » 0CR ) <4-2) under 41 o" = c u r r e n t e f f e c t i v e mean normal s t r e s s e = void OCR = over c o n s o l i d a t i o n r a t i o m ratio here OCR i s d e f i n e d a s : _ Maximum past ma.jor p r i n c i p a l s t r e s s C u r r e n t major p r i n c i p a l s t r e s s The G following non-dimensional equation i s widely used for max' G max * K Ga t ^ j " P (4-3) 2 i n which, K Q = a non-dimensional P = atmospheric fl The value d e n s i t y of the s o i l , roughness history. (1972) sandy of of K Q depends soils mainly on et f o r dynamic soil. void g r a i n contact c h a r a c t e r i s t i c s the s o i l Seed, f o r a given pressure. particles An e q u a t i o n s i m i l a r and constant e t c . , and ratio such also on max = K_ G relative as a n g u l a r i t y and previous loading to (4.3) has been proposed by H a r d i n , e t a l . a l . (1970) analyses. f o r the computation The e q u a t i o n given (1972) i n c l u d e s the e f f e c t of p r e v i o u s s t r e s s h i s t o r y . G or P fm/P ) a ^ a ; (OCR) of G by H a r d i n , M „ O for et a l , They proposed, (4.4) ' v 42 i n which the exponent n depends on et a l . 1 9 7 2 ) . the p l a s t i c index of the s o i l (Hardin, Values of n are g i v e n i n T a b l e 4 . 1 . Table 4 . 1 V a r i a t i o n of Exponent, r\ w i t h P l a s t i c Index, PI PI% 0 0 20 0.18 40 0.30 60 0.41 80 0.48 >100 For condition normally typical 0.5 consolidated values for KQ non-plastic varies between soils 200 under drained 800 (Byrne, and 1979). For related clayey soils to the undrained G M A X = K S under strength S U undrained conditions, through equation, U where K i s a c o n s t a n t f o r a g i v e n c l a y . an (4.5) G M A X can be 43 4.1.2.2 Estimation For o f max T soils under drained conditions, t m a x > is the maximum i shear stress that can be applied by keeping axial stresses o , and x i a at v and their respective the y - a x i s the (a j) ( of presented assuming m a x where the x - a x i s v e r t i c a l f o r convenience. estimation analysis values, maximum by that the horizontal I n p r i n c i p l e , t h i s i s s i m i l a r to deviatoric Kulhawy, i s taken et stress a l . (1969). minor principle v i n the They estimate (o"j) m Q stress remains constant. Let and shear 4.1). us c o n s i d e r stresses a case where the i n i t i a l i a, y are v i a, and v x The Mohr envelope i s d e f i n e d represent the i n i t i a l stress state. one i s tangent to the Mohr e n v e l o p e . OA x = of the Mohr c i r c l e OP radius of diagram and the Mohr The p o i n t s L and M The a p p l i c a t i o n of shear s t r e s s the s i z e a xy by c' and 0 ' . increase that horizontal respectively ( F i g . F i g . 4.2, shows the c o r r e s p o n d i n g Mohr c i r c l e envelope. i -r vertical, and the l a r g e s t the largest Mohr c i r c l e circle keeping will is a i y the and c o n s t a n t and i t i s g i v e n by, i OP = r c "•tan0' i f x g + q 2 y ) i sin0' ' (4.6) From t r i a n g l e ABO, A B - plane S a x = maximum shear stresses on the horizontal Fig. 4.2. Mohr Circle Diagram . 45 = /(OA - OB ) 2 2 and so, x This Drnevich f-*} sin 0' + max equation (1972) reduces i f a H^} ] 2 to the equation a and v - 2 2 1/2 presented are (4.7) by H a r d i n replaced by and K a o vo Q u and a,vo* The of T max estimation for a c o n d i t i o n s can be made based on standard may be based on e s t i m a t i o n of i n s i t u 4.1.2.3 soil field element tests, under undrained l a b o r a t o r y t e s t s or effective stress conditions. Influence of Over-Consolidation Compaction i s g e n e r a l l y used construction. compaction So some p a r t s pressure. The to o b t a i n a c e r t a i n d e n s i t y i n dam of a dam effect are over of over consolidated consolidation due to the on G__„ i s Illcl A. already on the soils shown i n the e q u a t i o n value would of look x m a x like (4.4). also. that A Over c o n s o l i d a t i o n has an i n f l u e n c e typical i n F i g . 4.3. should be used depending on whether the s o i l 4.1.2.4 Mohr envelope Different c 1 for plastic and 0' values i s i n NC s t a t e or OC s t a t e . E f f e c t s of Unloading In geotechnical problems which involve excavation or r e d u c t i o n 46 co CO Q> CO u cd u N«C Region CO O-C Region Normal Stress Fig. CO CO 0) tH * J Slope 4.3. Mohr Envelope. G2 (Loadin CO M Rt CO Slope G G > G x 4 2 (Unloading^ Shear Strain Fig. 4*4 . Effects of Loading and Unloading, 47 in applied load, some soil elements modulus, G^, d u r i n g u n l o a d i n g from will a strain, h i g h e r than the modulus, G 2 , c o r r e s p o n d i n g level experience y> a s unloading. The shown i n F i g . 4.4, i s to l o a d i n g from the same s t r a i n i f the mean normal s t r e s s remains a c o n s t a n t . Unloading presented small, i n Chapter 3. the d i f f e r e n c e circumstances 4.1.3 and r e l o a d i n g can be modelled However, i f the s t r a i n between G and 2 using ranges the procedures of i n t e r e s t a r e i s not l a r g e . Under these changes i n modulus need not be modelled. Tangent Bulk Modulus t B It i s assumed that the tangent bulk modulus (B ) i s elastic t f o r any s o i l under d r a i n e d c o n d i t i o n s and i t i s a f u n c t i o n o f the c u r r e n t t mean normal s t r e s s a B t where, only. L that, A}n (4 *8) = exponent values and the exponent of bulk varies r e l a t i v e d e n s i t y of the s o i l , In elastic modulus stress will constant vary between 0.3 and 0.6. soil (or incrementally not r e s u l t between 300- depends m a i n l y on type and p r e v i o u s l o a d i n g h i s t o r y . homogeneous m a t e r i a l s , a change normal suggested = K, P b a P ' Typical and Duncan, e t a l . (1978), = Bulk Modulus constant n* 1000 m elastic) i n shear i n any volume analysis stress change. with in isotropic constant But, s o i l s mean under 48 constant mean-normal shearing stress. stress exhibit volume I n the a n a l y s i s presented made t o i n c l u d e the v o l u m e t r i c strains change when herein, t h a t occur subjected allowance to has been due to shear s t r e s s e s . A d e t a i l e d d e s c r i p t i o n of how t h i s i s done i s g i v e n i n S e c t i o n 4.4. It here includes only estimation test for should be noted that the e f f e c t consolidation tests equipment must be used. mean-normal from stress defined T h e r e f o r e , f o r the performed in conventional i s not h e l d triaxial constant. test Values data of triaxial procedure i n which the determined in approximate. PHYSICAL MODELLING The domain of i n t e r e s t number of elements, connected finite constant Duncan, et a l . (1978) d e s c r i b e d a t h i s manner must be c o n s i d e r e d 4.2 modulus of mean normal s t r e s s . of K^, i s o t r o p i c determining the b u l k element equations i s assumed to be an assembly of a f i n i t e a t the nodal p o i n t s . including ( C h r i s t i a n , e t a l , 1970) i s presented the effect The f o r m u l a t i o n o f the of i n Appendix I . {P} = [ K ] {A} + [ K * ] {U} porewater The e q u a t i o n s a r e , (4.9) T where, {p} = g l o b a l column v e c t o r of i n c r e m e n t a l a p p l i e d l o a d s [ K ] = g l o b a l tangent F C stiffness pressure matrix {A} = g l o b a l column v e c t o r of i n c r e m e n t a l displacements [K*] = g l o b a l s t i f f n e s s m a t r i x d e f i n e d i n the Appendix I 49 {u} = column v e c t o r the The of i n c r e m e n t a l porewater p r e s s u r e s i n elements. tangent element stiffness matrix [k ] factors; the tangent moduli and the shape f u n c t i o n s element formulation. The shape functions depends t on two adopted i n the f i n i t e give the variation of d i s p l a c e m e n t s w i t h i n an element i n terms o f the nodal d i s p l a c e m e n t s . The simplest displacements, gives shape f u n c t i o n , which assumes a l i n e a r v a r i a t i o n o f constant e x p e r i e n c e has shown that predict stresses and strain soil structures strains such and can be used quite accurately. isoparametric 4.3. element. But Therefore, quadrilateral s t r a i n v a r i a t i o n w i t h i n an element a r e used. as dams, to model The triangular from such elements do not accurately. layered a r b i t r a r y q u a d r i l a t e r a l shape a r e v e r y simple a the r e s u l t s obtained elements which have a l i n e a r For within deposits appropriate the geometry element e t c . , elements because they a r e f a i r l y of these stiffness of soil matrix, structures [k ] t f o r an q u a d r i l a t e r a l element i s g i v e n i n Appendix 1. SIMULATION OF CONSTRUCTION SEQUENCE Dams a r e c o n s t r u c t e d materials computation are non-linear of stresses sequence be m o d e l l e d . gravity switch sequentially. and and stress strains An a n a l y s i s on, w i l l give final based Since path the behaviour dependent, requires that a the of dam realistic construction on s i n g l e stage c o n s t r u c t i o n or stresses those c a l c u l a t e d by f o l l o w i n g the c o n s t r u c t i o n and s t r a i n s sequence. d i f f e r e n t from 50 4.3.1 Method of A n a l y s i s Fig. procedure elements 4.5 shows involved in on which a schematic dam construction. subsequent lifts are compacted u n t i l layer by l a y e r representation layers There will be is may placed. the r e q u i r e d d e n s i t y c o n s t r u c t i o n procedure of is the be sequential pre-existing The c o n s t r u c t i o n obtained. T h i s type of out u n t i l the r e q u i r e d carried dimensions of the dam are o b t a i n e d . In stresses, added. strains This caused modelling by is of incremental values mode elements stress deformations a the fresh or dam are computed (4.9) for layer f o r the i n c r e m e n t a l loads algebraic applicable, stress-strain these elements sum strains and of the all Pressure can combination be carried of both. When the e f f e c t i v e out In stress in the a the elements the relationship. should be set effective c o m b i n a t i o n mode, principle is the t o t a l stiffness matrix Furthermore, to or used i n the some total finite i n t r o d u c e d as shown i n (4.9). f o r which element total mode and some may be i n a stress the r i g h t hand s i d e of the e q u a t i o n be every stresses, element f o r m u l a t i o n , the porewater p r e s s u r e term i s For incremental new final the the computed f o r a l l the l a y e r s . analysis a sequence, The simply may be i n an e f f e c t i v e mode. are layer. I n c r e m e n t a l Porewater Static stress construction done by s o l v i n g e q u a t i o n placing displacements 4.3.2 and the zero. stress [kt] mode i s is the porewater However, i f based assumed on a to total pressure, uQ, in the e f f e c t i v e stress 52 mode is elements value selected for should b a s e d on for u Q It is to in should Estimates In the of elements, an effective be the remembered in-situ view of equation u Q for this, (4.9) the using to in the element field porewater c o m p u t e s t r e s s e s and 4.3.3 the given changes i n shape of their the own strains weight elements also of ways schematically strains due i s the these and a and in to porewater be u analogy can the by the S t r e s s e s and finite reference Abel, load to applied. moduli a "long term" {Ti] vector term number of Measured used analysis long {IT} , matrix be value. methods porewater formulated i n equation (4.9) Strains element a n a l y s i s are elements a f t e r state adding incremental The Estimates increment stress-strain are level corresponding to state. they measure I t i s assumed have s e t t l e d strains of under The caused total by the state. a n a l y s i s can 1972). a (Ozawa, et a l . 1 9 7 3 ) . reference elastic F i g . 4.6. a a etc. The now is pressure made u s i n g used. Q to static displacements. by incremental corresponding New for relationship f o r the which correspond be pressures reason global electrical newly placed obtained (Desai the can may c o n s t r u c t i o n l a y e r s about t h i s An stiffness stress-strain t h e e l e m e n t s f r o m some r e f e r e n c e c o n d i t i o n of are element condition should Computation of Incremental Strains that that static s u c h as h y d r a u l i c m o d e l t e s t s , pressures the i s required. estimate condition. used be some be carried approach of the adopted increments determined before the using the average load of the out in a here is number shown i n stresses moduli increment strains and values was before Estimated I n i t i a l Estimates or Computed ^o 't Values t Layer Loading (incremental) S t r e s s e s and S t r a i n s M W - W + M = K I +M Calculate K —1> Moduli New Moduli G' , B[ from T Average S t r a i n s and S t r e s s e s 1 — r V. i I n i t i a l Estimates or computed Values with modified G' T and Layer Loading (incremental) F i n a l Stresses Next Layer and Strains Loading {e} = {e} + { A E } {a} = {a } + 0 {Aa} B£ Fig. 4.6. Schematic Diagram Showing the Incremental Analytical Procedure 54 the l o a d increment i s a p p l i e d and the s t r a i n s computed a f t e r the are incremental stresses used to compute more c o r r e c t same l o a d initial load increment. stresses These i n c r e m e n t a l s t r e s s e s and s t r a i n s to o b t a i n s t r e s s and stresses from section are computed In equilibrium is equilibrium correction condition. to the doing not condition using the this, as forces known. placing elements the added f o r the f o r the load out may be Therefore, at by the end Desai Under applied and these to can y Tl xy be to next 0 to the next l o a d a fresh layer are the i n these elements. estimated using the shear the Abel load (1972), circumstances equilibrium that correspond procedure o u t l i n e d increment. stress known. f r e s h l y placed the of satisfy the shear condition Therefore, in pre-existing in previously the initial elements elements, moduli must be are based Ozawa, et a l . (1973) suggested on that equations, 's = 0.5 which d i s the <= computed u s i n g increment, moduli f o r the However, f o r the stresses surface, r e l a t i o n s h i p between satisfied. (pre-existing) estimated stresses and the strains pointed shear s t r e s s e s Appendix I, are a p p l i e d analysis that shear necessarily changes i n the placed 4.1.2, In the method adopted here, c o r r e c t i o n f o r c e s Before in s t r a i n s are shear s t r a i n i s assumed to be h y p e r b o l i c . increment. the initial strains for increment. Recall in the and and increment is the v d s i n <* is o depth of slope (4.10) center of the of g r a v i t y of top surface the and y element from the s is the unit top weight 55 of soil. For totally submerged u n i t weight 4.4. elements Ys should be replaced by the y'« SHEAR-VOLUME COUPLING The an submerged tangent increment in bulk modulus B volumetric defined fc strain, Ae^, to But soils i n s e c t i o n 4.1.3, r e l a t e s an increment in effective i mean due normal stress, Aa « m to changes i n shear s t r e s s e s . be accounted f o r i n any The dense sand Initially characteristic samples they in a dilation volume range, initially and region of be But e v occur a d d i t i o n a l volumetric of s o i l behaviour shear both the of s t r a i n must initially is shown l o o s e and fixed the the initial of volume For volumetric e y values in loose and Fig. 4.7. the dense samples vs Y compaction both strain at plot region also behaviour. of apparatus (dilation). approach in strains l a t e r , over a c o n s i d e r a b l e range of variation interest problems would simple expansion the then drained shear s t r a i n s y, f o r small exhibit This volumetric r e a l i s t i c modelling undergo volume r e d u c t i o n . The in e v samples vs very high in typical and the Y ^ strain, in s strain the linear levels. geotechnical linear dilation region. The larger for rate the suggested u s i n g defined v dense sand dilation change than angle v Q for in the the linear loose dilation sand. to c h a r a c t e r i z e the region, Hansen, dilation is (1958) rate. He as, sinv o = dy~ = t a n Po (4.11) 56 F i g . 4.7. (b). T y p i c a l Plots for £ V s T for Dense and Loose Sands . 57 where 8 i s the s l o p e Q compressive angle v a given type of the r e l a t i v e increases Q The n e g a t i v e with of sand density with constant i s the s l o p e i s found the a n g l e the r e l a t i v e vertical Ottawa sand, (C-109), which i s introduced v Q was found and c o n f i n i n g p r e s s u r e . c l e a r l y shown by V a i d , et a l . (1981). tests sign since v o l u m e t r i c s t r a i n i s c o n s i d e r e d to be p o s i t i v e . For function angle. density They performed stress a y at various r e l a t i v e of the l i n e a r of = 0 200 densities. dilation to be a The the s o i l . dilation This d r a i n e d simple kPa, shear ( F i g . 4.8) The d i l a t i o n p o r t i o n of the p l o t e and y v to i n c r e a s e w i t h the r e l a t i v e d e n s i t y of the s o i l . T h i s was shown by Lee (1965) who performed Sacramento Fig. River sand samples drained t r i a x i a l of constant D r = 100%. Several 4.9. important of the t e s t data t e s t s on dense 0.1 MPa to 13.7 can be noted in dense samples a t h i g h c o n s o l i d a t i o n p r e s s u r e s behave l i k e l o o s e samples; s e c o n d l y , f a i l u r e i n terms of maximum p r i n c i p a l stress ratio Firstly, features level. F i g . 4.9 shows a of t e s t s w i t h c o n s o l i d a t i o n p r e s s u r e s v a r y i n g from MPa. on angle, The d i l a t i o n i s a l s o a f u n c t i o n of the mean normal s t r e s s series was occurs increases; at increasing and t h i r d l y , (compaction) strain levels as the c o n s o l i d a t i o n p r e s s u r e the d i l a t i o n angle decreases and becomes n e g a t i v e with i n c r e a s i n g c o n s o l i d a t i o n pressure. Fig. 4.10 shows the v a r i a t i o n of the d i l a t i o n angle v mean normal s t r e s s f o r a number of sands which were a t an i n i t i a l d e n s i t y of 80 p e r c e n t . v Q lies It i s interesting w i t h i n a narrow band (Robertson, l i n e a r w i t h l o g a r i t h m of mean normal Based approximation with relative t h a t the v a r i a t i o n of 1982) and a l s o the v a r i a t i o n i s stress. on the e x p e r i m e n t a l for analytical to observe Q purposes data presented can be made above, the f o l l o w i n g f o r medium dense and 58 Fig. 4 . 8 . Stress-Strain Behaviour of Ottawa Sand in Drained Simple Shear. (After Vaid et. al., 1981) Fig. 4.9 T y p i c a l D r a i n e d T r i a x i a l T e s t R e s u l t s on Dense Sacramento R i v e r Sand. (a) P r i n c i p a l S t r e s s R a t i o V s A x i a l S t r a i n (b) Volumetric Strain V s A x i a l Strain ( A f t e r L e e , 1965) ^'/(degree*) ~ 20 « TJ Ul -I o z < Vesie and Clough 1968 OeBeer 1965 ViKet ond Mitchell 1981 Hirshfteld ond Pbuloe 1963 Boldi et ol. 1981 Cole 1967 Chattahoochee Sand Mol Sond Monterey Sond Glacial Sand x SATAF Leigh ton Buzzard Sond V 32 32 35 •37 33 35 V Dj.- 802 10 z o 0.1 0.5 I 5 10 MEAN NORMAL STRESS, cr 50 100 500 1000 .13 m Kg/cm Fig. 4.10. Variation of Dilation Angle, with Mean Normal Stress for Various Sands. (After Robertson, 1982). ON o 61 dense sands: negligibly small shear strain means that where y shear stress Q the volumetric (Varadarajan, levels the and plot i s the is strain et al. above t h i s of shear e strain important. It shear be the be (e ) is at low v l i n e a r l y w i t h y. idealised which should stresses Byrne, et a l . 