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UBC Theses and Dissertations

Stability of buried pipelines subjected to wave loading Siddharthan, Rajaratnam 1981

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STABILITY OF BURIED PIPELINES SUBJECTED TO WAVE LOADING by RAJARATNAM SIDDHARTHAN B.S. (Hons.), U n i v e r s i t y o f S r i Lanka, P e r a d e n i y a Campus, S r i Lanka, 1977 A THESIS'SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May 1981 (c) R a j a r a t n a m S i d d h a r t h a n , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t 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 g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date Aprf H 8 I ABSTRACT The r e s p o n s e of b u r i e d o f f s h o r e p i p e l i n e s t o wave l o a d i n g i s shown t o depend m a i n l y on t h e s t a b i l i t y of t h e s e a f l o o r . Two a n a l y s e s t o i n v e s t i g a t e t h e s t a b i l i t y of t h e s e a f l o o r a r e p r e s e n t e d ; one i s based on B i o t ' s t h e o r y of a p o r o - e l a s t i c s o l i d , t h e o t h e r on t h e t h e o r y of r e s i d u a l wave i n d u c e d p o r e w a t e r p r e s s u r e s . P o r o - e l a s t i c a n a l y s i s g i v e s t h e d i s -t r i b u t i o n of t h e t r a n s i e n t p o r e w a t e r p r e s s u r e s and t h e e f f e c t i v e s t r e s s f i e l d i n d u c e d by t h e wave l o a d i n g . The s t a b i l i t y of t h e s e a f l o o r c a n be e s t i m a t e d by a p p l i c a t i o n of t h e Mohr-Coulomb f a i l u r e c r i t e r i o n t o t h e computed s t r e s s f i e l d . For a l l deep d e p o s i t s e x c e p t f o r h a r d f i n e sands, i t i s shown t h a t a much s i m p l e r a n a l y s i s c a n be u s e d . The e f f e c t i v e s t r e s s e s c a n be d e t e r m i n e d by computing t h e t o t a l s t r e s s f i e l d e l a s t i c a l l y and p o r e p r e s s u r e f i e l d by s i m p l e s o l u t i o n s of t h e L a p l a c e e q u a t i o n . R e s i d u a l p o r e p r e s s u r e a n a l y s i s i s based on computing t h e r e s i -d u a l p o r e p r e s s u r e s g e n e r a t e d by t h e c y c l i c a c t i o n of wave i n d u c e d s h e a r s t r e s s e s and t h e n u s i n g t h e Mohr-Coulomb f a i l u r e c r i t e r i o n t o i n v e s t i g a t e t h e s t a b i l i t y . I n many c a s e s , r e s i d u a l p o r e p r e s s u r e s c a n l e a d t o i n s t a -b i l i t y o r even l i q u e f a c t i o n of t h e s e a f l o o r . Because of t h e d u r a t i o n of t y p i c a l s t o r m s , d i s s i p a t i o n of r e s i d u a l p o r e p r e s s u r e s d u r i n g a s t o r m has been c o n s i d e r e d i n t h i s a n a l y s i s . The d e g r a d a t i o n i n e f f e c t i v e s t r e s s dependent p r o p e r t i e s of t h e d e p o s i t i s a l s o t a k e n i n t o a c c o u n t . Two computer programs, STAB-MAX and STAB-W, have been d e v e l o p e d t o p e r f o r m t h e p o r o - e l a s t i c and r e s i d u a l p o r e p r e s s u r e a n a l y s e s r e s p e c -t i v e l y . An example p r o b l e m i n p i p e l i n e s t a b i l i t y has been p r e s e n t e d t o i l l u s t r a t e t h e f a c t o r s t h a t s h o u l d be c o n s i d e r e d i n t h e a n a l y s i s of t h e s t a b i l i t y of b u r i e d o f f s h o r e p i p e l i n e s . i i i TABLE OF CONTENTS Page ABSTRACT n TABLE OF CONTENTS i i i LIST OF FIGURES v i LIST OF TABLES v i i i NOMENCLATURE i x ACKNOWLEDGEMENTS x i i CHAPTER 1 1 1.1 I n t r o d u c t i o n 1 1.2 Scope 2 1.3 L i t e r a t u r e Review 4 1.4 O r g a n i s a t i o n o f t h e T h e s i s 6 CHAPTER 2 GENERAL ASPECTS OF OFFSHORE PIPELINE STABILITY 7 2.1 Statement o f t h e P r o b l e m 8 2.2 E s t i m a t i o n o f t h e F o r c e s 10 2.2.1 E f f e c t i v e bouyancy f o r c e U 10 2.2.2 E f f e c t i v e w e i g h t o f t h e mass of s o i l I n v o l v e d : Ws 11 2.2.3 Shear r e s i s t a n c e component, Ry 12 2.2.4 C o n t r i b u t i o n o f s o i l a d h e s i o n , C^ 12 2.2.5 C o n t r i b u t i o n o f s o i l s u c t i o n f o r c e , P w 13 2.3 Pro p o s e d A n a l y s i s f o r t h e P i p e l i n e 13 CHAPTER 3 EVALUATION OF DESIGN STORM WAVE 18 3.1 O b t a i n i n g D e s i g n Storm Waves 19 3.2 Wave T h e o r i e s 21 i v Page 3.2. 1 L i n e a r wave t h e o r y 22 3.2. 2. S t o k e s ' h i g h e r o r d e r t h e o r i e s 24 3.2. 3 C n o i d a l wave t h e o r y 24 3.2. 4 S o l i t a r y wave t h e o r y 25 3.2. 5 S t r e a m - f u n c t i o n n u m e r i c a l wave t h e o r y 25 3.2. 6 Theory used i n t h i s t h e s i s 26 3.3 E q u i v a l e n t U n i f o r m Storm Wave 26 CHAPTER 4 PORO-ELASTIC ANALYSIS 31 4.1 I n t r o d u c t i o n 31 4.2 G o v e r n i n g E q u a t i o n s 31 4.3 Boundary C o n d i t i o n s 34 4.4 S o l u t i o n T e c h n i q u e 35 4.4. .1 H o r i z o n t a l l y l a y e r e d d e p o s i t 37 4.5 A n a l y s i s o f t h e Depth o f I n s t a b i l i t y 38 4.6 V a l u e s f o r E l a s t i c C o n s t a n t s 39 4.7 Example P r o b l e m 42 4.8 Comparison of Theory w i t h F i e l d Data 52 4.9 Some P r a c t i c a l S o l u t i o n s 55 CHAPTER 5 RESIDUAL PORE PRESSURE ANALYSIS 58 5.1 Theory 59 5.2 A n a l y s i s o f t h e Depth o f I n s t a b i l i t y 63 5.3 Changes i n S o i l P r o p e r t i e s and P o r e P r e s s u r e G e n e r a t i o n 63 5.4 Example P r o b l e m 65 5.4, .1 E q u i v a l e n t u n i f o r m storm . 67 5.4. .2 D i s c u s s i o n o f r e s u l t s 67 V Page CHAPTER 6. ANALYSIS OF PIPELINE FLOTATION 71 CHAPTER 7 CONCLUSIONS 75 REFERENCES 78 APPENDIX I 83 APPENDIX I I 89 v i LIST OF FIGURES Page FIGURE 1 F o r c e s on a B u r i e d P i p e l i n e 9 FIGURE 2 G e n e r a l S l i p S u r f a c e f o r S h a l l o w A n c h o r s 9 FIGURE 3 Flo w C h a r t t o C a l c u l a t e G e n e r a l Storm Wave Data 20 and t h e E q u i v a l e n t Storm Wave System FIGURE 4 R e g i o n s o f V a l i d i t y of Wave T h e o r i e s 23 FIGURE 5(a) M o d i f i e d P e n e t r a t i o n v s x / a ^ 0 29 FIGURE 5(b) E q u i v a l e n t Number of C y c l e s 29 FIGURE 6 Wave P r e s s u r e s on Ocean F l o o r 32 FIGURE 7 The V a r i a t i o n o f Secant Shear Modulus w i t h 40 Shear S t r a i n FIGURE 8 S o i l D e p o s i t used i n Example P r o b l e m 43 FIGURE 9 F i n i t e Element D i s c r e t i z a t i o n 43 FIGURE 10(a) P o r e w a t e r P r e s s u r e i n a D e p o s i t of F i n i t e Depth 45 FIGURE 10(b) Induced V e r t i c a l E f f e c t i v e S t r e s s e s i n a D e p o s i t 46 of F i n i t e Depth FIGURE 10(c) Induced Shear S t r e s s e s i n a D e p o s i t o f F i n i t e 48 Depth FIGURE 10(d) Induced H o r i z o n t a l E f f e c t i v e S t r e s s e s i n a 49 D e p o s i t o f F i n i t e Depth FIGURE 11 C o n t o u r s o f Developed F r i c t i o n A n g l e , <j> 51 FIGURE 12 B o r e h o l e Data 53 FIGURE 13 P o r e P r e s s u r e D i s t r i b u t i o n i n Sand D e p o s i t o f 54 C a l i f o r n i a Coast FIGURE 14 Rate o f P o r e P r e s s u r e G e n e r a t i o n d u r i n g C y c l i c 62 L o a d i n g FIGURE 15(a) B a s i c E q u a t i o n and S o l u t i o n Domain 64 FIGURE 15(b) S t r e s s C o n d i t i o n s B e f o r e and D u r i n g Storms 64 v i i Page FIGURE 16 Residual Pore Pressure D i s t r i b u t i o n i n S o i l 70 Deposit FIGURE 17 Wave Forces on a H o r i z o n t a l l y Layered Deposit 90 v i i i L IST OF TABLES Page TABLE I G a v i t y B r e a k t h r o u g h F a c t o r s 16 TABLE I I D e s i g n Wave Data 66 TABLE I I I L i q u e f a c t i o n P o t e n t i a l Curve 66 TABLE I V C a l c u l a t i o n o f E q u i v a l e n t Number of U n i f o r m 68 C y c l e s TABLE V R e s u l t s o f R e s i d u a l P o r e P r e s s u r e A n a l y s i s 68 TABLE VI C a l c u l a t i o n of B r e a k o u t S o i l Mass 73 i i x NOMENCLATURE B = d i a m e t e r o f t h e p i p e B = b u l k modulus of t h e d e p o s i t m C„ = c o r r e c t i o n f a c t o r f o r SPT v a l u e s N C = a d h e s i o n f o r c e between p i p e and s o i l a C = c o r r e c t i o n f a c t o r f o r l i q u e f a c t i o n d a t a Y c = c o h e s i o n o f t h e d e p o s i t c = c o e f f i c i e n t of c o n s o l i d a t i o n v c ^ - * , c ^ = a r b i t r a r y c o n s t a n t s a s s o c i a t e d w i t h t h e d e p o s i t D = T h i c k n e s s o f t h e d e p o s i t D, = d e p t h of embedment of t h e p i p e b = d e p t h o f i n s t a b i l i t y i n t h e d e p o s i t = r e l a t i v e d e n s i t y o f t h e d e p o s i t d = mean d e p t h o f w a t e r e = v o i d r a t i o F ,F = c a v i t y b r e a k o u t f a c t o r s c' q J G = shear modulus o f t h e d e p o s i t G = shear modulus of t h e d e p o s i t a t low s t r a i n l e v e l max H = wave h e i g h t H = e q u i v a l e n t wave h e i g h t i = seepage g r a d i e n t K x , K z = p e r m e a b i l i t y i n x and z d i r e c t i o n s K = 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 o L = wave l e n g t h m = b u l k modulus c o n s t a n t m = c o e f f i c i e n t o f volume c o m p r e s s i b i l i t y N = e q u i v a l e n t number of c y c l e s of a g i v e n r e f e r e n c e wave eq N = number of c y c l e s to l i q u e f a c t i o n N = c o r r e c t e d SPT v a l u e P n = p o r o s i t y of t h e sea-bed n w = t o t a l number of wave components i n t h e wave OCR = over c o n s o l i d a t i o n r a t i o = a t m o s p h e r i c p r e s s u r e P = s o i l s u c t i o n f o r c e w P = t o t a l s t a t i c p o r e p r e s s u r e p = induced t r a n s i e n t p o r e p r e s s u r e p^ = a m p l i t u d e of wave p r e s s u r e on top of t h e sea-bed q = weight of p i p e l i n e per u n i t l e n g t h R = domain of i n t e g r a t i o n R = v e r t i c a l component o f shear r e s i s t a n c e a l o n g t h e s l i p s u r f a c e R g = shear r e s i s t a n c e a l o n g s l i p s u r f a c e r u = pore p r e s s u r e r a t i o S = degree of s a t u r a t i o n T = wave p e r i o d = t o t a l d u r a t i o n o f .ithe storm = bouyancy f o r c e per u n i t l e n g t h U^jU^ = t o t a l p o r e p r e s s u r e a t top and bottom of t h e p i p e l i n e U = e f f e c t i v e bouyancy f o r c e on t h e p i p e l i n e per u n i t l e n g t h u = r e s i d u a l p o r e p r e s s u r e Ug = pore p r e s s u r e due to c y c l i c shear s t r e s s e s v = d i s p l a c e m e n t i n x d i r e c t i o n W = t o t a l weight of p i p e l i n e i n c l u d i n g i t s c o n t e n t s per u n i t P l e n g t h x i W = e f f e c t i v e weight of s o i l mass i n v o l v e d i n breakout t o g e t h e r w i t h t h e o b j e c t w = d i s p l a c e m e n t i n z d i r e c t i o n B = c o m p r e s s i b i l i t y of porewater a',B' = c o n s t a n t s Y g = s a t u r a t e d u n i t weight of t h e d e p o s i t Y = u n i t weight of water w -y' = e f f e c t i v e u n i t weight of t h e d e p o s i t Y M = d e f i n e d as (|' - y i ) w e = t o t a l v o l u m e t r i c s t r a i n e i r,e ,£ = normal s t r a i n s i n x,y,z d i r e c t i o n s d e f i n e d to be p o s i t i v e x y z vo i n e l o n g a t i o n d i s t a n c e measured i n z d i r e c t i o n from top of a l a y e r e m p i r i c a l c o n s t a n t wave number = 2TT/L P o i s s o n ' s r a t i o v e r t i c a l e f f e c t i v e s t r e s s a',a' = wave induced e f f e c t i v e s t r e s s e s x z x c = c y c l i c shear s t r e s s T = wave induced shear s t r e s s e s on x-z p l a n e ( h o r i z o n t a l ) X2 A>, \ ~ f r i c t i o n a n g l e d e v e l o p e d by s t r e s s s t a t e i n an element ( x , y , t ) <f>' =. a n g l e of i n t e r n a l f r i c t i o n x i i ACKNOWLEDGEMENTS The a u t h o r wishes to thank h i s a d v i s o r , P r o f e s s o r W.D. Liam F i n n , f o r h i s c o n t i n u e d i n t e r e s t , u n f a i l i n g h e l p and f o r making many v a l u a b l e s u g g e s t i o n s t o improve t h e p r e s e n t a t i o n of t h e t h e s i s . He i s a l s o i n d e b t e d t o P r o f e s s o r M. de St.Q. I s a a c s o n f o r r e a d i n g t h i s t h e s i s and f o r making c o n s t r u c t i v e s u g g e s t i o n s i n t h e p r e p a r a t i o n of Chapter 3. The a u t h o r wishes to exp r e s s h i s g r a t i t u d e to M i s s D e s i r e e Cheung f o r t y p i n g t h i s t h e s i s and f o r making v a l u a b l e comments. The aut h o r i s a l s o g r a t e f u l t o Mrs. 0. C u t h b e r t , who was r e s p o n s i b l e f o r p r o v i d i n g a p l e a s a n t and p e a c e f u l home d u r i n g h i s s t a y i n Canada. The f i n a n c i a l a s s i s t a n c e p r o v i d e d by Fugro, I n c . , Long Beach, C a l i f o r n i a i s g r a t e f u l l y acknowledged. 1 CHAPTER 1 1.1 I n t r o d u c t i o n The s t a b i l i t y o f t h e s e a f l o o r i s of major i m p o r t a n c e f o r o f f s h o r e i n s t a l l a t i o n s , such as p i p e l i n e s and g r a v i t y s t r u c t u r e s . The o c c u r r e n c e of s e a f l o o r i n s t a b i l i t y due t o wave a c t i o n has been r e p o r t e d by H e n k e l ( 1 9 7 0 ) . The i n f l u e n c e of s e a f l o o r i n s t a b i l i t y c aused by wave a c t i o n on o f f s h o r e i n s t a l l a t i o n s , such as p i p e l i n e s (Beckmann, 1970; C h r i s t i a n e t a l , 1974; N a t a r a j a , 1978), g r a v i t y s t r u c t u r e s ( L e e , 1975; Rahman, 1976), sea w a l l s ( H e y b i c h , 1968; S a t o , 1968) and j a c k e t - t y p e s t r u c t u r e s ( W r i g h t , 1972; Bea, 1971) has been e x t e n s i v e l y s t u d i e d . The o c c u r r e n c e of s e a f l o o r s l i d e s and l a r g e s o i l movements caused by waves d u r i n g t h e p e r i o d s o f s e v e r e s torm a c t i v i t y may have two e f f e c t s on an o f f s h o r e f a c i l i t y : (a) i t may l e a d t o l o s s o f s t r e n g t h i n f o u n d a t i o n s o i l ; (b) i t may pro d u c e s i g n i f i c a n t a d d i t i o n a l l a t e r a l l o a d . These two e f f e c t s r e p r e s e n t a v e r y s e v e r e l o a d i n g c o n d i t i o n f o r t h e d e s i g n of t h e s e s t r u c t u r e s . The problem of p r e d i c t i n g when, where, t o what e x t e n t a s e a f l o o r s l i d e w i l l o c c u r and t h e e f f e c t s o f t h i s s l i d e on t h e above o f f s h o r e i n s t a l l a t i o n s d u r i n g a storm a c t i v i t y i s e x t r e m e l y d i f f i c u l t . P i p e l i n e s a r e p r o b a b l y t h e most e f f i c i e n t and e c o n o m i c a l means of t r a n s p o r t i n g p e t r o l e u m p r o d u c t s from o f f s h o r e s o u r c e s t o ons h o r e f a c i l i t i e s . One of t h e major d e s i g n c o n s i d e r a t i o n s i n t h e d e s i g n of o f f s h o r e p i p e l i n e s would be t h e i n s t a b i l i t y due t o c u r r e n t , wave and f l o t a t i o n e f f e c t s , w h i c h may l e a d t o e v e n t u a l p i p e l i n e f a i l u r e . The e f f e c t s of s u r f a c e waves on a s e a f l o o r can be a n a l y s e d u s i n g wave t h e o r i e s . Because of many v a r i a b l e s i n f l u e n c i n g wave geometry and water k i n e m a t i c s , a g e n e r a l t h e o r y f o r t h e mechanics of wa t e r waves i s 2 v e r y d i f f i c u l t , i f n o t i m p o s s i b l e . I t i s common t o assume t h a t t h e sea s u r f a c e waves can be i d e a l i s e d a s a p l a n e t r a v e l l i n g s u r f a c e waves. These s u r f a c e t r a v e l l i n g waves e x e r t p r e s s u r e l o a d i n g on t h e s e a f l o o r , w i t h a m p l i t u d e v a r y i n g h a r m o n i c a l l y i n t i m e and h o r i z o n t a l d i s t a n c e . The h a r m o n i c a l l y v a r y i n g s t r e s s f i e l d i n d u c e d by t h i s p r e s s u r e l o a d i n g on t h e s e a f l o o r , may be l a r g e enough t o cause s e a f l o o r s l i d e s . H e r e , t h e Mohr-Coulomb c r i t e r i o n c an be employed s u c c e s s f u l l y t o a n a l y s e a p o t e n t i a l f a i l u r e zone. F u r t h e r , t h i s harmonic p r e s s u r e l o a d i n g w i l l p r o d u c e c y c l i c s h e a r s t r e s s e s i n t h e ocean bed w h i c h may g i v e r i s e t o t h e b u i l d u p o f r e s i d u a l p o r e p r e s s u r e ( F i n n , 1976) i n some s o i l s . T h i s i n c r e a s e i n p o r e p r e s s u r e w i l l l e a d t o s t r e n g t h l o s s o r even l i q u e f a c t i o n o f t h e s o i l s u r r o u n d i n g t h e p i p e l i n e . The t e c h n i q u e s o f s e i s m i c r e s p o n s e a n a l y s i s can be used t o e v a l u a t e t h i s r e s i d u a l p o r e p r e s s u r e . B u t , i n t h e f o l l o w -i n g a s p e c t s , s e i s m i c l o a d i n g d i f f e r s f r o m wave l o a d i n g . (a) The d u r a t i o n o f l o a d i n g and wave p e r i o d s a r e v e r y l o n g i n t h e c a s e of wave l o a d i n g and, t h e r e f o r e , p o r e p r e s s u r e d i s s i p a t i o n s h o u l d be c o n s i d e r e d . (b) Wave l o a d i n g i s e s s e n t i a l l y u n i d i r e c t i o n a l . (c) S e i s m i c l o a d i n g i s a p p l i e d a t t h e l o w e r boundary o f t h e d e p o s i t . As t h e r e s i d u a l p o r e p r e s s u r e i n c r e a s e s d u r i n g t h e s e i s m i c l o a d i n g , t h e s o i l d e p o s i t s o f t e n s and t h i s m i g h t l e a d t o l o w e r i n e r t i a f o r c e s on t h e s t r u c t u r e s c o n s t r u c t e d on t o p of t h e d e p o s i t . But wave l o a d i n g w h i c h i s a p p l i e d e x t e r n a l l y and i n d e p e n d e n t l y r e m a i n s u n a l t e r e d and t h e major e f f e c t o f d e g r a d a t i o n o r s o f t e n -i n g o f s o i l p r o p e r t i e s i s t o i n c r e a s e t h e d e p t h o f i n s t a b i l i t y under t h e g i v e n wave l o a d i n g . 1.2 Scope I n t h i s t h e s i s an a n a l y s i s o f t h e r e s p o n s e o f a b u r i e d p i p e l i n e t o wave l o a d i n g i s p r e s e n t e d . The seabed i s c o n s i d e r e d t o be a h o r i z o n -t a l l y l a y e r e d d e p o s i t . F u r t h e r , a common a s s u m p t i o n i n s o i l - s t r u c t u r e 3 i n t e r a c t i o n problems i s made, v i z . t h e p r e s e n c e of t h e p i p e l i n e i n no way a f f e c t s e i t h e r t h e s t r e s s f i e l d d i s t r i b u t i o n o r t h e i n s t a n t a n e o u s or r e s i d u a l p o r e p r e s s u r e s i n d u c e d by t h e wave l o a d i n g i n t h e d e p o s i t . W i t h t h e s e a s s u m p t i o n s t h e r e s p o n s e of a b u r i e d p i p e l i n e t o wave l o a d i n g can be deduced from t h e r e s p o n s e of t h e s o i l d e p o s i t a l o n e t o t h i s wave l o a d i n g . I t has been mentioned e a r l i e r t h a t t h e s u r f a c e waves have t h e f o l l o w i n g e f f e c t s on t h e seabed: 1) I t i n d u c e s a harmonic s t r e s s f i e l d . 