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

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S T A B I L I T Y OF BURIED P I P E L I N E S  SUBJECTED  TO WAVE LOADING  by  RAJARATNAM  SIDDHARTHAN  B.S. ( H o n s . ) , U n i v e r s i t y o f S r i L a n k a , P e r a d e n i y a Campus, S r i L a n k a , 1977  A THESIS'SUBMITTED I N P A R T I A L FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE  in  THE FACULTY OF GRADUATE STUDIES Department o f C i v i l  We a c c e p t  Engineering  t h i s t h e s i s as  to the required  conforming  standard  THE U N I V E R S I T Y OF B R I T I S H  COLUMBIA  May 1981  (c)  Rajaratnam  Siddharthan,  1981  In p r e s e n t i n g requirements  this thesis  British  it  freely available  for  f u l f i l m e n t of the  f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y  of  agree that  in partial  Columbia,  I agree  f o r reference  permission  scholarly  that  the L i b r a r y  shall  and s t u d y .  I  f o r extensive  p u r p o s e s may  for  that  shall  permission.  Department o f  Date  of this  Iti s thesis  n o t be a l l o w e d w i t h o u t my  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Aprf H 8 I  thesis  be g r a n t e d by t h e h e a d o f my  copying or p u b l i c a t i o n  f i n a n c i a l gain  further  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood  make  Columbia  written  ABSTRACT  The  r e s p o n s e of b u r i e d  shown t o d e p e n d m a i n l y on  the  i n v e s t i g a t e the  of  Biot's  stability  t h e o r y of  offshore  stability the  a poro-elastic  field  the  transient  i n d u c e d by  e s t i m a t e d by  the  solid,  it  stresses and  can  be  For  Residual  the  and  bility typical  or  o t h e r on  the  the of  can  computing the  simple solutions  the  be u s e d . total  the  seafloor.  Two  of  the  deposit  The  c o m p u t e r p r o g r a m s , STAB-MAX and  to perform the  poro-elastic  and  can  of  the  Laplace  be  the sands,  elastically  equation.  computing the  resi-  a c t i o n o f wave i n d u c e d  shear  Because of  to  investigate  lead  the  taken into  to  insta-  duration  of  a storm  degradation i n e f f e c t i v e  i s also  stress  effective  storms, d i s s i p a t i o n of r e s i d u a l pore p r e s s u r e s d u r i n g in this analysis.  dis-  to  stress f i e l d  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  dependent p r o p e r t i e s  the  seafloor  The  on  residual  effective  the  to  i s based  gives  t h e Mohr-Coulomb f a i l u r e c r i t e r i o n  even l i q u e f a c t i o n of  been c o n s i d e r e d  analyses  except f o r hard f i n e  i s b a s e d on  cyclic  is  t h e o r y of  analysis  stability  pore pressure a n a l y s i s  then using  stability.  The  analysis  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 stresses  p r e s e n t e d ; one  a l l deep d e p o s i t s  d e t e r m i n e d by by  the  Two  t h e Mohr-Coulomb f a i l u r e c r i t e r i o n  a much s i m p l e r  pore pressure f i e l d  are  t o wave l o a d i n g  seafloor.  p o r e w a t e r p r e s s u r e s and  a p p l i c a t i o n of  i s shown t h a t  the  Poro-elastic  wave l o a d i n g .  computed s t r e s s f i e l d .  of  seafloor  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 . t r i b u t i o n of  pipelines  has  stress  account.  STAB-W, h a v e b e e n d e v e l o p e d  r e s i d u a l pore pressure analyses  respec-  tively. An illustrate stability  example problem i n p i p e l i n e s t a b i l i t y the  of  factors that  buried  offshore  s h o u l d be  considered  pipelines.  