1982) i t varies can above to 1980, level y vs y due as in volumetric noted that This Fig. strain the 4.11, due value to of v Q should be m o d i f i e d f o r the changes i n mean normal s t r e s s a c c o r d i n g to some v a r i a t i o n such as shown i n F i g . 4.4.1 Analytical 4.10. Formulation There are a number of ways of m o d e l l i n g shear-volume c o u p l i n g . One i s to modify Ae depend y type of also 1979, shear will simpler incorporate elasticity in is structural Byrne, et a l , 1982). The incremental the increment b) v Q can 4.10, c) be and I) (Appendix increment to an I) such (Verruijt, unsymmetrical that A e 1977). x But stiffness and this matrix, computations. to keep the volume change the analysed a) rise the way matrix stress give complicates A are on approach which unduly the the same way mechanics I) m a t r i x as strains it is temperature ( Z i e n k i e w i c z , et T h i s i s accomplished shear the as and to variations a l . 1967; Byrne, i n the f o l l o w i n g manner. i n a l l elements are computed f o r i n l o a d , n e g l e c t i n g shear-volume c o u p l i n g . estimated then A e y f o r the new mean normal s t r e s s , u s i n g F i g . i s computed u s i n g e q u a t i o n The volumetric strain Ae ^ft. Here i t i s assumed t h a t , v then is 4.11. split into Ae x and 63 Aex = d A e vy T vector 0*5 = Ae "0 such dilational A e vv* that strains. L e t u define s i t s components Then Ae T is are a strain the given estimated by; JAe^, ~0 A e , 0}. y d) The nodal forces corresponding to t h i s s t r a i n v e c t o r Ae^ can be computed as, /// V I*?. A lo d (4.12) v (see Appendix I ) Now these f o r c e s can be added to the a p p l i e d i n c r e m e n t a l l o a d i n a) and new strains and s t r e s s e s can be computed. F o r computing i n c r e m e n t a l s t r e s s e s , the f o l l o w i n g e q u a t i o n should be used, Aa = D_(Ae - A_e ) (4.13) Q where, A£ = strain vector computed f o r the m o d i f i e d applied load. e) Now steps b) ->• d) can be c a r r i e d out u n t i l convergence o c c u r s i n stress and strain increments under the applied incremental load. 4.5 INTERFACE REPRESENTATION It the interface may be n e c e s s a r y between two to a l l o w r e l a t i v e finite displacement elements to model slip to occur a t s u r f a c e s i n the 64 field. S l i p elements, which a r e sometimes referred to as elements of zero t h i c k n e s s , can be used to model t h i s r e l a t i v e d i s p l a c e m e n t . can be assumed to be placed along the boundaries Slip between elements the two- d i m e n s i o n a l elements r e p r e s e n t i n g s o i l and s t r u c t u r a l elements o r wherever it i s anticipated may occur. that The s l i p relative movements or s e p a r a t i o n i s assumed to occur o n l y along t h i s o c c u r s when the s h e a r i n g f o r c e s i n the s l i p between elements this d i r e c t i o n and element exceeds the shear s t r e n g t h a t the i n t e r f a c e . Goodman, element with e t a l . (1968) eight degrees have of freedom behaviour i n rock mechanics problems. nodes a developed to element expressed are per unit the shear area of joint and slip fault F i g . 4.12 shows a s l i p element w i t h force the two-dimensional represent I,J,K and L, i n g l o b a l and element axes. slip a f„ The f o r c e s at any p o i n t i n and element. the The normal force f„ force-displacement r e l a t i o n s h i p i s assumed to be: (4.14) n where n K , K g n n = joint stiffness directions w, g w n = Shear and per u n i t length i n shear and normal respectively. normal displacement at the point of interest. The Imagine unit definition a direct thickness. of u n i t shear t e s t At f i r s t , joint stiffness needs clarification. being performed along an i n t e r f a c e when a normal force i s applied, element of the element 65 X Shear/Normal Displacement v^,*^ a Fig. 4.13. Plot of Typical Shear/Normal Stress vs Shear/Normal Displacement. 66 shortens as the a s p e r i t i e s d e f o r m a t i o n a t the j o i n t Fig. to 4.13. line purposes and the s l o p e i n the t a n g e n t i a l be o b t a i n e d . A t y p i c a l p l o t of normal the r e l a t i o n s h i p i s given direction by 1^. and a p l o t l e n g t h i s shown i n can be approximated Similar between tests can be f„ and w„ can The s l o p e of t h i s curve w i l l g i v e K „ . Using displacement deform. and the f o r c e a p p l i e d per u n i t For a n a l y t i c a l a straight performed i n the j o i n t the e q u a t i o n 4.14 and a l s o assuming a l i n e a r v a r i a t i o n of within the j o i n t element, a stiffness matrix K _ can be —-oil obtained i n local or element c o - o r d i n a t e s . the n o d a l f o r c e s and the n o d a l The u T This s t i f f n e s s matrix relates displacements. displacement v e c t o r here i s s i m p l y , = {u I f V-p U j , vj, u , v , u , v } K K I t has been shown i n Appendix I I that K 0 K 2K n L K 0 0 s 2 K " 2 K s -2K 0 K 0 - *n 2 0 2 K n 0 S " n s 0 n 2 K sym 0 ~ s 0 2 K i s g i v e n by, s n 0 s L -2K, n " s 0 0 ~ n K K K s 0 0 Kn s 0 2 K 2 K n (4.15) 67 where, L i s the l e n g t h of the element. To get the stiffness matrix in global co-ordinates a simple t r a n s f o r m a t i o n i s used, where T= given transformation matrix containing direction cosines and is by; (4.16b) i n which, and PQ = a = cosec sin<* r o o-i '--sinoc cos« J o o angle of i n c l i n a t i o n of the s l i p horizontal. Even of the though element directly, have been c o n s i d e r e d . slip this The type this finite degrees of freedom. evaluated simply formulation i s taken displacement element i s c o n s i s t e n t w i t h quadrilateral of element. into ., . (4.16c) r element w i t h does not account the include since rotation all 8 d.o.f. f i e l d v a r i a t i o n assumed w i t h i n the displacement Furthermore, field isoparametric the same Thus e s t a b l i s h i n g a g l o b a l s t i f f n e s s m a t r i x can be element l i k e any elements the have t r e a t i n g the s l i p both i n an N other element. 68 4.5.1 Method of A n a l y s i s of S l i p Elements It was relationship (f plastic and the Mohr-Coulomb analysis, value be is kept is negative, values at K linear and average Af n of the where yield, as and the K_ residual value. the But, stress-displacement element slip occurs be shear elastic-perfectly i s d e f i n e d by stress in of the = K Af n = 1 ^ (Aw ) yielding element yield small value. This small the The joint value the occurs, i s p o s s i b l e or should be of K^ element then point of to shear can be estimated point stress element, within Af not, the determined. i s assumed w i t h i n an from values g s slip displacement s t r e s s e s vary the and g from e q u a t i o n the shear Since the shear element. normal a The stress (4.14) as, (Aw ) g (4.17) n and elastic until normal f o r c e on of simple constant stiffness. i f the a incrementally kept to a v e r y separation is For should i s set i n v e s t i g a t e whether Af displacement slip criterion. K„ that incremental 5 K„ tangential IC^ should be s e t to a s m a l l v a l u e . variation Aw. the yield of viewed the r e g i o n where f o r a l o a d increment and Now plastic and normal g meaning g normal in w) i t s original To and vs values After can that type the occurs. g assumed Aw are incremental XI i n the element e x p r e s s i o n s f o r Aw and respectively. Aw n are, average shear and normal 69 A w s < = A u top> " ( a v e A u bottom> a v e = (Au +Au )/2 - (Au +Au )/2 K and Aw = (Av n t Q p L )ave - I (Av b o t t o m (4.18) J )ave = (Av +Av )/2 - (A +Avj)/2 K From equations 4.17 L to (4.19) Vl 4.19, Af„ and Af_ can now be written as, Af and = K s [(Au +Au )/2 - s K Af~ = 1^ shear and I J [(Av +Av )/2 - (Av +Av )/2] n Total (Au +Au )/2] L K normal L I stresses f and g (4.20) J f can be obtained by adding up the i n c r e m e n t a l s t r e s s e s f o r a l l the l o a d s t e p s . Mohr-Coulomb f m a i n the element x f where the of c max = s c and s failure f g then mentioned 0 s failure at any + are f n t a n the criterion. by f g and from f c o h e s i o n and If f f i n i t e element strength a x is greater than the K g to a small required the to d e f i n e absolute this value i s modelled residual value. as The i s i n d i c a t e d by n e g a t i v e f . in analysis m f r i c t i o n angle nodes c o u l d separate and reducing performed shear (4.21) s h o u l d be noted are the < s e p a r a t i o n of a s l i p element It gives time as, the s l i p element above, criterion n that a l l c a l c u l a t i o n s the local axes. f o r the computation Since the of displacements are g i v e n w i t h r e s p e c t to g l o b a l axes, they 70 must be transformed to get displacements i n the l o c a l axes by u s i n g the i n v e r s e of the t r a n s f o r m a t i o n m a t r i x T. 4.5.2 F a c t o r s t h a t I n f l u e n c e J o i n t Parameters In parameters the a n a l y t i c a l were introduced. element. (2) K , strength, f R behaviour m a of s l i p values stiffness by elements of c of roughness Details on how K„, K_ distinct stiffness joint along the element. These f and the (3) shear parameters model the parameters will mav depend on (1) the 111 cL 2v II elements and (3) c o n t a c t these unit across g three adequately. of the a d j a c e n t the a s p e r i t i e s , above, the 0 . and g S surface presented (1) K „ , the u n i t defined x The model area (2) shape and c h a r a c t e r i s t i c s ratio between can be o b t a i n e d the j o i n t walls. i n the l a b o r a t o r y a r e g i v e n by Goodman, e t a l . (1968). 4.6 SELECTION OF SOIL The properties It should i s probably be emphasized not i m p o r t a n t . parameters of soil process PARAMETERS of obtaining representative one of the d i f f i c u l t that individual tasks values i n stress e s t i m a t i o n of s o i l for soil analysis. properties i s But the s t r e s s - s t r a i n v a r i a t i o n g i v e n by the s e l e c t e d s o i l should give the best f i t t o the observed laboratory samples i n the s t r e s s ( o r s t r a i n ) range of i n t e r e s t . behaviour 71 4.6.1 O b t a i n i n g Shear S t r e s s - S t r a i n Parameters A f t e r d e c i d i n g what drainage c o n d i t i o n i s l i k e l y field, a laboratory conditions. strain F o r example relationship, performed test a Simple shear the test remains can be series tests to performed simulating parameters of drained simple shear are i d e a l constant. obtain A simple, values for G trial the corresponding to a test, parameters K Q , C' and 0 ' can be o b t a i n e d drained loading can be s t r e s s e s i n simple conditions can stress and e r r o r best method that f i t Then knowing the estimates f o r the easily. be shear at the b e g i n n i n g assumed t to be o" vo of the and K Q t o" . Then v o a /3 V Q and test. then to tests and m o v stress effective stress- s i n c e the mean normal s t r e s s d u r i n g of i n t e r e s t . The drainage f o r an e f f e c t i v e these curves i n the s t r e s s or s t r a i n ranges levels the % vs y f o r v a r i o u s c o n s t a n t mean normal reasonably employed be to obtain to obtain p l o t s levels. can to occur i n the the this mean normal i s i n general But i f c o n v e n t i o n a l Ojjj varies as the a x i a l the above procedure assumed triaxial o b t a i n these parameters. again stress to tests load v a r i e s , If triaxial a i s m remain given constant a r e performed by (1+2K ) D during the on the samples, and t h e r e f o r e , i t i s not easy t e s t data o n l y a r e a v a i l a b l e can be c a r r i e d then out by c o n s i d e r i n g the shear i s t r e s s and the shear s t r a i n on the f a i l u r e p l a n e . However, s i n c e o" i n c r e a s e s d u r i n g s h e a r i n g the shear s t r e s s - s t r a i n curve o b t a i n e d by t h i s m i procedure may be interpreted for a constant i o"m average am- The average i can be assumed to be t h e mean beginning and the end of the t e s t . of om, corresponding to the 72 4.6.2 O b t a i n i n g B u l k Modulus Parameters The tangent bulk modulus B was t assumed to be (equation 4.8) g i v e n by, ^ B da' " ~A t de vm a' n K, P {—) b a V ' a = Integrating ' v t h i s e q u a t i o n one g e t s , f K 1-n (4.22) K vm t a k i n g l o g a r i t h m s both (4.23) p b a ' sides, (l-n)log(o-' ) = log {K P^ (l-n)} + log (e _ n m b f f l ) (4.24) i.e., log( e v m ) = (l-n)log(a' ) - log {K p( The from drained m slope of isotropic (e.yjjj) axis, parameters, can the plot triaxial t h i s n can be determined. be Using log test h-n)} (e results v m ) vs will log give (a ) m (1-n), obtained and from the v a l u e of n, and the i n t e r c e p t on l o g calculated. and n, the tangent g i v e n mean normal s t r e s s . 1 _ n b bulk Now knowing modulus the bulk can be computed modulus a t any 73 CHAPTER 5 TWO-DIMENSIONAL DYNAMIC ANALYSIS 5.1 FORMULATION OF THE PROBLEM The g e n e r a l dynamic e q u i l i b r i u m equations f o r a l i n e a r system a t any time a r e g i v e n by (Clough and Penzien; 1975), [M] {X} + [C] {X} + [K] {X} = {P} (5.1) i n which {x}, X^, X^, {£}> and X^ { } x give column = the vectors relative whose acceleration, components velocity and displacement w i t h r e s p e c t to the base motion r e s p e c t i v e l y of a node, [M] = mass m a t r i x [c] = damping matrix [K] = s t i f f n e s s m a t r i x {p} where, = inertia {i} i s a vector force with vector, which a l l components as, -[MJ{l}x i s defined unity and X^ b i s the base acceleration. In components: written two-dimensional horizontal for acceleration the problems and v e r t i c a l . horizontal i n the h o r i z o n t a l the base a c c e l e r a t i o n I f the i ' direction t 1 then equation X^ may have two i n (5.1) i s is d i r e c t i o n , and i f the e q u a t i o n the base i s f o r the 74 vertical direction then X^ i s the base acceleration i n the vertical direction. 5.1.1 I n c r e m e n t a l E q u i l i b r i u m E q u a t i o n s f o r Non-Linear Systems In change with model the a n a l y s i s time. An i n c r e m e n t a l l y the n o n - l i n e a r equations of n o n - l i n e a r f o r any material time systems, elastic behaviour. interval, in matrices Incremental dynamic values. [M], relevant [C] and [K] are the to the time i n t e r v a l mass, f o r which equilibrium by r e p l a c i n g the This leads t o , {AX} + [C] {AX} + [K] {AX} = {AP} which, properties approach has been adopted t o A t , can be obtained v a r i a b l e s i n e q u a t i o n (5.1) by i n c r e m e n t a l [M] the m a t e r i a l (5.2) damping and stiffness the above e q u a t i o n s a r e written. It i s always assumed t h a t the mass m a t r i x i s c o n s t a n t . m a t r i x can be o b t a i n e d method. the matrix, finite by two ways: lumped mass method and c o n s i s t e n t mass In the lumped mass method, the mass m a t r i x i s o b t a i n e d mass of a f i n i t e i s obtained element element using formulation. equally the same The compute and i t has o n l y d i a g o n a l is a t i t s nodes. interpolation lumped mass method terms g r e a t l y i n c r e a s e s i s more by lumping The c o n s i s t e n t functions mass m a t r i x used i s very mass i n the simple t o terms, whereas the c o n s i s t e n t mass m a t r i x somewhat harder to compute and has o f f - d i a g o n a l terms. consistent The mass accurate, the presence the computation time r e q u i r e d Even though the of off-diagonal to s o l v e the dynamic 75 equilibrium equations. For the accuracy level required in typical g e o t e c h n i c a l problems, the lumped mass method i s c o n s i d e r e d a p p r o p r i a t e . In [K] which depend on are the introduced the d i s t r i b u t i o n Appropriate be general, values determined f o r the only by an time incremental of damping v a l u e s . a appropriate stiffness c o r r e c t i o n s to stiffness matrix displacement between any time because depend of i t e r a t i v e correspond results. to It w i l l t and the on t+At velocity the solution time be (5.2) i n the s t r u c t u r e . t are explained can and initial scheme f o r T h e r e f o r e , i n p r a c t i c e tangent which the the e q u i l i b r i u m equation increment T h i s type matrices and procedure, time time step i s v e r y e x p e n s i v e . tangent and interval iteration stiffness and the [c] matrix of v e l o c i t y at every end in displacement and the damping damping used with later how c o r r e c t i o n f o r c e s can be i n t r o d u c e d i n t o the s o l u t i o n scheme. The matrix at matrix will and stress-strain time be t, [K] = [K ] assumed the procedure law is FC to be a used to determine described constant in Section throughout to e v a l u a t e t h i s i s presented the the tangent 5.2. stiffness The dynamic a n a l y s i s i n S e c t i o n 5.3. With, [Kt]t (5.3) and [C] = [C] the dynamic i n c r e m e n t a l e q u i l i b r i u m e q u a t i o n s [c] (5.4) can be r e - w r i t t e n as, 76 [M] {AX} + [C] {AX} + [ K ] t When computing (5.5) have to [K ] and {AP} t t be response solved t {AX} = {AP} f o r a random l o a d i n g h i s t o r y , f o r every a r e updated. (5.5) The time step step. by step During equations the procedure integration procedure proposed by Newmark (1959) or Wilson's 0-method ( W i l s o n , e t a l . 1973) have been adopted to i n t e g r a t e the e q u a t i o n s . These procedures are described i n Appendix I I I . 5.1.2. Correction Forces The significant numerical assumptions: integration procedure (1) the v a r i a t i o n is based on three of a c c e l e r a t i o n w i t h i n a time step i s assumed to v a r y i n some known f a s h i o n e.g. l i n e a r or c o n s t a n t (2) the damping and (3) the response response and s t i f f n e s s a t time c o r r e c t , even If errors t. though accumulate instabilities may properties a t time t + remain c o n s t a n t d u r i n g a time At can be In general neither evaluated from of these assumptions step the known i s entirely the e r r o r s may be s m a l l when the time step i s s h o r t . from occur. step to step gross errors and even solution These problems can be avoided by imposing the c o n d i t i o n of g l o b a l e q u i l i b r i u m a t each step of the a n a l y s i s . The g l o b a l e q u i l i b r i u m equations a t time t i n terms of a l l f o r c e components a r e , {flit in which f inertia, system at any { f slt + = I^D^t' {^i}t» representing mass { Dlt + a damping, time P n {fg} d {p} t is are t spring and t and (5.6) { lt the forces the column acting inertia on force vectors the discrete vector at time t. Since the mass m a t r i x constant during dynamic and the damping m a t r i x were assumed to be {fj} analysis, t and {felt a r simply e given by, {flit = [M] {f } = [C] (X} (5.7) t and D t The element spring stresses elements. The i n an element {X} (5.8) t forces i n terms nodal {f^elem* of forces ^ r o m {^ lt» c a n s nodal that forces, correspond computed e applied to the at by the dynamic expressing nodes of the stresses Appendix I a r e , i n which _B i s the m a t r i x that t h i s manner n o d a l f o r a l l the f i n i t e forces ^ relates strain to nodal d i s p l a c e m e n t s . elements can be computed In and 78 the vector sum o f a l l these nodal forces will give the g l o b a l vector {f }f S If and left general the s o l u t i o n s hand sides i t will o b t a i n e d a t time t a r e a c c u r a t e then the r i g h t (5.6) o f the e q u a t i o n n o t be so. will be i d e n t i c a l . The c o r r e c t i o n force vector But, in { } p c o r r i s g i v e n by, ( corr} P = l }t " { llt " { p f From e q u a t i o n s (5.6) { corr> P to ( 5 . 