2) The c y c l i c s h e a r s t r e s s e s may g i v e r i s e t o t h e b u i l d u p of r e s i d u a l . p o r e p r e s s u r e s . The harmonic s t r e s s f i e l d has been a n a l y s e d by Yamamoto (1978) and Madsen ( 1 9 7 8 ) . They assumed an e l a s t i c a l l y c o u p l e d s o i l - w a t e r system and p r e s e n t e d an e f f e c t i v e s t r e s s a n a l y s i s based on B i o t ' s e q u a t i o n s ( 1 9 4 1 ) . They a n a l y s e d t h e i n s t a b i l i t y of t h e seabed f o r g i v e n extreme waves, u s i n g t h e Mohr-Coulomb f a i l u r e c r i t e r i o n . The wave l o a d i n g was assumed t o be q u a s i - s t a t i c . The s o i l d e p o s i t c o n s i d e r e d by them was u n i -form and deep. I n t h i s t h e s i s an e f f i c i e n t computer program based on t h e f i n i t e d i f f e r e n c e method has been p r e s e n t e d t o a n a l y s e t h e r e s p o n s e of t h e seabed t o wave l o a d i n g under more g e n e r a l f i e l d c o n d i t i o n s . T h i s method w i l l be r e f e r r e d t o as t h e p o r o - e l a s t i c method o f s t a b i l i t y a n a l y s i s . P r e -d i c t i o n s made by t h i s method have been compared w i t h a v a i l a b l e f i e l d d a t a . The p o s s i b i l i t y o f l i q u e f a c t i o n of t h e s o i l d e p o s i t due t o r e s i -d u a l p o r e p r e s s u r e g e n e r a t i o n d u r i n g a d e s i g n s t o r m was t a k e n i n t o a c c o u n t by Seed e t a l ( 1 9 7 7 ) , and N a t a r a j a ( 1 9 7 8 ) . They a n a l y s e d t h e s e a f l o o r as an u n c o u p l e d s o i l - w a t e r system. They d i d n o t c o n s i d e r t h e e f f e c t s of d e g r a d a t i o n of t h e s o i l p r o p e r t i e s d u r i n g a storm a c t i v i t y as t h e r e s i d u a l p o r e p r e s s u r e s a r e g e n e r a t e d . I t has been m e n t i o n e d e a r l i e r t h a t t h e 4 d e g r a d a t i o n o f t h e s o i l p r o p e r t i e s s h o u l d be c o n s i d e r e d as t h i s s o f t e n i n g of s o i l p r o p e r t i e s l e a d s t o h i g h e r r e s i d u a l p o r e p r e s s u r e s . I n t h i s t h e s i s t h e e f f e c t o f s o f t e n i n g has been i n c l u d e d . F u r t h e r , Seed,':s a n a l y s i s has been extended t o i n c l u d e e s t i m a t i n g t h e maximum d e p t h of i n s t a b i l i t y d u r i n g a storm u s i n g t h e Mohr-Coulomb f a i l u r e c r i t e r i o n . A l l r e s e a r c h e r s , t o d a t e , have used t h e l i n e a r wave t h e o r y t o compute t h e p r e s s u r e l o a d i n g on t h e s e a f l o o r . The l i n e a r wave t h e o r y i s n o t a p p l i c a b l e t o s h a l l o w water c o n d i t i o n s . So, s h a l l o w w a t e r t h e o r i e s l i k e t h e c n o i d a l t h e o r y may be a p p r o p r i a t e when a s s e s s i n g s t a b i l i t y i n n e a r s h o r e r e g i o n s . The r e l a t i v e m e r i t s of t h e v a r i o u s wave t h e o r i e s and t h e i r r a n g e s of a p p l i c a b i l i t y a r e d i s c u s s e d . An a p p r o p r i a t e example i l l u s t r a t i n g a l l t h e p r i n c i p l e s of t h e proposed methods of a n a l y s i s i s p r e s e n t e d . 1.3 L i t e r a t u r e Review L a r g e a m p l i t u d e s u r f a c e waves have t h e f o l l o w i n g e f f e c t s on t h e seabed: 1) t h e y i n d u c e p r e s s u r e l o a d i n g on t h e s e a f l o o r w h i c h i s harmonic i n space and t i m e , and 2) r e s i d u a l p o r e p r e s s u r e s a r e g e n e r a t e d due t o c y c l i c s h e a r s t r e s s e s . The e f f e c t s of p r e s s u r e l o a d i n g on t h e s t a b i l i t y of s o f t c l a y d e p o s i t s were a n a l y s e d by H e n k e l ( 1 9 7 0 ) . He t r e a t e d t h e s e a f l o o r as an u n c o u p l e d s o i l - w a t e r system and p r e s e n t e d an a n a l y t i c a l model based on t h e p r i n c i p l e of l i m i t i n g e q u i l i b r i u m , e m p l o y i n g t h e a s s u m p t i o n of a c i r c u l a r f a i l u r e s u r f a c e . F u r t h e r , he assumed t h e p e r i o d s of i m p o r t a n t waves were l o n g enough t o n e g l e c t t h e t r a n s i e n t e f f e c t s of t h e p r e s s u r e l o a d i n g . T h e r e -f o r e , t h e e f f e c t s of t h e h i g h e s t wave l o a d i n g s can be a n a l y s e d s t a t i c a l l y . An e x t e n s i o n of H e n k e l ' s method was p r o p o s e d by W r i g h t e t a l (1972) w h i c h employs t h e f i n i t e element method and i n c l u d e s n o n l i n e a r 5 p r o p e r t i e s o f s o i l . A h y p e r b o l i c s t r e s s - s t r a i n r e l a t i o n s h i p and a v a l u e of 0.495 f o r P o i s s o n ' s r a t i o ( t o r e p r e s e n t u n d r a i n e d c o n d i t i o n ) were assumed i n t h e c a l c u l a t i o n o f s t r e s s e s . B o t h t h e above methods a r e t o t a l s t r e s s methods w h i c h do n o t r e q u i r e p o r e p r e s s u r e d i s t r i b u t i o n and a r e s u i t a b l e f o r d e p o s i t s o f v e r y l o w p e r m e a b i l i t y , c h i e f l y , c l a y e y d e p o s i t s . An e f f e c t i v e s t r e s s a n a l y s i s c a n be p e r f o r m e d i n two ways: 1) by e s t i m a t i n g t h e t o t a l s t r e s s and t h e p o r e p r e s s u r e d i s t r i b u t i o n i n t h e d e p o s i t i n d e p e n d e n t l y by u n c o u p l e d a n a l y s e s or 2) by t r e a t i n g t h e d e p o s i t as a c o u p l e d s o i l - w a t e r system. I n t h e u n c o u p l e d e f f e c t i v e s t r e s s a n a l y s i s t h e e s t i m a t i o n of t o t a l s t r e s s d i s t r i b u t i o n f o r a wave l o a d i n g i n t h e bed can be made u s i n g a p p r o p r i a t e t o t a l s t r e s s s o i l p r o p e r t i e s . The d i s t r i b u t i o n o f i n s t a n t a n e o u s pore p r e s s u r e s has been p r e d i c t e d by a number of i n v e s t i g a t o r s based on d i f f e r e n t a s s u m p t i o n s : 1) L i u ( 1 9 7 3 ) , M a s s e l ( 1 9 7 6 ) , Putnam (1949) and R e i d e t a l (1957) assumed t h a t t h e p o r o u s bed i s r i g i d and non-d e f o r m a b l e and t h e p o r e w a t e r i s i n c o m p r e s s i b l e . I f t h e f l u i d m o t i o n i n t h e porous bed i s d e f i n e d by D a r c y ' s Law, t h e n f o r i s o t r o p i c p e r m e a b i l i t y , t h e g o v e r n i n g e q u a t i o n l e a d s t o t h e L a p l a c e e q u a t i o n f o r p o r e p r e s s u r e . Putnam's s o l u t i o n s l e a d t o p o r e p r e s s u r e r e s p o n s e b e i n g independent o f t h e p e r m e a b i l i t y of t h e bed m a t e r i a l . 2) Moshagen and T«5rum (1975) assumed t h a t t h e w a t e r i s com-p r e s s i b l e w h i l e t h e p o r o u s bed i s n o n d e f o r m a b l e , and t h i s l e a d s t o t h e heat c o n d u c t i o n e q u a t i o n f o r p o r e p r e s s u r e d i s t r i b u t i o n . They c o n c l u d e d t h a t t h e p o r e p r e s s u r e r e s p o n s e i s a f u n c t i o n o f p e r m e a b i l i t y of t h e bed m a t e r i a l . I t s h o u l d be n o t e d t h a t w h i l e compres-s i b i l i t y o f s o i l s o l i d s i s i n d e e d n e g l i g i b l e , t h e com-p r e s s i b i l i t y of t h e s o i l s k e l e t o n i s s i g n i f i c a n t l y h i g h e r t h a n t h e w a t e r c o m p r e s s i b i l i t y and, t h e r e f o r e , Moshagen and Thrum's a s s u m p t i o n i s n o t a c c e p t a b l e . Yamamoto (1978) and Madsen (1978) p r e s e n t e d t h e c o u p l e d e f f e c t i v e s t r e s s a n a l y s i s f o r a deep d e p o s i t based on B i o t ' s e q u a t i o n s ( 1 9 4 1 ) . T h e i r a n a l y s i s t a k e s i n t o a c c o u n t t h e e l a s t i c d e f o r m a t i o n of t h e porous 6 medium and t h e c o m p r e s s i b i l i t y o f p o r e f l u i d . T h i s i s a c o u p l e d a n a l y s i s and, t h e r e f o r e , t h i s method g i v e s s o l u t i o n s f o r e f f e c t i v e s t r e s s e s and p o r e p r e s s u r e s . They i n v e s t i g a t e d f a i l u r e zones u s i n g t h e Mohr-Coulomb f a i l u r e c r i t e r i o n . On t h e o t h e r hand, p o r e p r e s s u r e s g e n e r a t e d by c y c l i c s h e a r s t r e s s e s were c o n s i d e r e d by Seed e t a l (1977) and N a t a r a j a ( 1 9 7 9 ) . Seed's a n a l y s i s w h i c h i n c l u d e s d i s s i p a t i o n of r e s i d u a l p o r e p r e s s u r e , was c o n -c e r n e d w i t h t h e p o s s i b l e l i q u e f a c t i o n o f t h e seabed. A s i m p l e p o r e p r e s -s u r e model pr o p o s e d by Seed e t a l (1976) was used t o g e n e r a t e p o r e w a t e r p r e s s u r e under c y c l i c l o a d i n g . 1.4 O r g a n i s a t i o n o f t h e T h e s i s The g e n e r a l a s p e c t s o f o f f s h o r e p i p e l i n e s t a b i l i t y a r e e x t e n s i v e l y d i s c u s s e d i n Cha p t e r 2. The main a s s u m p t i o n s and t h e d e s i g n c o n s i d e r a -t i o n s a r e c r i t i c a l l y r e v i e w e d . C h a p t e r 3 d e a l s w i t h t h e f o l l o w i n g t o p i c s : (a) Wave T h e o r i e s : a b r i e f d e s c r i p t i o n of a v a i l a b l e wave t h e o r i e s and t h e i r v a l i d i t y ; (b) E q u i v a l e n t U n i f o r m Storm: t h e method based on M i n e r ' s r u l e t o e v a l u a t e an e q u i v a l e n t s t o r m from d e s i g n storm wave d a t a i s e x p l a i n e d . C h a p t e r s 4 and 5 d i s c u s s t h e p o r o - e l a s t i c and r e s i d u a l p o r e p r e s -s u r e methods, r e s p e c t i v e l y . A s s u m p t i o n s , t h e o r e t i c a l f o r m u l a t i o n , l i m i t a -t i o n s and u s e f u l n e s s o f t h e s e t h e o r i e s a r e p r e s e n t e d . S o l u t i o n s f o r an example pr o b l e m a r e d i s c u s s e d . F u r t h e r m o r e , p r e d i c t i o n s based on t h e s e approaches have been compared w i t h a v a i l a b l e f i e l d d a t a . S t a b i l i t y o f a b u r i e d p i p e l i n e i s checked u s i n g t h e p r o c e d u r e o u t l i n e d i n t h i s t h e s i s i n Cha p t e r 6. A b r i e f summary of t h e more impor-t a n t c o n c l u s i o n i s p r e s e n t e d i n Cha p t e r 7. 7 CHAPTER 2 GENERAL ASPECTS OF OFFSHORE PIPELINE STABILITY I n o r d e r t o p r e v e n t damage o c c u r r i n g due t o f i s h i n g g e a r , s h i p s ' a n c h o r s , e t c . , o f f s h o r e p i p e l i n e s a r e o f t e n b u r i e d . T h e r e f o r e , one of t h e main d e s i g n r e q u i r e m e n t s i s t h a t t h e p i p e l i n e s h o u l d n o t f l o a t up d u r i n g i t s o p e r a t i o n . I n g e n e r a l , t h e s p e c i f i c g r a v i t y of t h e f l u i d t h a t i s t o be t r a n s p o r t e d by t h e p i p e l i n e v a r i e s from 0.99 i n t h e c a s e o f s a n i t a r y w a t e r , t o 0.0007 i n t h e c a s e of n a t u r a l gas. I n c a s e s where low s p e c i f i c g r a v i t y f l u i d i s b e i n g t r a n s p o r t e d , t h e r e w i l l be p o s i t i v e n e t buoyancy f o r c e s w h i c h t r y t o f o r c e t h e p i p e l i n e t o t h e s u r f a c e . Net buoyancy f o r c e i s d e f i n e d as t h e upward f o r c e on a p i p e l i n e due t o s u r r o u n d -i n g w a t e r above i t s own w e i g h t . I n t h e c a s e o f b u r i e d p i p e l i n e s t h e s t a b i l i t y o f c o v e r s h o u l d be a n a l y s e d f o r t h e f o l l o w i n g d e s i g n c o n s i d e r a t i o n s : 1) s e a f l o o r s l i d e s s h o u l d n o t o c c u r below t h e b u r i e d d e p t h of t h e p i p e l i n e o r i n t h e c o v e r m a t e r i a l ; 2) i f p o s i t i v e n e t buoyancy f o r c e s a r e p r e s e n t t h e c o v e r s h o u l d p r o v i d e adequate r e s i s t a n c e so t h a t t h e p i p e l i n e would n o t f l o a t ; 3) i n t h e c a s e s where n e t buoyancy f o r c e s a r e n o t p r e s e n t a d equate b e a r i n g r e s i s t a n c e s h o u l d be p r o v i d e d by t h e s o i l . T h i s t h e s i s i s c o n c e r n e d w i t h c a s e s where p o s i t i v e n e t buoyancy f o r c e s a r e p r e s e n t . B u r i a l of p i p e l i n e s a t adequate d e p t h , t h e p r o v i s i o n of a d equate s t r o n g c o v e r and a d d i t i o n of a r t i f i c i a l w e i g h t a r e t h e common methods used t o c o u n t e r a c t p i p e l i n e f l o t a t i o n p r o b l e m s . MacPherson (1978) and L i u e t a l (1979) p r e s e n t e d t h e o r e t i c a l a p p r oaches t o e v a l u a t e f o r c e s d e v e l o p e d , due t o seepage f o r c e s on a b u r i e d p i p e l i n e . T h e i r a n a l y s i s was based on Putnam's a s s u m p t i o n s . When t h e 8 c r e s t of t h e wave i s a l i g n e d w i t h t h e c e n t r e l i n e o f t h e p i p e , t h e wave f o r c e a c t s v e r t i c a l l y downward ( s t a b i l i z i n g f o r c e ) and, on t h e o t h e r hand, i t a c t s v e r t i c a l l y upwards when t h e t r o u g h i s i n l i n e w i t h t h e c e n t r e o f t h e p i p e ( d e s t a b i l i z i n g f o r c e ) . MacPherson c o n c l u d e d t h a t f o r a deep d e p o s i t t h e magnitude of t h e n e t wave i n d u c e d f o r c e w h i c h c o u l d be as l a r g e as 30% of t h e buoyancy f o r c e , i s independent o f t h e d i s t a n c e between t h e wave c r e s t and t h e c e n t r e o f t h e p i p e . He f u r t h e r s u g g e s t e d t h a t t h i s p e r i o d i c seepage f o r c e may be r e s p o n s i b l e f o r t h e " j a c k i n g e f f e c t " w h i c h has been r e p o r t e d f o r b u r i e d p i p e l i n e s . The a n a l y s i s p r e s e n t e d i n t h i s t h e s i s i g n o r e s t h e i n f l u e n c e o f t h i s p e r i o d i c seepage f o r c e . Now l e t us assume t h a t t h e i n f l u e n c e of t h e p i p e l i n e on e i t h e r t h e s t r e s s f i e l d d i s t r i b u t i o n o r t h e r e s i d u a l p o r e p r e s s u r e g e n e r a t i o n -d i s s i p a t i o n can be n e g l e c t e d . We c o u l d t h e n a n a l y s e t h e r e s p o n s e o f a b u r i e d p i p e l i n e t o a wave l o a d i n g by a n a l y s i n g t h e r e s p o n s e of t h e seabed a l o n e t o t h i s wave l o a d i n g . F u r t h e r m o r e , s i n c e t h i s t h e s i s i s c o n c e r n e d w i t h h o r i z o n t a l l y l a y e r e d d e p o s i t s , a c o m p l e t e s e a f l o o r s l i d e would n o t o c c u r . So, t h e o n l y d e s i g n c o n s i d e r a t i o n t h a t has t o be a n a l y s e d i s t h e s t a b i l i t y o f t h e p i p e l i n e a g a i n s t f l o t a t i o n . 2.1 Statement o f t h e P r o b l e m F i g u r e 1 shows t h e f o r c e s w h i c h must be c o n s i d e r e d when e s t a b -l i s h i n g a c r i t e r i o n f o r s t a b i l i t y a g a i n s t f l o t a t i o n . The components a r e 1) The n e t buoyancy f o r c e on t h e p i p e l i n e , U. Here, U = U b - Wp where Wp = t o t a l w e i g h t o f p i p e l i n e and i t s c o n t e n t s per u n i t l e n g t h ; = buoyancy f o r c e caused by s u r r o u n d i n g w ater p r e s s u r e p er u n i t l e n g t h . 2) The e f f e c t i v e w e i g h t o f t h e mass of s o i l , Wg, i n v o l v e d i n b r e a k o u t t o g e t h e r w i t h t h e p i p e l i n e . 9 Still Water Level Sea Bottom FIG. 1 F o r c e s on a B u r i e d P i p e l i n e FIG. 2 G e n e r a l S l i p S u r f a c e f o r S h a l l o w Anchors 10 E f f e c t i v e weight of t h e i n v o l v e d s o i l mass can be determined e a s i l y under calm c o n d i t i o n s i f t h e e f f e c t i v e u n i t weight, y'» a n Q t n e volume of t h a t mass a r e known. Should t h e r e be ste a d y v e r t i c a l upward seepage of g r a d i e n t , i , i n t h e s o i l mass i n q u e s t i o n , t h e app a r e n t s o i l u n i t weight has to be changed t o Y" = Y' - Y w i ( 2 . 2 ) 3) The v e r t i c a l component, R^, of t h e f o r c e s of s h e a r i n g r e s i s t a n c e , R g, on t h e overburden s o i l a l o n g t h e s l i p s u r f a c e s e p a r a t i n g t h e p a r t of t h e s o i l i n v o l v e d i n breakout from t h e r e s t of t h e s o i l mass. 4) The v e r t i c a l component, C a, due to a d h e s i o n between t h e p i p e l i n e and t h e a d j a c e n t s o i l . 5) The s o i l s u c t i o n f o r c e s , P w, r e s u l t i n g from d i f f e r e n c e s i n porewater s t r e s s e s above and below t h e p i p e l i n e caused by attempted upward movement. For i n s t a b i l i t y o r upward movement one s h o u l d have U b > w p + W s + R v + C a + p w ( 2 . 3 ) 2.2 E s t i m a t i o n o f t h e F o r c e s 2.2.1 E f f e c t i v e buoyancy f o r c e U Weight per u n i t l e n g t h of p i p e l i n e , and i t s c o n t e n t s (Wp) can be e v a l u a t e d p r e c i s e l y . The average buoyancy f o r c e per u n i t l e n g t h , U^, can be e s t i m a t e d u s i n g t h e f o r m u l a , V u i U.b = TTB ( 2 . 4 ) where B = t h e diameter of t h e p i p e ; UjV U2 = t h e t o t a l p o r e p r e s s u r e s , i . e . , s t a t i c and r e s i d u a l pore p r e s s u r e a t t h e top and bottom of t h e p i p e , r e s p e c t i v e l y . T h i s e q u a t i o n i s o b t a i n e d by i n t e g r a t i n g t h e t o t a l p o r e p r e s s u r e s around the c i r c u m f e r e n c e o f t h e p i p e s assuming t h a t t h e v a r i a t i o n between and U2 i s l i n e a r . Note t h a t t h e v a l u e s of c o r r e s p o n d i n g t o the buoyancy 11 J f o r c e on a p i p e i n water and l i q u e f i e d s o i l d e p o s i t c a n be r e p r e s e n t e d by t h i s e q u a t i o n a s , 2 w a t e r w 4 and I L . f • a = Y "~7~ < 2 - 4 b ) l i q u e f i e d s 4 s o i l where Y w = t h e u n i t w e i g h t of w a t e r , and Y S = t h e u n i t w e i g h t of s a t u r a t e d s o i l d e p o s i t s u r r o u n d i n g t h e p i p e . Now knowing Wp and U^, e q u a t i o n (2.1) can be used t o e s t i m a t e U. 2.2.2 E f f e c t i v e w e i g h t of t h e mass of s o i l i n v o l v e d : Ws The volume of s o i l mass i n v o l v e d depends, i n g e n e r a l , on t h e d e p t h of embedment, D b, and r e l a t i v e d e n s i t y , D r. Based on e x t e n s i v e e x p e r i m e n t a l d a t a , V e s i c (1971) r e p o r t e d t h a t s h a l l o w a n c h o r s f a i l a l o n g t h e g e n e r a l s l i p s u r f a c e as shown i n F i g . 2. Deep a n c h o r s move v e r t i c a l l y f o r a c o n s i d e r a b l e d i s t a n c e p r o d u c i n g a f a i l u r e p a t t e r n s i m i l a r t o p u n c h i n g shear f a i l u r e and t h e n f a i l a l o n g t h e g e n e r a l s l i p s u r f a c e . The c r i t i c a l r e l a t i v e d e p t h , D^/B, above w h i c h t h e embedded o b j e c t s s h o u l d behave as s h a l l o w a n c h o r s , depends on t h e r e l a t i v e d e n s i t y o f t h e s o i l and, p o s s i b l y , some o t h e r as y e t u n c l a r i f i e d f a c t o r s . A v a i l a b l e e x p e r i m e n t a l e v i d e n c e ( V e s i c , 1971) s u g g e s t s t h a t t h i s l i m i t i n g d e p t h , D^/B, i n sand may i n c r e a s e from p e r h a p s 2 f o r a v e r y l o o s e d e p o s i t t o o v e r 10 i n a v e r y dense d e p o s i t . I n v e r y s o f t b e n t o n i t e c l a y t h e l i m i t i s about D^/B = 2 w h i l e i n a s t i f f c l a y i t i s a p p r o x i m a t e l y 10. A f t e r assuming t h e volume i n v o l v e d , an a v e r a g e e f f e c t i v e w e i g h t of t h e s o i l mass w h i c h a c c o u n t s f o r seepage f o r c e s has t o be e v a l u a t e d . E f f e c t i v e u n i t w e i g h t of t h e s o i l mass i s a f f e c t e d by t h e o c c u r r e n c e of seepage f o r c e s and c a n be c a l c u l a t e d u s i n g e q u a t i o n ( 2 . 2 ) . 2.2.3 Shear r e s i s t a n c e component, R v, a l o n g t h e assumed s l i p s u r f a c e The v e r t i c a l component of t h e s h e a r i n g r e s i s t a n c e , R v, c a n be d e t e r m i n e d by a p p r o p r i a t e a n a l y s e s f o r any t r i a l s l i p s u r f a c e . Even a s i m p l e y i e l d c r i t e r i o n s u c h as Mohr-Coulomb l e a d s t o r i g o r o u s c o m p u t a t i o n s u n l e s s some o t h e r a s s u m p t i o n s a r e made r e g a r d i n g t h e d i s t r i b u t i o n of s t r e s s e s a l o n g t h e s l i p s u r f a c e . An a n a l y t i c a l a p p r o a c h to t h i s p roblem was' p r o p o s e d by V e s i c ( 1 9 7 1 ) , who compared t h i s p r o b l e m w i t h t h e e x p a n s i o n of c a v i t i e s n ear t h e s u r f a c e of a s e m i - i n f i n i t e p l a s t i c s o l i d mass. The s o l u t i o n i s g i v e n i n terms o f an u l t i m a t e r a d i a l p r e s s u r e r e q u i r e d t o b r e a k o u t a c y l i n d r i c a l or s p h e r i c a l c a v i t y of a g i v e n r a d i u s and d e p t h of embedment below t h e s u r f a c e of t h e s o l i d mass. T h i s p r e s s u r e i s e q u i -v a l e n t t o W s p l u s t h e component, Rv. P h i l l i p s e t a l (1979) c a r r i e d out an e x t e n s i v e e x p e r i m e n t a l program t o s t u d y t h e s t a b i l i t y o f p i p e l i n e s i n sand, and c o n c l u d e d t h a t V e s i c ' s t h e o r y o v e r e s t i m a t e d t h e s t a b l e s o i l r e s i s t a n c e because t h e t h e o r y d i d not c o n s i d e r wave-induced e x c e s s p o r e p r e s s u r e s and t h e c o r r e s p o n d i n g r e d u c t i o n i n e f f e c t i v e s t r e s s e s . 