has  been p r e s e n t e d  i n the  analysis  of  to the  i i i  TABLE OF CONTENTS  Page  ABSTRACT  n  TABLE OF CONTENTS  i  L I S T OF FIGURES  i  i v  L I S T OF TABLES  i  viii  NOMENCLATURE  ix  ACKNOWLEDGEMENTS  x i i  CHAPTER 1  1  1.1  Introduction  1  1.2  Scope  2  1.3  L i t e r a t u r e Review  4  1.4  Organisation  6  CHAPTER 2  of the Thesis  GENERAL ASPECTS OF OFFSHORE P I P E L I N E S T A B I L I T Y  7  2.1  Statement of t h e Problem  8  2.2  Estimation  2.2.1  E f f e c t i v e bouyancy f o r c e U  2.2.2  E f f e c t i v e w e i g h t o f t h e mass o f s o i l  2.2.3  S h e a r r e s i s t a n c e c o m p o n e n t , Ry  12  2.2.4  Contribution of s o i l  a d h e s i o n , C^  12  2.2.5  Contribution of s o i l  suction force, P  2.3  Proposed A n a l y s i s f o r t h e P i p e l i n e  13  EVALUATION OF DESIGN STORM WAVE  18  3.1  Obtaining  19  3.2  Wave T h e o r i e s  CHAPTER 3  of the Forces  D e s i g n S t o r m Waves  10 10 Involved:  w  W  s  11  13  21  iv  Page  3.2. 1  L i n e a r wave t h e o r y  22  3.2. 2.  Stokes'  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  26  3.3  Equivalent Uniform  higher order t h e o r i e s  i nthis  thesis  26  S t o r m Wave  PORO-ELASTIC A N A L Y S I S  31  4.1  Introduction  31  4.2  Governing  Equations  31  4.3  Boundary C o n d i t i o n s  34  4.4  Solution  35  4.4..1  Horizontally  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  Values  39  4.7  Example  4.8  Comparison of Theory w i t h F i e l d  4.9  Some P r a c t i c a l  CHAPTER 4  Technique  37  layered deposit  forElastic  Constants  42  Problem  52  Data  55  Solutions  RESIDUAL PORE PRESSURE A N A L Y S I S  58  5.1  Theory  59  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  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 Generation  5.4  Example  5.4,.1  Equivalent uniform  5.4..2  Discussion of r e s u l t s  CHAPTER 5  Pressure  63  65  Problem storm  . 67 67  V  Page  CHAPTER  6.  CHAPTER 7  ANALYSIS OF P I P E L I N E FLOTATION  71  CONCLUSIONS  75  REFERENCES  78  APPENDIX I  83  APPENDIX I I  89  vi  L I S T 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  General S l i p S u r f a c e f o r Shallow Anchors  9  FIGURE  3  F l o w C h a r t t o C a l c u l a t e G e n e r a l S t o r m Wave D a t a a n d t h e E q u i v a l e n t S t o r m Wave S y s t e m  20  FIGURE  4  R e g i o n s o f V a l i d i t y o f Wave T h e o r i e s  23  FIGURE  5(a)  Modified Penetration vs x/a^  29  FIGURE  5(b)  E q u i v a l e n t Number o f C y c l e s  29  FIGURE  6  Wave P r e s s u r e s o n O c e a n F l o o r  32  FIGURE  7  The V a r i a t i o n Shear S t r a i n  40  FIGURE  8  S o i l D e p o s i t used i n Example Problem  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 1 0 ( a )  Porewater P r e s s u r e i n a Deposit of F i n i t e Depth  45  FIGURE 1 0 ( b )  Induced V e r t i c a l of F i n i t e D e p t h  46  FIGURE 1 0 ( 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 Depth  48  FIGURE 1 0 ( d )  Induced H o r i z o n t a l E f f e c t i v e Deposit o f F i n i t e Depth  49  FIGURE 11  C o n t o u r s o f D e v e l o p e d 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  Pore Pressure D i s t r i b u t i o n C a l i f o r n i a Coast  FIGURE 14  Rate of Pore Pressure Generation during Loading  FIGURE 1 5 ( 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 1 5 ( 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  0  o f Secant Shear Modulus w i t h  Effective  Stresses i n a Deposit  Stresses  ina  i n Sand D e p o s i t o f  Cyclic  54  62  vii  Page  FIGURE 16  R e s i d u a l Pore P r e s s u r e D i s t r i b u t i o n i n S o i l Deposit  70  FIGURE 17  Wave F o r c e s on a H o r i z o n t a l l y  90  Layered  Deposit  viii  L I S T OF TABLES  Page  TABLE  I  Gavity Breakthrough Factors  16  TABLE  II  D e s i g n Wave D a t a  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  IV  C a l c u l a t i o n of Equivalent Cycles  68  TABLE  V  TABLE  VI  i  Number o f U n i f o r m  Results of Residual Pore Pressure A n a l y s i s  68  C a l c u l a t i o n o f B r e a k o u t S o i l Mass  73  ix  NOMENCLATURE  B  =  diameter  =  b u l k modulus of t h e d e p o s i t  C„ N  =  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  C  =  adhesion  =  correction factor for liquefaction  =  cohesion  =  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  