9 ) , - ^ f D }t " {fsk <5-10) e q u a t i o n (5.10) can be r e - w r i t t e n a s , W - t " M t - [C] {*} J /// B elements all To impose equilibrium, the c o r r e c t i o n force vector C O j dv (5.11) {P rr^ c a ^ n C O e added to the i n c r e m e n t a l e q u i l i b r i u m e q u a t i o n s f o r m u l a t e d a t time t . T h i s is accomplished equation 5.2 { p c o r r } to the r i g h t hand side of the (5.5). DYNAMIC STRESS-STRAIN RELATIONSHIP In domain, selected and by adding B t t h e proposed the tangent shear incrementally e l a s t i c modulus as the two " e l a s t i c for static analysis G t and tangent parameters". were presented analysis bulk Some reasons i n Chapter i n t h e time modulus B t were for selecting G 4 . f u r t h e r v e r y important reason f o r a d o p t i n g these parameters There t is i n the dynamic a 79 analysis. In dynamic analysis, the moduli have to be changed f o r every time s t e p . T h i s means that time. and B This t the element s t i f f n e s s time consuming m a t r i x has procedure can be to be r e - f o r m u l a t e d each simplified somewhat if G t are used as the " e l a s t i c " c o n s t a n t s . The elasticity matrix D_ (Appendix I) under plane strain c o n d i t i o n s , i s g i v e n by, D = B t B t where and Q From t % t G " 3 0 = B. B 3 + G t B t 1 1 0 0 0 0 + G the - 3 + 3 G G t t 0 (5.12) G are two 2 t B + G. 4. 3 -2 3 0 -2 3 4. 3 0 (5.13) t*2 (5.14) constant m a t r i c e s . f o r m u l a t i o n of stiffness matrix presented i n Appendix I, the element s t i f f n e s s m a t r i x i s , k=J//B V Now as, substituting D B dv for D from (5.15) (5.14) the equation (5.15) can be r e w r i t t e n 80 J/J^ V k. = B £ B_ dv + G /// B V 1 L B dv (5.16) i.e. k fc - B R t x + G R2 t ( 5 > 1 7 ) where R != / / / I * B £L V dv (5.18) x (5.19) The c o n s t a n t m a t r i c i e s R^ The current lc t matrix m a t r i c e s R, and R 5.2.1 9 be obtained by the constant behaviour d u r i n g dynamic loading, t multiplying o n l y once. and G . t Volume Change Behaviour s o i l deposits includes induced now by the c u r r e n t v a l u e s of B With regards the may and R^ have to be computed to volume can be d i v i d e d change into s o i l s which can undergo volume by the base excitation and two b a s i c groups. The f i r s t group changes under the l o a d increments the second group includes soils which cannot. S a t u r a t e d g r a v e l s and d r y d e p o s i t s belong to the f i r s t soils. R e c a l l e q u a t i o n (4.8) which r e l a t e s B,. to a', group of 81 nun 3 This equation to be m o d i f i e d (5.20) J may be used f o r every to compute B . This t time s t e p . means t h a t B t has However, i t i s known t h a t the changes i n mean normal s t r e s s e s i n the s o i l elements, due to s e i s m i c e x c i t a t i o n i s small and furthermore response of s o i l B t may B t f o r elements using be kept equation loading induce the volume change behaviour does not i n f l u e n c e the structures constant significantly. throughout can be e v a l u a t e d (5.20). This stresses such the dynamic based their for simplicity, analysis. on i n - s i t u i s because that Therefore, the l o a d An average mean normal pulses mean v a l u e s stresses during seismic are i n i t i a l in-situ i stresses. are I t should considered, be noted for typical here values that of even n i f the changes the changes in B in a will t m be small. Laboratory occur bulk during cyclic modulus. effect results have loading, This increase can be modelled revealed the s o i l the same way by F i n n , et a l . i n t r o d u c i n g hardening constants saturated, modulus as soils harden volume leading changes to due t o s t r a i n the i n c r e a s e (1977). (equation sandy as p l a s t i c samples i n bulk modulus was modelled Loose that, higher hardening i n maximum shear T h i s was accomplished by 3.14). and s a t u r a t e d c l a y s belong t o the second group of s o i l s . I n s a t u r a t e d s o i l s volume change can occur o n l y by porewater Within seismic drainage. excitation, questionable the occurrence in soils the of short duration appreciable of low p e r m e a b i l i t y . of amount typical of In view of t h i s , a n a l y s i s proposed here assumes that no drainage occurs during drainage is the dynamic the dynamic 82 loading. In s a t u r a t e d gravels appreciable does not develop because of i t s h i g h For the second type r e s i d u a l porewater pressure permeability. of s o i l s , to s i m u l a t e the c o n d i t i o n of no volume change, the b u l k modulus i s s e t to a very h i g h v a l u e d u r i n g dynamic loading conditions. 5.2.2 Dynamic Shear S t r e s s - S t r a i n Behaviour In the f o r m u l a t i o n of a complete dynamic e f f e c t i v e r e l a t i o n s h i p the f o l l o w i n g b a s i c a s p e c t s 1) soil behaviour under initial should stress-strain be c o n s i d e r e d ; loading, unloading and r e l o a d i n g phases. 2) r e s i d u a l porewater p r e s s u r e 5.2.2.1 dynamic loading dynamic (or c y c l i c ) assumed to be h y p e r b o l i c . d e f i n e d by G m a x Seed ^ and i t s e f f e c t s . S k e l e t o n Curve f o r Dynamic L o a d i n g Under ^max generation o r s a n dy shear and - c m a x stress, % c > the and dynamic The h y p e r b o l i c relationship shear between strain, r e l a t i o n s h i p (equation y » I s c 4.1) i s . e t . a l , (1970) soils conditions is a proposed function of that maximum effective mean shear normal modulus, stress o n l y , and g i v e n by, (5.21) 83 in which, density (K ) 2 (K ) 2 for a c a is x constant a soil. Seed v a r i e s between 20 and m a x max a given Hardin T m ^ n that for loading, al, on suggested the relative that for the using the level sands 100. insitu shear s t r e n g t h parameters such as out et. depends e t . a l , (1972) suggested t h a t the u l t i m a t e shear calculated e which of c' and dynamic hyperbolic effective 0' (equation strain curve stresses in (y <l%) and static They pointed by seismic induced c terms 4.7). of strength static T M O V is in ci x s a t i s f a c t o r y f o r dynamic l o a d i n g . Unlike shear stress, c and Y direction c I T , is G analyses present in in one-dimension, the analyses in an initial two-dimensions. influence vs y c shown s of in Fig. initial static 5.1. The shear stress direction i t is ( x In dynamic of x on g m a x + analysis, a v a i l a b l e shear is (T . „ - strength T„) Unloading All the have available shear curve KL 3 are been the in the neglect the and s g the current practice strength about both x and Q and is to to assume Y axes. C that the % c Reloading b a s i c assumptions used phases i n Chapter equations and in x )• a v a i l a b l e shear r e l a t i o n s h i p i s symmetrical 5.2.2.2. The s mcix opposite static i causes the a v a i l a b l e shear s t r e n g t h to be different, the d i r e c t i o n of s h e a r i n g . A t y p i c a l r e l a t i o n s h i p between presence of depending on % dynamic a l s o adopted modified strength. from K i n F i g . 5.2 to For f o r dynamic a n a l y s e s . reflect example i s given to model u n l o a d i n g by, the the effect equation of and However, static for reloading the shear the on unloading 84 Fig. 5.1. Dynamic Shear Stress-Strain Relationship: Skeleton Curve. Fig. 5.2. Dynamic Shear Stress-Strain Relationship: Unloading and Reloading. 85 u where t shear m a t i + G ; K i x = T m ax + T K s» a n '''K d s t r e s s and s t r a i n c o r r e s p o n d i n g The equation " L where T shear M T A X 1 + G } = = 2 T M A X max ? - ( Modelling One n ^YL d a r e t n dynamic e to r e v e r s a l p o i n t K. max - V IY (5.22b) max2 -c , and s s t r e s s and s t r a i n c o r r e s p o n d i n g 5.2.3. a f o r the r e l o a d i n g curve LM, i s g i v e n by, G ( T (5.22a) Y - Y | / maxl max T , y l a e t h dynamic e to r e v e r s a l p o i n t L. the E f f e c t s o f R e s i d u a l Porewater of the important r L Pressure f a c t o r s i n s e i s m i c response s t u d i e s of s o i l d e p o s i t s comprised of s a t u r a t e d c o h e s i o n l e s s m a t e r i a l s i s the i n f l u e n c e of residual porewater porewater p r e s s u r e The generation of significantly pressure x . g pressure generation. The presence of residual reduces the r e s i s t a n c e to d e f o r m a t i o n . presence residual (Finn, porewater static by M a r t i n , pressure model shear pressure e t a l . 1978b, V a i d , model developed The porewater of i n i t i a l i n loose et a l . et a l . adopted stress, 1979). T , a f f e c t s the s saturated sands The porewater (1975), does not account f o r here i s an e x t e n s i o n model of M a r t i n , et a l . (1975) to account f o r the e f f e c t s of x . g of the 86 5.2.3.1 The Behaviour o f Samples w i t h The original procedure T s_ that accounts f o r the presence of x s was presented by Seed and b e h a v i o u r of an element x/a' is i n the f i e l d failure to the such behaviour that stress for typical the is field the equipment vertical /o' y *J soil initial is (45+0'/2) test load present ratio initial ratio on i s used, such that the the the It can be that the from Anisotropically triaxial initial behaviour <* on i t s r In the f o r earthquake T /o*y0, that effective which If simple shear then the the proper plane. and in s However radial stress initial ratio if « on = triaxial pressures ratio r a 0*3 plane from the above hypothesis of Seed and Lee of horizontally layered deposits can be isotropically laboratory under contrary, only very l i m i t e d sample r deduced samples c plane ( f a i l u r e p l a n e ) has the same r a t i o < r . consolidated Comprehensive a[ = r samples such horizontal axial « stress. representative ratio, history. plane the plane (Seed, e t a l . 1973). simply effective stress laboratory stress the f a i l u r e is that on i t s f a i l u r e representative can be a p p l i e d to the h o r i z o n t a l interpreted cyclic static history to be the h o r i z o n t a l vertical to and shear be (1969), a structures, stress i s used apparatus should of i t has the same e x c i t a t i o n can be assumed s hypothesized plane and i s s u b j e c t e d to the same shear s t r e s s field, Then They w i t h an i n i t i a l r consolidated t (1969). = o c and s u b j e c t e d to a shear identical Oy0 Lee consolidated triaxial results triaxial (ACT) t e s t s are a v a i l a b l e various consolidation have (ICT) t o be tests. used i f on the b e h a v i o u r of conditions. On the simple shear t e s t data are a v a i l a b l e . This i s 87 especially true when t e s t data are used and w i t h o u t T A x g * 0. number of researchers t e s t s and al. et and differences ACT section Selig, between the only triaxial of samples, w i t h plane. samples s u b j e c t e d to ICT 1978b; i n this to e x p l a i n the d i f f e r e n c e i n behaviour on the f a i l u r e g Therefore, al. have ACT studied tests 1981). behaviour of the (Seed, There samples behaviour of et a l , 1969; are a number subjected to ICT soil F i n n , et of basic tests and tests. In t y p i c a l c y c l i c triaxial t e s t s on s a t u r a t e d c o h e s i o n l e s s s o i l s the porewater p r e s s u r e and a x i a l s t r a i n s develop w i t h i n c r e a s i n g number of deviatoric time load i s the sum cyclic (or pressure cycles. The porewater of r e s i d u a l porewater t r a n s i e n t ) porewater i s an instantaneous pressure (U ) • of porewater r e s i d u a l v a l u e i s the porewater p r e s s u r e due porewater i s zero. The U Similarly, t p r e s s u r e may be The c f u n c t i o n of c u r r e n t changes i n the mean normal and residual recorded the total cyclic pressure shear to p l a s t i c at any (U) and the porewater which s t r e s s e s and is a the deformation. The when the a p p l i e d c y c l i c load t o t a l porewater p r e s s u r e i s then g i v e n by = U + U recorded t (or permanent) p r e s s u r e pressure increment (U ) (Fig. 5.3), (5.23a) c axial strain e l a s t i c and r e s i d u a l or p l a s t i c a t any time components, also can be separated into 88 8 (SI NUMBER OF CYCLES Fig. 5.3. Permanent and Cyclic Strains or Permanent and Cyclic Pore Pressure. COMPRESSION 0 2D I 50 i K» i ISO i 200 i NO. OF CYCLES . N Fig. 5.4. Cyclic and Residual Behaviour of Pore Pressure and Axial Strain for ICT and ACT Tests (after Selig, et. al, 1981). 89 e = e t e + p (5.23b) c R e s u l t s of Fig. 5.4 tests to et al. p o r o s i t y of and anisotropically (a) was and porewater 0.3 c y c l e s of kg/cm load The is axial number was of test in are presented behaviour samples The cyclic sample (b) response of record the This the was of in between these Ossterschelde deviatoric 0.45 these axial the residual two kg/cm . The 2 tests are two sand with (sample stress However, of cycles quite large. In than f o r the porewater a) for sample axial strain shown over and ACT that The the thereafter of ACT loading The ICT test pressure and at or the peak or ICT of e 200 a developed increased to about rapidly. axial small the cyclic component strain the end test the permanent s t r a i n test After for about 150 the ICT axial approached a l i m i t i n g with strain small. rapidly loading, rapidly value. be increased i n c r e a s e d more test of found to in this test c y c l e s of the end increased the of small of the s t r a i n was mean a x i a l of 60% At the is deviatoric are permanent test t applied values porewater p r e s s u r e (U/a^) r a t i o f o r ACT the c y c l i c s t r a i n remained r e l a t i v e l y ( F i g . 5.4b). ratio for average v a l u e residual z e r o , the test, residual test, the pressure a p p l i e d l o a d was in the Fig.5.4 c y c l i c component remained r e l a t i v e l y c y c l e s w h i l e the the w h i l e the The and in s t r a i n occurs when the strain. porewater pressure shown X axis means strain i s small. w i t h number of 100%, for about t e s t , when the small. ACT were c o n s o l i d a t e d i s o t r o p i c a l l y peak v a l u e s of a x i a l consolidation the Two b). strain maximum. axial until and an stress. components the 1981). (sample 2 and differences about 41.5% symmetrical value. test the pressure The almost ICT illustrate (Selig, initial an the approached 90 The cyclic of d e v i a t o r i c porewater stress. The increase i n deviatoric Secondly which the may net cyclic load dilation porewater p r e s s u r e . deviatoric load r e s u l t s deviatoric cause of the Only c s a t u r a t e d sands and roles. sample. stress This on the will ACT initial residual by of these two a f f e c t e d by porewater pressure to the pressure. a plane, drop effects w i l l I t i s accepted significantly to understand in g i v e the t h a t the behaviour the c y c l i c porewater affects the tests with the d i f f e r e n c e i n behaviour regards to (a| + a-j)/2. Points L s t r e s s s t a t e i n the (q,p*) p l o t porewater time, when no Firstly, lead the limit of behaviour of between ICT porewater p l o t , where q i s the p r i n c i p l e s t r e s s d i f f e r e n c e g i v e n by given magnitude failure g e n e r a t i o n i f both are compared i n e f f e c t i v e s t r e s s space, is the significantly. I t i s easy tests shear pressure. residual depending on i n an i n c r e a s e i n porewater T h e r e f o r e , the sum component U . the vary l o a d p l a y s two causes of of s a t u r a t e d sands are not pressure, pressures pressure deviatoric and M in such as a q, p' (o^ - o" ) and p' 3 F i g . 5.5 f o r the ICT and ACT i n c r e a s e s , the l o a d i s present effective i n the stress triaxial pressure represent tests. the As the s t a t e at any sample, i s g i v e n by, o{ = af 0-3 = a 3 c c - U (5.24a) - U (5.24b) then the c o r r e s p o n d i n g , P a! + ol 1 3 2 p' and q are g i v e n al + al lc 3c 2 " by, (5.25a) F i g . 5.5. q vs p Plot for ICT and A C T T e s t s . 92 and q = a{ This both to - o$ means (5.25b) c that p' axis cycles effective further stress and w i l l of path increments are s t r e s s path w i l l r e s i d u a l porewater &2 ( F i g . (Chern, zero failure cyclic stress close applied, pressure to the 5.5). 1981; line. the sample Kc is 3c* be on the f a i l u r e plotted are will be If failure will the envelope. If it that t h a t maximum Therefore, principles such such b 2 and f o r parallel applied, behave be for a number of T h i s means line. any ACT t e s t i s are they w i l l ratio recorded U s i n g simple geometric that the the maximum any ICT t e s t it can be shown that Chang, 1982), al where values pressure for U = b = "J2" max 2 2 0 the paths s t a t e s above OA and below OB cannot o c c u r . effective 'lc/ high stress load i s towards may move v e r y porewater a effective deviatoric sufficiently stress load the move residual is if t e s t s when the a p p l i e d the load c the {l 1 + K - (K c c anisotropic When Kc = - l)/sin0*} ' consolidation 1 (i.e. ICT , . (5.26) ratio test) defined U m a as is x Kc = simply g i v e n by c r ^ . It experience path It is is should reversal below the be of noted shear positive easy to show that that the stresses q axis reversal ACT t e s t at any time. then r e v e r s a l will occur if, reported If here the i n shear does total stress not stress occurs. 93 °-dy > °3c < c ~ 1) Byrne have K Finn and (1976) c o n d i t i o n of i n i t i a l However, whether consolidated contrast (5.27) shown t h a t there is reversal strain subjected to c y c l i c relative density, ratio, of porewater loading / characteristics of an ACT test the performed under s i m i l a r c o n d i t i o n s . of the porewater which U m cycle, N pressure cause the porewater p r e s s u r e the rate tests. of porewater ratio cyclic loading in in a given sample and applied cyclic stress porewater pressure development ICT the have test, tests to be U /cr3 m and vs c N/N recorded N^Q of 50%. in at any time d u r i n g i s the number The development (Fig.5.6) 5 0 of c y c l e s figure clearly a to shows t h a t i s d i f f e r e n t i n ACT and be done w i t h ICT care, i s a l s o a f u n c t i o n of K . C In saturated presenting laboratory cohesionless liquefaction l c above, liquefaction soil versus in defined as ACT U/o^ i t on dynamic is customary These curves are a p l o t number some results samples p o t e n t i a l curves. °'dy/2o' 3 explained anisotropically F i n n , et a l , (1978b) p r e s e n t e d a p l o t cycles pressure during The i n t e r p r e t a t i o n of these t e s t r e s u l t s should s i n c e N^Q ratio an ratio load the apparatus i s m a i n l y governed by i s the maximum porewater p r e s s u r e i s the number not, pressures, and if a i n anisotropic tests. development in a triaxial compare 2a, i s required samples. pressure consolidation °'dy ' 3c* or progressively to 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 rate reversal l i q u e f a c t i o n i s to be achieved samples The stress c of cycles tests = i . it properties to provide of c y c l i c to liquefaction, N^. is not to Therefore, possible the of stress As reach definition of uAu ufao—mkm~ tubas Number of Cycles to Liquefaction, N Fig. 5.7. L Strength Curves for (a) Ekofisk Sand with D = 85%; (b) Sand with D = 77% (after r Rahman et. al, 1977). r 95 has to be changed cause some s p e c i f i e d amplitude Fig. axial 5.4 equivalent to 5% is fairly in tests. ACT tests double = U m a x be Two liquefaction curves with a single amplitude. about This = U and of without strain g be potential are tests when t curves made from very 1 2 axial potential However, strain curves f o r curves f o r some figures. test second for two (Rahman, et a l . i n shape. f o r ICT the is or the c o n d i t i o n where potential these similar from However, t h i s i s not t r u e are g i v e n i n F i g . 