2.2.4 C o n t r i b u t i o n of s o i l a d h e s i o n , Ca C o h e s i v e s o i l s c o n t a i n i n g a c t i v e m i n e r a l s w i l l d e v e l o p a d h e s i o n when i n c o n t a c t w i t h a l m o s t a l l m a t e r i a l s . The p r o c e s s i s of a p h y s i c o -c h e m i c a l n a t u r e and r e q u i r e s some t i m e . E x p e r i e n c e s w i t h s t e e l , c o n c r e t e , and wood p i l e s seem t o i n d i c a t e , .at l e a s t i n s o f t c l a y s , t h a t t h e a d h e s i o n e q u a l s o r exceeds t h e u n d r a i n e d shear s t r e n g t h a f t e r a p e r i o d of a few days t o , p e r h a p s , a few months of b u r i a l . Most of t h e known a d h e s i o n s t u d i e s were c o n c e r n e d w i t h measurements of r e s i s t a n c e t o s h e a r . However, i n t h i s b r e a k o u t p r o b l e m r e s i s t a n c e t o t e n s i o n between t h e b u r i e d o b j e c t 13 and t h e u n d e r l y i n g s o i l must be d e a l t w i t h . S i n c e t h e a s s u m p t i o n t h a t C a i s e q u a l t o z e r o l e a d s t o a c o n s e r v a t i v e d e s i g n , C a i s assumed t o be z e r o i n t h i s a n a l y s i s . 2.2.5 C o n t r i b u t i o n o f s o i l s u c t i o n f o r c e , P w D u r i n g t h e p r o c e s s o f a t t e m p t e d upward p i p e l i n e movement, t h e o v e r b u r d e n s o i l a d j a c e n t t o t h e p i p e i s h e a v i l y compressed, w h i l e t h e u n d e r l y i n g s o i l i s r e l i e v e d from s t r e s s e s . T h i s w i l l l e a d t o an i n c r e a s e i n p o r e w a t e r p r e s s u r e s above t h e o b j e c t and a d e c r e a s e i n p o r e w a t e r p r e s -s u r e s below t h e o b j e c t . I f t h e s o i l i s v e r y p o r o u s t h e s e p o r e w a t e r p r e s -s u r e s w i l l v a n i s h as t h e y a p p e a r . The d i f f e r e n c e i n p o r e w a t e r p r e s s u r e s above and below t h e p i p e l i n e r e s u l t s i n a s u c t i o n f o r c e . V e s i c (1971) s u g g e s t e d a p o s s i b l e way of a n a l y s i n g t h i s s u c t i o n f o r c e by p e r f o r m i n g u n d r a i n e d t e s t s o f s o i l samples. The e f f e c t i v e s t r e s s e s and p o r e p r e s s u r e s d u r i n g b r e a k o u t have been s t u d i e d by B y r n e and F i n n (1978) . S i n c e t h e a n a l y s i s i s r e s t r i c t e d t o sandy or s i l t y s o i l s , w h i c h have f a i r l y h i g h p e r m e a b i l i t y , i t i s assumed t h a t P w i s v e r y n e a r l y z e r o . T h i s a s s u m p t i o n a l s o w i l l l e a d t o a c o n s e r v a t i v e d e s i g n . So, f i n a l l y , e q u a t i o n (2.3) can be now w r i t t e n as U >W +W +R - (2.4) b - p s v f o r t h e upward movement. 2.3 P r o p o s e d A n a l y s i s f o r t h e P i p e l i n e D u r i n g s t o r m s , r e s i d u a l p o r e p r e s s u r e s a r e i n d u c e d i n some s o i l d e p o s i t s and t h e s e w i l l have t h r e e e f f e c t s on t h e r e s p o n s e of a p i p e l i n e b u r i e d i n t h e d e p o s i t , t o wave l o a d i n g . 1) I t may l e a d t o i n c r e a s e i n bouyancy f o r c e , Uij. T h i s i s because t h e p o r e p r e s s u r e s around t h e p i p e i n c r e a s e . 1 2) I t w i l l r e d u c e t h e e f f e c t i v e w e i g h t o f t h e s o i l mass ( W s ) . T h i s happens because of upward seepage of p o r e w a t e r c r e a t e d by r e s i -d u a l p o r e p r e s s u r e s . 3) I t w i l l r e d u c e t h e shear s t r e n g t h a l o n g an assumed s l i p s u r f a c e and, t h e r e f o r e , r e d u c t i o n i n Ry w i l l r e s u l t . The f i r s t two e f f e c t s n o t e d above can be a n a l y s e d i f r e s i d u a l p o r e p r e s s u r e d i s t r i b u t i o n i s known i n t h e d e p o s i t . T h i s c a n be done u s i n g r e s i d u a l p o r e p r e s s u r e a n a l y s i s e x p l a i n e d i n d e t a i l i n C h a p t e r 5. The s h e a r i n g r e s i s t a n c e o f f e r e d by t h e d e p o s i t a g a i n s t upward movement .of t h e p i p e l i n e w i l l depend on t h e s t r e s s e s i n d u c e d i n t h e d e p o s i t by t h e wave l o a d i n g and shear s t r e n g t h of t h e d e p o s i t . I f t h e s t r e s s e s i n d u c e d by t h e wave l o a d i n g a r e g r e a t e r t h a n o r e q u a l t o t h e shear s t r e n g t h ( f a i l u r e ) o f t h e d e p o s i t , t h e n t h e r e s i s t a n c e o f f e r e d by t h e d e p o s i t f o r v e r t i c a l movement of t h e p i p e l i n e b u r i e d i n t h i s f a i l u r e r e g i o n w i l l be z e r o . The Mohr-Coulomb c r i t e r i o n c a n be used t o d e t e r m i n e f a i l u r e d e p t h (Df) i f i n d u c e d s t r e s s e s a r e known i n t h e d e p o s i t . P o r o - e l a s t i c method o s t a b i l i t y a n a l y s i s e x p l a i n e d i n Cha p t e r 4 c a n be u s e d t o d e t e r m i n e t h i s . H i g h e s t Df w i l l o c c u r as t h e extreme wave p a s s e s o v e r , when t h e bed has maximum r e s i d u a l p o r e p r e s s u r e . So, t h e p o r o - e l a s t i c method of s t a b i l i t y a n a l y s i s s h o u l d be used t o f i n d Df, knowing t h e maximum r e s i d u a l p o r e p r e s s u r e i n t h e d e p o s i t d u r i n g t h e storm a c t i v i t y . Three c a s e s o f p i p e l i n e f l o t a t i o n p r o b l e m w i l l be c o n s i d e r e d . Case I T h i s i s c o n c e r n e d w i t h p o s s i b l e f l o t a t i o n d u r i n g c a l m sea s u r f a c e c o n d i t i o n s . Here, t h e t h e o r y o f V e s i c o r Reese (Reese e t a l , 1968) c a n be employed s u c c e s s f u l l y , i r r e s p e c t i v e of t h e d e p t h of b u r i a l . P h i l l i p s e t a l (1979) showed t h a t t h e r e s i s t a n c e t o f l o t a t i o n p r e d i c t e d by V e s i c i s s l i g h t l y h i g h e r t h a n t h a t p r e d i c t e d by Reese e t a l . U s i n g V e s i c ' s a p p r o a c h , e x p l a i n e d e a r l i e r 15 R +W B c b q where F c > F q = t h e c a v i t y b r e a k t h r o u g h f a c t o r s w h i c h depend on t h e shape and r e l a t i v e d e p t h of t h e c a v i t y , as w e l l as oh t h e a n g l e of i n t e r n a l f r i c t i o n , and c 1 = t h e e f f e c t i v e c o h e s i o n of t h e s o i l d e p o s i t . These f a c t o r s c a n be used d i r e c t l y f o r embedded s p h e r e s o r embedded h o r i -z o n t a l c y l i n d e r s . T h e r e f o r e , f o r s t a b i l i t y o r no upward movement of t h e p i p e l i n e U, < W + R + W b ~ p v s i . e . , r B 2 y <W + B ( c ' F + y ' V . F )'. (2.6) 4 w - p c b q ( F a c t o r s F c and F q a r e t a b u l a t e d i n T a b l e 1 ) . A l t e r n a t i v e l y , t h i s a n a l y s i s can be p e r f o r m e d a c c o r d i n g t o recommendations made by t h e P i p e l i n e F l o t a t i o n R e s e a r c h C o u n c i l (ASCE, R e p o r t , 1966). Case I I T h i s c a s e i s c o n c e r n e d w i t h t h e s t a b i l i t y o f a p i p e l i n e d u r i n g a storm c o n d i t i o n s u c h t h a t t h e d e p t h of b u r i a l , D^, i s l e s s t h a n t h e d e p t h o f i n s t a b i l i t y , . Under t h e s e c i r c u m s t a n c e s , where < D^, i t i s r e a s o n a b l e t o assume t h a t 1) t h e s l i p s u r f a c e i s e s s e n t i a l l y a l o n g two v e r t i c a l p l a n e s , and 2) R v, w h i c h i s t h e v e r t i c a l component of t h e shear f o r c e on t h e s l i p s u r f a c e , i s n e a r l y z e r o . U s i n g e q u a t i o n ( 2 . 4 ) , f o r no upward movement u, < w + w b p s Here, U b = f B O J ^ ) (2.7) TABLE 1 CAVITY BREAKTHROUGH FACTORS (AFTER VESIC, 1971) <J), i n d e g r e e s F i r s t number i s F c Second number i s F c 0.5 1.0 1.5 2.5 5.0 0 0.81 1.61 2.42 4.04 8.07 0.21 0.61 0.74 0.84 0.92 10 0.84 1.68 2.52 4.22 8.43 0.30 0.77 0.99 1.26 1.75 20 0.84 1.67 2.52 4.19 8.37 0.38 0.94 1.23 1.67 2.57 30 0.79 1.58 2.37 3.99 7.89 0.45 1.08 1.45 2.03 3.30 40 0.70 1.40 2.11 3.51 7.02 0.51 1.19 1.61 2.30 3.83 50 0.58 1.17 1.75 2.92 5.84 0.53 1.25 1.70 2.44 4.12 17 where u 2 ' ^ l = t n e maximum t o t a l p o r e p r e s s u r e s on t h e t o p and bottom of t h e p i p e a t any t i m e d u r i n g t h e s t o r m , w h i c h c a n be e s t i m a t e d u s i n g t h e r e s i d u a l p o r e p r e s s u r e method, and Wg = e f f e c t i v e s o i l mass i n v o l v e d w i t h c o r r e c t i o n f o r upward seepage f o r c e i n c l u d e d . T h i s c a s e has been c o n s i d e r e d i n t h e sample p r o b l e m i n C h a p t e r 6. Case I I I T h i s c a s e d e a l s w i t h t h e s i t u a t i o n where t h e d e p t h of b u r i a l , D^, i s g r e a t e r t h a n t h e d e p t h of f a i l u r e , Df. The f o l l o w i n g s t e p s a r e t o be c a r r i e d out t o a n a l y s e t h i s c a s e : a) A s l i p s u r f a c e has t o be assumed between l e v e l s Df and D b. G u i d e l i n e s o u t l i n e d i n S e c t i o n 2.2.2 c a n be used t o assume a s l i p s u r f a c e . The s l i p s u r f a c e w i t h i n t h e i n s t a b i l i t y r e g i o n does n o t o f f e r any r e s i s t a n c e and i s a l o n g two v e r t i c a l p l a n e s . b) The v e r t i c a l component of t h e shear r e s i s t a n c e o f f e r e d by t h e s u r r o u n d i n g s o i l a g a i n s t t h e s l i p a l o n g t h i s s l i p s u r f a c e has t o be e s t i m a t e d . T h i s can be done i n an a p p r o x i m a t e way by computing t h e e f f e c t i v e s t r e s s f i e l d on t h e s l i p s u r f a c e , and e s t i m a t i n g t h e s h e a r r e s i s t a n c e m o b i l i s e d i n t h e r e g i o n between Df and D^. c) An e s t i m a t i o n o f Ws c a n be made a f t e r c o r r e c t i n g t h e b r e a k o u t s o i l mass i n v o l v e d f o r upward seepage e f f e c t s . Now, e q u a t i o n (2.4) can be used t o c h e c k whether t h e upward movement i s p o s s i b l e . 18 CHAPTER 3 EVALUATION OF DESIGN STORM WAVE The d e s i g n e r of an o f f s h o r e f a c i l i t y i s a l w a y s f a c e d w i t h t h e prob l e m of e s t i m a t i n g d e s i g n storm wave d a t a f o r a storm t h a t w i l l o c c u r d u r i n g t h e f u t u r e l i f e o f t h i s f a c i l i t y . I n any l o c a t i o n , i t i s o f t e n u n e c o n o m i c a l t o d e s i g n an o f f s h o r e f a c i l i t y t o w i t h s t a n d t h e w o r s t p o s s i -b l e l o a d i n g c o n d i t i o n s . The d e s i g n , t h e r e f o r e , i s a l w a y s based on c o n d i -t i o n s t h a t have a s u i t a b l y s m a l l p r o b a b i l i t y of o c c u r r e n c e . A n u m e r i c a l v a l u e f o r t h i s p r o b a b i l i t y has t o be d e c i d e d based on a number of f a c t o r s such as t h e l i f e o f t h e s t r u c t u r e , r e l a t i v e c o s t o f r e p a i r s , c o n s t r u c t i o n , e x t e n t o f damages, e t c . D e s i g n storm wave d a t a f o r a p a r t i c u l a r l o c a t i o n a r e of t e n r e p o r t e d i n terms of e i t h e r extreme wave h e i g h t o r s p e c t r a ( i . e . , wave h e i g h t s and wave p e r i o d s ) . S p e c t r a a r e much more u s e f u l but i s u f f e r from t h e d i s a d v a n t a g e s o f b e i n g e x t r e m e l y t e d i o u s and e x p e n s i v e t o compute. C o m p l a i n t s a r e o f t e n made about t h e two ma j o r u n c e r t a i n t i e s i n v o l v e d i n t h e e s t i m a t i o n o f a d e s i g n storm wave d a t a , e i t h e r i n terms of maximum h e i g h t o r s p e c t r a , w i t h a c e r t a i n d e s i g n p r o b a b i l i t y . The two u n c e r t a i n t i e s a r e : 1) H i s t o r i c a l d a t a r e q u i r e d t o do p r o b a b i l i s t i c s t u d i e s a r e a v a i l a b l e o n l y i n l i m i t e d l o c a t i o n s and, o f t e n , i n t e r p o l a t i o n has t o be employed t o e s t i m a t e r e l e v a n t d a t a f o r a p a r t i c u l a r l o c a t i o n . 2) The a v a i l a b l e d a t a span o n l y a few y e a r s and t h e y have t o be used t o e s t i m a t e t h e d e s i g n storm w i t h a r e t u r n p e r i o d of perhaps.. 100 y e a r s . The p o r o - e l a s t i c method of s t a b i l i t y a n a l y s i s a n a l y s e s t h e r e s p o n s e o f a s o i l d e p o s i t t o extreme waves. So, a d e s i g n extreme wave w i t h a s m a l l p r o b a b i l i t y o f o c c u r r e n c e i s r e q u i r e d . On t h e o t h e r hand, 19 t h e r e s i d u a l p o r e p r e s s u r e a n a l y s i s computes r e s i d u a l p o r e p r e s s u r e s g e n e r a t e d I n t h e d e p o s i t due t o c y c l i c s h e a r s t r e s s e s d u r i n g a s t o r m a c t i v i t y . A u n i f o r m storm wave i n p u t i s r e q u i r e d f o r t h i s p u r p o s e . T h i s c h a p t e r e x p l a i n s , b r i e f l y , a s i m p l e b a s i s f o r o b t a i n i n g an " e q u i v a l e n t u n i f o r m storm wave" c o r r e s p o n d i n g t o t h e d e s i g n storm wave .data w h i c h was e v a l u a t e d u s i n g t h e p r o p e r p r o b a b i l i t y model f o r a c e r t a i n s m a l l s p e c i f i e d p r o b a b i l i t y ^ o f o c c u r r e n c e . Here, an e q u i v a l e n t u n i f o r m storm wave i s d e f i n e d as a s i n g l e wave of c e r t a i n wave h e i g h t (H eq) and p e r i o d ( T e q ) w i t h a c e r t a i n number of c y c l e s ( N e q ) a c t i n g f o r t h e d u r a -t i o n o f t h e s t o r m , w h i c h i s r e p r e s e n t a t i v e o f t h e d e s i g n s t o r m wave i n terms o f p o r e p r e s s u r e g e n e r a t i o n due t o t h e c y c l i c s h e a r s t r e s s e s i n t h e d e p o s i t . A g e n e r a l f l o w c h a r t t o e s t i m a t e t h e e q u i v a l e n t storm f r o m weather maps i s p r e s e n t e d i n F i g . 3. I n n e a r s h o r e l o c a t i o n s , d i r e c t storm wave d a t a may a l s o be a v a i l a b l e . Here, storm-produced waves o n l y a r e c o n s i d e r e d but t h i s a p p r o a c h c a n be extended t o c a s e s where waves a r e g e n e r a t e d by, f o r example, e a r t h q u a k e s o r l a n d s l i d e s . 3.1 O b t a i n i n g D e s i g n Storm Waves Storm wave d a t a f o r a p a r t i c u l a r s i t e may be p r e s e n t e d i n terms of a maximum wave ( i n p u t f o r p o r o - e l a s t i c a n a l y s i s ) and wave s p e c t r a . The t a s k o f c o l l e c t i n g t h e r e q u i r e d s t o r m d a t a f o r a s i t e i s , n e e d l e s s t o say, a b i g t a s k i n i t s e l f . The sea c o n d i t i o n s a r e h i g h l y v a r i a b l e and s e e m i n g l y u n p r e d i c t a b l e . So, a s t a t i s t i c a l a p p r o a c h f o r s e l e c t i n g d e s i g n c o n d i t i o n s i n w h i c h t h e v a r i a b i l i t y o f t h e sea s t a t e i s a c c o u n t e d f o r i s i m p o r t a n t . The s t a t i s t i c a l p r o p e r t i e s o f storm o c c u r r e n c e r a t e s and h i s -t o r i c a l d a t a a r e used t o d e t e r m i n e an e s t i m a t e o f a t o t a l p o p u l a t i o n o f s t o r m s . S i g n i f i c a n t wave f i e l d s c a n be e s t i m a t e d based on c o n d i t i o n s of wind speed, f e t c h , d e p t h , beach s l o p e and wind d u r a t i o n . T h i s method t o 20 Weather Maps Storm Data: Wind V e l o c i t y and D u r a t i o n P r o b a b i l i t y Model, D e s i g n Storm Data Storm Wave C h a r a c t e r i s t i c s H, T, number of waves Wave T h e o r i e s to C a l c u l a t e P r e s s u r e L o a d i n g E q u i v a l e n t U niform Storm Wave i FIG. 3 Flow Chart t o C a l c u l a t e G e n e r a l Storm Wave Data and t h e E q u i v a l e n t U niform Storm Wave System 21 e s t a b l i s h s i g n i f i c a n t wave f i e l d s f o r d i f f e r e n t c a s e s s u c h as deep w a t e r and s h a l l o w w a t e r c o n d i t i o n s i s d e s c r i b e d by W e i g e l ( 1 9 6 4 ) . The r e l a -t i v e d i s t r i b u t i o n o f wave h e i g h t s w h i c h o c c u r s d u r i n g p a s s a g e o f a h u r r i -c ane c a n be e s t i m a t e d t h r o u g h c o n s i d e r a t i o n o f t h i s s i g n i f i c a n t wave f i e l d a s w e l l as t h r o u g h t h e a p p l i c a t i o n o f an assumed R a y l e i g h d i s t r i -b u t i o n f o r wave h e i g h t s d u r i n g u n i f o r m t i m e i n c r e m e n t s ( N a t a r a j a e t a l , 1978). Harmonic p r e s s u r e l o a d i n g e x e r t e d by t r a v e l l i n g s u r f a c e waves can be now e v a l u a t e d u s i n g wave t h e o r i e s . 3.2 Wave T h e o r i e s Waves i n t h e ocean o f t e n appear as a c o n f u s e d and c o n s t a n t l y c h a n g i n g sea of c r e s t s and t r o u g h s on t h e wa t e r s u r f a c e because o f i r r e g u -l a r i t y o f wave shapes and t h e v a r i a b i l i t y i n t h e d i r e c t i o n of p r o p a g a t i o n . T h i s i s p a r t i c u l a r l y t r u e when t h e waves a r e under t h e i n f l u e n c e o f w i n d . A p r e c i s e d e s c r i p t i o n o f t h e sea s u r f a c e waves i s d i f f i c u l t b e cause of t h e i n t e r a c t i o n between i n d i v i d u a l waves. F a s t e r waves o v e r t a k e and pass t h r o u g h s l o w e r ones from v a r i o u s d i r e c t i o n s . Waves sometimes r e i n f o r c e o r c a n c e l each o t h e r by t h i s i n t e r a c t i o n , and o f t e n c o l l i d e w i t h each o t h e r and t r a n s f o r m i n t o t u r b u l e n c e and s p r a y . Wave energy d e r i v e d from s o u r c e s such as wind i s d i s s i p a t e d i n t e r n a l l y w i t h i n t h e f l u i d by i n t e r a c t i o n w i t h t h e a i r above, by t u r b u l e n c e on b r e a k i n g and a t t h e bottom i n s h a l l o w d e p t h s . I t i s because o f t h i s h i g h l y v a r i a b l e and s e e m i n g l y u n p r e d i c t a b l e n a t u r e o f sea s u r f a c e waves, t h a t a p r e c i s e m a t h e m a t i c a l model i s i m p o s s i b l e t o f o r m u l a t e . However, t h e r e a r e about e i g h t w e l l - k n o w n wave t h e o r i e s based on i d e a l t w o - d i m e n s i o n a l i n c o m p r e s s i b l e f l u i d f l o w c o n d i t i o n s w h i c h have been used s u c c e s s f u l l y f o r e n g i n e e r i n g p u r p o s e s . A l l t h e s e e i g h t wave t h e o r i e s c a n be used t o e s t i m a t e t h e p r e s s u r e l o a d i n g on t h e s e a f l o o r due t o s u r f a c e waves. I t has been e x p e r i m e n t a l l y v e r i f i e d t h a t a p a r t i -c u l a r wave t h e o r y i s most s u i t a b l e f o r a p a r t i c u l a r r a n g e o f wave 22 c h a r a c t e r i s t i c s and s t i l l w a ter c o n d i t i o n s . An e x t e n s i v e r e v i e w o f wave t h e o r i e s has been p r e s e n t e d by Dean ( 1 9 7 4 ) . These w e l l - k n o w n wave t h e o r i e s and a b r i e f d e s c r i p t i o n o f t h e i r r e l a t i v e m e r i t s a r e l i s t e d below. T h i s w i l l be h e l p f u l i n s e l e c t i n g t h e a p p r o p r i a t e wave t h e o r y f o r e v a l u a t i n g t h e p r e s s u r e l o a d i n g on t h e ocean f l o o r . F i g u r e 4 shows t h e r e g i o n s of v a l i d i t y f o r v a r i o u s wave t h e o r i e s . P r e l i m i n a r y s t u d i e s c a r r i e d o u t a t t h e U n i v e r s i t y of B r i t i s h C o l umbia s u g g e s t t h a t t h e e r r o r i n u s i n g i mproper wave t h e o r y t o e s t i m a t e t h e p r e s s u r e l o a d i n g on t h e s e a f l o o r c a n be even as much as 30%. A n a l y t i c a l v a l i d i t y o f t h e wave t h e o r i e s i s a n a l y s e d below. A n a l y t i c a l v a l i d i t y i s d e f i n e d as t h e d e g r e e t o w h i c h t h e t h e o r i e s s a t i s f y t h e d e f i n i n g boundary c o n d i t i o n s . There a r e f i v e d e f i n i n g e q u a t i o n s t o s a t i s f y : t h e d i f f e r e n t i a l e q u a t i o n and f o u r boundary c o n d i t i o n s . These a r e as f o l l o w s : 1) D i f f e r e n t i a l e q u a t i o n f o r t w o - d i m e n s i o n a l i d e a l f l o w . 2) No f l o w o c c u r s a c r o s s t h e bottom boundary. 3) K i n e m a t i c f r e e s u r f a c e boundary c o n d i t i o n w h i c h r e q u i r e s t h a t t h e components of f l o w a t t h e f r e e s u r f a c e be i n a c c o r d a n c e w i t h t h e geometry and m o t i o n o f t h e f r e e s u r f a c e . 