c^-*,c^  =  a r b i t r a r y constants  D  =  Thickness  D, b  =  d e p t h o f embedment o f t h e p i p e  =  depth of i n s t a b i l i t y  =  r e l a t i v e density of the deposit  d  =  mean d e p t h o f w a t e r  e  =  void  F ,F c' q  =  c a v i t y breakout  G  =  shear  =  shear modulus of t h e d e p o s i t a t low s t r a i n  H  =  wave h e i g h t  H  =  e q u i v a l e n t wave  i  =  seepage g r a d i e n t  =  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  =  lateral  L  =  wave l e n g t h  m  =  b u l k modulus  m  =  coefficient  B  m  C  a Y  c c  v  G  max  K ,K x  K  o  z  of the pipe  f o r c e between p i p e and  soil data  of the deposit  associated with the deposit  of the deposit  i n the deposit  ratio J  factors  modulus o f t h e d e p o s i t  height  earth pressure  at rest  constant of volume c o m p r e s s i b i l i t y  level  N  =  equivalent  =  number o f c y c l e s  =  corrected  =  porosity  =  total  =  over  =  atmospheric  =  soil  P  =  total  p  =  induced  p^  =  a m p l i t u d e o f wave p r e s s u r e o n t o p o f t h e s e a - b e d  q  =  weight  R  =  domain o f  R  =  vertical  eq  N N  P  n n  w  OCR  P  w  slip  number o f c y c l e s  wave  SPT v a l u e o f t h e sea-bed  number o f wave c o m p o n e n t s consolidation  i n t h e wave  ratio  pressure  suction static  force pore  pressure  transient  pore  of p i p e l i n e  pressure  per unit  length  integration component  of shear  resistance  along the  surface  g  =  shear r e s i s t a n c e  r  u  =  pore pressure  S  =  degree of  T  =  wave  =  total  =  bouyancy f o r c e  U^jU^  =  total  U  =  e f f e c t i v e bouyancy f o r c e  u  =  residual  Ug  =  p o r e p r e s s u r e due t o c y c l i c  v  =  displacement  in x  =  total  of p i p e l i n e  P  reference  to l i q u e f a c t i o n  R  W  of a given  along  slip  surface  ratio  saturation  period duration  o f .ithe  storm  per u n i t  length  pore p r e s s u r e a t t o p and bottom o f t h e p i p e l i n e  pore  weight  length  on t h e p i p e l i n e  per u n i t  length  pressure shear  stresses  direction including  i t s contents per u n i t  xi  W  =  e f f e c t i v e weight together  with  w  =  displacement  B  =  compressibility  a',B'  =  constants  Y  =  saturated  =  unit  -y'  =  effective unit  Y  =  defined  =  total volumetric  Y  g  w  M  e  e , e ,£ = x y z ir  normal in  mass i n v o l v e d  i n breakout  in z direction  unit  weight  of s o i l  the object  of porewater  weight  of the  deposit  of the  deposit  of water weight  a s (|' - y i ) w  strains  strain  i n x,y,z d i r e c t i o n s  defined  t o be  positive  elongation  d i s t a n c e measured empirical  in z direction  from  top of a  layer  constant  wave number = 2TT/L Poisson's vertical  vo  ratio effective  a',a' x z  =  wave i n d u c e d  x  =  cyclic  =  wave  A>, \ (x,y,t)  ~  friction  <f>'  =.  angle  c  T  shear  induced  stress  effective  stresses  stress shear  stresses  on x-z plane  (horizontal)  X2  angle developed  of i n t e r n a l  by s t r e s s  friction  state  i n an  element  xii  ACKNOWLEDGEMENTS  The Liam  Finn,  i s also  thesis Chapter  suggestions  indebted  to thank h i s a d v i s o r , P r o f e s s o r interest,  t o improve  to Professor  and f o r making  unfailing  help  W.D.  and f o r making  t h e presentation of t h e t h e s i s .  M. d e S t . Q .  Isaacson  c o n s t r u c t i v e suggestions  for reading  this  i n the preparation of  3. The  author  Cheung f o r t y p i n g author  wishes  f o rh i scontinued  many v a l u a b l e He  author  wishes  this  a pleasant The  California  t o M r s . 0. C u t h b e r t ,  and p e a c e f u l  financial  h i sgratitude to Miss  t h e s i s and f o r making v a l u a b l e  i s also grateful  providing  to express  home d u r i n g  assistance provided  i sgratefully  acknowledged.  Desiree  comments.  The  who was r e s p o n s i b l e f o r h i s stay  by F u g r o ,  i n Canada.  