5.7 are above the curves i s present. g the to double of 2 / % as number of c y c l e s liquefaction x cycles r I t can be seen i s because s t r a i n or double amplitude plots o m a x Often i n p r a c t i c e amplitude the X a x i s . to d e f i n e o b s e r v a t i o n s can f o r ACT resistance test, necessary Typical 1977). U Therefore, i n presenting l i q u e f a c t i o n « samples cause to peak) of 5% i s used. symmetrical i t may dense to (Seed, et a l . 1969). (peak maximum s i n g l e amplitude U cycles f o r the ICT record ACT strain strain that to Firstly, the Secondly, the indicating higher c o n c l u s i o n may not be t r u e f o r l o o s e samples ( V a i d , et a l . 1979). In T„, s only p r e s e n t i n g the triaxial number of simple results shear differences were used. tests between samples w i t h and V a i d , et a l . (1979) c a r r i e d w i t h and without % . The 5.2.3.2 The modified a are a l s o to those p r e s e n t e d above. R e s i d u a l Porewater porewater to out c o n c l u s i o n s drawn by t h e i r i n v e s t i g a t i o n and r e s u l t s r e p o r t e d by Seed, (Seed, 1983) very s i m i l a r without include P r e s s u r e G e n e r a t i o n Model p r e s s u r e model proposed the effects of t . g by M a r t i n , et a l . (1975) With regards to was g e n e r a t i o n of 96 porewater p r e s s u r e , presence the liquefaction potential of different in T , has been documented i n the the amount of i t has g residual when x 0. the curve The the porewater p r e s s u r e Recall basic porewater * s three attempts (3.5) components and horizontal the cyclic the equation But strains also component (5.28) a its made to rate account soil r C / — £ i n an Based showed on occur. shear can still which strain. analysis T d^ vo a a number of unidirectional stress cause generation is for these in only effects section. volumetric shear two-dimensions, be y y contributes X used. This adequate to seismic static that v d , is factor of quite is components slip. table porewater liquefaction then safety t e s t s , Pyke, et a l . (1975), to volumetric Seed, et a l . (1975) have a l s o r e v e a l e d conditions, only excitation in furthermore, only c y c l i c shaking strain vertical Ae assumption components of a c c e l e r a t i o n c o n t r i b u t e A n a l y t i c a l s t u d i e s by t o of However, i f i t i s assumed for grain under m u l t i - d i r e c t i o n a l shaking under the 4 vd strain have responsible t h a t a l l three to (5.28) analysis shear s t r a i n ( S e r f f , et a l . 1976), and of shear s t r a i n are limit J response of structures p o s i t i o n of incremental r e a s o n a b l e s i n c e the major s t r a i n t h a t o c c u r s d u r i n g typical is that amplitude, one-dimensional present. there which r e l a t e s the 0 are the model are presented i n t h i s , = C. ( y - C e .) + . vd 1 ' 2 vd Y + ' In and section They a r e : altered, pressure, s t r a i n to c y c l i c shear s t r a i n Ae is generation equation effects. preceeding pressures and under that b u i l d up the shear faster stress multi-directional that than ratio shaking 97 conditions shaking is about 10% conditions. obtained from account for this effect. / A horizontal plane of the soil component can be volumetric shear dynamic the v e r t i c a l accounted or V £ of can the 10% to only for the on the which third either dynamic shear ratio about account modifying the stress by acceleration of unidirectional the results effects by under reducing analysis The for Ae j required test components structure. strain that suggested simple two-dimensional and than They d 'o"vo T less act acceleration the incremental stress ratio V vo0 The porewater p r e s s u r e model of M a r t i n , et a l . (1975) was such that the l i q u e f a c t i o n r e s i s t a n c e curve and porewater p r e s s u r e matches the behaviour with a given obtained by x s / y ' a Here 0 defining condition of as residual the observed of porewater i n the l a b o r a t o r y sample cycles pressure r i s g i v e n by e q u a t i o n the g e n e r a t i o n r a t e of the liquefaction number modified resistance required U = to U ^, max' r curve is reach the where TJ „ max 5.26. R e c a l l the e q u a t i o n f o r E , (1-m) E The term r K adjusted r = mK r in to model procedure to ,, (a vo rn-m ) equation the noting that various K values are r (5.29) and this the also has been liquefaction very similar C± constants l a b o r a t o r y behaviour. accomplish worthwhile (5.29) A through proposed outlined potential in trial Chapter curves may and 3. error It i s generated i n shape to the curves be for obtained 98 for T /tr^ various ratios a r e a s o n a b l e matching of In the and 0^ are Oy and Oy 0 vertical static by are {P} = [Kj equation can in porewater pressure pressures in incremental procedure is Furthermore, the can to be current There that can because to strains. with be achieved the ultimate is D dynamic current known o"y yQ Here and analysis. a from is the obtained (5.30) by the case stresses and strains of the strains The current x m a to to can of the the {p} static and = given be by viewed the to {o}. The porewater analysis. s o f t e n i n g .of also deposit residual effective stress values, x The adopting this the elements. as permanent system can dynamic stress-strain relationship now be analysis compatible system. the triaxial stress from deposit of setting in as a hyperbolic in a response formed and limit the pressures effective stress is a stress o"y is incremental m a x effective obtain {u} elements, G respectively. {U} porewater response modify (5.29), stresses. [K*] matrix these continued w i t h the + displacements, the and y o the that (4.9), used residual components of the used {A} be increases before indicates equation respectively. effective R e c a l l equation using r a This possible. E" vertical stress performed of for initial effective laboratory. these curves i s substituted analysis the computation computing c u r r e n t This in state amount of apparatus. of a residual This sample has porewater i s easy to to be on pressure estimate, the Mohr 99 envelope and i n a t r i a x i a l apparatus the e f f e c t i v e s t r e s s path f o l l o w e d by the sample i s predictable. effective stress hand, and therefore developed cannot path However, followed i n two-dimensional by an element the maximum residual cannot porewater analysis the be p r e d i c t e d before pressure that can be be known b e f o r e the dynamic l o a d i n g . The procedure used to c a l c u l a t e e f f e c t i v e s t r e s s e s can be used to impose l i m i t s on the amount of residual porewater porewater pressure. This can be p r e s s u r e g e n e r a t i o n to occur d u r i n g the Mohr c i r c l e drawn f o r the c u r r e n t effective accomplished by dynamic l o a d i n g stress system allowing only until touches the Mohr envelope. DAMPING MATRIX [ c ] 5.3 Two with soils, fundamentally namely m a t e r i a l different damping damping phenomena and r a d i a t i o n are damping. associated The material damping can be viewed as a measure of energy d i s s i p a t i o n when waves t r a v e l through s o i l s . The l o s s r e g i o n of i n t e r e s t The of energy i s known as r a d i a t i o n material damping v i s c o u s and h y s t e r e t i c damping. depends on dependent. that displacement most which an the v e l o c i t y Whereas i s largely independent of the energy damping. be divided two groups: The energy d i s s i p a t i o n i n v i s c o u s damping or damping strain broadly into rate and i t i s frequency involves f r i c t i o n a l L a b o r a t o r y t e s t s on s o i l in soils samples l o s s of energy approach, have shown t h a t o c c u r s through i n t e r n a l When m o d e l l i n g the n o n - l i n e a r elastic away from the of frequency but depends on the magnitude of dissipation is hysteretic. can of motion hysteretic or s t r a i n . incrementally due to waves t r a v e l l i n g the e f f e c t friction behaviour of s o i l of h y s t e r e t i c damping by has 100 been i n c l u d e d a l r e a d y i n the a n a l y s i s . take into account the e f f e c t Furthermore, a pseudo frequency high small of flow amount V i s c o u s damping i s s t i l l of water i n s i d e of v i s c o u s response damping introduced by needed to the s o i l s t r u c t u r e . i s necessary the n u m e r i c a l to c o n t r o l integration procedure. The damping use of R a y l e i g h due to v i s c o u s e f f e c t s can be accounted f o r through the damping. The damping matrix [c] i s given by a linear of [M] and [ K ] g i v i n g , combination [C] = a [ M ] + b [ K ] (5.31) i n which a and b a r e c o n s t a n t s . The stiffness matrix a n a l y s i s and t h e r e f o r e [c] But knowing hysteretic that and t=0. can be varies with time would have to be computed the amount of v i s c o u s damping, time s t e p s may [K] damping during the f o r every i s very time s t e p . s m a l l compared the time consuming procedure of computing [c] be u n n e c e s s a r y . evaluated using Therefore, the tangent [c] i s assumed stiffness dynamic to at a l l to be a c o n s t a n t matrix [K ] T at time Then, [C] = a [ M ] + b [ K ] T (5.32) T = 0 T h i s w i l l g i v e a damping ratio X f o r the n t h mode a s , bo) n 2(JO n 2 (5.33) 101 where the i s the n t h mode f r e q u e n c y . mass-proportional the frequency proportional while the of the damping stiffness i s inversely proportional shows that the p r o p o r t i o n a l to component is (1975) proposed be used and suggested that only s t i f f n e s s - p r o p o r t i o n a l damping using, a = 0 and b = 0 . 0 0 5 = 0.0025u> n It lower should modes of unnecessary implies to that Therefore, using remembered vibration include govern higher ratio due that the response, mode increases to h i g h soil and components. structures therefore, The linearly frequency be damped out s i g n i f i c a n t l y . proportional i n typical equation with only i t is (5.35) the f r e q u e n c y . components of the input T h i s i s one of the advantages of damping r a t i o s f o r s o i l s . From the e q u a t i o n the damping r a t i o a t the fundamental frequency i s g i v e n by, \ Typical be the damping stiffness (5.35) (5.35) n the response motion w i l l (5.34) f o r a and b i n ( 5 . 3 3 ) one g e t s , In s u b s t i t u t i n g these v a l u e s X directly to the f r e q u e n c y . Lee should component (5.33) Equation L = 0.0025 periods 1.5 s e c . This (5.36) W l of v i b r a t i o n of s o i l means that the t y p i c a l structures damping may ratio vary between 0 . 5 to f o r the fundamental mode at the s t a r t of the dynamic l o a d i n g v a r i e s between 1% - 3%. 102 The average s t i f f n e s s less when than the s t i f f n e s s using the e f f e c t i v e damping 5.4 of a s o i l at the s t a r t damping matrix r a t i o may structure d u r i n g dynamic of the dynamic [c] computed loading. using loading i s Therefore, [K] t = 0 , the be h i g h e r than the range shown above. BOUNDARY CONDITIONS Appropriate boundary conditions in terms of forces or excitation, two d i s p l a c e m e n t s have to be s p e c i f i e d a t a l l b o u n d a r i e s . In the dynamic analyses involving earthquake types of bottom boundary c o n d i t i o n s are o f t e n used: 1. A f i x e d bottom boundary l o c a t e d at the top of a r i g i d general, base rock or a s t i f f soil layer layer. can be assumed In to be rigid. 2. A bottom boundary located at the top of a rock with constant e l a s t i c p r o p e r t i e s . g e n e r a l l y known as t r a n s m i t t i n g For specified conditions rigid both the above a t the bottom i s used, the domain layer. leading This procedure reduces Three soft the earthquake motion i s I f the second type of bottom of i n t e r e s t or boundary. boundary need not be extended down to a the number to a r e d u c t i o n i n c o m p u t a t i o n a l c o s t s . presented here o n l y the f i r s t layer T h i s type of boundary i s boundary c o n d i t i o n s boundary. soil of degrees of freedom In the method of a n a l y s i s type of boundary c o n d i t i o n i s c o n s i d e r e d . types of l a t e r a l boundary c o n d i t i o n s a r e commonly used i n two-dimensional dynamic problems, i n v o l v i n g f i n i t e element p r o c e d u r e s : 103 1) boundaries a r e l o c a t e d s u f f i c i e n t l y that wave reflection minimized. On does these not f a r away from a s t r u c t u r e so occur boundaries, during forces, analysis or i s displacements or a combination of f o r c e s and d i s p l a c e m e n t s can be s p e c i f i e d . 2) viscous boundaries radiating waves by are a used which series of c o n s t a n t or v a r i a b l e p r o p e r t i e s 3) attempt dashpots (Lysmer to and absorb the springs with and Kuhlemeyer; 1969) c o n s i s t e n t boundaries can be p r o v i d e d c l o s e to the f o u n d a t i o n of structures. response used t o model which a the dynamic frequency can be o b t a i n e d problem to reproduce the f a r f i e l d i n a way c o n s i s t e n t w i t h the f i n i t e element f o r m u l a t i o n formulating Of These boundaries attempt problem. dependent by s o l v i n g This boundary the e l a s t i c i n a l a y e r e d h a l f - s p a c e (Lysmer the t h r e e types of l a t e r a l stiffness wave and Wass; boundaries, by matrix propagation 1972). the a n a l y s i s w i t h the consistent boundaries accuracy. I n the a n a l y s i s w i t h c o n s i s t e n t b o u n d a r i e s , o n l y a s m a l l r e g i o n needs i s by f a r s u p e r i o r i s accomplished to be c o n s i d e r e d , thus to the o t h e r s w i t h r e d u c i n g the number of degrees But u n f o r t u n a t e l y , the f o r m u l a t i o n i s s t r i c t l y (or iterative applicable linear) problems and f o r s o l u t i o n s lateral boundaries r e s p e c t to of freedom. o n l y to i n the f r e q u e n c y linear domain only. The the time i n an i n c r e m e n t a l l y e l a s t i c domain t h e r e f o r e should be l o c a t e d as p r a c t i c a b l e . approach i n as f a r away from a s t r u c t u r e The u s u a l way to model the l a t e r a l boundaries i s to a l l o w the nodes on these boundaries to move o n l y i n the h o r i z o n t a l direction. 104 5.5 ANALYSIS OF The the the SOIL - STRUCTURE SYSTEMS response of a s t r u c t u r e local soil spatial affected. conditions. distribution By be of i n c l u d i n g the response a n a l y s i s , may The the founded on a s o i l d e p o s i t peak a c c e l e r a t i o n , the response structure coupled i n the seismic i s affected frequency characteristics finite content may response of the soil and and all element domain f o r by be the structure determined. The presence of the increases the deposit. It additional inertia stress forces. dependent structural analysis are predict has effective two major e f f e c t s on stresses and it also T h e r e f o r e , f o r s o i l s which e x h i b i t behaviour, systems cannot structure an uncoupled uncoupled the may response not of analysis be buried in which applicable. structures the soil provides non-linear soil and Uncoupled where strong s o i l - s t r u c t u r e i n t e r a c t i o n occur. 5.5.1 S l i p Elements i n Dynamic Relative during slip strong d i s p l a c e m e n t which may shaking can be element model d e s c r i b e d dynamic loading element, the acceleration computing equation bottom and nodes. Ideally, bottom nodes v e l o c i t i e s (Nadim accelerations (5.5), modelled and different In order occur between s o i l and using i n Chapter 4 has conditions. top and Analysis the and will overcome by elements. However, to be m o d i f i e d f o r use the slippage element Whitman; velocities values to of once slip structure stops should 1982). numerically have the under in a slip the same However, when integrating be unavoidable for the this difficulty, when no top the and slippage 105 occurs In a s l i p element, the v e l o c i t i e s and a c c e l e r a t i o n s of the bottom nodes were made equal to those of the top nodes. 5.6 SOLUTION SCHEME A step by step dynamic response solution scheme i n the time domain. i s carried A brief out to o b t a i n the o u t l i n e of the procedure i s g i v e n below: 1) based at on time the the current t the tangent elements initial values shear of modulus using the c u r r e n t loading, unloading G G t , max' max and i s calculated stress-strain or T v. ' t for all curve, f o r e i t h e r reloading whichever is appropriate. 2) the g l o b a l s t i f f n e s s m a t r i x [ K ] 3) knowing t {x}, {ACT} the base {x}, acceleration {x} at a r e computed methods to s o l v e time t i s determined. v a l u e a t ( t + A t ) , new v a l u e s f o r (t + At) and increments, Ay and by employing any of the d i r e c t integration the e q u a t i o n s (5.5) as d e t a i l e d i n Appendix III 4) i f stress time or s t r a i n interval, s h o r t e r time 5) reversal occurs i n any element A t , the dynamic analysis during i s repeated this for a interval. using the shear strain and then strain porewater (5.28) and ( 5 . 2 9 ) . increment, increments i n volumetric p r e s s u r e a r e computed u s i n g e q u a t i o n 106 6) using loads increments static effective i n residual analysis stresses. update and A program computer is porewater pressure performed These e f f e c t i v e to U, as determine stresses virtual current a r e then used to values. X ] n a x TARA-2 has been developed incorporating a l l these b a s i c s t e p s . 