4) Dynamic f r e e s u r f a c e boundary c o n d i t i o n w h i c h r e q u i r e s t h a t t h e p r e s s u r e i m m e d i a t e l y below t h e f r e e s u r f a c e be u n i f o r m and e q u a l t o a t m o s p h e r i c p r e s s u r e . 5) M o t i o n i s p e r i o d i c i n x w i t h s p a t i a l p e r i o d i c i t y o f t h e wave l e n g t h , L. 3.2.1 L i n e a r wave t h e o r y The l i n e a r wave t h e o r y i s t h e s i m p l e s t o f a l l t h e t h e o r i e s . I t s a t i s f i e s k i n e m a t i c and dynamic f r e e s u r f a c e boundary c o n d i t i o n s t o t h e f i r s t o r d e r o n l y . I t has been r e p o r t e d t h a t t h i s t h e o r y can 23 0.05 H 0.00005 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 d QT' Here, H = Wave H e i g h t T "= Wave P e r i o d d = Mean Water Depth FIG. 4 Regions of V a l i d i t y of Wave T h e o r i e s 24 be used s u c c e s s f u l l y i n b o t h s h a l l o w and deep w a t e r c o n d i t i o n s w i t h r e a s o n a b l e a c c u r a c y , a l t h o u g h i t i s b e s t s u i t e d f o r t r a n s i t i o n a l w a t e r d e p t h s . F u r t h e r m o r e , t h e l i n e a r t h e o r y i s r e a d i l y a p p l i e d t o r e p r e s e n t a -t i o n s o f random waves. 3.2.2 S t o k e s h i g h e r o r d e r t h e o r i e s S t o k e s t h e o r i e s s a t i s f y t h e f r e e s u r f a c e boundary c o n d i t i o n s t o t h e i r r e s p e c t i v e o r d e r s o f a p p r o x i m a t i o n . Each e x t e n s i o n i n t h e o r d e r of a p p r o x i m a t i o n u s u a l l y p r o v i d e s b e t t e r agreement w i t h t h e o r e t i c a l and o b s e r v e d wave b e h a v i o u r . These extended wave t h e o r i e s c a n e x p l a i n pheno-mena, s u c h as mass t r a n s p o r t , t h a t cannot be e x p l a i n e d by t h e l i n e a r wave t h e o r y . G e n e r a l l y , h i g h e r o r d e r S t o k e s t h e o r i e s do n o t improve t h e a c c u -r a c y commensurate w i t h t h e i n c r e a s e d c o m p u t a t i o n s i n v o l v e d . The r a n g e of a p p l i c a b i l i t y i s l i m i t e d t o 1/25 < d/L. Here, d and L a r e as p r e v i o u s l y d e f i n e d (Sen, 1971). 3.2.3 C n o i d a l wave t h e o r y The c n o i d a l wave t h e o r y i s a n o n l i n e a r t h e o r y . The t e r m , c n o i d a l , i s u sed s i n c e t h e wave p r o f i l e i s g i v e n by t h e J a c o b i a n e l l i p t i c a l c o s i n e f u n c t i o n u s u a l l y d e s i g n a t e d by c n . There a r e h i g h e r o r d e r c n o i d a l t h e o r i e s and t h e y t o o do n o t improve t h e a c c u r a c y commensurate w i t h t h e i n c r e a s e d c o m p u t a t i o n s . F o r s h a l l o w w a t e r r e g i o n s , t h e c n o i d a l wave t h e o r y p r e d i c t s r a t h e r w e l l t h e waveform and a s s o c i a t e d m o t i o n s . However, c n o i d a l wave t h e o r y has had l i m i t e d a p p l i c a t i o n t o t h e s o l u t i o n s o f e n g i n e e r i n g p roblems due t o t h e d i f f i c u l t i e s i n making t h e n e c e s s a r y c o m p u t a t i o n s ( S h o r e P r o t e c t i o n Manual, 1977) . The work i n v o l v e d i n u s i n g t h e c n o i d a l wave t h e o r y has been sub-s t a n t i a l l y r e d u c e d by t h e i n t r o d u c t i o n of g r a p h i c a l and t a b u l a r forms of 25 f u n c t i o n s (Masch and W i e g e l , 1961). However, a p p l i c a t i o n of t h i s t h e o r y i s s t i l l q u i t e i n v o l v e d . Long, f i n i t e a m p l i t u d e waves of p e r -manent form p r o p a g a t i n g i n s h a l l o w w a t e r w i t h a r a n g e of a p p l i c a b i l i t y d/L < 1/8 a r e f r e q u e n t l y b e s t d e s c r i b e d by t h i s t h e o r y . 3.2.4 S o l i t a r y wave t h e o r y Waves c o n s i d e r e d so f a r a r e o s c i l l a t o r y or n e a r l y o s c i l l a t o r y waves. The w a t e r p a r t i c l e s move backwards and f o r w a r d s w i t h t h e p a s s a g e of each wave and a d i s t i n c t wave c r e s t and wave:.trough i s e v i d e n t . A s o l i t a r y wave i s n e i t h e r o s c i l l a t o r y n o r does i t e x h i b i t a t r o u g h . I n t h e p u r e sense, t h e s o l i t a r y wave l i e s e n t i r e l y above t h e s t i l l w a t e r l e v e l . Long waves, such as t s u n a m i s and waves r e s u l t i n g f r o m l a r g e d i s -p l a c e m e n t s o f w a t e r caused by su c h phenomena as l a n d s l i d e s and e a r t h q u a k e s , sometimes behave s i m i l a r l y t o s o l i t a r y waves. T h i s t h e o r y i s b r o a d l y v a l i d f o r . r e l a t i v e l y s h a l l o w w a t e r . * . '' 3.2.5 S t r e a m - f u n c t i o n n u m e r i c a l wave t h e o r y N u m e r i c a l a p p r o x i m a t i o n s t o s o l u t i o n s of t h e hydrodynamic equa-t i o n s of wave m o t i o n e x p r e s s e d i n terms of s t r e a m f u n c t i o n s were proposed and d e v e l o p e d by Dean (1965) . The n o n l i n e a r s t r e a m - f u n c t i o n t h e o r y i s s i m i l a r t o t h e h i g h e r o r d e r S t o k e s ' t h e o r i e s . The t h e o r y i s used t o d e t e r -mine t h e c o e f f i c i e n t o f each h i g h e r o r d e r term so t h a t a b e s t f i t , i n t h e l e a s t s q u a r e s sense, i s o b t a i n e d t o t h e dynamic f r e e - s u r f a c e boundary c o n d i t i o n s . T h i s t h e o r y has been shown t o p r o v i d e a b e t t e r f i t t o l a b o -r a t o r y measurements of p a r t i c l e v e l o c i t y t h a n o t h e r s . I t r e q u i r e s a number of t a b u l a t e d p a r a m e t e r s and a d i g i t a l computer w i t h a r e a s o n a b l y l a r g e memory. The s t r e a m - f u n c t i o n t h e o r y p r o v i d e s b e s t f i t o v e r a w i d e r a n g e i n c l u d i n g a l l of t h e t r a n s i t i o n a l and deepwater wave r e g i o n s and 26 a l s o a s i g n i f i c a n t p a r t o f t h e s h a l l o w w a t e r r a n g e . B u t , as one can e x p e c t , t h i s i s n o t a s t r a i g h t f o r w a r d t h e o r y and cannot be r e a d i l y a p p l i e d . 3.2.6 Theory used i n t h i s t h e s i s A l l t h e t h e o r i e s d e t a i l e d above assume t h a t t h e s e a f l o o r i s impermeable and r i g i d . M a l l a r d and D a l r y m p l e (1977) d e v e l o p e d a u s e f u l a n a l y s i s showing t h e e f f e c t s o f a d e f o r m a b l e s e a f l o o r on wave p r e s s u r e a m p l i t u d e s . T h e i r a n a l y s e s c o n c l u d e d t h a t p r e s s u r e s on a d e f o r m a b l e i m p e r -meable s e a f l o o r a r e h i g h e r t h a n t h o s e on a r i g i d base by up t o 15% f o r v e r y s o f t c o h e s i v e s o i l s . But t h i s e f f e c t may be i g n o r e d f o r most sands. I n t h i s t h e s i s , t h e l i n e a r wave t h e o r y i s used because of i t s s i m p l i c i t y and r e l i a b i l i t y o v e r a l a r g e segment of t h e whole wave r e g i m e . S o l u t i o n s of l i n e a r wave t h e o r y f o r p r e s s u r e l o a d i n g on t h e s e a f l o o r i s g i v e n by Y H AP = W 9 , Cos {2TT(^ - £ ) } = pn Cos {2Tr(f - f ) } (3.1) 2 C o s h ( ^ ) ° where 2 L = - f ^ - t a n h ( ^ ) (3.2) ZTT L 3.3 E q u i v a l e n t U n i f o r m Storm Wave To u s e t h e r e s i d u a l p o r e p r e s s u r e a n a l y s i s , we have t o f i n d an e q u i v a l e n t u n i f o r m storm wave r e p r e s e n t a t i v e of t h e d e s i g n storm wave i n terms of p o r e p r e s s u r e g e n e r a t i o n i n t h e d e p o s i t . The d e s i g n s t o r m wave i s a s e t o f s p e c i f i e d waves each c h a r a c t e r i s e d by i t s wave h e i g h t , wave p e r i o d and number o f c y c l e s . T h i s t h e s i s u s e s a s i m p l e p o r e p r e s s u r e g e n e r a t i o n model pro p o s e d by Seed e t a l ( 1 9 7 6 ) , ( s e e e q u a t i o n ( 5 . 6 ) ) . A c c o r d i n g t o t h i s e q u a t i o n , r e s i d u a l p o r e p r e s s u r e g e n e r a t e d due t o t h e 27 c y c l i c e f f e c t o f t h e shear s t r e s s e s i s a f u n c t i o n of N/N L. He r e , N i s t h e number o f a p p l i e d c y c l e s and i s t h e number of c y c l e s r e q u i r e d t o cause i n i t i a l l i q u e f a c t i o n under g i v e n s t r e s s c o n d i t i o n s . N L can be o b t a i n e d f r o m a l i q u e f a c t i o n p o t e n t i a l c u r v e w h i c h i s a p l o t between Tc/°vo and N^. He r e , T c / a ^ 0 i s t h e c y c l i c s h e a r s t r e s s r a t i o . S i n c e N L i s u n i q u e l y d e f i n e d by TC/O^Q f o r a g i v e n s o i l d e p o s i t , we can c o n c l u d e t h a t t h e e x c e s s p o r e p r e s s u r e r a t i o i s u n i q u e l y governed by t h e number of a p p l i e d c y c l e s and TC/O^Q. Now, an e q u i v a l e n t u n i f o r m s t o r m wave c a n be e s t i m a t e d by r e d u c i n g t h e T C / O v o of a l l wave components t o z -> 0 and u s i n g t h e method pro p o s e d by Lee and Chan ( 1 9 7 2 ) . I t w i l l be seen i n Cha p t e r 4 t h a t t h e s o i l - w a t e r system c a n be u n c o u p l e d and t h e shear s t r e s s e s c a n be d e t e r m i n e d w i t h good a c c u r a c y u s i n g s i m p l e s o l i d m e c h a n i c a l p r i n c i p l e s as e x p l a i n e d i n Appendix I I . Lee and Chan su g g e s t e d t h a t N „ n , t h e e q u i v a l e n t number of c y c l e s o f a g i v e n r e f e r e n c e wave, c a n be g i v e n by nw N N = I - ^ N r (3.3) e c* r = l L r r where ^Leq = n u m D e r of c y c l e s r e q u i r e d t o c a u s e i n i t i a l l i q u e f a c t i o n f o r t h e shear s t r e s s r a t i o c o r r e s -p o n d i n g t o t h e s e l e c t e d r e f e r e n c e wave a t z + 0; ^ L r = number of c y c l e s r e q u i r e d t o cause i n i t i a l l i q u e f a c t i o n f o r t h e shear s t r e s s r a t i o c o r r e s -p o n d i ng t o t h e - r t n wave component a t z -> 0; N. = number of waves of t h e r t n wave component; n w = t o t a l number of wave components. To c a l c u l a t e t h e number of c y c l e s t o c a u s e i n i t i a l l i q u e f a c t i o n o r l i m i t e d s t r a i n f o r a g i v e n shear s t r e s s r a t i o , one has t o e x p e r i m e n t a l l y e s t a b l i s h t h e c u r v e known as t h e l i q u e f a c t i o n p o t e n t i a l c u r v e . T h i s can be done by p e r f o r m i n g c y c l i c t e s t s on ' u n d i s t u r b e d ' samples. The t e s t s can be 28 c a r r i e d out i n e i t h e r t r i a x i a l o r s i m p l e shear a p p a r a t u s . S i n c e ocean wave l o a d i n g i s e s s e n t i a l l y u n i - d i r e c t i o n a l , a c o r r e c t i o n f a c t o r , Cy, of a v e r a g e v a l u e , 0.60 (Seed, 1979; N a t a r a j a e t a l , 1979) c a n be used f o r t h e r e s u l t s of c y c l i c t r i a x i a l t e s t s t o r e d u c e t o p r e v a i l i n g s i m p l e s h e a r c o n d i t i o n s o r s i m p l e shear t e s t s can be used w i t h o u t any c o r r e c -t i o n (Seed, 1979). L i q u e f a c t i o n p o t e n t i a l c u r v e s can a l s o be o b t a i n e d from s t a n d a r d p e n e t r a t i o n t e s t s . T h i s method i s based on d a t a p r e s e n t e d by Seed ( 1 9 7 9 ) . C a l c u l a t e d f i e l d v a l u e s of c y c l i c s t r e s s r a t i o can be p r e s e n t e d as a f u n c t i o n of c o r r e c t e d a v e r a g e p e n e t r a t i o n r e s i s t a n c e , Np. Here, Np i s t h e p e n e t r a t i o n r e s i s t a n c e c o r r e c t e d t o an e f f e c t i v e o v e r b u r d e n p r e s s u r e of one t o n per s q u a r e f o o t and i s g i v e n by t h e f o l l o w i n g e q u a t i o n . NP = ( V N f where C N = 1 - 1 . 2 5 l o g (|^) (3.4) where 5Q = e f f e c t i v e o v e r b u r d e n p r e s s u r e i n t o n s per s q u a r e f o o t a t t h e p o i n t where p e n e t r a t i o n r e s i s t a n c e has a v a l u e of N j ; a\ = one t o n p e r s q u a r e f o o t ; Nf = s t a n d a r d p e n e t r a t i o n r e s i s t a n c e measured i n t h e f i e l d , blow count p e r f o o t . I n F i g . 5(a) t h e r e l a t i o n s h i p between t h e m o d i f i e d p e n e t r a t i o n r e s i s t a n c e , Np, and t h e c o r r e s p o n d i n g c y c l i c s t r e s s r a t i o w h i c h c a u s e s i n i t i a l l i q u e f a c t i o n o r a s p e c i f i e d l i m i t e d s t r a i n i s shown. C o r r e s p o n d -i n g i n t e r p o l a t e d e a r t h q u a k e m a g n i t u d e s a r e a l s o shown i n F i g . 5 ( a ) . U s i n g F i g s . 5(a) and (b) i t i s p o s s i b l e t o g e t an a v e r a g e l i q u e f a c t i o n p o t e n t i a l c u r v e f o r v a r i o u s N p v a l u e s ( N a t a r a j a e t a l , 1979). N a t a r a j a et a l s u g g e s t i n c r e a s i n g by 10 p e r c e n t the. l i q u e f a c t i o n p o t e n t i a l c u r v e , 0.6 (a) UJ < O H Q_ 00 Solid, points indicate sites and test conditions showing liquefaction O7.0 • Based on field data Extrapolated from results large score laboratory tests 50 MODIFIED PENETRATION RESISTANCE. N -BLOWS/ FT. which was based on f i e l d d a t a d u r i n g e a r t h q u a k e s . T h i s i s because t h e s t r e n g t h of t h e s o i l under two- d i m e n s i o n a l p l a n e s t r a i n c o n d i t i o n s of ocean wave l o a d i n g i s somewhat h i g h e r t h a n t h a t under t h e t h r e e -d i m e n s i o n a l c o n d i t i o n s of earthquake l o a d i n g . 31 CHAPTER 4  PORO-ELASTIC ANALYSIS 4.1 I n t r o d u c t i o n The e f f e c t s o f waves on f o u n d a t i o n s o i l s and s t r u c t u r e s have been i m p o r t a n t c o n s i d e r a t i o n s i n t h e d e s i g n o f o f f s h o r e i n s t a l l a t i o n s . I n t h e d e s i g n o f o f f s h o r e s t r u c t u r e s s u c h as p i p e l i n e s , sea w a l l s , e t c . , i t i s common t o i d e a l i s e t h e s u r f a c e waves as p l a n e t r a v e l l i n g waves. When t h e s e waves p r o p a g a t e o v e r a porous seabed, f l u i d f l o w i s i n d u c e d i n t h e bed and t h e po r o u s medium i t s e l f i s f o r c e d t o deform. Thus, t h e r e s p o n s e o f t h e bed t o s u r f a c e waves i s a c t u a l l y a c o m b i n a t i o n o f f l u i d and s o l i d m e c h a n i c a l e f f e c t s . T h e r e f o r e , t h e r e s p o n s e a n a l y s i s o f a s e a -bed t o wave l o a d i n g s h o u l d i n c l u d e p o r e w a t e r f l o w , volume change and d e f o r m a t i o n c h a r a c t e r i s t i c s o f t h e bed. Yamamoto (1978) and Madsen (1978) p r e s e n t e d a c o u p l e d e f f e c t i v e s t r e s s a n a l y s i s based on B i o t ' s e q u a t i o n (1941) f o r a deep s o i l d e p o s i t . 4.2 G o v e r n i n g E q u a t i o n s The f o l l o w i n g b a s i c a s s u m p t i o n s a r e made w h i l e f o r m u l a t i n g t h e t h e o r y : 1) The p e r i o d s o f i m p o r t a n t waves a r e l o n g enough so t h a t t r a n s i e n t r e s p o n s e of t h e p r e s s u r e l o a d i n g caused by s u r f a c e waves c a n be n e g l e c t e d . T h e r e f o r e , t h e p r e s s u r e l o a d i n g can be a n a l y s e d q u a s i - s t a t i c a l l y . 2) Theory o f l i n e a r e l a s t i c i t y i s a p p l i c a b l e . 3) P l a n e s t r a i n c o n d i t i o n s p r e v a i l . 4) D a r c y ' s l a w i s a p p l i c a b l e . F i g u r e 6 shows a s o i l d e p o s i t o f c o n s t a n t t h i c k n e s s D. The 32 Sea-bed Porous Bed Free Surface y s = y Cos (Xx -a)t) Mean Water Level Pressure Loading Ap = pQ Cos (Xx-ajt) A Element A ^Rigid Bottom FIG. 6 Wave P r e s s u r e s on Ocean F l o o r 33 x - a x i s i s t a k e n on t h e bed s u r f a c e ; t h e p o s i t i v e z - a x i s i s shown as b e i n g v e r t i c a l l y downward from t h e bed s u r f a c e . C o n s i d e r element A shown i n F i g . 6. D a r c i a n f l o w o f c o m p r e s s i b l e p o r e w a t e r i n a compres-s i b l e p orous medium l e a d s t o t h e f o l l o w i n g form of t h e c o n s o l i d a t i o n e q u a t i o n ( B i o t , 1941) under p l a n e s t r a i n c o n d i t i o n s : s 2 a 3(£ +e,) k " ^ - T n e | £ = Y * Z ( 4 . D x . 2 z . 2 w 3t w 3t 3x 3z where ^x»^z = t* i e p r i n c i p a l p e r m e a b i l i t i e s i n t h e x and z d i r e c t i o n s , r e s p e c t i v e l y ; p = t h e e x c e s s p o r e p r e s s u r e ; Y W = t h e u n i t w e i g h t of p o r e w a t e r ; n = t h e p o r o s i t y o f t h e bed; e x and e z = t h e n o r m a l s t r a i n s (ey=0) d e f i n e d t o be p o s i t i v e as e l o n g a t i o n s , and t = t i m e . The c o m p r e s s i b i l i t y o f .porewater, 3, i s g i v e n by If - >»H where p = t h e d e n s i t y o f p o r e w a t e r . 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 i n x and z d i r e c t i o n s c a n be w r i t t e n as 3o' 3T . 3a' 3T x x z 3p , z zx 3p ,, ~ N ~Z— + — 5 — = ~ 2 • » a n d " a — + ~ ^ — = ~ (4.3) 3x 3,z 3x 3z 3x 3z where a x , a z and T x z = t h e i n c r e m e n t a l e f f e c t i v e s t r e s s e s and shear s t r e s s e s . H ere, i n e r t i a terms a s s o c i a t e d w i t h a c c e l e r a t i o n s a r e n e g l e c t e d and a q u a s i - s t a t i c l o a d i n g i s assumed. Assuming t h a t t h e s o i l s k e l e t o n i s an i d e a l i s o t r o p i c e l a s t i c m a t e r i a l t h e n , under t h e p l a n e s t r a i n c o n d i t i o n 34 _ _3v _ (1-y) , , v ,N ( 4 4 N £ x " 3x _ 2G ( C Jx 1-v V ^ ' ^ and 3w (1-v) , , v , N £ z = ~3z" = ~ ^ G - ( ° z " ° x ) where v and w = d i s p l a c e m e n t v e c t o r s ; G and v = shear modulus and P o i s s o n ' s r a t i o . E q u a t i o n (4.4) can be r e w r i t t e n as °x A l - 2 v ) G C 3 x + 1-v a V a' = - 2 ( t - )G (^- + ^ -5-) (4.5) z l - 2 v 3z l ^ v 3x and a l s o , shear s t r e s s _,3v , 3w. xz 3z 3x By s u b s t i t u t i n g e q u a t i o n (4.5) i n e q u i l i b r i u m e q u a t i o n s (4.3) we g e t two e q u a t i o n s i n terms of t h e unknowns v,w and p. E q u a t i o n (4.1) g i v e s a t h i r d e q u a t i o n i n t h e b a s i c v a r i a b l e s v,w and p. T h e r e f o r e , a s o l u t i o n i s , i n p r i n c i p l e , p o s s i b l e . 4.3 Boundary C o n d i t i o n s To s o l v e t h e above boundary v a l u e problem, we need t h r e e i n d e p e n -dent boundary c o n d i t i o n s per boundary. At t h e s e a f l o o r s u r f a c e t h e boun-d a r y c o n d i t i o n s a r e : (z=0) 1) t h e v e r t i c a l e f f e c t i v e s t r e s s , a z = 0; 2) shear s t r e s s x x z i s n e g l i g i b l e s m a l l or x x z = 0; 3) p o r e p r e s s u r e p i s g i v e n by p Q Cos(Ax-iot) . T h e r e f o r e , = 0, p = p Cos( Ax-cot) a t z = 0 (4.6) zx r r o 35 Here, A and to a r e d e f i n e d as 2TT/L and 2TT/T, r e s p e c t i v e l y , w i t h L b e i n g t h e wave l e n g t h and T b e i n g t h e wave p e r i o d ; p Q i s t h e a m p l i t u d e of t h e p r e s s u r e l o a d i n g w h i c h can be c a l c u l a t e d u s i n g t h e l i n e a r wave t h e o r y . The boundary c o n d i t i o n s a t t h e bottom boundary of t h e seabed a r e 1) d i s p l a c e m e n t s i n x and z d i r e c t i o n s a r e z e r o ; 2) no f l o w o c c u r s a c r o s s t h i s boundary. I n o t h e r words, i t i s assumed t h a t t h e bottom boundary i s r i g i d and impermeable, i . e . , v = w = - | £ = 0 a t z = D ( 4 . 7 ) 3z 4.4 S o l u t i o n T e c h n i q u e S i n c e t h e t h i r d boundary c o n d i t i o n i n e q u a t i o n ( 4 . 6 ) i s p e r i o d i c i n b o t h t i m e and space, i t i s r e a s o n a b l e t o assume t h a t v,w and p a r e a l s o p e r i o d i c i n t i m e and space. S i n c e d i f f e r e n t i a t i o n w i t h r e s p e c t t o x and t o c c u r f r e q u e n t l y , i t i s c o n v e n i e n t t o s o l v e t h i s boundary v a l u e p r o b l e m i n terms of complex v a r i a b l e s . The s o l u t i o n i s t a k e n t o be t h e r e a l p a r t o f t h e complex s o l u t i o n . Now, e q u a t i o n ( 4 . 6 ) becomes , i ( Ax-cot) a' = x = 0 , p = p e z x z c ro E q u a t i o n ( 4 . 1 ) can now be w r i t t e n as 2 - K A 2p + K M r + i Y ngcop = - ly to(iAv + -^) ( 4 . 8 ) X Z _ 2 W W 3z 3z From e q u a t i o n ( 4 . 3 ) we o b t a i n 3x i A a ' + = - p i A ( 4 . 9 ) x 3z and 36 3a' z , + i A x = - I2 (4-10) 3z x z 3z i n t r o d u c i n g t h e . e x p r e s s i o n f o r p i n e q u a t i o n s (4.8) and (4.10) 2 3 3 a ' 3 x 9 , 8T K ( 1 + 1^) - (K A - i Y n0u>) (a'' + - ^ ) z -2 i A „ 3 x 'w x i A 3z 3z 3z = i y co(iAv + |^) (4.11) W oZ and 3a' 3a' . 3 2 x Z + i A x = P (4.12) 3z x z 3z i A . 2 3z S u b s t i t u t i n g s t r e s s e s i n terms of d i s p l a c e m e n t s from e q u a t i o n (4.5) i n ( 4 . 1 1 ) , (4.12) and e l i m i n a t i n g w, we o b t a i n 6 „ K 2 4 . K a 2 3 v . A 2 { 2 + J c _ J i } i v + A 4 { 1 + 2 J i . 2 J i } l z K ,2 . 4 K , 2 J . 2 3z z A 3z z A 3z K 2 - A 6 {-£ - \ } v = 0 (4.13) K z A where 2 . { n B + ( l - 2 v ) / ( 2 G ( l - v ) ) } y = Z The g e n e r a l s o l u t i o n t o t h i s d i f f e r e n t i a l e q u a t i o n w i t h c o n s t a n t c o e f f i -c i e n t s c a n be o b t a i n e d by s u b s t i t u t i n g . r A z i(Ax-cot) . . . v = Ae e i n e q u a t i o n (4.13) 2 T h i s l e a d s t o t h e c h a r a c t e r i s t i c e q u a t i o n w h i c h i s c u b i c i n r and has r o o t s K,. 2 r 2 = 1,1, ^ - (4.14) z A R e a l i z i n g t h a t r 2 = 1 i s a r e p e a t e d r o o t , we o b t a i n t h e g e n e r a l s o l u t i o n 37 v = { ( A 1 + A o Z ) e A z + ( A , + A A z ) e - X z + A , e T l A z + A , e - r i A 2 } e«**-at) (4.15) 1 2 3 4 D o J K 2 ^ "72 z A and A^ ->- Ag a r e a r b i t r a r y c o n s t a n t s t o be d e t e r m i n e d u s i n g t h e boundary c o n d i t i o n s . I t s h o u l d be n o t e d t h a t c o n s t a n t s A^ t h r o u g h Ag, i n g e n e r a l , w i l l be complex and o n l y t h e r e a l p a r t of t h e c o m p l e t e complex s o l u t i o n c o n s t i t u t e s t h e s o l u t i o n t o our problem. 4.4.1 H o r i z o n t a l l y l a y e r e d d e p o s i t s I n t h e c a s e of h o r i z o n t a l l y l a y e r e d d e p o s i t s , an e q u a t i o n of t h e t y p e (4.15) e x i s t s f o r e v e r y l a y e r . The number of a r b i t r a r y c o n s t a n t s w i l l t h e n be 6 x N, where N i s t h e number of l a y e r s . These a r b i t r a r y c o n -s t a n t s c a n be s o l v e d by s e t t i n g up 6 x N s i m u l t a n e o u s e q u a t i o n s , i n t h e f o l l o w i n g manner. S i x e q u a t i o n s a r e o b t a i n e d from t h e t h r e e boundary c o n d i t i o n s a t t h e t o p and bottom of t h e d e p o s i t d e f i n e d by e q u a t i o n s (4.6) and ( 4 . 7 ) . Then a t i n t e r f a c e s between l a y e r s t h e f o l l o w i n g c o n d i -t i o n s a r e s a t i s f i e d , v = v w = w a' = cr1 , T = x , , v , n n-1 n n-1 z n z ( n - l ) x z n x z ( n - l ) dp dp _ 1 p = p ., K = K , . v (4.16) r n * n - l z n 9z z ( n - l ) 9z These c o n d i t i o n s a r e r e q u i r e d f o r c o n t i n u i t y i n d i s p l a c e m e n t s , s t r e s s e s , p o r e p r e s s u r e s and f l o w n o r m a l t o t h e i n t e r f a c e between two a d j a c e n t l a y e r s . An e q u a t i o n of t h e form of e q u a t i o n (4.16) w i l l l e a d t o 6 x (N-1) e q u a t i o n s . T h e r e f o r e , t h e t o t a l number of s i m u l t a n e o u s e q u a t i o n s a r e 6 x N. A f t e r s o l v i n g t h e s e e q u a t i o n s , i n c r e m e n t a l s t r e s s e s , p o r e p r e s -s u r e s , e t c . c a n be o b t a i n e d i n any l a y e r . These i n c r e m e n t a l v a l u e s f o r s t r e s s e s and p o r e p r e s s u r e a r e s i m p l e f o r t h e s p e c i a l c a s e , G3 -> 0 and 38 AD -> 0 (Yamamoto, 1978) ' . Under t h e s e c o n d i t i o n s , - a ' = a ' = p Aze ^ Z Cos(Ax - to t ) X z ro —Az x = p Aze Sin(Ax-cot) xz o —Az and p = p e Cos(Ax -co t ) ( 4 . 1 7 ) o 4 . 5 A n a l y s i s of t h e Depth of I n s t a b i l i t y An a n a l y s i s has been p r e s e n t e d to e v a l u a t e t h e wave induced i n c r e m e n t a l changes i n s t r e s s e s and pore p r e s s u r e s from the i n i t i a l e q u i -l i b r i u m s t a t e . The g r o s s e f f e c t i v e s t r e s s e s a t any p o i n t can be w r i t t e n as a ' = a ' + a ' x t X O X a ' = a ' + a ' ( 4 . 1 8 ) z t zo z T . = T x z t xz where ° xt> °zt a n a T x z t = t * i e e f f e c t i v e s t r e s s e s a t any p o i n t ; a ' and a ' = t h e i n i t i a l e f f e c t i v e s t r e s s e s ( i . e . , d u r i n g t h e calm p e r i o d ) . These i n i t i a l s t r e s s e s f o r a u n i f o r m d e p o s i t a r e a ' = y.z and a ' = K a ' ( 4 . 1 9 ) zo b xo o zo where = the buoyant u n i t weight of t h e s o i l ; z = the de p t h of t h e p o i n t c o n s i d e r e d ; K Q = t h e c o e f f i c i e n t of e a r t h p r e s s u r e assumed to be g i v e n by (1-Sin<j>'). Here, cb' i s t h e a n g l e of i n t e r n a l f r i c t i o n . For l a y e r e d d e p o s i t s t h e same i d e a c a n be extended t o f i n d a ' by u s i n g t h e a p p r o p r i a t e buoyant u n i t z o weight f o r each l a y e r . The s t r e s s s t a t e a t any p o i n t i s g i v e n by c r x t , O y t and T x v t . L e t 39 U o ' - o ' J +4x } S i n * ( x , z , t ) = , , (4.20) z t x t I f t h e Mohr-Coulomb f a i l u r e c r i t e r i o n i s assumed t h e n i n s t a b i l i t y w i l l o c c u r i f <j)(x,z,t) > (j) 1. F or t h e d e s i g n extreme wave, a r e g i o n o f i n s t a -b i l i t y w i t h i n w h i c h <j>(x,z,t) > <j>' c a n be d e f i n e d . A computer program, STAB-MAX, has been d e v e l o p e d based on t h i s method. T h i s program c a n compute i n d u c e d s t r e s s e s , p o r e p r e s s u r e s and p r e d i c t t h e r e g i o n o f i n s t a -b i l i t y due t o a wave l o a d i n g i n a h o r i z o n t a l l y l a y e r e d d e p o s i t of f i n i t e d e p t h . 4.6 V a l u e s f o r E l a s t i c C o n s t a n t s E x c e p t f o r some s p e c i a l c a s e s mentioned i n S e c t i o n 4.9, a p p r o -p r i a t e v a l u e s f o r t h e l i n e a r e l a s t i c c o n s t a n t s a r e r e q u i r e d t o compute s t r e s s e s u s i n g t h i s method. We r e q u i r e 2 l i n e a r e l a s t i c c o n s t a n t s and th e y a r e s e l e c t e d t o be shear and b u l k m o d u l i . I n g e n e r a l , t h e s t r e s s -s t r a i n r e l a t i o n s h i p i s i n e l a s t i c and h i g h l y s t r a i n - d e p e n d e n t . B u t , i t has been f o u n d t h a t s t r e s s e s can be e v a l u a t e d w i t h r e a s o n a b l e a c c u r a c y by assuming t h e s t r a i n dependent s e c a n t m o d u l i as t h e l i n e a r e l a s t i c c o n s t a n t s . Based on c y c l i c t e s t s i n l a b o r a t o r y , Seed and I d r i s s (1970) p r e s e n t e d t h e v a r i a t i o n o f a v e r a g e s e c a n t shear modulus f o r sands as a f r a c t i o n of t h e modulus a t low s t r a i n l e v e l ( 1 0 - 4 p e r c e n t ) w i t h s h e a r s t r a i n ( F i g . 7 ) . The shear modulus a t l o w s t r a i n l e v e l , G m a x , can be d e t e r m i n e d e i t h e r i n t h e f i e l d o r i n t h e l a b o r a t o r y . Because o f t h e i n f l u e n c e of sample d i s -t u r b a n c e on t h e measurement o f G m a x i n t h e l a b o r a t o r y and t h e i n c a p a b i l i t y of l a b o r a t o r y equipments t o measure v e r y l o w s t r a i n s , i t i s d e s i r a b l e t o measure G m a x i n t h e f i e l d . S e i s m i c methods a r e e x t e n s i v e l y used f o r t h i s p u r p o s e . Based on t h e number o f r e s o n a n t column t e s t s c a r r i e d o ut i n t h e l a b o r a t o r y , H a r d i n and D r n e v i c h (1972) proposed a s i m p l e e q u a t i o n f o r G m a x i n d i m e n s i o n l e s s form a s 10 4 10 3 10 2 10 1 S H E A R S T R A I N , 7 - P E R C E N T FIG. 7 V a r i a t i o n of Secant Shear Modulus w i t h Shear S t r a i n ( A f t e r Seed and I d r i s s , 1970) 41 G = 320.8 P a ( 2 ; ? ! i " e ) 2 (OCR) r S ) h max a (1+e) P a where e = v o i d r a t i o ; OCR = o v e r c o n s o l i d a t i o n r a t i o r = c o n s t a n t dependent on t h e p l a s t i c index of s o i l . For z e r o p l a s t i c index r has a v a l u e of z e r o . (Here, r i s assumed to be' zero) ; cjjjj = mean e f f e c t i v e normal s t r e s s g i v e n by 1+2K 3 vo K Q = c o e f f i c i e n t of e a r t h p r e s s u r e a t r e s t (assumed to be e q u a l to ( l - s i n c b ' ) ) ; <j>' = e f f e c t i v e a n g l e o f i n t e r n a l f r i c t i o n ; P a = a t m o s p h e r i c p r e s s u r e . (4.21) So, knowing G m a x , we r e q u i r e shear s t r a i n induced by t h e wave l o a d i n g to e s t i m a t e t h e s e c a n t shear modulus. T h i s can be done i n an i t e r a t i v e way by f i r s t assuming a v a l u e f o r secant modulus and then m o d i f y i n g i t f o r t h e s t r a i n d e v e l o p e d and p e r f o r m i n g t h i s a n a l y s i s u n t i l t h e r e i s no a p p r e -c i a b l e d i f f e r e n c e i n t h e i n p u t and m o d i f i e d shear m o d u l i . I n t h i s t h e s i s , i t e r a t i v e a n a l y s i s i s not performed. The l i n e a r e l a s t i c shear modulus was assumed to be g i v e n by e q u a t i o n (4.21), and c a l c u l a t e d u s i n g s t r e s s e s b e f o r e t h e a p p l i c a t i o n of t h e wave l o a d i n g . B u l k modulus v a l u e s were found t o be a f u n c t i o n of minor normal s t r e s s o n l y , by Duncan e t a l (1978) . They proposed B = K,P (^) m b a P a where = b u l k modulus c o n s t a n t ; m = b u l k modulus exponent c o n s t a n t ; (4.22) 42 = a v e r a g e e f f e c t i v e m i n o r p r i n c i p a l - s t r e s s (assumed t o be-K-b^).-. The same i t e r a t i v e a p p r o a c h used f o r shear modulus can be used f o r b u l k modulus t o g e t c o m p a t i b l e b u l k modulus. I n t h i s t h e s i s , t h e b u l k modulus was assumed t o be c o n s t a n t and was c a l c u l a t e d u s i n g e q u a t i o n (4.22) based on s t r e s s e s t h a t e x i s t e d b e f o r e t h e a p p l i c a t i o n of t h e wave l o a d i n g . 4.7 Example P r o b l e m The s i m p l e example shown i n F i g . 8 w i l l be a n a l y s e d t o i l l u s t r a t e t h e a p p l i c a t i o n of t h e method and t h e k i n d s of r e s u l t s t h a t c a n be o b t a i n e d . The d e p o s i t i s a u n i f o r m sand of r e l a t i v e d e n s i t y , D r = 54%. The mean wat e r d e p t h i s 12 f t . Other r e l e v a n t d a t a a r e g i v e n i n F i g . 8. The c h a r a c t e r i s t i c s of t h e d e s i g n maximum wave a r e a wave h e i g h t = 9.0 f t and a wave p e r i o d = 7.0 s e c . The c o m p r e s s i b i l i t y of water i s t a k e n t o be 0.1953 x 1 0 " 7 f t 2 / l b (Madsen, 1978). The f i n i t e element d i s c r e t i z a t i o n used f o r t h i s d e p o s i t i s shown i n F i g . 9. Madsen (1978) s o l v e d t h e wave i n d u c e d p o r e p r e s s u r e s and e f f e c t i v e s t r e s s e s f o r a u n i f o r m medium s t i f f d e p o s i t of i n f i n i t e t h i c k n e s s and c o n s t a n t p r o p e r t i e s . He c o n c l u d e d t h a t 1) h y d r a u l i c a n i s o t r o p y of t h e s o i l d i d n o t have any a p p r e c i a b l e e f f e c t on t h e s e s o l u t i o n s f o r s o i l s c o a r s e r t h a n s i l t ; 2) t h e s o l u t i o n s were v e r y s e n s i t i v e t o r e l a t i v e c o m p r e s s i b i l i t y of t h e s o i l s k e l e t o n and porewater;-3) n e i t h e r a n i s o t r o p i c r a t i o nor c o m p r e s s i b i l i t y of p o r e w a t e r had any i n f l u e n c e on t h e v a l u e of s h e a r s t r e s s e s . To examine t h e s e c o n c l u s i o n s f o r a d e p o s i t of f i n i t e t h i c k n e s s and s t r e s s dependent e l a s t i c c o n s t a n t s , t h e f o l l o w i n g a n a l y s i s was c a r r i e d o u t . Two s o i l t y p e s were assumed by c h o o s i n g t h e v a l u e of v e r t i c a l p e r m e a b i l i t y K z = 0.002 cm/s and 0.2 cm/s w h i c h c o r r e s p o n d t o t y p i c a l 43 Wave Height = 9.0' Wave Period =7.05 = 64.0 l b f / f t 3 60 Void r a t i o =0.7 _ E f f e c t i v e unit weight = 47.6 lb/ft"* Compressibility of porewater = 0.1953xl0 - /ft / l b ' Bulk, modulus constants K b = 950, n = 0.03 = 33.0L FIG. 8 S o i l Deposit used i n Example Problem H = Wave Height Node No. I 2 3 4 5 6 7 8 9 10 rE lements Nodes Depth - Ft o 2 4 6 8 10 15 20 25 30 11 12 36 46 13 FIG. 9 F i n i t e Element D i s c r e t i z a t i o n 60 44 v a l u e s f o r f i n e and c o a r s e sandy d e p o s i t s , r e s p e c t i v e l y . For e ach v a l u e of K z, s o l u t i o n s were o b t a i n e d f o r h y d r a u l i c a n i s o t r o p y KK/KZ v a l u e s of 1,2 and 5. The c a l c u l a t e d maximum i n c r e m e n t a l s t r e s s e s a r e p r e s e n t e d i n n o n - d i m e n s i o n a l form i n F i g s . 1 0 ( a ) , ( b ) , ( c ) and ( d ) . The c u r v e s a r e l a b e l l e d by a two d i g i t number where t h e f i r s t d i g i t i n d i c a t e s v e r t i c a l p e r m e a b i l i t y : 1 and 2 f o r K z = 0.002 and 0.2 cm/s, r e s p e c t i v e l y . The second d i g i t i n d i c a t e s t h e a n i s o t r o p y r a t i o : K x / K z = 1,2 and 5, r e s p e c -t i v e l y . The a t t e n u a t i o n of p o r e p r e s s u r e w i t h d e p t h i s i l l u s t r a t e d i n F i g . 1 0 ( a ) . The c u r v e s e x h i b i t e s s e n t i a l l y t h e same b e h a v i o u r as Madsen's s o l u t i o n s . S i n c e one of t h e bottom boundary c o n d i t i o n s had been 9p/9z = 0, t h e ' k i n k ' i n wave i n d u c e d p o r e p r e s s u r e and s h e a r s t r e s s e s were o b s e r v e d . The a t t e n u a t i o n i s u n a f f e c t e d by t h e h y d r a u l i c a n i s o t r o p y r a t i o i n t h e c a s e of f i n e sand, but i n t h e c a s e of c o a r s e sand, i n c r e a s e d p e r m e a b i l i t y i n t h e h o r i z o n t a l d i r e c t i o n l e d t o a f a s t e r decay i n p o r e p r e s s u r e . F i g u r e 10(b) shows t h e v a r i a t i o n of t h e maximum i n c r e m e n t a l v e r t i c a l e f f e c t i v e s t r e s s . The maximum e f f e c t i v e s t r e s s o c c u r s under t h e c r e s t of t h e wave. T h i s i s t o be e x p e c t e d as w a t e r i s f o r c e d i n t o t h e s o i l d e p o s i t under t h e c r e s t c a u s i n g v e r t i c a l seepage f o r c e s w h i c h c o n t r i -b u t e t o an i n c r e a s e i n t h e v e r t i c a l e f f e c t i v e s t r e s s under t h e c r e s t of t h e wave. I t c a n be i n f e r r e d from F i g . 10(b) t h a t , as i n t h e c a s e of p o r e p r e s s u r e a t t e n u a t i o n w i t h d e p t h , t h e i n c r e m e n t i n v e r t i c a l e f f e c t i v e s t r e s s i s i ndependent of t h e a n i s o t r o p y r a t i o f o r f i n e sands and i s a f u n c t i o n o f t h e r a t i o i n c o a r s e sand d e p o s i t s . I n c r e a s e s i n p e r m e a b i l i t y i n t h e h o r i z o n t a l d i r e c t i o n l e a d t o an i n c r e a s e i n a z , and a l s o c l o s e t o t h e s u r f a c e o f t h e seabed t h e v e r t i c a l e f f e c t i v e s t r e s s g r a d i e n t i n c r e a s e s w i t h i n c r e a s e i n t h e a n i s o t r o p i c p e r m e a b i l i t y r a t i o . The s h e a r s t r e s s , on t h e o t h e r hand, a p p e a r s t o be f a i r l y FIG. 1 0(a) Porewater P r e s s u r e i n a D e p o s i t of F i n i t e Depth 46 0 0.1 0.2 0.3 0.4 0.5 0 .6 0.7 • 25 FIG. 10(b) Induced V e r t i c a l E f f e c t i v e S t r e s s e s i n a D e p o s i t of F i n i t e Depth 47 i n s e n s i t i v e t o b o t h p e r m e a b i l i t y and a n i s o t r o p i c p e r m e a b i l i t y r a t i o s ( F i g . 1 0 ( c ) ) . Assuming t h a t t h e bed i s e l a s t i c and impermeable, i n d u c e d shear s t r e s s e s c a n a l s o be c a l c u l a t e d u s i n g s i m p l e s o l i d m e c h a n i c a l p r i n c i p l e s . T h i s can be done by c o n s t r u c t i n g a s t r e s s f u n c t i o n , s u c h t h a t i t s a t i s f i e s boundary c o n d i t i o n s . T h i s p r o c e d u r e i s e x p l a i n e d i n A p pendix I I . T h i s method w i l l g i v e c y c l i c s h e a r s t r e s s e s i n dependent of b o t h p e r m e a b i l i t y and c o m p r e s s i b i l i t y of w a t e r . P l o t of c y c l i c s h e a r s t r e s s e s o b t a i n e d by t h i s method i s l a b e l l e d as 00 i n F i g . 1 0 ( c ) . S i n c e t h i s c u r v e compares w e l l w i t h t h e o t h e r c u r v e s , t h e s i m p l e e l a s t i c method can be used w i t h o u t much e r r o r f o r a l l p r a c t i c a l p u r p o s e s . The r e s i d u a l p o r e p r e s s u r e a n a l y s i s e x p l a i n e d i n C h a p t e r 5 u s e s t h i s e l a s t i c method t o c a l c u l a t e i n d u c e d s h e a r s t r e s s e s . The maximum v a l u e of shear s t r e s s o c c u r s h a l f w a y between t h e c r e s t and t h e t r o u g h of t h e wave. The i n c r e a s e i n h o r i z o n t a l e f f e c t i v e s t r e s s i s a f f e c t e d by two s e p a r a t e mechanisms. One i s t h e i n c r e a s e , due t o t h e i n c r e a s e i n v e r t i c a l e f f e c t i v e s t r e s s . T h i s c o n t r i b u t i o n i s i n phase w i t h t h e v e r t i c a l e f f e c -t i v e s t r e s s . The o t h e r c o n t r i b u t i o n i s governed by t h e s o i l r e s p o n s e t o seepage f o r c e s i n d u c e d by t h e f l o w . T h i s c o n t r i b u t i o n i s a f u n c t i o n of t h e h o r i z o n t a l p r e s s u r e g r a d i e n t w h i c h d e c r e a s e s w i t h i n c r e a s e i n a n i s o -t r o p i c p e r m e a b i l i t y r a t i o s . The h o r i z o n t a l seepage f o r c e s a c t on t h e s o i l i n t h e d i r e c t i o n away from t h e wave c r e s t and towards t h e t r o u g h s . T h i s h o r i z o n t a l seepage f l o w w i l l r e s u l t i n t e n s i o n under t h e c r e s t . T h e r e -f o r e , n e t i n c r e a s e i n a x i s complex and i s g overned by b o t h t h e a n i s o -t r o p i c p e r m e a b i l i t y r a t i o and t h e p e r m e a b i l i t y . The e f f e c t of r e l a t i v e c o m p r e s s i b i l i t y has a g r e a t e f f e c t on t h e s e s o l u t i o n s . Yamamoto (1978) had shown t h a t , f o r a seabed of i n f i n i t e d e p t h , w i t h G3 -»• 0, t h e s o l u t i o n s f o r t h e wave-induced s t r e s s e s r e d u c e FIG. 10(c) Induced Shear S t r e s s e s i n a D e p o s i t of F i n i t e Depth O-x'/Po FIG. 10(d) Induced H o r i z o n t a l E f f e c t i v e S t r e s s e s i n a D e p o s i t of F i n i t e Depth 50 t o s i m p l e m a t h e m a t i c a l e q u a t i o n s , as shown i n e q u a t i o n ( 4 . 1 7 ) . For p a r t i a l l y s a t u r a t e d s o i l s , t h e e f f e c t i v e c o m p r e s s i b i l i t y of t h e p o r e -f l u i d s may exceed t h e c o m p r e s s i b i l i t y of p u r e w a t e r ( a i r f r e e ) by a c o n s i d e r a b l e amount depending on t h e d e g r e e of s a t u r a t i o n . A s i m p l e a n a l y s i s by V e r r u i t (1969) g i v e s an upper bound f o r t h e c o m p r e s s i b i l i t y of a w a t e r - g a s m i x t u r e as 1-S 3 = 3„ + — 5 ; 1 - S << 1 (4.23) t i n w h i c h S g i s t h e d e g r e e of s a t u r a t i o n , P t i s t h e t o t a l s t a t i c p o r e p r e s s u r e , 3 and 3 Q a r e t h e c o m p r e s s i b i l i t y o f p a r t i a l l y s a t u r a t e d and s a t u r a t e d p o r e w a t e r , r e s p e c t i v e l y . For S = 0.99, e f f e c t i v e c o m p r e s s i b i -—5 —1 —8 —1 l i t y d e f i n e d by y w3 i s 10 cm compared t o 4 x 10 cm f o r p u r e w a t e r . The e f f e c t o f p a r t i a l s a t u r a t i o n i s w e l l - d o c u m e n t e d by Madsen (1978) and, t h e r e f o r e , f u r t h e r a n a l y s i s i s n o t c o n s i d e r e d i n t h i s t h e s i s . I n t h e program, STAB-MAX, s t r e s s e s a r e computed a t t h e m i d d l e of t h e l a y e r s and t h e e f f e c t i v e a n g l e of f r i c t i o n r e q u i r e d f o r s t a b i l i t y i s p r i n t e d o u t . Based on t h i s , v a r i o u s c o n t o u r s of cf>' c a n be drawn and a r e shown i n F i g . 