5.7 COMPUTATION OF POST EARTHQUAKE It points on referred i s often the s o i l to as earthquake required record with saturated cyclic sands conditions. enough the end of an tailing zeros To should compute be used This i s these, an so t h a t the can be i n c l u d e d i n the a n a l y s i s . silts, are assumed to occur under undrained i n these s o i l s when the pressure d i s s i p a t e s . undrained silts the p o t e n t i a l slip, i s reflected pressure simple shear volumetric as r e s i d u a l dissipates, sample r e s u l t i n g i n settlement. dissipation residual of at various earthquake. displacements. There w i l l be a d d i t i o n a l deformations In follows. the d i s p l a c e m e n t s and permanent components of d i s p l a c e m e n t response f o r and r e s i d u a l porewater porewater at residual f r e e damped v i b r a t i o n response The to p r e d i c t structure dynamic DEFORMATION cyclic strain tests e j, porewater on s a t u r a t e d sands and which y < occurs pressure. volumetric strain, When e V ( due to grain the r e s i d u a l j occur i n the The p r e d i c t i o n of d e f o r m a t i o n s due to the porewater pressure can be accomplished as 107 One method is to treat this problem c o n s o l i d a t i o n problem w i t h known i n i t i a l porewater for A the d e f o r m a t i o n second method at discrete i s to compute time intervals deformations as a two-dimensional p r e s s u r e s , and to s o l v e as the d r a i n a g e using occurs. the v o l u m e t r i c strain is accumulated at accomplished volumetric Chapter 4. end of treating strain e this in y the dynamic loading, volumetric shear-volume e j» This V( strain the coupling same was can be way modelled the in In the program TARA-2 the second method i s used. The porewater by the final or post earthquake p r e s s u r e has d i s s i p a t e d deformation after the i s the sum of the d e f o r m a t i o n residual calculated ic from the modified e y ( j and the residual p r e d i c t e d a t the end of the dynamic a n a l y s i s . dynamic deformation 108 CHAPTER 6 VERIFICATION OF THE METHOD OF ANALYSIS The prototype validation data for of computational comparison. The techniques common shaking table r e p r e s e n t the range of i n - s i t u p r e s s u r e s e x p e r i e n c e d by in the data field. from those application centrifugal self-weight to For of of the the stress-strain models scale prototype Here i n - s i t u the and soil prototype. response scale from design similarity event p r o p e r t i e s of be matched i n model and deduce where prototype. full to f u l l can requires achieved t h e r e f o r e , the stress the scaled dependent cohesionless s o i l s ) can C e n t r i f u g e m o d e l l i n g laws are used model response. The to principles of (1981). CAMBRIDGE CENTRIFUGE TESTS by Lee (1983) u s i n g the Cambridge U n i v e r s i t y Cambridge c e n t r i f u g e equipment Schofield (1981). island 200mm. that require directly A number of c e n t r i f u g e t e s t s on submerged i s l a n d s were The elements with c e n t r i f u g a l m o d e l l i n g have been d i s c u s s e d i n d e t a i l by S c h o f i e l d 6.1 cannot problems we s t r e s s e s are (especially test the s o i l be good The the slopes was 3:1 Fig. 6.1 and test shows the a 90mm h i g h w i t h and has a prototype crest width procedures model island of 8m. 3:1 Full have island s i d e s l o p e s at c e n t r i f u g e a c c e l e r a t i o n used corresponding Centrifuge. used and The of height d e t a i l s of been given i n the a crest i n the t e s t s was is conducted by tests. width of 40g. T h i s means 3.6m, with structural l o a d i n g on side the I LVDT 82260 720mm • I I LVDT 82273 ACC 734 Mild steel PPT 2 338 ACC 90mm 1244 • plates 3 OPPT 2332 Concrete base PPT 2331 C o n t a i n e r Base ACC 258^ 750 mm 776 mm • LVDT Direction of positive a c c e l e r a t i o n measured on accelerometers Pore p r e s s u r e *• Fig. transducer Accelerometer 6.1. Submerged Island Showing Transducer Locations. o KO 110 island The was simulated island rests by on using a centrifuge container. mild concrete The steel base p l a t e s of which base shaking was in various turn generated is thicknesses. bolted to the when the r o t a r y arm on wheels f o l l o w s a t r a c k mounted on the w a l l of the c e n t r i f u g e chamber. The island accelerometers, was instrumented test soil p a s s i n g between B.S. was 60 - 70%, 6 DJB A23 are a l s o shown i n F i g . was fine Leighton-Buzzard s i e v e s i z e No. with e m a x = 1.03 120 and e m i n and No. = 0.63 N 2 times would pore f l u i d , faster only excess N times sand, mostly 200. The (Lee, fluid of s i l i c o n e o i l was In the N times Here N is relative density the scale wave-form was intended to be of However, the 0.5secs. complicated due i n t e r f e r i n g with that carried to i s the same as of water i n the out 12 the sinusoidal actual resonances on input and model i n Lee's island, to mechanical of the to use test. A a pore special tests. the pulses with motion factor T h e r e f o r e , to model c o n d i t i o n , i t i s necessary used as the model pore f l u i d tests size i n the p r o t o t y p e , w h i l s t the earthquake faster. porewater d r a i n a g e viscosity of 1983). c e n t r i f u g a l a c c e l e r a t i o n g i v e n as a r a t i o of g r a v i t y . the p r o t o t y p e The pore p r e s s u r e s would be a b l e to d i s s i p a t e i n the model than occur 2 LVDTs. 6.1. In c e n t r i f u g a l m o d e l l i n g , i f the model pore f l u i d the p r o t o t y p e piezoelectric 6 Druck PDCK81 pore p r e s s u r e t r a n s d u c e r s and l o c a t i o n of these instruments The by the theoretical a constant island linkage input period was more clearances the i n p u t motion, e s p e c i a l l y d u r i n g the i n i t i a t i o n of the base motion. The writer. two The results average of two contact centrifuge tests prototype t e s t s were 15kPa ( T e s t 1) and were made a v a i l a b l e pressure on the islands 31kPa ( T e s t 2) r e s p e c t i v e l y . to the for these The input Ill a c c e l e r a t i o n s measured by the a c c e l e r o m e t e r mounted on top of the c o n c r e t e base for recorded 0.17g Test 1 and Test are The The island comparative study F i g . 6.2(a), 1 and very out Test and The 2 were O . l l g differently to p r e d i c t (b). and i n these two performance of both chapter. COMPARATIVE STUDY density D The sand = 65%. r dense sand Table 6.1. of The UBC simple Fig. T , G can are not water zero curve. t /°" s The v 0 for static test and without liquefaction explained be in selected initial was at are dynamic an resistance. were analysis Chapter 5, the The T . are given (t =0) g matches and test i s shown i n the predicted pressure f o r the scaling the r of data on samples w i t h only f o r the laboratory pore- changes i n l i q u e f a c t i o n curves which correspond by K model behaviour to i g n o r e the changes i n the to account in o b t a i n e d by u s i n g porewater Since G decided to non liquefaction done by u s i n g a v a i l a b l e l a b o r a t o r y d a t a on medium dense Ottawa sand ( V a i d , et a l . resistance can curve bias curve relative when porewater p r e s s u r e model c o n s t a n t , rates, obtained s c a l i n g was static resistance stress, i t was generation any average c o n s i s t e n t f o r medium a p p r o p r i a t e l y to account static available, pressure liquefaction 65% model p r o p e r t i e s which r e s i s t a n c e curve As samples w i t h the apparatus This 0.012. in Typical = r shear 6.3. constants D used l i q u e f a c t i o n r e s i s t a n c e curve f o r the sand liquefaction = in responded carried the t e s t s are r e p o r t e d i n t h i s 6.2 shown maximum base a c c e l e r a t i o n s i n T e s t respectively. tests. 2 be 1979). specified by The changes i n the associating the liquefaction appropriate K r ~i 3.0 1 4.6 r 1 5.0 Time in Sees Fig. 6.2(a). Input Base Motion in Test 1. • rc l_ tu —i _| max = 0.17g —i 2.0 1 1 1 3.0 1 1 4.0 1 5.0 1 1 6.0 i r - 7.0 Time in Sees Fig. 6.2(b). Input Base Motion in Test 2. 8.0 113 T a b l e 6.1 Properties Total u n i t weight kN/m 3 Bulk modulus exponent c o n s t a n t n Angle of i n t e r n a l Effective K 2 m a x friction Static Dynamic 18.1 18.1 high 0.4 19.3 55.0 38.0 38.0 0.0 0.0 0.45 Q a, b v a l u e s used C^-K^ (K ) cohesion Coefficient Properties 800 B u l k modulus constant Shear modulus parameter Soil to compute [C] Constants Rebound modulus c o n s t a n t s 0.0,0.005 0.75,0.79,0.459,0.73 m,n 0.43,0.62 04\ Number of Cycles to Initial Liquefaction, Fig. 6 . 3 . Liquefaction Resistance Curve of Medium Dense Leighton-Buzzard Sand. 115 value with each static interpolation may be used intermediate -u /a . s For no slip 6.2.1 each t o get t h e K model test between ratio, r two a n a l y s e s soil The f o l l o w i n g s l i p value "^/^vo* Linear corresponding t o an" performed. One w i t h with slip element p r o p e r t i e s were used, g = 6.3 x 1 0 kN/m /m, % C g = 0.0 5 were and s t r u c t u r e and the o t h e r K » 6.3 x 1 0 kN/m /m 2 0 5 2 = 35° g Results of Test 1 The recorded acceleration have v e r y h i g h frequency type of h i g h testing (1983), suggested value noise due t o ambient i t i s necessary shown scheme, value histories o f t h e model island i s unavoidable sources such quantities. here suggested have as the e l e c t r i c out v e r y high The computed and r e c o r d e d been by Dean a t any p o i n t i n time the p r e v i o u s to f i l t e r smoothed (1983). once using In using This i n centrifuge the c e n t r i f u g e , and a l s o due to c e n t r i f u g e v i b r a t i o n s . histories current electrical as i t may o r i g i n a t e components from output average time components which c o n t a i n n e g l i g i b l e energy. frequency motor d r i v i n g time stress vo elements elements. shear Dean frequency acceleration a three this point scheme, the i s computed as the sum of 1/4 o f the p o i n t , 1/2 the v a l u e o f t h e c u r r e n t p o i n t and 1/4 o f the v a l u e o f the next p o i n t . Fig. acceleration During 6.4 t o F i g . 6.6 show time the f i r s t histories t h e smoothed of a c c e l e r o m e t e r s 1.5 seconds o f shaking recorded A1244, and computed A1225 and A734. low a c c e l e r a t i o n s w i t h v e r y high 116 ~i aO 1.0 2.0 1 3.0 1 1 4.0 r 5.0 6.0 Tine i n Sees Fig. 1.0 2-0 3.0 1 r 4.0 Time I n Fig. 8.0 6.4(a). Recorded Acceleration of ACC1244 in Test 1. -i Q.D r "5.0 5.0 r T "7.0 Sees e.c 6.4(b). Computed Acceleration of ACC1244 in Test 1. (with and without Slip Elements). 117 -r 6.0 ! "c r — T 7.0 \ 8.0 Tine in Sees Fig. 6.5(a). Recorded Acceleration of ACC1225 in Test 1. Acceleration of ACC1225 * Teat 1. Fig. 6.5(b). Computed (with and without SUp Elements). Fig. 6.6(a). Recorded Acceleration of ACC734 in Test 1. Pig. 6.6(b). Computed Acceleration of ACC734 in Test 1. (with and without Slip Elements). 119 frequency of were r e c o r d e d acceleration steadily 5.5 seconds and then The without slip i n c r e a s e d as i n the case acceleration t h a t the amplitude of i n p u t motion, upto time histories computed by TARA-2 with and elements a r e v e r y s i m i l a r and t h e r e f o r e , o n l y one of them i s the computed recorded After subsided. shown i n F i g . 6.4.(b), of i n a l l accelerometers. through acceleration response values. F i g . 6.6(b). response Table The frequency are very 6.2 shows similar and magnitude to corresponding the computed and recorded maximum a c c e l e r a t i o n of a l l three a c c e l e r o m e t e r s . T a b l e 6.2 Recorded and Computed Maximum A c c e l e r a t i o n s Maximum A c c e l e r a t i o n Instrument Location % 8 Computed by TARA-2 Recorded Without S l i p Elements With S l i p Elements A1244 13.3 11.6 11.6 A1225 15.9 12.5 12.5 A734 13.9 12.7 12.7 The comparison between recorded and computed maximum a c c e l e r a t i o n v a l u e s a r e v e r y good. Four porewater experimentally and computed (b), (d) . ( c ) and developed. During In pressure development by TARA-2 a r e presented this test very plots i n Figures low porewater the low l e v e l shaking of the f i r s t obtained 6.7 ( a ) , pressures second, were the response 120 Recorded Computed With and Without S l i p Elements I 1 I 1 4.0 5.0 Tine In Sees Fig. 1 1 1 1 "7.0 -6.0 1 1 6.0 1 at 6.7(a). Recorded and Computed Porewater Pressure of PPT2330 in Test 1. x i i.O x x— Recorded Computed Without S l i p Elements Computed With S l i p Elements r 5.0 1 Time In Seas Fig. 6.7(b). Recorded and Computed Porewater Pressure of PPT68 in Test 1. 121 Fig. 6.7(c). Recorded and Computed Porewater Pressure of PPT2338 in Teat 1. Recorded Computed With and Without S l i p Elements i e.c Fig. 6.7(d). Recorded and Computed Porewater Pressure of PPT2342 in Test 1. 122 of the i s l a n d the is essentially instantaneous pressures response develop from elastic to the and changes elastic porewater p r e s s u r e s in total coupling stress. of soil recorded Such and porewater water. Later d u r i n g the p e r i o d of more severe s h a k i n g , p l a s t i c v o l u m e t r i c s t r a i n s resulting in the development of independent of the i n s t a n t a n e o u s pressures during components. next two this instantaneous to have zero. porewater s t a t e s of s t r e s s . both A f t e r 6 seconds of shaking seconds During pressures The residual which recorded and occur are porewater instantaneous the i n p u t motion s u b s i d e s over the this time the magnitude of the depending on the component of porewater p r e s s u r e i s s m a l l . Drainage drainage time residual are may occur characteristics of during the the sand. excitation Since generation of porewater p r e s s u r e a f t e r 6 seconds of e x c i t a t i o n i s v e r y s m a l l , changes i n porewater pressure can examination reveals be caused of recorded t h a t the only by drainage during porewater p r e s s u r e s a f t e r porewater pressures a constant. indicating At this location movement of water s i n c e these changes are time. 6 seconds of A center s m a l l i t i s reasonable at the of the i s l a n d are more or porewater p r e s s u r e towards the close excitation i n a l l f o u r l o c a t i o n s except t r a n s d u c e r P2342, which i s l o c a t e d at the middle less this of the increases slightly island. However, to assume t h a t d r a i n a g e i n the i s l a n d i s n e g l i g i b l e d u r i n g the base e x c i t a t i o n . TARA-2 computes o n l y r e s i d u a l no fluctuations computed curves. excitation consistently. the due to changes Furthermore, computed in porewater p r e s s u r e s , instantaneous s i n c e no residual drainage porewater stress was so there levels assumed pressures are in the during the increase However, the r a t e of i n c r e a s e i n porewater p r e s s u r e s during 123 low l e v e l e x c i t a t i o n which occur b e f o r e 1.5 seconds and a f t e r 6 seconds i s r e l a t i v e l y small. When structural rigid element joint and an dynamic s t r a i n s developed to compatibility slip connection adjacent i s assumed soil i n the a d j a c e n t requirements may occur between s o i l soil soil, in a heavy, an analysis, element are v e r y due However, by i n t r o d u c i n g the r e l a t i v e movement which i n d i c a t e t h a t computed porewater p r e s s u r e s are d i f f e r e n t o n l y i n porewater pressures incorporates assumes slip rigid be computed and transducer 6.2.2 as the available structure are slightly between soil in higher and The the than the structure. slip predicted analysis which analysis The that comparison mean for recorded a l l six maximum porewater values after transducers. residual porewater the model tests p r e s s u r e s , which the Table excitation 6.3 pressures shows at can has the a l l the locations. R e s u l t s of T e s t 2 smoothed accelerometers 6.10. structures. without computed porewater p r e s s u r e s i s good. recorded The of the However, maximum r e s i d u a l interpreted are the f o u r porewater p r e s s u r e time h i s t o r i e s from available. subsided, below below elements, between r e c o r d e d and are just connection Only just and be elements, located TARA-2 a n a l y s e s , w i t h can for. transducers from the small s t r u c t u r e d u r i n g s t r o n g base e x c i t a t i o n results stiff accounted the The and element i n displacements. elements between the s t r u c t u r e and between recorded and computed acceleration time A1244, A1225 and A734 are shown i n F i g s . 6.8 histories through T a b l e 6.3 Recorded and Computed Maximum R e s i d u a l Porewater Pressures R e s i d u a l Porewater P r e s s u r e , kPa Computed by TARA-2 Transducer Location Recorded Without S l i p Elements With S l i p Elements P2330 0.4 0.4 0.4 P2331 0.4 0.3 0.4 P2332 4.0 2.9 3.7 P68 3.0 1.5 4.4 P2338 2.4 0.6 1.8 P2342 4.0 4.2 4.2 125 T I 2.0 3.D 4.0 T" 5.0 Time In Sees Fig. 6.8(a). Recorded Acceleration of ACC1244 in Test 2. Fig. 6.8(b). Computed Acceleration of ACC1244 in Test 2. (with Slip Elements). + 25% 1 13 1 1 1 1 2.0 LC 1 1 3.0 1 1 1 1 5.0 4.0 1 6.0 Time In Sees I 1 ">.0 I I 8.0 Fig. 6.9(a). Recorded Acceleration of ACC1225 in Test 2 . + 2 6% "1 a0 1 1 1.0 1 1 2.0 1 1 1 3.0 1 1 4.0 1 1 5.0 1 6.C 1 1 "7.0 1 ' 8.TJ Time i n Sees Fig. 6.9(b). Computed Acceleration of ACC1225 in Test 2 . (with Slip Elements). 127 0.0 2.0 "i i 1 3.0 1 4.0 1 r 5.0 i 6.0 1 1 r "7.0 s.c Time in Sees Fig. 6.10(a). Recorded Acceleration of ACC734 in Test 2. Time in Sees Fig. 6.10(b). Computed Acceleration of ACC734 in Test 2 (with Slip Elements). 128 The following second base e x c i t a t i o n characteristics. and then f o r Test Low the amplitude level f o r 2 seconds excitation of a c c e l e r a t i o n c o n s t a n t maximum v a l u e i n the next maintained 2, shown i n F i g . 6.2(b), two seconds. and then has the occurs i n the first increases steadily to a T h i s maximum amplitude i s i t subsides over the next 3 seconds. The a c c e l e r a t i o n r e c o r d s , except the r e c o r d o b t a i n e d i n a c c e l e r o m e t e r A734 which the i s located on top of acceleration amplitude being acceleration histories recorded structure, similar to t h a t show the v a r i a t i o n of the i n p u t motion. of The on the top of the s t r u c t u r e dropped to v e r y low v a l u e s a f t e r 4 seconds of e x c i t a t i o n . Unlike found Test 1, to be d i f f e r e n t response amount the response from computed a n a l y s i s a l s o showed t h a t when s l i p of slip accelerations occurs without between slip when s l i p elements a r e used histories shown using those computed slip i n Figs. elements soil elements and 6.8 without slip elements elements. structure. than were The significant The computed the v a l u e s computed I n the computed to F i g . 6.10, o n l y are presented. slip elements a r e used a r e lower i n the a n a l y s i s . with acceleration a c c e l e r a t i o n s computed L i n e s of c o n s t a n t a c c e l e r a t i o n s have been drawn i n F i g s . 6.8 to 6.10 t o a i d the i n t e r p r e t a t i o n of r e s u l t s . The similar observed spikes) accelerometers between closer of computed to t h a t of the i n p u t motion. which were sharp variation and look maximum at and computed the other frequencies computed accelerations a r e shown i n Table are very values suggests of 6.4. corresponding satisfactory ( i n d i c a t e d by f o r a l l three The a c c e l e r a t i o n s a r e not good. peak computed a c c e l e r a t i o n h i s t o r i e s histories The maximum r e c o r d e d a c c e l e r a t i o n s to be a s s o c i a t e d w i t h h i g h i n the i s l a n d recorded acceleration comparison However, recorded comparison. a and 129 The r e c o r d e d a c c e l e r a t i o n a t any time has two components: a c c e l e r a t i o n component t r a n s m i t t e d through the s o i l acceleration t r a n s m i t t e d to s o i l through the component centrifuge container due to the c o n t a i n e r computed a c c e l e r a t i o n h i s t o r y accounts transmitted through high the s o i l . from the base and the side walls itself and top of vibrating. The o n l y f o r the a c c e l e r a t i o n component The a c c e l e r a t i o n s t r a n s m i t t e d frequency the components. The presence through the container have o f these high frequency components may be r e s p o n s i b l e f o r the d i s c r e p a n c i e s between the recorded and computed a c c e l e r a t i o n h i s t o r i e s . T a b l e 6.4 Recorded and Computed Maximum A c c e l e r a t i o n s Maximum A c c l e r a t i o n , % g Accelerometer No. Computed Recorded level With S l i p Elements 24.0 15.1 18.2 A1225 42.5 15.5 23.1 A734 23.9 15.8 18.2 presented Test Without S l i p Elements A1244 Four are by TARA-2 1, very recorded i n Figs. high and computed porewater p r e s s u r e 6.11 ( a ) , ( b ) , ( c ) and ( d ) . porewater pressures of e x c i t a t i o n o f the f i r s t recorded. With the onset were second very of more severe development In t h i s developed. test During plots unlike the low low porewater p r e s s u r e s shaking, very high were porewater 130 Fig. 6.11(a). Recorded and Computed Porewater Pressure of PPT2330 in Test 2. Fig. 6.11(b). Recorded and Computed Porewater Pressure of PPT68 in Test 2. Fig. 6.11(c). Recorded and Computed Porewater Pressure of PPT2338 in Test 2. Fig. 6.11(d). Recorded and Computed Porewater Pressure of PPT2342 in Test 2. 