11 f o r t h e c a s e of K z = 0.002 cm/s and K x / K z = 1. Assuming t h e waves t r a v e l a l o n g t h e h o r i z o n t a l a x i s , t h e d e p t h of f a i l u r e c a n be e s t i m a t e d f o r t h e example p r o b l e m c o n s i d e r e d ( $ ' = 33°) and was found t o be 12.5'. The r e a l u n d e r w a t e r s e a f l o o r s t a b i l i t y p r o b l e m i s more c o m p l i -c a t e d t h a n t h e one w h i c h has been a n a l y s e d by t h e s i m p l i f i e d a p p r o a c h used h e r e . There i s a l w a y s energy t r a n s f e r between t h e wave and t h e moving s o i l because o f work done a g a i n s t t h e s h e a r i n g r e s i s t a n c e of t h e s o i l . T h i s c o n t r i b u t e s t o t h e damping of t h e waves. F u r t h e r , t h e p o r o - e l a s t i c m e t h o d , p r e d i c t e d t e n s i l e h o r i z o n t a l s t r e s s e s under t h e wave c r e s t . S i n c e 51 FIG. 11 C o n t o u r s of <j) Developed 52 s o i l c annot c a r r y t e n s i o n , t h e s t r e s s f i e l d p r e d i c t e d by t h i s method i s n o t a c c u r a t e . However, t h i s s i m p l i f i e d a p p r o a c h can be used t o p r e -d i c t t h e r e s p o n s e of t h e s e a f l o o r w i t h r e a s o n a b l e a c c u r a c y . 4.8 Comparison of t h e Theory w i t h F i e l d Data Wastewater Management M a s t e r P l a n of t h e C i t y and County of San F r a n c i s c o , C a l i f o r n i a d e a l s w i t h c o l l e c t i n g , t r e a t i n g and d i s p o s i n g of b o t h s a n i t a r y and s t o r m w a s t e w a t e r f l o w s . These f l o w s were t o be c o l l e c t e d and d i s p e r s e d t h r o u g h Ocean O u t f a l l a t d i f f u s e r s e c t i o n s i n t h e P a c i f i c Ocean. The O u t f a l l was t o be embedded t h r o u g h o u t most of i t s l e n g t h of a p p r o x i m a t e l y 4000 f t i n a t r e n c h - e x c a v a t e d as much as 25 f t below t h e e x i s t i n g s e a f l o o r . The a t t e n u a t i o n of wave-induced p o r e p r e s s u r e w h i c h i s r e q u i r e d i n t h e p l a n n i n g of o f f s h o r e t u n n e l i n g o p e r a t i o n s and i n t h e c o m p u t a t i o n of f o r c e s on b u r i e d o f f s h o r e s t r u c t u r e s , was measured i n t h e f i e l d by C r o s s e t a l (197 9 ) . They measured wave-induced p o r e p r e s s u r e s u s i n g a v e r t i c a l p i e z o m e t e r a r r a y i n s t a l l e d f rom a m u n i c i p a l p i e r i n P a c i f i c a , C a l i f o r n i a , l o c a t e d a p p r o x i m a t e l y 600 f t from s h o r e i n 16 f t of w a t e r . The p i e z o m e t e r s w h i c h were IRAD v i b r a t i n g w i r e t y p e were p l a c e d a t 30, 60, 86 and 120 f t below t h e s e a f l o o r . The s o i l m a t e r i a l s encoun-t e r e d were p r e d o m i n a n t l y l o o s e f i n e sands n e a r t h e s e a f l o o r , medium dense below 2 f t and dense t o v e r y dense below 8 f t . S o i l t y p e d a t a o b t a i n e d d u r i n g t h e i n s t a l l a t i o n p r o c e s s a r e p r e s e n t e d i n F i g . 12. The a n a l y s i s e x p l a i n e d i n t h i s c h a p t e r was used t o p r e d i c t t h e wave-induced p o r e p r e s -s u r e . F o r t h i s a n a l y s i s , I^/K^ was c h osen t o be 2 and t h e v a l u e s of K z were s e l e c t e d a c c o r d i n g t o p a r t i c l e s i z e d i s t r i b u t i o n d a t a g i v e n by C r o s s et a l . These a u t h o r s p r e s e n t e d p o r e p r e s s u r e f l u c t u a t i o n s as a f r a c t i o n o f o b s e r v e d s i g n i f i c a n t wave h e i g h t s d e f i n e d as p/2y wH i n F i g . 13. R e s u l t s o b t a i n e d u s i n g STAB-MAX a r e a l s o shown f o r c o m p a r i s o n . The DEPTH BELOW SEAFLOOR (FEET) MATERIAL DESCRIPTION •SEA FLOOR S A N D (SP) : Loose to medium dense, l i g h t brown, f i n e - g r a i n e d , sub-angular t o sub-rounded 10 20 • 30 4 0 5 0 6 0 70 80 9 0 100 110 120 —{ 130 n o —' GRAVEL (GP) : Very dense, l i g h t brown, medium t o " v ^ _ c o a r s e - g r a i n e d , sub-angular t o sub-rounded y S A N D (SP-SM): Very dense, l i g h t brown, f i n e -g r a i n e d , sub-angular t o sub-rounded ^GRAVEL (GP): Very dense, l i g h t brown, medium t o / co a r s e - g r a i n e d , sub-angular t o sub-rounded S A N D (SP) : Very dense, l i g h t brown, f i n e -g r a i n e d , sub-angular t o sub-rounded, with a small amount o f sub-angular g r a v e l Piezometer S I L T Y S A N D (SM) Very dense, brown, f i n e - g r a i n e d , s ub-angular to sub-rounded Piezometer • J Becomes l i g h t brown S A N D (SP) Very dense, brown, f i n e - g r a i n e d , sub-an g u l a r t o sub-rounded Piezometer S A N D (SP-SM) Very dense, l i g h t brown, f i n e - g r a i n e d , sub-angular t o sub-rounded — Piezometer J Becomes gray-green BOTTOM OF B O R I N G § 1 3 0 . 5 ' FIG. 12 B o r e h o l e Data ( A f t e r C r o s s e t a l , 1979) P o r e P r e s s u r e F l u c t u a t i o n p/(2y wH) 0 0.1 , OJZ (L3 OA ( L 5 0.6 p. 7 i —i 1 1 1 H r n FIG. 13 P o r e P r e s s u r e D i s t r i b u t i o n i n Sand D e p o s i t of C a l i f o r n i a Coast 55 a t t e n u a t i o n of wave-induced p o r e p r e s s u r e w i t h d e p t h below t h e s e a f l o o r a g r e e s r e m a r k a b l y w e l l w i t h t h e p r e d i c t i o n s made by t h i s method. Only a v e r a g e p o r e p r e s s u r e s were measured by t h e s e a u t h o r s , and n o t i n s t a n -t a n e o u s v a l u e s , when a g i v e n wave i s p a s s i n g by. The e r r o r i n p r e d i c t i o n c a n be t o a c e r t a i n e x t e n t a t t r i b u t e d t o t h i s . 4.9 Some P r a c t i c a l S o l u t i o n s The p o r o - e l a s t i c method e x p l a i n e d i n t h i s c h a p t e r i s a c o u p l e d a n a l y s i s , i n o t h e r words i t assumes t h a t t h e wa t e r and s o i l c o n s t i t u t e t o a c o u p l e d problem. T h i s t y p e of a n a l y s i s i s t e d i o u s and r e q u i r e s t h e computer f o r s o l v i n g ; . F i n n , S i d d h a r t h a n and M a r t i n (1980) who c a r r i e d out an e x t e n s i v e s t u d y u s i n g t h e program, STAB-MAX, s u g g e s t e d t h a t s i m p l e u n c o u p l e d a n a l y s i s can be pe r f o r m e d f o r a number o f u n i f o r m deep s a t u r a t e d d e p o s i t s . They c o n c l u d e d t h a t t h e n o r m a l e f f e c t i v e s t r e s s e s i n d u c e d i n a d e p o s i t can be o b t a i n e d by computing t o t a l wave-induced n o r m a l s t r e s s e s and p o r e p r e s s u r e s i n d e p e n d e n t l y of each o t h e r and u s i n g t h e p r i n c i p l e of e f f e c t i v e s t r e s s , i . e . , e f f e c t i v e s t r e s s e q u a l s t o t a l s t r e s s minus p o r e p r e s s u r e . The shear s t r e s s e s a r e t h e same whether computed by t o t a l o r e f f e c t i v e s t r e s s e s . T o t a l wave-induced s t r e s s e s c a n be o b t a i n e d by a n a l y s i n g t h e d e p o s i t as a s e m i - i n f i n i t e e l a s t i c medium, and s o l u t i o n s a r e g i v e n by Fung ( 1 9 6 5 ) . The a m p l i t u d e of t h e s e s t r e s s e s a r e g i v e n by az = P 0 ( e " A z + z A e " A z ) (4.24) / ~Az , -Az. ,, °x = p o ( e ~ zXe ) (4.25) a n d T x z = p o X z e ~ A Z (4.26) I t s h o u l d be n o t e d t h a t t h e s e s o l u t i o n s do n o t c o n t a i n t h e p r o p e r t i e s o f t h e e l a s t i c medium. 56 Now knowing t o t a l i n d u c e d s t r e s s e s one needs t o know s i m p l e s o l u -t i o n s f o r p o r e p r e s s u r e a t t e n u a t i o n t o compute e f f e c t i v e s t r e s s e s . To do t h i s , F i n n , S i d d h a r t h a n and M a r t i n (1980) c o n s i d e r e d 4 d i f f e r e n t s o i l t y p e s w i t h extreme s o i l p r o p e r t i e s . The p r o p e r t i e s c o n s i d e r e d were PROPERTY SAND TYPE Hard S o f t C o a r s e F i n e G, N/m2 K , m/sec z 1 0 9 5 x l 0 6 i o - 3 i o " 6 Four t y p e s o f s o i l s v i z . h a r d - c o a r s e , h a r d - f i n e , s o f t - c o a r s e , and s o f t -f i n e sands were c o n s i d e r e d . The f o l l o w i n g a d d i t i o n a l p r o p e r t i e s were a l s o assumed: K x / K z = 1,2, and 5; P o i s s o n r a t i o = 1/3; b u l k modulus of s a t u r a t e d w a t e r = 2.45 x 10^ N/m2. The r e s u l t s o b t a i n e d were somewhat s u r p r i s i n g . The a n i s o t r o p i c p e r m e a b i l i t y r a t i o has l i t t l e e f f e c t on t h e p o r e w a t e r p r e s s u r e i n s o f t - c o a r s e and s o f t - f i n e sands. A s i n g l e p o r e w a t e r p r e s s u r e a t t e n u a t i o n c u r v e w h i c h i s n o t a f u n c t i o n K x / K z r a t i o , g i v e n by -Az P e *o (4.27) can be used f o r t h e s e sands. I t was found t h a t f o r h a r d - c o a r s e sand, t h e po r e p r e s s u r e a t t e n u a t i o n was a f u n c t i o n o f a n i s o t r o p i c p e r m e a b i l i t y r a t i o and g i v e n by P o 6 - A ( K X / K z ) ^ z (4.28) The p o r e p r e s s u r e a t t e n u a t i o n i n h a r d - f i n e sands was o b s e r v e d t o be i n d e -pendent of a n i s o t r o p i c r a t i o s , and s i m p l e s o l u t i o n f o r a t t e n u a t i o n was not p o s s i b l e . T h i s s i m p l e method of d e t e r m i n i n g i n d u c e d e f f e c t i v e s t r e s s e s i s -57 r e s t r i c t e d t o c a s e s where t h e d e p o s i t c o n c e r n e d i s deep, s a t u r a t e d and u n i f o r m . F o r l a y e r e d , s h a l l o w d e p o s i t s a n u m e r i c a l s o l u t i o n of t h e t y p e ( c o u p l e d s o l u t i o n ) e x p l a i n e d i n t h i s c h a p t e r s h o u l d be u s e d . 58 CHAPTER 5 RESIDUAL PORE PRESSURE ANALYSIS The s u r f a c e waves i n d u c e c y c l i c s h e a r s t r e s s e s on t h e seabed. The c y c l i c s h e a r s t r e s s e s d e v e l o p e d d u r i n g s t o rm a c t i v i t y may be h i g h enough t o b u i l d up t h e r e s i d u a l p o r e p r e s s u r e s t o a s i g n i f i c a n t m a gnitude. The development o f t h e r e s i d u a l p o r e p r e s s u r e i s t h e r e s u l t of two oppos-i n g p r o c e s s e s . The volume c o m p a c t i o n t e n d e n c y o f t h e s o i l under c y c l i c l o a d i n g c a u s e s t h e p o r e p r e s s u r e t o r i s e , and t h e d i s s i p a t i o n o f r e s i d u a l p o r e p r e s s u r e w h i c h i s governed by t h e dynamic f o r m o f t h e c o n s o l i d a t i o n e q u a t i o n , d e c r e a s e s i t ( M a r t i n e t a l , 1975). I t i s common i n e a r t h q u a k e r e s p o n s e a n a l y s e s t o assume t h a t t h e u n d r a i n e d c o n d i t i o n p r e v a i l s s i n c e t h e l o a d i n g a c t s f o r a v e r y s h o r t t i m e . I n o t h e r words, t h e d i s s i p a t i o n of r e s i d u a l p o r e p r e s s u r e i s i g n o r e d . B u t , d u r i n g s t o r m a c t i v i t y , t h e d i s s i p a t i o n o f r e s i d u a l p o r e p r e s s u r e s h o u l d be used s i n c e t h e d u r a t i o n of t h e wave l o a d i n g i s v e r y l a r g e . An u n d r a i n e d a n a l y s i s h e r e w i l l l e a d t o v e r y h i g h r e s i d u a l p o r e p r e s s u r e s i n t h e d e p o s i t . The volume c o m p a c t i o n c h a r a c t e r i s t i c s w h i c h g o v e r n t h e g e n e r a t i o n of t h e r e s i d u a l p o r e p r e s s u r e i n t h e d e p o s i t depends p r i m a r i l y on 1) t h e r e l a t i v e d e n s i t y o f t h e s o i l , D r, and t h e t y p e of s o i l , 2) t h e i n d u c e d c y c l i c s h e a r s t r e s s r a t i o , and 3) t h e e x i s t i n g r e s i d u a l p o r e p r e s s u r e . On t h e o t h e r hand, t h e d i s s i p a t i o n w i l l depend on t h e p e r m e a b i l i t y com-p r e s s i b i l i t y and t h e d r a i n a g e c o n d i t i o n s a t t h e b o u n d a r i e s of t h e d e p o s i t . The n e t p o r e p r e s s u r e r e s p o n s e w i l l be t h e r e s u l t a n t o f t h e s e g e n e r a t i o n -d i s s i p a t i o n e f f e c t s . Seed e t a l (1977) and N a t a r a j a (1978) a n a l y s e d t h e 59 p o s s i b i l i t y o f seabed l i q u e f a c t i o n d u r i n g storm a c t i v i t y . 5.1 Theory A p l a n e t r a v e l l i n g wave i n d u c e s a c o n s t a n t shear s t r e s s a m p l i -t u d e a l o n g any h o r i z o n t a l p l a n e i n a h o r i z o n t a l l y l a y e r e d d e p o s i t . F u r t h e r , d r a i n a g e c o n d i t i o n s a r e a l s o t h e same f o r a l l t h e p o i n t s i n a h o r i z o n t a l p l a n e . Under t h e s e c o n d i t i o n s , t h e wave l o a d i n g p r o b l e m i s r e d u c e d e s s e n -t i a l l y t o a o n e - d i m e n s i o n a l p r o b l e m . U s i n g D a r c y ' s law, t h e c o n t i n u i t y e q u a t i o n c a n be w r i t t e n as J L f _ £ \ = l £ (5 1) w where u = r e s i d u a l p o r e p r e s s u r e ; K z = c o - e f f i c i e n t of p e r m e a b i l i t y i n z d i r e c t i o n ; Y w = u n i t w e i g h t of w a t e r ; e = i n c r e a s e i n v o l u m e t r i c s t r a i n b e i n g c o n s i d e r e d p o s i t i v e . F o r t h i s p r o b l e m e w i l l be a n e g a t i v e quant i t y . D u r i n g an i n t e r v a l , A t , t h e p o r e p r e s s u r e i n an element of s o i l w i l l undergo a change, A t , w h i l e t h e element w i l l a l s o be s u b j e c t e d t o c y c l e s of shear s t r e s s w h i c h w i l l c a u s e an a d d i t i o n a l i n c r e a s e i n p o r e p r e s s u r e , 3ug/3t At where 3 u g / 3 t i s t h e r a t e of p o r e p r e s s u r e i n c r e a s e . C o n s i d e r i n g t h a t t h e change i n b u l k s t r e s s i s n e g l i g i b l e , t h e volume change, Ae, of t h e element i n t i m e , A t , i s g i v e n by 3u Ae = m (Au - — & 3t) V at where m^ = t h e c o e f f i c i e n t o f volume c o m p r e s s i b i l i t y . As At -> 0 ~ „ 3u I t ~ m v ( ^ ( 5 ' 2 ) 60 c o m b i n i n g e q u a t i o n s (5.1) and (5.2) w The c o m p r e s s i b i l i t y o f s o i l has been f o u n d t o i n c r e a s e w i t h an i n c r e a s e i n p o r e p r e s s u r e , and t h e f o l l o w i n g e q u a t i o n f o r t h i s v a r i a t i o n has been g i v e n by Seed (1976) e A r u m = 0 o p m (5.4) v .2 2B vo tj A r 1 + Ar + u 2 where A = 5(1.5 - D r ) ; B = 3/2< 2 Dr>; r u = p o r e p r e s s u r e r a t i o d e f i n e d as u / a ^ Q ; iiiy = c o m p r e s s i b i l i t y a t p o r e p r e s s u r e r a t i o , r u ; m v o = c o m p r e s s i b i l i t y a t z e r o p o r e p r e s s u r e r a t i o ; o\^0 = i n i t i a l v e r t i c a l e f f e c t i v e s t r e s s . I n o r d e r t o s o l v e e q u a t i o n ( 5 . 3 ) , i t i s n e c e s s a r y t o e s t a b l i s h t h e s o i l p r o p e r t i e s s u c h as m^, K z, D r, e t c . and t h e r a t e of p o r e p r e s s u r e gene-r a t i o n , 9ug/3t. The r a t e o f p o r e p r e s s u r e g e n e r a t i o n c a n be w r i t t e n as (Seed e t a l , 1976) g_ = g. _9N 9t 9N 9t where N i s t h e number of s t r e s s c y c l e s d u r i n g t h e s t o r m . The v a l u e of (5.5) 9u /9N can be e i t h e r e v a l u a t e d by a l a b o r a t o r y t e s t s i m u l a t i n g t h e f i e l d ••conditions o r by an a p p r o x i m a t e m a t h e m a t i c a l f o r m u l a . I t has been found t h a t f o r many s o i l s t h e r e l a t i o n s h i p between U g and N can be e x p r e s s e d f o r p r a c t i c a l p u r p o s e s i n terms of t h e number of c y c l e s , N-^ , r e q u i r e d t o c a u s e 61 i n i t i a l l i q u e f a c t i o n under a g i v e n s t r e s s c o n d i t i o n i n t h e f o l l o w i n g form (Seed e t a l , 1976) J _ g 2 . . 20. rs 7 = — a r c s i n (x ) (5.6) a " vo where Q = an e m p i r i c a l c o n s t a n t ; x = c y c l i c r a d i o , d e f i n e d as (N/N-^) . T h i s r e l a t i o n s h i p i s p r e s e n t e d i n F i g . 14. D i f f e r e n t v a l u e s of G may be used t o r e p r e s e n t d i f f e r e n t r a t e s o f p o r e p r e s s u r e g e n e r a t i o n . A v a l u e o f 0 = 0.7 has been found t o r e p r e s e n t t h e a v e r a g e c u r v e f o r many s o i l s . By d i f f e r e n t i a t i n g e q u a t i o n (5.6) w i t h r e s p e c t t o N, one g e t s _g_ _ _ V O 1_ 3N GUNT . 26-1,11 v ,U . ^ s i n ("2 r u ) cos{j r u ) (5.7) I n g e n e r a l , t h e d e s i g n s t o r m wave d a t a w i l l be i r r e g u l a r and f o r p r a c t i c a l p u r p o s e s i s u s u a l l y r e p r e s e n t e d by N e q c y c l e s o f a u n i f o r m wave. The t o t a l d u r a t i o n o f t h e u n i f o r m wave storm i s t h e same as t h a t o f t h e i r r e g u l a r s t o r m and i s d e s i g n a t e d as T D. Then i S = _§a ( 5 . 8 ) 9t T D Combining e q u a t i o n s (5.7) and ( 5 . 8 ) , t h e r a t e of p o r e p r e s s u r e g e n e r a t i o n , 3ug/3t, a t any t i m e c a n be d e t e r m i n e d f r o m t h e e q u a t i o n : a = g. . M 3t 3N 3t 3u a 1 N o r _ £ = ^ o _ _ e a 1 3t GUTT) N T . 2 0 - L H . J , K ' u s m (-j r u ) cos(-^ r u ) So, now knowing 3ug/3t, e q u a t i o n (5.3) can be used t o d e t e r m i n e t h e r e s i -d u a l p o r e p r e s s u r e d i s t r i b u t i o n . F i n i t e element method i s used t o s o l v e FIG. 14 R a t e of P o r e P r e s s u r e G e n e r a t i o n d u r i n g C y c l i c L o a d i n g 63 f o r u. The domain and boundary c o n d i t i o n s of t h i s p r o b l e m a r e shown i n F i g . 1 5 ( a ) . The f o r m u l a t i o n of f i n i t e element method i s e x p l a i n e d i n d e t a i l i n A ppendix I . 5.2 A n a l y s i s of t h e Depth o f I n s t a b i l i t y The p r i n c i p l e e x p l a i n e d i n - S e c t i o n 4.5 c a n be extended t o p r e d i c t t h e d e p t h of i n s t a b i l i t y . L e t t h e i n i t i a l v e r t i c a l and h o r i z o n t a l e f f e c -t i v e s t r e s s e s i n an element d u r i n g a c a l m p e r i o d be g i v e n by o^ 0 and o x o . Shear s t r e s s T v „ w i l l be z e r o . B u t , d u r i n g t h e s t o r m r e s i d u a l p o r e p r e s -s u r e s d e v e l o p i n t h e d e p o s i t and t h e new s t r e s s s t a t e a t any p o i n t can be e x p r e s s e d as o' = a1 - u (5.10) z t vo a' = K (a' - u) ( s e e F i g . 15(b)) x t o vo b and T „,.. Here, T i s t h e maximum c y c l i c s h e a r s t r e s s i n d u c e d a t t h a t X Z L X Z u p o i n t when t h e h i g h e s t wave p a s s e s o v e r . As b e f o r e , d e f i n i n g a n g l e , cj>, d e v e l o p e d by i n d u c e d s t r e s s e s i n t h e d e p o s i t , as 2 2 h . , z t x t 7 x z t f c. . sin<£, . = 7—j ; r (5.11) V ( x , z , t ) ( a ^ + a' t) i n s t a b i l i t y c a n be e x p e c t e d up t o a. d e p t h w i t h i n w h i c h <t>^ x z t^ > <j)1. I t c a n be seen t h a t t h e h i g h e r t h e u and x x z t , t h e h i g h e r t h e v a l u e of ^ ( x z t ) b e ' So, t h e w o r s t c a s e would be t h e extreme waves p a s s i n g over t h e bed when t h e r e a r e h i g h r e s i d u a l p o r e p r e s s u r e s i n t h e d e p o s i t . 5.3 Changes, i n S o i l P r o p e r t i e s and P o r e P r e s s u r e G e n e r a t i o n The a m p l i t u d e of c y c l i c shear s t r e s s e s have t o be computed i n t h e d e p o s i t t o u s e t h e p o r e p r e s s u r e g e n e r a t i o n e q u a t i o n ( 5 . 9 ) . I t has 64 -u = 0 at z = 0 Soil Deposit _3_[l<z du\ [du du£\ dz|_7w 3 z J ' m v [ a t - a t J | ^ = 0 at z - D Basic Equotion and Solution  Domain (a) vo ° " v o - u K o °"vo K 0 ( C T v o - u ) Stress Condition at Calm Periods Stress Condition at Storm Periods (b) FIG. 15 S t r e s s C o n d i t i o n s B e f o r e and D u r i n g Storms 65 been o b s e r v e d i n S e c t i o n 4.7 t h a t t h e shear s t r e s s e s i n d u c e d i n a d e p o s i t due t o wave l o a d i n g c a n be computed assuming t h a t t h e bed i s impermeable. A s i m p l e p r o c e d u r e u s i n g s o l i d m e c h a n i c a l p r i n c i p l e s t o compute c y c l i c s hear s t r e s s e s i s p r e s e n t e d i n A ppendix I I . I t i s shown t h a t f o r a deep u n i f o r m d e p o s i t , ^c/o^Q i s i n d e p e n d e n t of e l a s t i c c o n s t a n t s of t h e s o i l d e p o s i t . B u t , i n t h e c a s e of a n o n - u n i f o r m o r s h a l l o w d e p o s i t , T c/a^. 0 depends on t h e e l a s t i c c o n s t a n t s . When t h e r e s i d u a l p o r e p r e s s u r e i n c r e a s e s d u r i n g a storm i n a d e p o s i t , t h e e l a s t i c c o n s t a n t s w h i c h were assumed t o be s t r e s s dependent r e d u c e ( e q u a t i o n s (4.21) and ( 4 . 