132 pressures P2330. 6 were developed A c l o s e examination seconds of excitation transducer P2342, significant dissipation in in this test a l l transducers is that, except located of porewater at the pressure The the transducer porewater middle of occurred. This and after pressure the island, i s because leading t r a n d u c e r P2342 behaved d i f f e r e n t l y too f a r from the f r e e d r a i n i n g boundaries to h i g h because i t i s at t h i s l o c a t i o n inward flow water o c c u r s . The are well and f o r the very h i g h porewater p r e s s u r e s were developed pressure g r a d i e n t s . of in of the recorded porewater p r e s s u r e p l o t s reveals which except very comparison between computed and good inside the without pressures elements comparison When s l i p developed for the island. slip very At these gave under the low. between elements were used in the maximum transducer when by slip computed analysis porewater response elements below the i n those elements. porewater the structure Table 6.5 pressures are were The located porewater slip provided the improved. h i g h shear resulting in strains higher shows the computed at with without pressures analysis pressures the a n a l y s i s results. elements i n the residual locations similar structure and porewater (P2330, P2342) which very But, predicted porewater p r e s s u r e s recorded transducers elements predicted are two recorded a l l the and transducer locations. 6.3 APPLICABILITY OF THE The steady recorded increase in METHOD OF ANALYSIS r e s i d u a l porewater p r e s s u r e s are i n t e r p r e t e d as porewater n o i s e from ambient sources pressures. Therefore, any on r e c o r d e d v a l u e s do not a f f e c t high the the frequency comparison T a b l e 6 .5 Recorded and Computed Maximum R e s i d u a l Porewater Pressures Maximum R e s i d u a l Porewater P r e s s u r e , kPa Transducer No. Computed by TARA-2 Recorded Without S l i p Elements With S l i p Elements P2330 1.0 1.5 1.5 P2331 0.9 1.1 1.1 P2332 10.5 12.0 12.1 P68 38.0 6.4 38.1 P2338 18.0 2.2 . 18.9 P2342 22.0 19.8 21.3 134 between recorded comparison frequency in and computed porewater a c c e l e r a t i o n response noise affects the pressures. is made comparison. However, point by Therefore, since point, caution any the high should be e x e r c i s e d when comparing maximum a c c e l e r a t i o n s . When a sand sample i s s u b j e c t e d change in the approaches the various Any line, effective low to stress-strain applicable only transformation for as abruptly few degrees the The are less stress to The than path p o i n t s at which assumed line. abrupt the l i e on slope of failure s t r e s s e s i n very dilation sands and i n medium dense or relationship the loose region assumed of i n this stress dense line. sands. below the may increased thesis i s space a The strictly the phase line. Figs. 6.12 t h a t were f o l l o w e d by which porewater (a) here are and (b) f o r the followed time h i s t o r i e s during validity of the response s t r e s s paths are by the dynamic computed elements two the f a i l u r e l i n e i s q u e s t i o n a b l e . q, p' plot The s t r e s s paths elements were i n c l u d e d . i n Test 1 are well below the elements are on the f a i l u r e l i n e f o r loading. in in a to the l o c a t i o n s f o r available. a n a l y s i s i n which s l i p l i n e , where as i n T e s t 2, sometime show the f o u r elements which correspond pressure s t r e s s paths failure occurs, transformation is a effective s t r e s s e s due hyperbolic The direction phase line path ( I s h i h a r a , et a l . 1975). change the stress l o a d i n g , an (or monotonic) l o a d i n g beyond the phase t r a n s f o r m a t i o n l i n e i n very reported of line called transformation cyclic result failure s t r e s s paths straight phase direction to undrained Test Under 2 after these circumstances elements have the reached Fig. 6.12(a). Effective Stress Paths in Test 1 Fig. 6.12(b). Effective Stress Paths in Test 2. ON 137 CHAPTER 7 APPLICATION OF THE METHOD OF ANALYSIS: TANKER ISLAND RESPONSE 7.1 INTRODUCTION Man extensively Beaufort complex made as Sea. islands drilling of cohesionless p l a t f o r m s f o r o i l and R e c e n t l y , as exploration has i s l a n d (De Jong, et a l . are examples. two moved typical reduce down water These the amount of f i l l newer type material depth of suction dredges berms are c o n s t r u c t e d from an o f f s h o r e and/or as a s l u r r y through a p i p e l i n e d i r e c t l y in the The tanker i s l a n d procedures a l s o reduce some The maximum s e t the tanker i s l a n d s Therefore, i n the case of up to the s e t down water by dumping sand excavated onshore borrow p i t and pumped onto the l o c a t i o n of the i s l a n d . Once the sand berm i s ready, a s e r i e s of c a i s s o n s or t a n k e r i s brought to l o c a t i o n and b a l l a s t e d onto the berm, and b a c k f i l l e d w i t h sand, g r a v e l or water. The d r i l l i n g Because sand and i s l a n d and sand berm i s c o n s t r u c t e d Most of the sand steel i s l a n d beaches. f o r the c a i s s o n - r e t a i n e d deep water a underwater the used to deep waters, more construction required depth i s f i x e d , g e n e r a l l y around 6 to 9 metres. by been exploration (1978), and of the hazards of wave l o a d i n g on exposed depth. gas have forms of i s l a n d c o n s t r u c t i o n procedures have been i n t r o d u c e d . caisson-retained greatly soils is often of loose i s then c a r r i e d out from the upper the nature and of the therefore island the structure. construction, deformation, the dumped stability and 138 liquefaction potential of the i s l a n d berm d u r i n g earthquakes a r e of major concern. 7.2 ANALYSIS OF A TYPICAL TANKER ISLAND Fig. 7.1, shows, s c h e m a t i c a l l y a tanker i s l a n d . p r o v i d e d w i t h a cover of about Typical given i n Table 2m of rock p r o p e r t i e s of rock f i l l This island i s fill. and sand f o r s t a t i c a n a l y s i s a r e 7.1. TABLE 7.1. Static Soil Properties Properties Rock T o t a l u n i t weight kN/m 3 Bulk modulus c o n s t a n t K^ * Fill Sand 18.7 18.1 1000 800 0.40 0.40 B u l k modulus exponent c o n s t a n t n Shear modulus parameter ( K ) 24.0 16.0 Angle of i n t e r n a l 38.0 32.0 0.0 0.0 0.45 0.45 2 Effective ballasted K Q tanker i s assumed w i t h p l a n dimensions In dynamic friction cohesion Coefficient The m a x the case stress-strain to weigh 200,000 metric tons when fully 170m and x 60m and 21m h i g h . presented here, relationship i t i s assumed has equal shear that the h y p e r b o l i c strength in both Fig. 7.1. Schematic of Tanker Island 140 d i r e c t i o n s of s h e a r i n g . T y p i c a l p r o p e r t i e s used f o r the dynamic a n a l y s i s are g i v e i n T a b l e 7.2 TABLE 7.2. Dynamic S o i l P r o p e r t i e s Properties Rock F i l l Bulk modulus c o n s t a n t Sand 1300 ft V. High 0.4 Bulk modulus parameter n 70 45 Shear modulus parameter ( K ) 2 0.8, 0.79, 0.459 C^ •* CK Constants and Rebound modulus Constants m,n During no drainage B t shaking 0.43, 0.62 the rock f i l l i s assumed t o be f r e e d r a i n i n g and i s assumed i n dumped sand. to s i m u l a t e very 0.730 A very h i g h v a l u e was a s s i g n e d to low c o m p r e s s i b i l i t y imparted by the water i n the pores which i s not allowed to the s a t u r a t e d to d r a i n . L i q u e f a c t i o n r e s i s t a n c e curves are r e q u i r e d f o r d i f f e r e n t stress ratios appropriate K i n the i s l a n d . r The v a l u e s used value with These each are s p e c i f i e d static sand shear by stress i n the example a r e presented i n T a b l e 7.3. static associating ratio the t /o" . s v o TABLE 7.3. s/ vo T T The considered static here shear g The K K /a* s vo r 0.0 0.004 0.10 0.015 0.20 0.05 stress ratios f o r t h e example problem between 0.0 t o 0.13 and t h e r e f o r e , the v a l u e s to r a t i o s T / o corresponding component vary and r v a l u e s a y o r , above 0.15 a r e not n e c e s s a r y . i n p u t motion used o f the I m p e r i a l of K f o r the a n a l y s e s Valley Earthquake i s the S00E a c c e l e r a t i o n o f May 18, 1940 s c a l e d t o O.lg. The i n p u t motion was a p p l i e d a t the bottom boundary o f the i s l a n d . Three dynamic analyses were s t r u c t u r e w i t h and without were s e l e c t e d island. C s = 0 and 0 by d i s c r e t i z i n g since tanker alone, island The p r o p e r t i e s o f s l i p c o u l d occur = 6.3 x 1 0 kN/m /m, K 5 s elements. island plus elements between the s t r u c t u r e and t h e element p r o p e r t i e s were assumed to be, A complete out slip so that some s l i p The s l i p K performed: the s t i f f n e s s g 2 = 6.3 x 1 0 , kN/m /m 5 N 2 = 30° response study the e n t i r e of tanker and i t s c o n t e n t s would o f the tanker domain wall into i s very respond like island finite much elements. higher a rigid c o u l d be c a r r i e d than box. However, s o i l , the I n view o f 142 this, The the tanker and i t s c o n t e n t s were modelled as a u n i f o r m s t i f f n e s s of s t r u c t u r a l elements were s e l e c t e d as 1 0 3 r i g i d box. of the rock fill elements. 7.2.1. R e s u l t s f o r Tanker I s l a n d Problem One of the f a c t o r s which i n f l u e n c e the development porewater pressure generation the i s cyclic of r e s i d u a l maximum dynamic porewater shear s e c t i o n T-T which runs a r e developed f o r c e s on the t a n k e r . slip i s allowed higher than magnitudes when i s p o s s i b l e only induced of shear elements that by the shear the i s l a n d may be expected very along shear i s i n place shear due to the i n e r t i a s t r e s s e s i n the dumped sand when were p r o v i d e d . soil are s l i g h t l y When s l i p o c c u r s , the can be t r a n s m i t t e d to the s t r u c t u r e i s s t r e n g t h of the s l i p elements. Therefore, s l i p o c c u r r i n g between the s t r u c t u r e and to d i f f e r . high c y c l i c island sand t h a t h i g h e r dynamic shear s t r e s s e s generated i s i n p l a c e , the g r e a t e s t porewater p r e s s u r e the unloaded i n the sand, i n the dumped indicates the tanker The induced stress S i n c e the the c e n t r e of the i s l a n d , a r e shown i n F i g . i s l a n d responses w i t h and without in strain) . t o occur between s t r u c t u r e and a d j a c e n t limited, dictated tanker (or c y c l i c pressure This figure when no s l i p Despite stress stresses through 7.2 f o r a l l t h r e e c a s e s . stresses shear of r e s i d u a l (Fig. 7.4). This i n the sand ratios i s because when a r e developed the vertical over-burden p r e s s u r e s a r e v e r y much g r e a t e r when the tanker i s p r e s e n t , so that, cyclic the important shear stress parameter c o n t r o l l i n g g i v e n sand, a r e a c t u a l l y ratios, T c v / O y 0 , which is the most the development of porewater p r e s s u r e i n a s m a l l e r ( F i g . 7.3). I t can be r e a d i l y seen that Maximum Dynamic Shear Stress (kN/m ) 2 0 Fig. I 20 1 40 1 60 1 7.2. Distribution of Maximum Dynamic Shear Stress in Sand. Fig. 7.3. Distribution of Maximum Dynamic Shear Stress Ratio in Sand. 144 the d i s t r i b u t i o n of maximum c y c l i c shear s t r e s s r a t i o s are p r o p o r t i o n a l to the d i s t r i b u t i o n of r e s i d u a l porewater p r e s s u r e Fig. distribution ratios 7.4, for obtained island. The viewed as to higher a of The elements. analyses. unloaded obtained The island porewater residual are higher ratio porewater than i n the a n a l y s i s without structure i s i n place w i l l response slip residual H/aL f o r the the Q pressure loaded tanker T h e r e f o r e , the d i s t r i b u t i o n of r e s i d u a l porewater were with the can be the s o l u t i o n s at a s e c t i o n where the i n f l u e n c e of the s t r u c t u r e conclusions table. the results when the middle a l l three for is negligible. ratios shows ratios. values drawn rigid as by one moves Yoshimi vary from away and from lower the values at structure. Tokimatsu (1977) who to base excitation structure subjected pressure a r e s i d u a l porewater p r e s s u r e d i s t r i b u t i o n g i v e n i n the elements i s consistently T h i s i s because lower higher shear than the a n a l y s i s s t r e s s e s are induced Same studied on the the shaking analysis without i n the slip latter case. The 1-1 distribution i s shown i n F i g . 7.5. unloaded island have developed Firstly, the are softer. any Even though the shear s m a l l e r than i n the in-situ much s m a l l e r and of maximum dynamic shear unloaded overburden the loaded island. when an T h i s i s because of curve stress-strain i n c r e a s e i n r e s i d u a l porewater p r e s s u r e w i l l curve. shear two island have c o n t r i b u t e d to h i g h shear i n the strains factors. are very f o r a g i v e n element i s relationship s o f t e n the strains is used, stress-strain The g e n e r a t i o n of h i g h e r r e s i d u a l porewater p r e s s u r e s and burden p r e s s u r e s island. higher s t r e s s e s i n the unloaded effective for section s t r e s s e s induced island, t h e r e f o r e the s t r e s s - s t r a i n Secondly, strains i n the low over unloaded Fig. 7.4. Distribution of Residual Porewater Pressure Ratio in Sand. Fig. 7.5. Distribution of Maximum Dynamic Shear Strain in Sand. 146 The earthquake maximum f o r the horizontal section 1-1 displacements are shown which occur i n F i g . 7.6. during Much dynamic d i s p l a c e m e n t s are computed when the tanker i s i n p l a c e . elements were p r o v i d e d , s l i p the displacements elements. and Y are about twice the results The residual post earthquake displacement and the smaller When s l i p and s t r u c t u r e and obtained F i g . 7.7 a and b show the post earthquake directions. dynamic o c c u r r e d between the s o i l the without deformations i n the X displacement i s the sum displacement due to slip of the volumetric * strain component results presented earthquake e ^ • in same order displacement e observations figure. The comes Firstly, can be made the amount from the the post of i n the X - d i r e c t i o n are p r o p o r t i o n a l to the maximum Secondly, the X-component of magnitude as (settlement). settlement this deformations dynamic d e f o r m a t i o n s . the Two y main from of the Y-component contribution the dynamic the main c o n t r i b u t i o n of the displacement to residual i s from the of the displacement X-component displacement i s of and the v o l u m e t r i c s t r a i n of the f o r the component ft vd* In acceleration the dynamic response i n the s t r u c t u r e maximum induced without slip accelerations elements of structures the i s one of the main d e s i g n given are 0.15g by TARA-2 and 0.17g. maximum induced concerns. i n the s t r u c t u r e T h i s means t h a t , with The and i f slip is p r e v e n t e d , the a c c e l e r a t i o n induced may be h i g h e r by as much as 15% of the a c c e l e r a t i o n when s l i p top of the unloaded i s allowed. island i s 0.15g. The maximum a c c e l e r a t i o n computed on Fig. 7.6. Distribution of Maximum Dynamic Displacement. Post Earthquake X Disp.. (cm). Fig. 7.7. Post Earthquake Y Disp., (cm) Post Earthquake X and Y Displacements. 00 149 One of the ways of p r e s e n t i n g dynamic response i s to p r e s e n t i t i n terms of response s p e c t r a . and acceleration structure. behaviour shear. that These of at structures the used presented are and shows the berm s u r f a c e example using the f o r the motions then used also by of the the e n g i n e e r s to p r e d i c t the spectrum a was problem 3%. greater The predominant for acceleration than spectrum 0.5 The I n s p e c t i o n of t h i s f i g u r e c o n s i d e r e d here, response period such as f o r a l l t h r e e cases c o n s i d e r e d . of sec, the unloaded base of the damping suggests the a c c e l e r a t i o n the h i g h e r f o r s t r u c t u r e s w i t h a v e r y low p e r i o d . with at the base to compute d e s i g n f o r c e s , response i n the computation f o r the predicted often results F i g . 7.8 motion ratio are Response s p e c t r a f o r d i s p l a c e m e n t , v e l o c i t y response island will be However, f o r the s t r u c t u r e s response predictions will be similar. and r o c k i n g . comparing The which were dimensional the island results obtained show as that 30% of ratio reported response the by F i g . 7.9, pressure with of a tanker d u r i n g e x c i t a t i o n are r e l a t i v e importance response a n a l y s i s . porewater motion earlier and tanker, without those U/Oy predicted 0 by of considered the i n this However, example and the distribution The one-dimensional case any slip The elements. elements may be by one-dimensional two-dimensional two-dimensional residual analyses from a considered results predicted response onewas clearly as analysis. low This n e g l e c t s the r o c k i n g mode of v i b r a t i o n i s it should i s very r o c k i n g mode of v i b r a t i o n may three also some a means a response a n a l y s i s which non-conservative. two-dimensional from y o analysis. maximum of these two modes can be s t u d i e d shows the computed d i s t r i b u t i o n of /o u a sliding tall be mentioned (21m) and, that rigid, the and tanker therefore have been more important than u s u a l . 600 1 r- r —i 500 — Island Alone 400 |t "Jl II 300 •*—. Island + Structure (Slip) - \f\ Island + Structure ?A (No Slip) - 200 - 100 0 10 2-0 3-0 4-0 5-0 Period, (sec) Fig. 7 . 8 . Acceleration Response Spectrum for the Motion at Berm Surface. 7.9. Distribution of Residual Porewater Pressure Ratio in Sand. 151 7.3 SOME PRACTICAL CONCLUSIONS In computing practice the a response procedure o u t l i n e d by steps. step The alone first for the predicting of the given response of the In number National is to performance pressure, induced step are considered. suggests the also Building excitation. on second the level This etc., means made when deposits. of step structure, are Canada has response base a c c e l e r a t i o n s of soil Code of the The strain s o i l - structure assumptions founded compute s t r u c t u r e , to the the simplifying structures porewater one of the is The two soil to basic deposit compute the o b t a i n e d i n step the results which that are the systems be uncoupled and one. such obtained code in as from essence analysed independently. Figures computed Canada using may be influences it also on the in soil the code may of based on not uncoupled be demonstrates analysis. inertia be the increases forces. dependent not of the Because tall, importance of the two Therefore, the for soils uncoupled of basic e f f e c t i v e stresses basic conclusions systems. of the domain which n e g l e c t s the has Code and which analysis applicable. soil-structure to structures Building structure behaviour, t y p i c a l example, three applicable response of National a l l i t r a i s e s q u e s t i o n s about the m e r i t representation may It the i n the presence deposit. stress show that outlined The additional From t h i s First clearly procedures non-linear p r o p o s e d by to 7.8 error. the provides exhibit 7.2 heavy of great and of any the r o c k i n g rigid the drawn. response a n a l y s i s Secondly, incorporating weight of can be one-dimensional degrees of structures. slip caissons Thirdly i t elements or freedom in the tankers, and 152 their large always be i s l a n d and lateral important. dimensions, In these soil-structure interaction effects will type of problems a coupled a n a l y s i s of the structure i s required. 153 CHAPTER 8 SUMMARY AND CONCLUSIONS 8.1 SUMMARY The main dimensional including in either linear static and this seismic The forstatic new m e t h o d effective or total elastic modulus were behaviour taken i n shear was was the B , t shear of tangent developed an a static soil t o be element shear modulus, during stresses. T„, „, t m 0 a r e kept G , x > ^max The e f f e c t a n c * of soil modelled shear a twodeposits performed of both. by and s t r e s s with using behaviour G tangent v , When t B of d i l a t i o n a r and throughout e d u r i n g shear from mode, modulus, the analysis. stress computed stress bulk f o r corresponding effective stress dependent. i n the t o t a l a an The m a t e r i a l Masing i s performed m Non- modulus and t a n g e n t parameters. hyperbolic constant the analysis. a was tangent i s modified t develop R e s p o n s e t o c h a n g e s i n mean n o r m a l analysis strength, parameters, into of a s s u m e d t o be n o n - l i n e a r , e l a s t i c When analysis as t h e two " e l a s t i c " d u r i n g u n l o a d i n g and r e l o a d i n g . to modes o r a c o m b i n a t i o n i n which assumed was a n d d y n a m i c a n a l y s e s c a n be stress approach research response interaction. stress-strain response of soil-structure incrementally bulk purpose shear The strains mode i s u s e d , t h e the effective on v o l u m e c h a n g e i s t a k e n account. In the s t a t i c analysis proposed h e r e , g r a v i t y may be s w i t c h e d o n 154 at once f o r the be modelled by the static completed soil structure layer analysis. analysis give the The or the construction s t r e s s - s t r a i n conditions in-situ stress conditions sequence can determined by b e f o r e the dynamic analysis. Slip to represent elements. or contact the The elements have been i n c o r p o r a t e d interface properties characteristics of the slip the dynamic porewater pressures proposed by max» The a and structural i n t e r f a c e g i v e n by elastic the Mohr Coulomb criterion. In T soil analysis element were assumed to be p e r f e c t l y p l a s t i c , w i t h f a i l u r e at the failure between i n the r e m are Martin °dified dynamic effective calculated et.al, for the stress response using a (1975). effects includes modification The residual the prediction study extensive study c a r r i e d out the G porewater of residual of parameters, of response analysis, post m a x model > and pressure. earthquake deformations. An analysis be using centrifuge t e s t data suggests that s u c c e s s f u l l y used to p r e d i c t Seismic to v e r i f y the response of seismic a the response of typical tanker proposed method proposed method of can structures. island computed by this method i s p r e s e n t e d . 8.2 CONCLUSIONS The following 1. work that has been presented in this thesis leads to the transient and conclusions. A consistent and r e l i a b l e method for computing 155 permanent deformations in two-dimensional soil structures is needed. 2. A two-dimensional account soils, dynamic response analysis which takes into the n o n - l i n e a r h y s t e r e t i c s t r e s s - d e p e n d e n t p r o p e r t i e s of has been developed i n terms of both total and effective stresses. 3. The method has been v e r i f i e d models with predicted predictions and by comparing of measured the response data from method. centrifuged Comparison parameters between i s generally very good. 4. Allowing for s l i p is very important. consistently lead h i g h e r porewater 5. The method typical it to occur between s o i l has Analyses which structural allow p r e s s u r e s i n the s o i l slip have been applied that The to compute results and deposit. s e i s m i c response of t h i s of study suggests the response of s t r u c t u r e s d e p o s i t s be a n a l y s e d as a coupled s o i l - s t r u c t u r e 6. for elements to h i g h e r displacements i n the s t r u c t u r e tanker i s l a n d . i s important and founded on a that soil systems. The v a l i d i t y of a one-dimensional response a n a l y s i s i n s t e a d of a two-dimensional questionable. dimensional analysis for The porewater response tanker type of structures p r e s s u r e s p r e d i c t e d by u s i n g a analysis model may be those p r e d i c t e d by a two-dimensional response as low as analysis. 30% is oneof 156 SUGGESTIONS FOR FURTHER STUDY 1. Additional greater comparative confidence method. Comparative studies could studies c e n t r i f u g e t e s t s or f i e l d 2. Sandy m a t e r i a l s exhibit pressures to due be should placed may be on carried out so the v a l i d i t y be performed with that of this data from studies. partial dilation. e v a l u a t i o n near l i q u e f a c t i o n , stabilization Therefore, a t low c o n f i n i n g in the response i t i s important t h a t the method of a n a l y s i s I n c l u d e the d i l a t a n t behaviour of sands. 3. I n the response during evaluation the s e i s m i c l o a d i n g of more may be should be developed to take t h i s i n t o permeable soils, significant and account. drainage procedures 157 REFERENCES 1. Byrne, P.M., (CE573). 2. Byrne, Elastic (1979), C l a s s Notes: N u m e r i c a l Methods i n S o i l U n i v e r s i t y of B r i t i s h Columbia, P.M., and E l d r i d g e , T.L., Stress-Strain Model Vancouver, B.C., Mechanics Canada. 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Kulhawy, F.H., Duncan, J.M., Analysis Stresses of Construction", Department and and Seed, H.B., Movements (1969), " F i n i t e in Embankments G e o t e c h n i c a l E n g i n e e r i n g Research Report No. of C i v i l Engineering, Unviersity Element of C a l i f o r n i a , During TE-69-4, Berkeley, Nov. 28. Lee, F.H., Embankment (1983), "Partial Liquefaction in Centrifuge Model i n an Earthquake", M . P h i l . T h e s i s , E n g i n e e r i n g Department Cambridge U n i v e r s i t y , Cambridge, J u l y . 29. Lee, K.W., Under (1965), " T r i a x i a l Seismic California, 30. Lee, K.W., of Loading Compressive Conditions", S t r e n g t h of S a t u r a t e d Sands Ph.D Thesis, University of B e r k e l e y , ppl-521. (1975), "Mechanical Model f o r the A n a l y s i s of L i q u e f a c t i o n Horizontal Engineering, Canada, Sept. Soil Deposits", University of Ph.D British Thesis, Department Columbia, of Vancouver, Civil B.C., 161 31. L e e , K.W., and F i n n , Effective Stress Liquefaction Civil W.D.L., (1975), "DESRA-1: Program Response Analysis Evaluation", Engineering, Soil University of Soil Mechanics of B r i t i s h f o r Dynamic Deposits Series, Including No.36, Dept. o f Columbia, Vancouver, B.C., Canada. 32. Lysmer, J . , and Kuhlemeyer, Infinite Media", Journal R.L., (1969), " F i n i t e o f the E n g i n e e r i n g Dynamic Model f o r Mechanics, Division, ASCE, V o l . 95, EM4, Sept. pp859-877. 33. Lysmer, J . , and Waas, Structures", Journal G., (1972), "Shear of Engineering Waves Mechanics i n Plane Infinite D i v i s i o n , ASCE, V o l . 98, EMI, Feb., pp85-105. 34. Martin,. G.R., F i n n , W.D.L., and Seed, H.B., (1975), "Fundementals o f Liquefaction Engineering 35. Under Cyclic Loading", Journal o f the Geotechnical D i v i s i o n , ASCE, V o l . 101, GT5, May, pp423-438. Masing, G., (1926), "Eigenspannungen and V e r f e s t i g u n g P r o c e e d i n g s , 2nd I n t e r n a t i o n a l Congress o f A p p l i e d beim Messing", Mechanics, Z u r i c h , Switzerland. 36. Mroz, Z., N o r r i s , U.A., and Z i e n k i e i c z , O . C , (1979), " A p p l i c a t i o n o f an Anisotropic Hardening Model i n the A n a l y s i s of E l a s t o - P l a s t i c Deformations of S o i l s " , Geotechnique, 29(1), p p l - 3 4 . 37. Nadim, F., and Whitman, R.V. (1982), " S l i p Response", Proceedings, 4th International Methods i n Geomechanics, 38. Newmark, Dynamics", N.M., (1959), Journal 85, EM3, J u l y . Elements Conference Edmonton, A l b e r t a , Canada, "A Method on N u m e r i c a l pp61-67. o f Computation o f the E n g i n e e r i n g f o r Earthquake f o r Structural Mechanics D i v i s i o n , ASCE, V o l . 162 39. Newmark, N.M., (1965), "Effects of Earthquakes Embankments", 5th Rankine L e c t u r e , Geotechnique 40. Newmark, N.M., Earthquake and 'Rosenblueth, E., Engineering", Prentice-Hall on 15, No.2, (1971), Dams and ppl39-160. "Fundementals Inc., Englewood, C l i f f , of N.J., ppl62-163. 41. Ozawa, Y., and Analysis of Duncan, J.M., Static (1973), "ISBILD: Stresses and G e o t e c h n i c a l E n g i n e e r i n g Research Civil 42. A Computer Program f o r Movements Report No. in Embankments", TE-73-4, Department of E n g i n e e r i n g , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , Dec. P r e v o s t , J.H., Strength (1978), Behaviour", Numerical "Mathematical M o d e l l i n g of S o i l Proceedings, 3rd Stress-Strain International Conference Methods i n Geomechanics, Aachen, Germany, A p r i l 6, on pp347- 361. 43. Pyke, R., Under Seed, H.B., and Multidirectional Chan, C.K., Shaking", Journal E n g i n e e r i n g ASCE, V o l . 101, No. GT4, 44. Rahman, M.S., Development Seed, Under Geotechnical H.B., and Offshore Engineering (1975), April, Booker, Gravity Division, "Settlement of the of Sands Geotechnical pp379-398. J.R., (1977), Structures", ASCE, Vol. "Pore Pressure Journal 103, of GT12, the Dec, ppl419-1436. 45. Robertson, its to 46. P.K., Application Department of Vancouver, B.C., Schnabel, P.B., Computer Program (1982), "In-situ to L i q u e f a c t i o n Civil of Assessment", Soil Ph.D Engineering, University Canada, Lysmer, for Testing w i t h Emphasis on Thesis, submitted of B r i t i s h Columbia, Dec. J., and Earthquake Seed, Response H.B., (1972), Analysis of "SHAKE: A Horizontally 163 Layered S i t e s " , Report No. EERC 72-12, Earthquake E n g i n e e r i n g Reseach Center, U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , Dec. 47. 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Seed, H.B., Pyke, R. and M a r t i n , G.R. (1975), of Multi-directional Sands", Report Shaking NO. EERC " A n a l y s i s of the E f f e c t on the L i q u e f a c t i o n 75-41, Earthquake C h a r a c t e r i s t i c s of Engineering Research C e n t e r , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , C a l i f . Dec. 52. Seed, Design H.B., of Geotechnique 53. Seed, H.B., (1979), Earth "Considerations and Rockfill in Dams", the Earthquake-Resistant 19th Rankine Lecture, 29, No.3, pp215-263. (1983), "Earthquake-Resistant Design of E a r t h Proceedings of Seismic Design of Embankments and Caverns, Dams", pp41-64. 164 54. S e l i g , E.T., Cyclic and Loading", ASCE, V o l . 107, 55. Chang, S.C. Serff, N., Journal GT5, Seed, "Earthquake of May, 56. United the Geotechnical Engineering Division, pp539-551. H.B., Makdisi, F.I., and Chang, C.Y., Induced Deformations of E a r t h Dams", Report No. 4_, Earthquake E n g i n e e r i n g Berkeley, (1981), " S o i l F a i l u r e Modes i n Undrained Research Center, U n i v e r s i t y of States National Research Research-1982", Y.P., and Liquefaction 58. Report Finn, W.D.L., Potential", Council by GT10, V a i d , Y.P., and Byrne, P.M., Liquefaction Potential", D i v i s i o n , ASCE, V o l . 107, 59. Varadarajan, A., and Stress-Strain International (1979), Journal D i v i s i o n , ASCE, V o l . 105, and 76- California, (1982), Committee "Earthquake on Earthquake E n g i n e e r i n g R e s e a r c h, N a t i o n a l Acadamy P r e s s , Washington, Vaid, EERC Sept. Engineering 57. (1976), GT7, OCT., the of Static Geotechnical Shear on Engineering ppl233-1246. Hughes, J.M.O., (1981), " D i l a t i o n J o u r n a l of the G e o t e c h n i c a l Angle Engineering July. Mishra, Volume of "Effect D.C. S.S., Change (1980), Behaviour "Stress-Path of Symposium on S o i l s under C y c l i c and Dependent Granular Soil", Transient Loading, Swansea, ppl09-119. 60. Verruijt, Pressures", John Wiley 61. A., (1977), "Generation and Dissipation of F i n i t e Elements i n Geomechanics, E d i t e d by, and Porewater Gudehus, G., Sons, pp293-319. Wilson, E.L., Farhomand, Dynamic Analysis of I., Complex Earthquake E n g i n e e r i n g and and Bathe, Structures", K.J., (1973), "Non-Linear International Journal S t r u c t u r a l Dynamics, V o l . 1, pp241-252. of 165 62. Wedge, N.E. Soils", (1977), "Problems In Non-Linear M.A.Sc. T h e s i s , Submitted to the Dept. of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, 63. Yoshimi, Saturated Y., and Sand Tokimatsu, During K. (1977), Earthquakes", S o c i e t y of S o i l Mechanics A n a l y s i s of Movements i n B.C., Canada. "Settlement Soils and of B u i l d i n g s Foundations, on Japanese and Foundation E n g i n e e r i n g , V o l . 17, No. 1, March, pp23-38. 64. Z i e n k i e w i c z , O.C., and Cheung, Y.K., (1967), F i n i t e Element Method i n S t r u c t u r a l and Continuum Mechanics", McGraw H i l l Book Company. 166 APPENDIX I FINITE ELEMENT FORMULATION In divided the f i n i t e into a finite within a f i n i t e element = element displacement vector, giving here TJ*" = x = interpolation the both the element element The of same displacement geometrical y displacements at any point y d i s p l a c e m e n t s of the nodes, and used term the analysis unknown d i s p l a c e m e n t s the is isoparametric implies interpolation a in f u n c t i o n s N^ geometry number of differentiations and of the common and the are used element. 4 a node (iso-) geometry to of express Isoparametric advantages; i t offers efficient i t can handle curved and arbitrary shapes. The expressed and the and f o r m u l a t i o n has integrations, be of element. The and function. description element. displacement {u,v} [N] parametric of (Al.l) = displacement v e c t o r , g i v i n g x and isoparametric variation where {6} type The i s assumed to be g i v e n by, w i t h i n an element, The the e n t i r e domain of i n t e r e s t i s number of elements. {tf} = [N] {6} {tf} analysis interpolation in ordinates i n t r i n s i c natural function [N] can be coordinates to an element selected ( s , t ) which (Fig. A l . l ) . is such t h a t a system i t can of co- 168 In the i s o p a r a m e t r i c concept, the c o o r d i n a t e s of a p o i n t i n the element i s g i v e n by, {X} = [ N ] {X } (A1.2) ± I f we c o n s i d e r a four-node q u a d r i l a t e r a l i s o p a r a m e t r i c element, the m a t r i x [ N ] i s composed o f , „ . d-s(l-t) 4 1 N _ W (i+s)(i+t) 4 = 3 Now equations 2 4 (l+s)(l-t) 4 _ q - s ^ (i+t) 4 ( A l . l ) and (A1.2) can be r e w r i t t e n as, l l u v 2 v u u jU, 0 N rN, M L 0 N 0 N 2 0 x N 0 N 3 0 2 3 N^ 0 0 -I N^ v u v 3 4 4 and, X l yi x y, x 2 x V ( } = f N L l 0 0 N N x 2 0 0 N N 2 3 0 0 N S l f 0 , 0 N^- 1 3 3 ?3 4 X 2 3 (A1.3) 169 I t can be seen t h a t the t r a n s f o r m a t i o n shown i n (A1.4), maps q u a d r i l a t e r a l element i n t o a square as If simply normal plane given s t r a i n conditions by strains _e and y = t 1 S _ M e s t r a i n m a t r i x e from e ax x aN Y 'xy x .e. ey s t r a i n vector strain. These e x and strains e are y e is are given y A1.3 and ax 0 aN ax ay ay (A1.5) ax A1.5, 0 ay aN1 8u ax 0 ay 2 8N 3N 2 w 0 ay u. 0 ax u„ 3 0 ay 2 ax k ax a\ ay at^ ay ax 2 (A1.6) > {e} But 1 the where x y Y ' xy aN 0 Y } y av ay 0 = assumed as, ox X are shear t n e shown i n F i g . A l . l . e , x x y i n terms of d i s p l a c e m e n t s e {e , the = = [B] f ( s , t ) and also should be computed u s i n g aN as ± aN ± ax In m a t r i x form, the (A1.7) {6} x,y are functions the following ax aN ay as ay ± g l o b a l and of s,t. So any derivative relationship, (A1.8) as local derivatives can be w r i t t e n as, 170 5N —— ax as ± ax as ± as BN. 3x at a ay. t 8 8N± aN - — - — ax r - i ax (AI.9) 3N, L J . l A i . y ; J ay t o N ay where, [ j ] = Jacobian matrix The d e r i v a t i v e s of with r e s p e c t to x, y can now be o b t a i n e d from A1.9, as, 8N ± aN ± 5 ± • [ ] J 3N, _ 1 a y a t where, [j] - 1 = I n v e r s e of J a c o b i a n m a t r i x J , which i s s i m p l y , a t Now as from A1.4, the J a c o b i a n m a t r i x can be w r i t t e n a s , _ L J " -I R ax as By. as J a x ay. a t a t t^± 1^1 _ as . 8N i _ ^ 5 x i i u i t • fc x i A as 8N f \ o t y i y± ... (A1.12) 171 The oN^/Qt and components of m a t r i x ( x ^ , y^) a r e known. can be computed. be noted oN that - 1 also (A1.13) these I ^ j are f ( s , t ) . Therefore, substituting yields, 8N. tl11 jL L The m a t r i x as" aN.i at 22 21 (A.14) [ B ] ( e q u a t i o n A1.6) which r e l a t e s displacements, respect J, [ j ] 22 ± By" nodal the m a t r i x oN^/os, i2) this i n equation ( A L I O ) , oN since T •21 should So, knowing evaluated Say, = [hi It [ j ] can be to x and y. has Now derivatives knowing of the s t r a i n v e c t o r to interpolation these d e r i v a t i v e s function from e q u a t i o n with (A1.4), [ B ] can be r e w r i t t e n a s , aN, •11 + 0 S 12 X aN i 9 t 3N | X aN, o [B] = 3x8 aN 2i a s aN I + 22 1 at 2i as I A n bi n 2 + i2 a~T x aN, +i . 22 at aN, aN. | 3N 2 + i 12 at aN. | x2i a£ 9N„ + i 22 at | 172 0 3N hi aN as + as + aN 12 n o as 3N aN 2 I 21 m 3 + as + s i 2 T a I 2 i n + as aN 3 at 2 21 3N 3 as I 22 at B 3 + h2 3 N 3 at | t aN„ A 2i aN, + a as~ *22 stresses IT | x n x s aN The X ai^ | + 21 at 0 o I 0 3 2 at 1 aN^ [ + at 22 I aN 3 22 m IT h 4 as (A1.15) at - + x i2 and s t r a i n s a r e connected through elasticity matrix g i v e n by, {a'} = [D] {e} (A1.16) where, [a'] = effective conditions, io'}*- [D] - G X O , T y matrix x y For 2D plane strain } f o r 2D p l a n e B - f G 4 B + j Sym vector. i t i s g i v e n by, {0 , = elasticity B + ^ [D] = stress condition, 0 0 G strain (A1.17) 173 Using the virtual work applying infinitesimal W M internal work (Wj^) done by {£} i s , dv (A1.18) where {e} = v i r t u a l s t r a i n s and {a} = t o t a l s t r e s s v e c t o r The t o t a l the v i r t u a l nodal displacement = /// IN principle, due to v i r t u a l displacement {£} (A1.19) s t r e s s v e c t o r can be s p l i t into effective s t r e s s and porepressure v e c t o r s . i.e.: {a} = {a'} Effective + {u } Q stress Porepressure Vector Here {a'= and {xxj*- i n which u Q = {u , D y u , D i s the porewater Now, Vector {a , a , T x substituting (A1.19) x y } o} pressure. {a} from equations (Al.19) and (A1.18), one gets, W = /// {e}* [{o'} + {u }] dv V IN S u b s t i t u t i n g f o r {a'} from A1.16 WIN = /// v M (A1.20) t h i s reduces t o , + K) D V (AI.21) 174 using equation W But, (A1.7), {e} can be r e p l a c e d by [ B ] {£}, then, - J/J V I N [B]' [ D ] [B] {6} + J J J {6} V e x t e r n a l work done by the l o a d v e c t o r displacement W EX [B] {p} r i d i n g t { u J dv (A1.21) through the v i r t u a l {?