2 2 ) ) . D u r i n g t h e p r e s e n c e of r e s i d u a l e x c e s s p o r e p r e s s u r e , t h e mean n o r m a l s t r e s s e s and minor p r i n c i p a l s t r e s s e s s h o u l d be c a l c u l a t e d a s , 1+2K a' = =-5- (a' - u) (5.12) m 3 vo a' = K (a' - u) 3 o vo T h i s w i l l v a r y t h e T c / a ^ 0 i n d u c e d by t h e waves i n s h a l l o w o r non-u n i f o r m d e p o s i t s d u r i n g a s t o r m a c t i v i t y . I t has been found t h a t t h e m o d i f i c a t i o n of e l a s t i c c o n s t a n t s t o be c o m p a t i b l e w i t h t h e c u r r e n t l e v e l s o f e f f e c t i v e s t r e s s e s l e a d s t o h i g h e r r e s i d u a l p o r e p r e s s u r e s and a g r e a t e r d e p t h of i n s t a b i l i t y i n t h e seabed. An e f f i c i e n t computer program, STAB-W, w h i c h i n c o r p o r a t e s m o d i f i c a t i o n of e l a s t i c c o n s t a n t s has been d e v e l o p e d t o p e r f o r m s t a b i l i t y a n a l y s i s . 5.4 Example P r o b l e m The s i m p l e example shown i n F i g . 8 i s a n a l y s e d t o i l l u s t r a t e t h e a p p l i c a t i o n of t h i s method. T h i s u n i f o r m d e p o s i t was d i v i d e d i n t o 12 l a y e r s and t h e e l a s t i c s o i l c o n s t a n t s , G and B, were s e l e c t e d t o be c o n -s i s t e n t w i t h t h e l e v e l s of e f f e c t i v e s t r e s s . A v a l u e of 0 = 0.7, w h i c h 66 i s a r e p r e s e n t a t i v e v a l u e f o r medium dense sands was used i n p o r e p r e s -s u r e g e n e r a t i o n ( e q u a t i o n ( 5 . 6 ) ) . The d e s i g n s t o r m wave d a t a i s g i v e n i n T a b l e I I . The TQ/a^0 and N^ v a l u e s c o r r e s p o n d i n g t o l i q u e f a c t i o n p o t e n -t i a l c u r v e i s g i v e n i n T a b l e I I I . The r e s i d u a l p o r e p r e s s u r e d i s t r i b u t i o n was a n a l y s e d f o r two TABLE I I D e s i g n Wave Data Wave H e i g h t f t (H) Wave P e r i o d sec (T) No. o f Waves 9.0 7.0 50 8.0 6.5 80 6.0 6.0 155 4.0 5.0 180 2.0 4.0 200 TABLE I I I L i q u e f a c t i o n P o t e n t i a l Curve Shear S t r e s s R a t i o No. of C y c l e s N L 0.06 100,000 0.081 10,000 0.1 1,000 0.13 24 0.163 7.2 0.198 3.2 d i f f e r e n t p e r m e a b i l i t y v a l u e s , K z = 2 x 10 J cm/s and K z = 2 x 10 1 cm/s. To s t u d y t h e i n f l u e n c e o f m o d i f i c a t i o n o f m o d u l i on t h e d e p o s i t , two 67 t y p e s of a n a l y s i s were c o n d u c t e d . I n one, t h e m o d u l i were n o t m o d i f i e d f o r t h e changes i n r e s i d u a l p o r e p r e s s u r e and, i n t h e o t h e r , t h e y were m o d i f i e d . 5.4.1 E q u i v a l e n t u n i f o r m s t o r m The r e s i d u a l p o r e p r e s s u r e method r e q u i r e s e s t a b l i s h i n g an " e q u i v a l e n t u n i f o r m s t o r m " w h i c h i s r e p r e s e n t a t i v e of t h e d e s i g n s t o r m wave i n terms of p o r e p r e s s u r e g e n e r a t i o n i n t h e d e p o s i t . The method pro p o s e d by Lee and Chan (1972) was u s e d t o o b t a i n t h e e q u i v a l e n t storm. T h i s p r o c e d u r e has been e x p l a i n e d i n S e c t i o n 3.3. T a b l e I V shows t h e p r o c e d u r e c a r r i e d out t o c a l c u l a t e t h e e q u i v a l e n t storm of " u n i f o r m wave h e i g h t s f o r t h e d e s i g n storm g i v e n i n T a b l e I I . 5.4.2 D i s c u s s i o n of R e s u l t s F i g u r e 16 shows t h e d i s t r i b u t i o n of r e s i d u a l p o r e p r e s s u r e i n t h e d e p o s i t a t t h e end of t h e s t o r m f o r two c a s e s , K z = 2 x 1 0 - ^ cm/s and K z = 2 x l O - ^ cm/s. The maximum p o r e p r e s s u r e r a t i o s and d e p t h of i n s t a b i l i t y , c a l c u l a t e d u s i n g e q u a t i o n ( 5 . 1 1 ) , a r e p r e s e n t e d i n T a b l e V. M o d i f i c a t i o n of s o i l p r o p e r t i e s f o r t h e i n c r e a s e i n r e s i d u a l p o r e p r e s s u r e has n o t i c e a b l e impact on t h e p r e d i c t e d r e s i d u a l p o r e p r e s -s u r e . When t h e d e g r a d a t i o n i n m o d u l i i s t a k e n i n t o a c c o u n t , t h e p r e d i c -t e d p o r e p r e s s u r e d i s t r i b u t i o n i s about 30% h i g h e r i n t h e l e s s permeable sand. I n t h e c a s e of K z = 2 x 10-''' cm/s, t h e r e s i d u a l p o r e p r e s s u r e s were so low t h a t t h e changes i n t h e e l a s t i c c o n s t a n t s f o r changes i n e f f e c t i v e s t r e s s l e v e l s had no a p p r e c i a b l e impact on t h e r e s u l t s . The d e p t h of i n s t a b i l i t y a l s o i n c r e a s e d when m o d i f i c a t i o n of p r o p e r t i e s was c o n s i d e r e d . T h i s example c l e a r l y d e m o n s t r a t e s t h a t t h e e f f e c t of d e g r a -d a t i o n i n m o d u l i s h o u l d be c o n s i d e r e d f o r p r o p e r a n a l y s i s . TABLE IV C a l c u l a t i o n of E q u i v a l e n t Number of U n i f o r m C y c l e s Wave H e i g h t H ± ( f t ) Number of Waves Nwi Wave P e r i o d TJL ( s e e s ) Wave L e n g t h L ± ( f t ) Wave P r e s s u r e P ± ( p s f ) Shear S t r e s s R a t i o a t z=0 Number of C y c l e s t o L i q u e f . N ^ i E q u i v a l e n t No. of C y c l e s N e q = N w i / % i - N r e f 9 50 7.0 130.7 245.9 0.202 3.0 61.0 8 80 6.5 120.3 212.8 0.190 3.66 80.0 6 155 6.0 109.9 154.2 0.151 7.79 72.8 4 180 5.0 88.6 92.5 0.112 229.5 2.9 2 200 4.0 66.6 37.4 0.059 100,000. 0.01 TABLE V R e s u l t s o f R e s i d u a l P o r e P r e s s u r e A n a l y s i s K z = 0.002 cm/s K z = 0. 2 cm/ s w i t h o u t m o d i f i c a t i o n w i t h modif i c a t i o n w i t h o u t m o d i f i c a t i o n w i t h mod i f i c a t i o n Maximum R e s i d u a l P o r e P r e s s u r e R a t i o 0.34 0.46 0.012 0.012 Maximum Depth o f I n s t a b i l i t y ( f t ) 3.0 5.0 0.0 0.0 69 The d r a i n a g e c h a r a c t e r i s t i c s of t h e sands a r e governed m a i n l y by c o e f f i c i e n t c o n s o l i d a t i o n c v g i v e n by Kz/(va^y^) . The h i g h e r t h e c v t h e l o w e r t h e r e s i d u a l p o r e p r e s s u r e d i s t r i b u t i o n w i l l be. I t i s i n t e r e s -t i n g t o n o t e t h a t an a n a l y s i s based on u n d r a i n e d c o n d i t i o n s p r e v a i l i n g d u r i n g t h e s t o r m a c t i v i t y w i l l i n d i c a t e t h a t t h e d e p o s i t would l i q u e f y t o a d e p t h of 10 f t . A c o v e r of a few f e e t of h i g h c v m a t e r i a l ( g r a v e l ) w i l l r e d u c e t h e p o r e p r e s s u r e d r a m a t i c a l l y . 70 FIG. 16 R e s i d u a l P o r e P r e s s u r e D i s t r i b u t i o n 71 CHAPTER 6 ANALYSIS OF PIPELINE FLOTATION A n u m e r i c a l example p r o b l e m i s c o n s i d e r e d t o e x p l a i n t h e p r i n -c i p l e s i n Cha p t e r 2. A p r e - s t r e s s e d c o n c r e t e p i p e l i n e o f 4 f t d i a m e t e r i s b u r i e d a t a d e p t h of 8 f t below t h e s e a f l o o r . The p r o p e r t i e s o f t h e d e p o s i t a r e shown i n F i g . 8. The p i p e l i n e i s t o be used t o t r a n s m i t a f l u i d of s p e c i f i c g r a v i t y S Q = 0.70 and i t weighs q = 80 l b f / f t r u n . The r e s p o n s e o f t h i s d e p o s i t u s i n g r e s i d u a l p o r e p r e s s u r e a n a l y -s i s t o a g i v e n d e s i g n storm i s p r e s e n t e d i n S e c t i o n 5.4. The maximum d e p t h o f i n s t a b i l i t y , , w i l l o c c u r when t h e extreme wave p a s s e s o v e r w h i l e t h e bed has h i g h e s t r e s i d u a l p o r e p r e s s u r e s . The p r e s e n c e o f r e s i -d u a l p o re p r e s s u r e s r e d u c e t h e s t i f f n e s s p r o p e r t i e s of t h e d e p o s i t arid l e a d t o h i g h e r d e p t h o f i n s t a b i l i t y . The program,- STAB-MAX, was used t o a n a l y s e t h e r e s p o n s e of t h e d e p o s i t f o r t h e extreme wave l o a d i n g w i t h p r o p e r t i e s m o d i f i e d f o r r e s i d u a l p o r e p r e s s u r e . T h i s a n a l y s i s gave a v a l u e o f 17.5 f t f o r Df. T h e r e f o r e , a n a l y s i s of t h i s p i p e l i n e c o r r e s p o n d s t o Case I I , s t a t e d i n S e c t i o n 2.3, where D^ < Df. Then, u s i n g e q u a t i o n (2.7) f o r no upward movement IL < W + W b p s where U b = t h e buoyancy f o r c e p er u n i t l e n g t h on t h e p i p e l i n e . From e q u a t i o n ( 2 . 4 ) , U b = | B ( l ^ - U p Buoyancy f o r c e U>, U2 = maximum t o t a l p o r e p r e s s u r e ( s t a t i c and r e s i d u a l ) a t t h e bottom of t h e p i p e l i n e ; 72 = 10 x 64 + 0.263 x ( o ^ o ) z = 1 0 i = 765.2 l f b / f t 2 U-^  = maximum t o t a l p o r e p r e s s u r e a t t h e t o p of t h e p i p e l i n e ; = 6 x 64 + 0.370 x ( o ^ 0 ) z = 6 , = 489.7 l b f / f t 2 U b = x 4 x (765.2 -489.7) = 865.5 l b f / f t r u n Weight o f p i p e l i n e and i t s c o n t e n t s (Wp) «P - 1 + 4^ Here, Y j = u n i t w e i g h t of f l u i d = S Q x 62.4 l b f / f t 3 W • = 80 + T x 0.7 x 62.4 x 16 P 4 = 628.9 l b f / f t r u n C o r r e c t e d b r e a k o u t s o i l mass Ws U s i n g e q u a t i o n (2.2) n (W ) = I (Y' - Y i ) V s ave | w r r where ( Y ' - Y w i ) r = a v e r a g e e f f e c t i v e u n i t w e i g h t of r l a y e r ; V r = a v e r a g e volume i n v o l v e d i n r l a y e r ; n = t o t a l number of l a y e r s ; i = h y d r a u l i c g r a d i e n t = d h s / d z ; h s = h y d r a u l i c head = u / Y W « For t h i s example p r o b l e m , f r o m T a b l e VI ( W s ) a v e = 789.8 l b f / f t r u n T h e r e f o r e , t h e f a c t o r o f s a f e t y a g a i n s t f l o t a t i o n TABLE VI C a l c u l a t i o n o f C o r r e c t e d B r e a k o u t S o i l Mass Depth'Below S e a f l o o r E x c e s s P o r e P r e s s u r e R a t i o u/-<40 a' vo H y d r a u l i c Head h s ( f t ) = u / y w i = d h s / d z ( Y ' - Y w i ) Volume V r f t 3 ( Y ' - Y w i ) V r 2 0.464 '95.2 0.690 0.345 25.5 8 204.0 4 0.421 190.4 1.25 0.280 29.7 8 237.6 6 0.370 285.6 1.65 0.20 34.8 8 278.4 8 0.315 380.8 1.87 0.110 40.6 1.72 69.8 ( W s ) a v e = 7 8 9 . 8 74 s t a b i l i z i n g f o r c e d i s t u r b i n g f o r c e W + W _E s U b 628.9 + 789.8 865.5 = 1.64 I t s h o u l d be n o t e d t h a t t h e f a c t o r o f s a f e t y would have a v a l u e of 1.82 i f s o f t e n i n g e f f e c t o f t h e d e p o s i t has n o t been c o n s i d e r e d and 4.19 i f a n a l y s i s was performed assuming c a l m sea s u r f a c e c o n d i t i o n s . CHAPTER 7 CONCLUSIONS T r a v e l l i n g s u r f a c e waves e x e r t harmonic p r e s s u r e l o a d i n g on t h e s e a f l o o r . T h i s p r e s s u r e l o a d i n g has two e f f e c t s on t h e d e p o s i t . I t i n d u c e s a harmonic s t r e s s f i e l d and i t may g i v e r i s e t o r e s i d u a l p o r e p r e s s u r e s i n t h e d e p o s i t . The d e p t h of i n s t a b i l i t y can be e s t i m a t e d f o r a g i v e n wave l o a d i n g u s i n g a f a i l u r e c r i t e r i o n s u c h as Mohr-Coulomb i f t h e e f f e c t i v e s t r e s s f i e l d i n t h e d e p o s i t i s known. The most s e v e r e c o n d i t i o n f o r i n s t a b i l i t y w i l l o c c u r when t h e seabed has t h e maximum r e s i d u a l p o r e p r e s s u r e s and t h e extreme wave p a s s e s by. The e f f e c t i v e s t r e s s f i e l d can be o b t a i n e d . e i t h e r by a c o u p l e d a n a l y s i s based on B i o t ' s t h e o r y o f p o r o - e l a s t i c s o l i d o r by an a p p r o p r i a t e u n c o u p l e d a n a l y s i s . I n t h i s t h e s i s , a c o u p l e d a n a l y s i s f o r a h o r i z o n t a l l y l a y e r e d d e p o s i t o f f i n i t e d e p t h i s p r e s e n t e d . A computer program, STAB-MAX, has been d e v e l o p e d t o implement t h e a n a l y s i s . STAB-MAX g i v e s p o r e p r e s s u r e s and e f f e c t i v e s t r e s s e s i n t h e d e p o s i t f o r a g i v e n wave l o a d i n g . For deep, u n i f o r m , s a t u r a t e d d e p o s i t s , an u n c o u p l e d e f f e c t i v e s t r e s s f i e l d c a n be computed by i n d e p e n d e n t l y e v a l u a t i n g t o t a l s t r e s s and p o r e p r e s s u r e s i n t h e d e p o s i t and t h e n u s i n g e f f e c t i v e s t r e s s p r i n c i p l e . The u n c o u p l e d a n a l y s i s i s s i m p l e and does n o t r e q u i r e a computer. Based on d a t a from an e x t e n s i v e s t u d y u s i n g t h e program, STAB-MAX, F i n n , S i d d h a r t h a n and M a r t i n (1980) e s t a b l i s h e d g u i d e l i n e s f o r u n c o u p l e d a n a l y s e s . They showed t h a t t h e t o t a l s t r e s s i n c r e m e n t s f o r a g i v e n wave l o a d i n g can be o b t a i n e d u s i n g t h e t h e o r y of e l a s t i c i t y (Fung, 1965). They found t h e p o r e p r e s s u r e f i e l d t o be i ndependent of h y d r a u l i c a n i s o t r o p h y f o r s o f t 76 c o a r s e and f i n e sands. I t can be e s t i m a t e d u s i n g -Xz p = p e r r o I n s t i f f c o a r s e sand, t h e d i s t r i b u t i o n of p o r e p r e s s u r e i s a f u n c t i o n of t h e a n i s o t r o p i c p e r m e a b i l i t y r a t i o and c a n be computed u s i n g - A /K x/K z Z p = p e x ^ The d i s t r i b u t i o n i n s t i f f f i n e sands i s independent o f t h e a n i s o t r o p i c p e r m e a b i l i t y r a t i o , but i t c o u l d n o t be e x p r e s s e d i n s i m p l e f u n c t i o n a l form. The g e n e r a l i s a t i o n made f o r deep, u n i f o r m , s a t u r a t e d d e p o s i t s can not be made f o r s h a l l o w o r l a y e r e d d e p o s i t s . So, f o r a s h a l l o w o r l a y e r e d d e p o s i t one s h o u l d do a c o u p l e d a n a l y s i s t o d e t e r m i n e t h e e f f e c t i v e s t r e s s e s i n t h e d e p o s i t , u s i n g a program s u c h as STAB-MAX. The r e s i d u a l p o r e p r e s s u r e s caused by c y c l i c a c t i o n o f t h e wave i n d u c e d shear s t r e s s e s i s v e r y i m p o r t a n t i n l o o s e sandy s o i l s . The e v a l u a -t i o n o f r e s i d u a l p o r e p r e s s u r e s s h o u l d t a k e i n t o a c c o u n t p o r e p r e s s u r e d i s s i p a t i o n because o f t h e r a t h e r l o n g d u r a t i o n of most storm wave l o a d -i n g s . An example c o n s i d e r e d i n t h i s t h e s i s r e v e a l s t h a t t h e m o d i f i c a t i o n of s o i l p r o p e r t i e s f o r t h e p r e s e n c e o f r e s i d u a l p o r e p r e s s u r e s , g i v e about 30% h i g h e r r e s i d u a l p o r e p r e s s u r e s t h a n t h a t o b t a i n e d w i t h o u t t h e m o d i f i -c a t i o n . T h e r e f o r e , s t a b i l i t y a n a l y s e s s h o u l d i n c l u d e m o d i f i c a t i o n o f s o i l p r o p e r t i e s f o r t h e i n c r e a s e i n r e s i d u a l p o r e p r e s s u r e s . A computer program, STAB-W, has been d e v e l o p e d t o e v a l u a t e t h e r e s i d u a l p o r e p r e s s u r e s i n a l a y e r e d d e p o s i t o f f i n i t e d e p t h t a k i n g t h e e f f e c t s o f p o r e p r e s s u r e s on s o i l p r o p e r t i e s i n t o a c c o u n t . The c o v e r p r o v i d e d f o r a b u r i e d p i p e l i n e w i t h p o s i t i v e buoyancy s h o u l d be such t h a t f a i l u r e does n o t o c c u r w i t h i n t h e c o v e r and i t s h o u l d p r o v i d e a d equate r e s i s t a n c e t o p i p e l i n e f l o t a t i o n d u r i n g wave l o a d i n g . 77 E f f e c t i v e w e i g h t o f t h e s o i l mass (W s) and v e r t i c a l component of t h e shear f o r c e (RyO a l o n g t h e s l i p s u r f a c e a r e t h e two f o r c e components t h a t r e s i s t upward movement of a b u r i e d p i p e l i n e . There c a n be a r e g i o n i n s t a b i l i t y i n t h e seabed w i t h i n w h i c h t h e a p p l i e d s t r e s s e s caused by t h e wave l o a d i n g i s h i g h e r t h a n t h e s t r e n g t h o f t h e d e p o s i t . T h i s means t h a t any a d d i t i o n a l s h e a r r e s i s t a n c e c a n n o t be m o b i l i s e d a l o n g a s l i p s u r f a c e s e l e c t e d f o r p i p e l i n e b r e a k o u t p r o b l e m w i t h i n t h i s r e g i o n . The i n s t a b i l i t y r e g i o n w i l l be deeper when t h e d e p o s i t has h i g h e r r e s i d u a l p o r e p r e s s u r e s as t h e extreme wave p a s s e s by. The r e g i o n of i n s t a b i l i t y c a n be o b t a i n e d by a n a l y s i n g t h e r e s p o n s e o f t h e d e p o s i t w i t h i t s p r o p e r t i e s m o d i f i e d f o r t h e p r e s e n c e o f h i g h e r r e s i d u a l p o r e p r e s s u r e s and t h e n u s i n g t h e p o r o -e l a s t i c method of i n s t a b i l i t y a n a l y s i s . F u r t h e r m o r e , t h e p r e s e n c e of r e s i d u a l p o r e p r e s s u r e s may g i v e r i s e t o an i n c r e a s e i n p o s i t i v e bouyancy and r e d u c e t h e e f f e c t i v e mass (W s) of t h e d e p o s i t due t o t h e upward seepage. An a p p r o p r i a t e example has been c o n s i d e r e d i n d e t a i l t o e x p l a i n how t h e p r i n c i p l e s o u t l i n e d i n t h i s t h e s i s a r e used t o a n a l y s e p i p e l i n e f l o t a t i o n p r o b l e m s . The a n a l y s i s p r e s e n t e d h e r e i s r e s t r i c t e d t o h o r i z o n t a l l y l a y e r e d d e p o s i t . B u t t h e same a p p r o a c h can be extended t o a n a l y s e s l o p i n g seabeds. I n t h e s e c a s e s , t h e i n f l u e n c e o f s t a t i c s h e a r s t r e s s e s on t h e f a i l u r e c r i t e r i o n and on p o r e p r e s s u r e g e n e r a t i o n c h a r a c t e r i s t i c s of t h e d e p o s i t must be c o n s i d e r e d . I n t h i s c a s e , p o s s i b l e d i s p l a c e m e n t o f t h e d e p o s i t , d u r i n g and a f t e r t h e wave l o a d i n g needs t o be c o n s i d e r e d . The p r e s e n c e o f s t r u c t u r e s d i s t o r t s t h e p o r e p r e s s u r e f i e l d . T h i s e f f e c t has n o t been c o n s i d e r e d h e r e and may r e s u l t i n an i n c r e a s e i n bouyancy f o r c e s . Q u a n t i t a t i v e v e r i f i c a t i o n o f t h e p o r o e l a s t i c method has been r e p o r t e d . The p o r e p r e s s u r e d i s t r i b u t i o n p r e d i c t e d by t h e t h e o r y was found t o be i n good agreement w i t h t h e f i e l d measurements. 78 REFERENCES 1. ASCE P r e l i m i n a r y R e s e a r c h on P i p e l i n e F l o t a t i o n ( 1 9 6 6 ) , "Report of t h e P i p e l i n e F l o t a t i o n R e s e a r c h C o u n c i l , " J o u r n a l of t h e  P i p e l i n e D i v i s i o n , ASCE, V o l . 92, No. P L 1 , P r o c . 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( 1 9 7 4 ) , " L a r g e Diameter Underwater P i p e l i n e f o r N u c l e a r Power P l a n t D e s i g n e d A g a i n s t S o i l L i q u e f a c t i o n , " S i x t h A n n u a l O f f s h o r e T e c h n o l o g y  C o n f e r e n c e , H ouston, Texas, Paper No. OTC 2094, May, pp. 597-606. 7. C o r n f o r t h , D.H. ( 1 9 7 3 ) , " P r e d i c t i o n of D r a i n e d S t r e n g t h of Sands from R e l a t i v e D e n s i t y Measurements," ASTM STP 523. E v a l u a t i o n of r e l a t i v e d e n s i t y and i t s r o l e i n g e o t e c h n i c a l p r o j e c t s i n v o l v i n g c o h e s i o n l e s s s o i l s , pp. 281-303. 8. C r o s s , R., Huntsman, S.R., T r e a d w e l l , D.D. and B a k e r , V.A. ( 1 9 7 9 ) , " A t t e n u a t i o n o f Wave-Induced P o r e P r e s s u r e s i n Sand," P r o c e e d i n g s , 4 t h I n t e r n a t i o n a l C o n f e r e n c e on C i v i l E n g i n e e r i n g i n t h e Oceans, San F r a n c i s c o , pp. 745-757. 9. 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( 1 9 7 0 ) , "The R o l e o f Waves i n C a u s i n g Submarine L a n d s l i d e s , " G e o t e c h n i q u e 20, No. 1, pp. 75-80. 17. H e y b i c h , J.B., Hugh, D.M. and B r i a n , V.W. ( 1 9 6 5 ) , "Scour o f F l a t Sand Beaches Due t o Wave A c t i o n i n F r o n t o f Sea W a l l s , " P r o c e e d i n g s , S p e c i a l t y C o n f e r e n c e on C o a s t a l E n g i n e e r i n g , S a n t a B a r b a r a , O c t o b e r , pp. 705-728. 18. Lambe, W.T. and Whitman, R.V. ( 1 9 7 9 ) , S o i l M e c h a n i c s , S I V e r s i o n , John W i l e y & Sons, I n c . , New Y o r k . 19. Lee, K.L. and Chan, K. ( 1 9 7 2 ) , "Number of E q u i v a l e n t S i g n i f i c a n t C y c l e s i n S t r o n g M o t i o n E a r t h q u a k e s , " P r o c e e d i n g s , I n t e r n a t i o n a l  C o n f e r e n c e on M i c r o z o n a t i o n , S e a t t l e , V o l . I I , pp. 609-627. 20. Lee, K.L. and A l b e i s a , A. ( 1 9 7 4 ) , " E a r t h q u a k e Induced S e t t l e m e n t s i n S a t u r a t e d Sands," J o u r n a l o f t h e S o i l M e c h a n i c s and F o u n d a t i o n s  D i v i s i o n , ASCE, V o l . 100, No. GT4, P r o c . Paper 10496, A p r i l , pp. 387-406. 21. Lee, K.L. and F o c h t , J.A. ( 1 9 7 5 ) , " L i q u e f a c t i o n P o t e n t i a l a t E k o f i s k Tank i n N o r t h Sea," J o u r n a l o f t h e G e o t e c h n i c a l E n g i n e e r i n g  D i v i s i o n , ASCE, No. GT1, P r o c . Paper 11054, pp. 1-18. 80 22. L i u , P.L.F. ( 1 9 7 3 ) , "Damping of Water Waves Over P o r o u s Bed," J o u r n a l o f t h e H y d r a u l i c s D i v i s i o n , ASCE, V o l . 99, No. HY12, P r o c . Paper 10218, December, pp. 2263-2271. 23. L i u , P.L.F., Timothy, P. and O ' D o n n e l l , ( 1 9 7 9 ) , Wave Induced F o r c e s on B u r i e d P i p e l i n e s i n Permeable Seabeds," P r o c e e d i n g s , 4 t h C o n f e r e n c e on C i v i l E n g i n e e r i n g i n t h e Oceans, San F r a n c i s c o , pp. 111-121. 24. McPherson, H. ( 1 9 7 8 ) , "Wave F o r c e s on P i p e l i n e B u r i e d i n Permeable Seabed," J o u r n a l o f t h e Waterway P o r t C o a s t a l and Ocean D i v i s i o n , WW4, November, pp. 407-419. 25. Madsen, O.S. ( 1 9 7 8 ) , "Wave Induced P o r e P r e s s u r e s and E f f e c t i v e S t r e s s e s i n a Po r o u s Bed," G e o t e c h n i q u e 28, No. 4, December, pp. 377-393. 26. M a r t i n , G.R., F i n n , W.D. Liam and Seed, H.B. ( 1 9 7 5 ) , "Some Funda-m e n t a l A s p e c t s i n L i q u e f a c t i o n Under C y c l i c L o a d i n g , " J o u r n a l o f  t h e G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 101, No. GT5, May, pp. 423-438. 27. Masch, F.D. and W i e g e l , R.L. ( 1 9 6 1 ) , " C n o i d a l Waves: T a b l e s of F u n c t i o n s , " C o u n c i l o f Wave R e s e a r c h , The E n g i n e e r i n g F o u n d a t i o n , Richmond, C a l i f . 28. M a s s e l , S.R. ( 1 9 7 6 ) , " G r a v i t y Waves P r o p a g a t e d Over Permeable Bottom," J o u r n a l o f Waterways, Harbour and C o a s t a l E n g i n e e r i n g , ASCE, V o l . 102, No. WW2, May, pp. 111-121. 29. Moshagen, H. and T^rum, A. ( 1 9 7 5 ) , "Wave Induced P r e s s u r e s i n Permeable Sea Beds," J o u r n a l o f Waterways, Harbour and C o a s t a l  E n g i n e e r i n g , ASCE, V o l . 101, No. WWl, F e b r u a r y , pp. 49-58. 30. N a t a r a j a , M.S., Wong, I.H. and T o t o , J.V. ( 1 9 7 8 ) , " L i q u e f a c t i o n P o t e n t i a l a t an Ocean O u t f a l l i n P u e r t o R i c o , " P r o c e e d i n g s , ASCE  G e o t e c h n i c a l C o n f e r e n c e on E a r t h q u a k e E n g i n e e r i n g and S o i l Dynamics, Pasadena, C a l i f . , June, V o l . I I , pp. 685-703. 31. N a t a r a j a , M.S. and S i n g h , H. ( 1 9 7 9 ) , " S i m p l i f i e d P r o c e d u r e f o r Ocean Wave-Induced L i q u e f a c t i o n A n a l y s i s , " P r o c e e d i n g s , 4 t h C o n f e r e n c e  on C i v i l E n g i n e e r i n g i n t h e Oceans, San F r a n c i s c o , pp. 948-963. 81 32. P h i l l i p s , B.A., G h a z z a l y , O.I. and K a l a j i a n , E.H. ( 1 9 7 9 ) , " S t a b i l i t y of P i p e l i n e i n Sand Under Wave P r e s s u r e , " P r o c e e d i n g s , 4 t h C o n f e r e n c e  on C i v i l E n g i n e e r i n g i n t h e Oceans, San F r a n c i s c o , pp. 122-136. 33. Putnam, J.A. ( 1 9 4 9 ) , " L o s s of Wave Energy Due t o P e r c o l a t i o n i n a Permeable Sea Bottom," T r a n s a c t i o n s , A m e r i c a n G e o p h y s i c a l U n i o n , V o l . 30, No. 5, June, pp. 349-356. 34. Rahman, M.S., Seed, H.B. and B o o k e r , J.R. ( 1 9 7 7 ) , " P o r e P r e s s u r e Development Under O f f s h o r e G r a v i t y S t r u c t u r e s , " J o u r n a l of t h e  G e o t e c h n i c a l D i v i s i o n , ASCE, V o l . 103, No. GT12, P r o c . Paper 13411, December, pp. 1419-1436. 35. Reese, L.C. and C a s b a r i a n , A.O.P. ( 1 9 6 8 ) , " P i p e S o i l I n t e r a c t i o n f o r a B u r i e d O f f s h o r e P i p e l i n e , " S o c i e t y of P e t r o l e u m E n g i n e e r s  J o u r n a l , Paper No. SPE 2343. 36. R e i d , R.O. and K a j i u r a , K. ( 1 9 5 7 ) , "On t h e Damping of G r a v i t y Waves Over a Permeable Sea Bed," T r a n s a c t i o n s , A m e r i c a n G e o p h y s i c a l U n i o n , V o l . 30, No. 5, O c t o b e r , pp. 662-666. 37. S a t o , S., Tanaka, N. and I r i e , I . ( 1 9 6 8 ) , "Study on S c o u r i n g a t t h e Feed o f C o a s t a l S t r u c t u r e s , " P r o c e e d i n g s , 1 1 t h C o n f e r e n c e on  C o a s t a l E n g i n e e r i n g , pp. 579-598. 38. Sen, A.K. ( 1 9 7 1 ) , " F l u i d - D y n a m i c E f f e c t s on t h e Response of O f f s h o r e Towers t o Wave and E a r t h q u a k e F o r c e s , " M.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h C o l u m b i a , Vancouver, J u l y . 39. Seed, H.B. and I d r i s s , I.M. ( 1 9 7 0 ) , " S o i l M o d u l i and Damping F a c t o r s f o r Dynamic Response A n a l y s i s , " EERI~ R e p o r t No. 70-10, C o l l e g e 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 , December. 40. Seed, H.B., M a r t i n , P.P. and Lysmer, J . ( 1 9 7 6 ) , " P o r e - w a t e r P r e s s u r e Changes D u r i n g S o i l L i q u e f a c t i o n , " J o u r n a l of t h e G e o t e c h n i c a l  E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 102, No. GT4, P r o c . Paper 12074, A p r i l , pp. 323-346. 41. Seed, H.B. and Rahman, M.S. ( 1 9 7 7 ) , " A n a l y s i s f o r Wave-Induced L i q u e f a c t i o n i n R e l a t i o n t o Ocean F l o o r S t a b i l i t y , " R e p o r t on  r e s e a r c h s p o n s o r e d by t h e N a t i o n a l S c i e n c e F o u n d a t i o n , R e p o r t No. UCB/TE-77/02, May. 82 42. Seed, H.B. ( 1 9 7 9 ) , " S o i l L i q u e f a c t i o n and C y c l i c M o b i l i t y E v a l u a -t i o n f o r L e v e l Ground D u r i n g E a r t h q u a k e s , " J o u r n a l of t h e G e o t e c h - n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, V o l . 105, No. GT2, P r o c . Paper . 14380, F e b r u a r y , pp. 201-255. 43. U.S. Army C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r ( 1 9 7 7 ) , "Shore P r o t e c t i o n Manual," V o l . I , Ch. 2, T h i r d E d i t i o n . 44. V e r r u i j t , A. ( 1 9 6 9 ) , " E l a s t i c S t o r a g e of A q u i f e r s , " Flow Through  P o r o u s M e d i a , Chap. 8, R.J.M. De W i e s t , E d i t o r , Academic P r e s s P u b l i s h e r s , New Y o r k . 45. V e s i c , A.S. ( 1 9 7 1 ) , " B r e a k o u t R e s i s t a n c e of O b j e c t s Embedded i n Ocean Bottom," J o u r n a l of t h e S o i l M e c h a n i c s and F o u n d a t i o n s D i v i s i o n , ASCE, V o l . 97, No. SM9, P r o c . Paper 8372, September, pp. 1183-1205. 46. W i e g e l , R.L. ( 1 9 6 4 ) , " O c e a n o g r a p h i c a l E n g i n e e r i n g , " P r e n t i c e - H a l l , I n c . , Englewood C l i f f s , New J e r s e y . 47. W r i g h t , S.G. and Dunham, R.S. ( 1 9 7 2 ) , "Bottom S t a b i l i t y Under Wave Induced L o a d i n g , " 4 t h A n n u a l O f f s h o r e T e c h n o l o g y C o n f e r e n c e , V o l . I , Paper No. OTC 1603, pp. 853-862. 48. Yamamoto, T. ( 1 9 7 8 ) , "Sea Bed I n s t a b i l i t y from Waves," P r o c e e d i n g s , 1 0 t h A n n u a l O f f s h o r e T e c h n o l o g y C o n f e r e n c e , H o u s t o n , Texas, Paper No. 3262, V o l . I , pp. 1819-1824. 83 APPENDIX I FORMULATION OF FINITE ELEMENT EQUATIONS FOR  ANALYSIS OF RESIDUAL PORE PRESSURES The g o v e r n i n g e q u a t i o n i s K 3u 3 , z 3u> ,3u g. H (7~ ^  = mv ( ^ " w ( A . I - l ) At any i n s t a n t o f t i m e , ( 3 u / 3 t - 3 u g / 3 t ) may be c o n s i d e r e d a s b e i n g a f u n c t i o n o f z o n l y . Now, say m „ (• 9u g_ ) = Q(z) v v 3 t 3t the n e q u a t i o n ( A ; I - 1 ) r e d u c e s t o '3 , K z 3vu , v ^ = Q ( z ) w A . I - l C o n s t r u c t i n g F u n c t i o n a l F u n c t i o n a l J f o r a d i f f e r e n t i a l e q u a t i o n o f t h e fo r m Au = f i s J = (Au,u) - 2 ( f , u ) U s i n g t h i s , t h e f u n c t i o n a l J f o r t h i s d i f f e r e n t i a l e q u a t i o n i s D 0 ^ ( f | ) U ' 2 Q ( Z ) u } d Z w r K z 3 u i° .y 3z •w r D o w 4 The boundary c o n d i t i o n s a t t h e t o p and bottom o f t h e s o i l d e p o s i t a r e u = 0 a t z = 0 3u 3z 0 a t z = D (No f l o w o c c u r s a t t h e bottom boundary) 84 Under t h e s e c i r c u m s t a n c e s t h e boundary term i n t h e f u n c t i o n a l v a n i s h e s . Then, ? K 2 - J = — A dz + 2 Q(z) udz (A.1-2) Y 9z 0 w A.1-2 F i n i t e Element T e c h n i q u e L e t u s c o n s i d e r t h a t t h e e n t i r e s o i l d e p o s i t as an assemblage of a f i n i t e number of e l e m e n t s . Then, ^ ^ t o t a l n i ^ ^ e l e m e n t s ( A . I 3) a l l e l e m e n t s To u s e t h i s method ( f i n i t e element) one must s e l e c t an i n t e r p o l a t i o n f u n c t i o n f o r u such t h a t t h e f u n c t i o n a l c a n be computed. S i n c e t h e f u n c -t i o n a l has 9u/9z term, t h e i n t e r p o l a t i o n f u n c t i o n f o r u must be a t l e a s t o r d e r 1 i n z. T h i s c r i t e r i o n i s r e f e r r e d t o as " c o m p l e t e n e s s c r i t e r i o n " i n f i n i t e element l i t e r a t u r e . So, l e t us choose t h e s i m p l e l i n e a r v a r i a -t i o n o f u o v e r an element as u - ( 1 -i7> u i + i : u i + i ( A - x - 4 ) i i = N T u 6 (A.1-5) - l where d^ i s t h e t h i c k n e s s of t h e i t ^ 1 element, e u . -x u. u i + l ; u i ' u i + l = t* i e r e s i d u a l p o r e p r e s s u r e s a t nodes i and i + 1 . These node numbers a r e a s s o c i a t e d w i t h element i . n = d i s t a n c e measured f r o m t h e t o p of t h e element p a r a l l e l t o z d i r e c t i o n . rn N = .{1 - n/d^ n/cL] = n o d a l i n t e r p o l a t i o n f u n c t i o n . Now, from e q u a t i o n (A.1-2) element = t h e f u n c t i o n a l f o r element 0 i K 2 -5- A dn +• 2Q(n)udn w say where e e = T + I 1 2 K z ,3u. 2 , ^ w ^ " w d . l and I , 2Q (n)udn Now c o n s i d e r 31 3{u*} ^ z ^ u ^ fl7 Y 3n 3n l l j w ^ ' dn S u b s t i t u t i n g f o r from e q u a t i o n ( A . 1 - 5 ) , 3 { u e } - l K u. . -u . z 1+1 x Y d. w 1 d. l _1_ d. K = 2 _ z J _ Y d. w 1 1 -1 2 [ K e ] {u?} where [ K e ] = m -m -m m ] and K Y d. w 1 86 Now c o n s i d e r d. x 2 Q(z) udn d. x 0 3u 2 m v (IE " 1^ ^ S i m p l e t r a p e z o i d a l i n t e g r a t i o n scheme can be used t o e v a l u a t e t h i s i n t e g r a l , i . e . , I * = 2m.. {•£[(• 1 r , 3U g. 9u v L 2 - 3 t - 3t ) u ] i + " T t ^ ) u ] i + 1 } d i Here, s u b s c r i p t i and i+1 r e f e r t o t h e v a r i a b l e s a t nodes i and i+1 i . e . , 3u = 2 31, 3 { u e } -x \ 3u 3t 3t ' i ) . = 2[D e] m d. . 3u v i f 3 u _ 2 Kdt 3t ; i + l . V 3u v 3 t 3t ' i J u _ _ j > v 3 t 3t 'i+1 where and p Now, 3J m d. v x element 3{u^} 3{u?} 3 { u 6 } - X - X (A.1-8) 87 from e q u a t i o n s (A.1-7) and (A.-8) •in &> [K ]{u?} + [D ] I v 3t 3t ' i ^ u _ | V 3 t 3t J±+l U s i n g v a r i a t i o n a l p r i n c i p l e s , # T = { 0 } 3 i u } 3J x. e. ^ element _ a l l 3 { u e } elements ( i E ) _ (_§.) v 3 f x v 3 t ; i X [K ] { u 6 } + [D ] -( . 3u f = {0} e x e J \ ._3u. _ , g. | ^ 3 t ; i + l ^ 3 t ; i + l Summing up f o r a l l t h e e l e m e n t s , one w i l l g e t a 3u [K]{u} + [ D ] ( { - ^ } - { ^ } ) = {0} (A.1-9) where m a t r i x [K] i s a c o n s t a n t m a t r i x f o r a g i v e n p r o b l e m , m a t r i x [D] i s a f u n c t i o n o f c o m p r e s s i b i l i t y m v, w h i c h w i l l v a r y w i t h p o r e p r e s s u r e r a t i o , and {u}, {3u/3t} and {3ug/3t} d e n o t e t h e v e c t o r s o f n o d a l v a l u e s o f 3u/3t and 3ug/3t. For nodes a t t h e i n t e r f a c e o f two d i f f e r e n t m a t e r i a l s , {3u / 3 t } i s t a k e n as t h e maximum of i t s v a l u e j u s t above and below t h e node. E q u a t i o n (A.1-9) can be r e g a r d e d as a s e t of o r d i n a r y d i f f e r e n -t i a l e q u a t i o n s and may be a p p r o x i m a t e l y i n t e g r a t e d o v e r t h e i n t e r v a l t , t+At as f o l l o w s : 3u [ K ] ( 3 ' { u t + A t + a ' { u t } ) A t + [ D ] ( { u t + A t > - " ^ > A t ) = <0> (A.I-10) 88 where a'+B 1 = 1 and s u b s c r i p t s t , t+At i n d i c a t e t h e v a l u e s of a v a r i a b l e a t t i m e s t , t+At r e s p e c t i v e l y , and t h e bar d e n o t e s an a v e r a g e v a l u e o v e r t h e i n t e r v a l t , t+At ( c a l c u l a t e d from t h e a v e r a g e p o r e p r e s s u r e r a t i o o v e r t h a t i n t e r v a l ) . D i f f e r e n t v a l u e s of a' c o r r e s p o n d t o d i f f e r e n t a p p r o x i m a t i o n s ; i f B'>0.5 t h e i n t e g r a t i o n by t h i s p r o c e d u r e i s a l w a y s s t a b l e . I n t h i s program, B 1 i s assumed t o have a v a l u e 0.5 ( C r a n k -N i c h o l s o n method). Then e q u a t i o n (A.I-10) c a n be w r i t t e n as [ A Q ] { u t + A t } = {BQ1} + {BQ2} (A.I-11) where [AQ] = [K]B'At + [D] [BQ1] = (-[K]a'At + [ D ] { u t } ) 3u [BQ2] = [ D ] { ^ } A t A f t e r i n c o r p o r a t i n g e s s e n t i a l boundary c o n d i t i o n s , e q u a t i o n (A.I-11) i s m o d i f i e d as [ A Q * ] { u t + A t } = {BQ1*} + {BQ2*} = {BQ*} (A.I-12) H e r e , t h e a s t e r i s k d e n o t e s v a l u e s of t h e v a r i a b l e s a f t e r s u i t a b l e m o d i f i -c a t i o n f o r t h e boundary c o n d i t i o n s . The above e q u a t i o n i s s o l v e d by t h e G a u s s i a n e l i m i n a t i o n p r o c e -d u r e . The program has t h e o p t i o n f o r t r e a t i n g t h e c o m p r e s s i b i l i t y v a l u e s e i t h e r c o n s t a n t o r v a r y i n g . I t i s assumed t h a t t h e c o m p r e s s i b i l i t y v a l u e s of t h e s o i l p r o f i l e does n o t v a r y w i t h p o r e p r e s s u r e r a t i o , t h e m a t r i x [D] i s c o n s t a n t and e q u a t i o n (A.I-12) can be used t o march t h e s o l u t i o n f o r w a r d i n t i m e . I f t h e c o m p r e s s i b i l i t y v a l u e s a r e t r e a t e d as v a r i a b l e s t h e n t h e m a t r i x [D] becomes v a r i a b l e and e q u a t i o n (A.I-12) has t o be s o l v e d i t e r a -t i v e l y by u s i n g t h e b e s t c u r r e n t e s t i m a t e of p o r e p r e s s u r e r a t i o t o c a l c u -l a t e [D] and r e p e a t i n g t h i s p r o c e d u r e u n t i l t h e p r o c e s s c o n v e r g e s . 89 APPENDIX I I CALCULATION OF CYCLIC SHEAR STRESSES IN AN ELASTIC  SEABED UNDER WAVE LOADING A h o r i z o n t a l l y l a y e r e d s o i l d e p o s i t s u b j e c t e d t o a t r a v e l l i n g h armonic p r e s s u r e i n c r e m e n t i s shown i n F i g . 17. The maximum c y c l i c s hear s t r e s s e s i n t h e d e p o s i t a r e r e q u i r e d i n o r d e r t o d e t e r m i n e t h e b u i l d - u p i n r e s i d u a l p o r e p r e s s u r e s . The f o l l o w i n g b a s i c a s s u m p t i o n s a r e used i n s o l v i n g f o r t h e s t r e s s e s . 1) The p e r i o d s o f i m p o r t a n t waves a r e l o n g enough t h a t t h e t r a n s i e n t r e s p o n s e t o p r e s s u r e l o a d i n g c a n be n e g l e c t e d and t h e wave p r e s s u r e s t r e a t e d as s t a t i c l o a d s . 2) Theory o f l i n e a r e l a s t i c i t y i s a p p l i c a b l e . 3) P l a n e s t r a i n c o n d i t i o n p r e v a i l s . 4) The s o i l d e p o s i t i s assumed t o be a con t i n u u m and no f l u i d f l o w t a k e s p l a c e i n t h e bed. These a s s u m p t i o n s r e d u c e t h e prob l e m t o a c l a s s i c a l s o l i d m e c h a n i c a l p r o b l e m . I n t r o d u c i n g an A i r y s t r e s s f u n c t i o n , <j>, t h e prob l e m o f d e t e r m i n i n g s t r e s -ses r e d u c e s t o s o l v i n g , 4 + 2 7 ^ r + ^ - 0 (A-II-1> 3x 3x 3z 3z L e t <J> = f ( z ) CosAx where A = 2ir/L and f ( z ) i s a f u n c t i o n o f z. S u b s t i t u -t i n g t h i s i n e q u a t i o n ( A . I I - 1 ) A 4 f ( z ) - 2A 2 f U ( z ) + f I V ( z ) = 0 The g e n e r a l s o l u t i o n o f t h i s e q u a t i o n i s f ( z ) = C i e A z + C 2 e _ A z + C 3 z e A z + ^ z e - * 2 ( A . I I - 2 ) -«a i L 'H »• V 1 I \ y s = 2 Sin 27r( r ) d 1 Ap = P 0 Sin 2 T T ( - L - ) Ist Layer (N- I ) t h Layer N t h Layer (N + I) th Layer ^ ^ ^ R i g i d Bottom ^ X z FIG. 17 Wave F o r c e s on a H o r i z o n t a l l y L a y e r e d D e p o s i t 91 where C^ = any a r b i t r a r y c o n s t a n t s t o be d e t e r m i n e d by a p p r o p r i a t e boundary c o n d i t i o n s , i . e . , <j> = ( C ^ 2 + C 2 e " A z + C 3 z e X z + C 4 z e ~ A z ) CosAx ( A . I I - 3 ) I n g e n e r a l , t h e s o i l d e p o s i t i s d i v i d e d i n t o a number of l a y e r s (NM). Then, t h e A i r y s t r e s s f u n c t i o n f o r t h e N l a y e r w i l l be of t h e f o r m :f v r Az , -Az , Az - i z , 4>(n) = [ c , ..e + c. „.e + c, „.ze + c, M z e JCosAx ( n , l ) (n,2) (n,3) (n,4) ( A . I I - 4 ) The a r b i t r a r y c o n s t a n t s c a n be found u s i n g t h e f o l l o w i n g boundary c o n d i -t i o n s . Y H (1) A t z=0, a = — C o s A x = p CosAx ( A . I I - 5 ) z z 2Cosh(Ad) o and a = 0 x z (2) z=D, v=w=0 where v,w a r e d i s p l a c e m e n t s i n x and z d i r e c t i o n s , r e s p e c t i v e l y . a v a , and a a r e g i v e n by X X 7 zz x z 2 2 2 3 <)> 3 <r 3 ( j ) / A T T C \ a =• — , a = — , a = - . ^ ( A . I I - 6 ) X X n 2 z z „ 2 x z 9x3z 9z 9x : S p e c i a l Case: U n i f o r m i n f i n i t e d e p o s i t . I f D -> °° t h e n t h e c o e f f i c i e n t s a s s o c i a t e d w i t h e ^ z terms s h o u l d v a n i s h . ± r , -Az. i . e . , <j> = ( c ^ e + C2ze )CosAx U s i n g boundary c o n d i t i o n s , i t c a n be e a s i l y p r o v e d t h a t t h e maximum c y c l i c s hear s t r e s s , x = (a ) c x z max Y H , w , -Az 2Cosh(Ad) -Az Aze ( A . I I - 7 ) p Aze o 92 I t i s i n t e r e s t i n g t o n o t e t h a t Yamamoto ( 1 9 7 8 ) , who p e r f o r m e d an e l a s t i c c o u p l e d a n a l y s i s , showed t h a t when G3 -> 0, x c f o r an i n f i n i t e t h i c k u n i f o r m d e p o s i t r e d u c e s t o t h e same e x p r e s s i o n . H e r e , G i s t h e shear modulus of s o i l and g i s t h e c o m p r e s s i b i l i t y of p o r e w a t e r . H o r i z o n t a l l y L a y e r e d D e p o s i t The A i r y s t r e s s f u n c t i o n f o r t h e n l a y e r i s . / \ r Az , —Az , Az —Az -, <Kn) = [ c , .,e + c, O N e + c. , z e + c. , . z e JCosAx ( n , l ) (n,2) (n,3) (n,4) The number of t o t a l a r b i t r a r y c o n s t a n t s a s s o c i a t e d w i t h a l l l a y e r s i s 4*NM and t h e y c a n be d e t e r m i n e d by s o l v i n g s i m u l t a n e o u s e q u a t i o n s s e t up i n t h e f o l l o w i n g manner. At t h e s e a f l o o r , z=0 Y W H CosAx and a = 0 (2 Eqns.) zz 2Cosh(Ad) xz At t h e i n t e r f a c e of t h e n t h and ( n + l ) t h l a y e r , s t r e s s e s o z z , 0 x z > w, v s h o u l d be c o m p a t i b l e . H e r e , i . e . , z z ( n ) a z z ( n + l ) x z ( n ) CTxz(n+l) (n) = W(n+1) (n) = V ( n + 1 ) T h i s l e a d s t o 4 e q u a t i o n s p e r i n t e r f a c e , A t t h e bottom, z = D, W(NM) " ° „ _ , (2 Eqns.) (NM) T o t a l number of e q u a t i o n s and unknowns i s 4ANM. These s i m u l t a n e o u s equa-t i o n s c a n be s o l v e d f o r t h e a r b i t r a r y c o n s t a n t s c ^ j ^ ; i=l,NM; j = l , 4 . The c y c l i c s h e a r s t r e s s a t a d e p t h z i s g i v e n by T • = a ( z ) c x z max A 2 Az .2 -Az , 2 , = c , ..A e - c , O N A e + c, , N l A z+A>£ ( n , l ) (n,2) (n,3) • . + c. ,\ {A-A^z }e - A z" (n,4) H e re, n i s t h e number of t h e l a y e r i n v o l v e d . 

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