>}, i s simply, t J = (A1.22) p The v i r t u a l work p r i n c i p l e W fc IN W gives, EX i .e. {*}*{*} = {*}'/// [ B ] [ D ] [ B ] t v + {«}'/// V Noting dv {6} (A1.23) [ B ] ' {u } dv Q that, dv = dxdydz and also where = |j| dz = t h i c k n e s s of the element. After {?} t dsdtdz one g e t s , substituting this i n (A1.23), and d i v i d i n g both s i d e s by 175 {P} = [k]{6} + [k*] {u } (A1.24) D where, [k] = element s t i f f n e s s l i [/ = J t [] B matrix [ ][ ] l l D B (assuming u n i t J d s d t ] (A1.25) thickness) and, [k ] = porewater p r e s s u r e 11 t = [/ / [B] -1-1 The The (A1.26) i n t e g r a t i o n s shown above used are 2 x 2 . and | j | dsdt] Gauss i n t e g r a t i o n technique material. matrix The have to be has been employed and f o r m u l a t i o n presented should i n c r e m e n t a l s t r e s s e s and After be simply tangent matrix estimating the [k ], incremental i s f o r any linear the d i s p l a c e m e n t s , elastic stresses displacements, moduli. porewater t numerically. the number of p o i n t s r e p l a c e d by i n c r e m e n t a l e v a l u a t i n g the i n c r e m e n t a l stiffness porewater l o a d v e c t o r j ) , element pressure pressure the g l o b a l i n c r e m e n t a l l o a d - d i s p l a c e m e n t will here For i n c r e m e n t a l l y e l a s t i c a n a l y s i s , moduli v a l u e s evaluated matrix u , Q for [k*], a l l the tangent and also elements r e l a t i o n s h i p can be formed. This lead to, {P} = [K ]{A} + t [K*]{U} (A1.27) 176 in which global axes. obtained, using {P}, and By [ K T ] , {A}, solving {u} are relevant variables e q u a t i o n the displacement f i e l d be used to c a l c u l a t e (A1.6.) and f u n c t i o n gives l i n e a r therefore, stresses stress strain and and this i t can equations [K ] (A1.16) element strains respectively. within of an element an element. For and Since s t r a i n v a r i a t i o n w i t h i n an element, vary {A} can be stresses the shape the s t r a i n s convenience, in and average are computed at the c e n t r e of g r a v i t y of the element. In Chapter and stresses 4, i n terms and Chapter 5, of n o d a l f o r c e s . n o d a l f o r c e s are g i v e n i t i s required to e x p r e s s strains R e c a l l from e q u a t i o n (A1.24) the by, {P} = J / J !>]' [D] [ B ] dv {£} V (A1.28) But, s t r a i n s are connected to the m a t r i x [ B J i n e q u a t i o n (A1.7), {e} Therefore, = [B] {6} from (A1.29) (A1.28) and (A1.29), the nodal forces can be written i n terms of s t r a i n s as, {P} " J J J [ B ] V Now, 1 [D] {e} dv (A1.30) from e q u a t i o n (A1.16), k ' l = [D] { e } (A1.31) 177 and from e q u a t i o n s , (A1.30), and (A1.31) the nodal f o r c e s in terms of s t r e s s e s a s , can be w r i t t e n {P} = /// [ B f {a'} dv V The equation (A1.30) and (A1.32) can now be used s t r e s s e s i n terms o f element nodal f o r c e s . (A1.32) to express s t r a i n s and 178 APPENDIX I I STIFFNESS MATRIX FORMULATION FOR THE SLIP ELEMENT As relationship outlined in a t any p o i n t Chapter within 4, that a slip the element force-displacement has been assumed to be g i v e n by, f K (fS} i.e., f_ = = U ICQ 0 w K ] [ S S W ] (A2.1) W where, f g and f Kg, ^ w, g The w n = shear and normal = joint n stiffness stresses i n shear and normal d i r e c t i o n s = shear and normal d i s p l a c e m e n t s elastic stored energy, 0 E in a slip element due (Fig, (A2.1) to a p p l i e d f o r c e s can be o b t a i n e d by, E 2t 0 = in which included L i s the t o t a l because * f dZ (A2 length of the s l i p the r e l a t i o n s h i p between element. A factor f_ and w i s assumed linear. From (A2.1), 0g now can be w r i t t e n a s , 2) half i s to be Fig. A2.1. Slip Element. s = tangential direction Uj = n tangential displacement o f node i = normal direction Vj = normal displacement of node i 180 0 Since the E 2 /o ' o = W k W d A ( A 2 the v a r i a t i o n of d i s p l a c e m e n t s displacement at any point ( u , v) w i t h i n an element which i s at a d i s t a n c e , A, * 3 ) i s linear from node I on the bottom edge I J of the element i s , U) u = £ U + j - (1 L> U I (A2-4> bottom In a s i m i l a r manner the f o l l o w i n g e q u a t i o n s can be w r i t t e n f o r , u (A), top v and v U ) , (1) bottom top l .e. u U) = L ° K+ = L J top v(A) bottom V + ~ L? " L ( 1 ( 1 L + } V (A2.5) (A2.6) I and v where, u^, = (x) top v^ refer L V K + ( 1 " L? V L to displacement ( i n tangential and A 2 ' 7 ) normal d i r e c t i o n of the nodes I, J , K and L . Shear and normal d i s p l a c e m e n t s at any p o i n t a r e , w = u (A) - u top and, (A) bottom (A2.8) 181 wn Now v = v (A) - v U) top bottom s u b s t i t u t i n g bottom and f r o f o r equations m (A2.9) u (A2.4) t o to p u > b o t (A2.7), t in o m , v t o equations p and (A2.8)- (A2.9), H 1 w = w_ = [-(1 -i> ~i i u - ft] - f) f f (1 (A.2.10) and, (A2.ll) From e q u a t i o n (A2.1), w i s , l v, U -A w = [ ] = S — L J w 0 [ 0 -A -B 0 0 B 0 A 0 -B 0 B 0 A J ] \ V K \ in which, A = 1 " L A N D B - L In m a t r i x form the e q u a t i o n (A2.12) i s , (A2.12) 182 w = C 6 2x1 2x§ 8X1 where, ( A 2 . 1 3 ) = the i n t e r p o l a t i o n m a t r i x and 6^ = n o d a l displacement S u b s t i t u t i o n of K = 9 E Performing in ( A 2 . 1 3 ) t T •'o 2 C o 0 2 A 0 ABK 0 - A K To perform B 0 0 A 0 ABK 0 n 0 0 - A 2 K - B n 2 K s 0 -ABK s 0 the i n t e g r a t i o n integrations of, 0 B2K n 0 -ABK 0 s K shown 0 s - B 2 K s ABK 0 2 K 0 B 0 A On - B 0 B 0 A-l - A -ABK ABK ABK i n equation A 2 K K -ABK ABK A 2 . 1 4 , (A2.15) n 0 s 0 i 0 s 0 n 2 0 s 0 n 0 s 0 n 0 B2K s - A 0 0 0 - B n -ABK 2 0 -A 0 n - B 2 K r-A 0 n - B B2K s K -ABK 0 n 0 8 0 B2K s 0 0 A s K [ 0 0 2 -ABK 2 B ABK K 0 -B ABK s 0 0 0 s 0 -ABK o - A -B T C k C o o o A ( A 2 . 1 4 ) the m a t r i x m u l t i p l i c a t i o n , 0 s d o - A K gives, ( A 2 . 3 ) l K J* c s matrix A 2 K one should know 183 / o L J These a r e Jo A 2 / B 2 J Now dA, 2 J f B L o dA and 2 J f AB dA L o simply, L and, A L dA = dA == 3 o J o 0 o J AB dA (1 (f) 2 ft 2 L 3 dA L dA J ( 1 - f t f dA =• L 6 o ; (A2.16) 3 the e q u a t i o n (A2.14) can be w r i t t e n a s , 0 E " 2 ° T sn K ( 6 A 2 ' 1 7 > where, 2K sn 0 s 2K 6 K n 0 2K 0 s s K 0 2K Sym -K n n 0 -2K 0 2K s 0 -K 0 s -2K 0 s -2K n 0 -K n -2K 0 s 0 -K K 0 s 2K n 0 s n 0 s that elastic f i n i t e element i s , 0 E elastic = j 6 stored T K 6 energy 0g i n the formulation n K 0 2K Recall n of 2K a linear (A2.18) 184 Where K i s the (A2.17) and as K „ _ . —sn stiffness m a t r i x of the f i n i t e element. (A2.18), the s t i f f n e s s m a t r i x f o r s l i p element Now, comparing can be deduced 185 APPENDIX III STEP BY STEP INTEGRATION For time domain step. the the The deposit. proposed "elastic" elastic equations every step. Newmark's popular and one are discrete used expressed and so depend displacement step. 5.5) (1959) method to on dynamic a n a l y s i s i n the be modified the field level in the f o r every of strain deposit time in the should be T h i s r e q u i r e s t h a t the i n c r e m e n t a l dynamic (equation e x t e n s i v e l y used provides numerical by the time equilibrium time p r o p e r t i e s have parameters Therefore, e v a l u a t e d at every incrementally e l a s t i c have of to step by be solved step i n dynamic a n a l y s e s . numerically integration T h i s method is for very basically s o l u t i o n i n time domain, where the s o l u t i o n i s advanced step that at a time. the velocity In t h i s and method, two displacement i n terms of a c c e l e r a t i o n , v e l o c i t y of the known a c c e l e r a t i o n a t time t+At. and at parameters <* and time displacement For convenience p t+At can be at time t, l e t us d e f i n e that, T = t+At Then the r e l a t i o n s h i p i n terms of <* and {X} and, T = {X} t + (1 - «) At {X} t 8 are, + ocAt {X} T (A3.1) 186 {X} T = {X} + At {X} t t + (± - 6) ( A t ) {X} 2 t (A3.2) + B At Newmark (1959) proposed unconditionally corresponds = = 1/2 stable that « acceleration within 1/2 integration to a c o n s t a n t average and 8 = 1/6 = {X} 2 are used T and B procedure, 1/4 be which used f o r an incidentally a c c e l e r a t i o n method of i n t e g r a t i o n . If then t h i s method g i v e s a l i n e a r v a r i a t i o n of the time s t e p . R e - w r i t i n g the i n c r e m e n t a l e q u i l i b r i u m [M] {AX} + [C] {AX} + [ K ] t Now = t {AX} = e q u a t i o n s from Chapter 5, (A3.3) {AP} substituting f o r , {AX} (A3.4) {AX} and, {AX} i n e q u a t i o n (A3.3), one g e t s , [M] { ^ From equations - X j + [C] {Xj, - X } t (A3.1) and + [K ] t (A3.2), expressed i n terms of o t h e r v a r i a b l e s as t {^ {x}^, follows, - X} t and = {AP} {^}^ (A3.5) can be 187 W T m = [ 2 { A X " } A t { x } i t " ~ P> ( A t 2 W J ( A 3 * 6 ) and, W - ( X } + ( ! - - ) At {X} T t (A3.7) + f o r {x}-£ Substituting W [{AX} - At{x}t - ^ [ {AX} 2 and {x},p - At{ } X B ) A t * {X}J i n e q u a t i o n (A3.5), - (j t - 6 ) A t * {X} - B A t + [C] [ ( 1 — ) A t {X} + ^ Collecting terms, and d e f i n i n g and, + [K ] t t (A3.8) t {AX} = {AP} following simplifying symbols, . Wt the {X}J - B) At2 {x}J [{AX} - At {X} fc - £ 2 t = f W t + ' ( 2 p " 1 } A t ^ t ( A 3 ' 1 0 ) e q u a t i o n (A3.8) can be reduced t o , [ | £ 2 + [ c ] p £ + [ * t ] t ] (AX} = {AP} + [M] { E } + [C] { F } t (A3.11) t 188 Recall be applied evaluated from to using 5, S e c t i o n 5.1.2, c o r r e c t i o n restore total equation (5.11) can be added e q u a t i o n (A3.11). ^+[c Chapter equilibrium at time to r i g h t f o r c e s should T. The hand {P c o r r } s i d e of the Then the e q u a t i o n (A3.11) i s , ] p tr+[K T ]T] w (A3.12) = {AP} + [M] {E} t + [C] { F } t where, {AP} = {AP} + { P The only c o r r unknown } (A3.13) i n the above e q u a t i o n i s {AX} and t h e r e f o r e , {AX} can be o b t a i n e d as, {AX} = [ D ] " [{AP} + [M] {E} 1 in t + [C] {F} ] (A3.14) t which, -> (A3 15 [»]-[{S.]*WJE+[" U t Now knowing evaluated. {AX}, {x} T {X} T the unknowns {x}^ and {x} T and {x} T can be i s simply, = {AX} + {X} t (A3.16) 189 (A3.2), an e x p r e s s i o n f o r {x} i s , From e q u a t i o n W T • "p^2 [{AX} - At {X} T Substituting for {E} from t t " (| - 8) A t (A3.9), the 2 {X} ] (A3.17) t equation (A3.17) can be s i m p l i f i e d as, W From "pit" 2 {AX} - { E } + {X} = T t equations (A3.1) and (A3.18) t (A3.2) an f o r {x}^, expression after r e a r r a n g i n g terms i s , {X} Knowing = {X} + ( I - ) At {X} + ccAt {X} T t t {AX} by s o l v i n g equation computed u s i n g e q u a t i o n s , In sequence 1. the the response Initial numerical step {x} velocity from v a l u e s initial {E} , by step integration, t equations (A3.9), With these values soil deposit, and t the T can be of step. {x} are known increment or as displacements and the problem. { p C O rr}» a Based r e following time at the end o f the p r e c e d i n g conditions {F} , t a t time (A3.18) and (A3.19). of c a l c u l a t i o n s have to be performed f o r every either 2. (A3.14), (A3.19) T on t these evaluated values, using (A3.10) and (5.11). and the known the damping matrix a c c o r d i n g to a p p r o p r i a t e equations non-linear [c] p r o p e r t i e s of the and [ K ] a r e e v a l u a t e d i n Chapter t 5. t 190 3. The m a t r i x 4. Using [D] i s then c a l c u l a t e d u s i n g e q u a t i o n the increment possible to (A3.15). i n base a c c e l e r a t i o n v a l u e at time evaluate the right hand side of the t, i t i s equation (A3. 12). 5. The equation (A3.12) can now be solved for {AX} and then d i s p l a c e m e n t , a c c e l e r a t i o n and v e l o c i t y v e c t o r s can be e v a l u a t e d from e q u a t i o n s When increment time any step 5 i s finished step. has and Obviously desired ( A 3 . 1 6 ) , ( A 3 . 1 8 ) and been completed, (A3.19) r e s p e c t i v e l y . the analysis the e n t i r e process may this process can number of time increments; be be r e p e a t e d carried thus for out this time f o r the next consecutively for the complete response history can be computed. Two important integration procedure. procedures. Accuracy the exact continuous solutions are decay, and limit that well are in the to how solution. to limit depends as on « and 10 times i s applied be accuracy well the Stability such a way considered a on the type of and refers that numerical stability they of solution to whether the matches extraneous increase rather than U s u a l l y t h e r e i s an upper stability, element i n any numerical come t o dominate the r e s u l t s . and stiffness and the v a l u e of mass m a t r i x as 8. linear acceleration assumption (<= = 1/2, 6 = 1/6) the g i v e good a c c u r a c y i f the s h o r t e s t p e r i o d of the d e p o s i t i s greater a c c e l e r a t i o n method it refers have to At that i s n e c e s s a r y to guarantee analysis w i l l to They introduced thus With 5 aspects than At (Clough i s only c o n d i t i o n a l l y to s o i l and Penzien stable, s t r u c t u r e s w i t h the s h o r t e s t 1.8 times the i n t e g r a t i o n i n t e r v a l . and 1975). i t will period less The linear blow up i f than about Thus the time increments must be made 191 short relative regardless In the of periods step in the period higher finite general may with be the required to avoid average simplest of good procedures several orders significant be Wilson 9-method stable method. method and methods than stable (« this = 1/2 is a Wilson In best is based of very of the are has of the cases, the short time step. of been r e p o r t e d an The the not to assumption. unconditionally linear acceleration a l l unconditionally stable 1975). 8-method on the assumption that the %, that, 0 = 1 , this method method. The presented above except replaced and time stable i s one x = 9 At where 0 > 1.37 When of than a c c e l e r a t i o n v a r i e s l i n e a r l y over an extended computation i n t e r v a l , such not. available. 1/4) also modification the less these the or periods acceleration is system unconditionally , 8 = linear method et a l . an assumption methods w i t h be magnitude methods a l . 1973), to shortest regardless et reported (Clough, The blow up (Wilson, This is the But the because of instead, a c c e l e r a t i o n method these methods. of i n the significantly response. used instability; unconditionally results v i b r a t i o n contained element i s r e q u i r e d which w i l l not constant of modes c o n t r i b u t e a c c e l e r a t i o n method cannot Several The the using associated method give least whether analysis vibration linear to {x}T by analysis T and have to be reverts procedure that also i n the the modified. (A3.20) to is the standard linear exactly the same equations the time step to evaluate equations as acceleration the At procedure has {x}-j,, to be {&} ^, 192 S i n c e t h i s i s e s s e n t i a l l y a l i n e a r a c c e l e r a t i o n method, here and 8=1/6. By inspection, the required equations can be = 1/2 rewritten as follows. The f e q u a t i o n (A3.12) can + 1[C] ^M] + [K ] f c t ] M t { A X } be r e w r i t t e n m { ~ p } + as, [ M ] [ c ] { p } t ( A 3 > 2 1 ) i n which, { *t E x ^ t = 3 + ( A 3 ' 2 2 ) and, {F} t = 3 After the W {x},p, The + \ {X} (A3.23) t {AX}, v e l o c i t y and which i s over an acceleration a c c e l e r a t i o n at t = T can = [* T t evaluating displacement, time t = T. {X} {AX} 2 + {E}J {X} from e q u a t i o n (A3.19) w i t h « = 1/2 {X} T = W t + f [ W t + {X} ] T be extended values should evaluated time be increment computed at using, (A3.24) t and 8 = 1/6 is, (A3.25) 193 and a l s o the displacement {x}^, from (A3.2) i s , {X} T = ( X } t + At ( X } t + *f - [ { x } T + 2 {X}J (A3.26) f It must does not guarantee artificial be remembered accuracy that or v i c e damping i n h i g h e r modes. stability versa. i n numerical The W i l s o n But knowing t h a t h i g h e r modes of v i b r a t i o n c o n t r i b u t e s very l i t t l e structures, Therefore, t h i s method i n a way filters t h i s method has been found number of dynamic a n a l y s e s . integration 0-method the response due to to the t r u e response of out the h i g h frequency to y i e l d imposes realistic response. results i n a
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A two-dimensional non-linear static and dynamic response analysis of soil structures Siddharthan, Rajaratnam 1984
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Title | A two-dimensional non-linear static and dynamic response analysis of soil structures |
Creator |
Siddharthan, Rajaratnam |
Publisher | University of British Columbia |
Date Issued | 1984 |
Description | A method of analysis in two-dimensions to predict static and dynamic response of soil structures, including soil-structure interaction has been presented herein. The static and dynamic analyses can be performed in either effective or total stress mode or a combination of both modes. Non-linear stress-strain behaviour of soil has been modelled by using an incrementally elastic approach in which tangent shear modulus and tangent bulk modulus were taken as the two "elastic" parameters. The material response in shear was assumed to be hyperbolic coupled with Masing behaviour during unloading and reloading. Responses to changes in mean normal stress was assumed to be non-linear, elastic and stress dependent. Slip or contact elements have been incorporated in the analysis to represent the interface characteristics between soil and structural elements. The properties of the slip elements were assumed to be elastic, perfectly plastic, with failure at the Interface given by the Mohr-Coulomb failure criterion. In the static analysis proposed here, gravity may be switched on at once for the completed soil structure or the construction sequence can be modelled by layer analysis. The stress-strain conditions determined by the static analysis give the in-situ stress condition before the dynamic analysis. In the dynamic effective stress analysis, the residual porewater pressures are calculated using a modification of the model proposed by Martin, et al. (1975). The parameters, G[sub max] and[sub max] are modified for the effects of residual porewater pressure. The dynamic response study includes the prediction of post earthquake deformations. The predictive capability of the new method of analysis has been verified by comparing the recorded porewater pressure and accelerations of two centrifuged models subjected to simulated earthquakes, to those computed by the new method. This method has also been used to compute response of an offshore drilling island supporting a tanker mounted drilling rig. Results suggest that the common practice of neglecting soil-structure interaction may not be appropriate for islands which support heavy tanker type of structures. At present one-dimensional methods are used for computing the response of these islands. Comparative studies are also reported to asses the validity of this procedure. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-06-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062642 |
URI | http://hdl.handle.net/2429/25677 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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