Inc.,  Long  Beach,  1  CHAPTER 1  1.1  Introduction The  s t a b i l i t y of the seafloor  installations, of  seafloor  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 i n f l u e n c e o f s e a f l o o r  offshore  installations,  al,  1974; N a t a r a j a ,  sea  walls  such as p i p e l i n e s  (Beckmann, 1970; C h r i s t i a n e t  1978), g r a v i t y s t r u c t u r e s  B e a , 1971) h a s b e e n e x t e n s i v e l y  ( L e e , 1 9 7 5 ; Rahman,  studied.  structures  s o i l m o v e m e n t s c a u s e d by w a v e s d u r i n g  i t may l e a d  to loss of strength  significant  additional l a t e r a l load. condition  i n foundation  soil;  storm a c t i v i t y  the periods  of  facility:  ( b ) i t may  f o r the design of these s t r u c t u r e s .  s l i d e on t h e above o f f s h o r e  i s extremely  (Wright,  (a)  produce  T h e s e two e f f e c t s r e p r e s e n t a v e r y  o f p r e d i c t i n g when, w h e r e , t o w h a t e x t e n t a s e a f l o o r e f f e c t s of t h i s  1976),  The o c c u r r e n c e o f s e a f l o o r  s e v e r e s t o r m a c t i v i t y may h a v e two e f f e c t s o n a n o f f s h o r e  severe loading  by H e n k e l  i n s t a b i l i t y c a u s e d by wave a c t i o n on  ( H e y b i c h , 1 9 6 8 ; S a t o , 1968) a n d j a c k e t - t y p e  s l i d e s and l a r g e  the  The o c c u r r e n c e  i n s t a b i l i t y d u e t o wave a c t i o n h a s b e e n r e p o r t e d  (1970).  1972;  i s of major importance f o r o f f s h o r e  The p r o b l e m  slide will  installations  o c c u r and  during  a  difficult.  P i p e l i n e s a r e p r o b a b l y t h e m o s t e f f i c i e n t a n d e c o n o m i c a l means o f transporting  petroleum products from o f f s h o r e  One o f t h e m a j o r d e s i g n c o n s i d e r a t i o n s  i n the design of o f f s h o r e  w o u l d 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 , may l e a d  to eventual The  pipelines  effects,  waves on a s e a f l o o r c a n be a n a l y s e d  B e c a u s e o f many v a r i a b l e s  water kinematics,  wave a n d f l o t a t i o n  facilities.  which  pipeline failure.  e f f e c t s of surface  wave t h e o r i e s .  sources t o onshore  a general  using  i n f l u e n c i n g wave g e o m e t r y a n d  t h e o r y f o r t h e m e c h a n i c s o f w a t e r waves i s  2  very  difficult,  i fnot impossible.  I t i s common t o a s s u m e t h a t t h e s e a  s u r f a c e waves c a n 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 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  with amplitude varying harmonically  varying  harmonically  stress field  surface  l o a d i n g on t h e s e a f l o o r ,  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 .  i n d u c e d by t h i s p r e s s u r e  s e a f l o o r , may b e l a r g e e n o u g h t o c a u s e s e a f l o o r s l i d e s .  The  l o a d i n g on t h e  Here, t h e Mohr-  Coulomb c r i t e r i o n c a n b e e m p l o y e d s u c c e s s f u l l y t o a n a l y s e failure  waves.  a potential  zone. Further,  t h i s harmonic pressure  loading w i l l  produce  cyclic  s h e a r s t r e s s e s i n t h e o c e a n b e d 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 pore pressure pressure  will  surrounding  This  increase  i n pore  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  the pipeline.  can  be u s e d t o e v a l u a t e  ing  aspects,  1.2  ( F i n n , 1 9 7 6 ) i n some s o i l s .  seismic  The t e c h n i q u e s  of seismic  t h i s r e s i d u a l pore pressure.  response a n a l y s i s But, i n the follow-  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)  T h e d u r a t i o n o f l o a d i n g a n d wave p e r i o d s a r e v e r y i n t h e c a s e o f wave l o a d i n g a n d , 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  (c)  Seismic l o a d i n g i s a p p l i e d a t t h e lower boundary o f t h e deposit. As t h e r e s i d u a l pore p r e s s u r e i n c r e a s e s d u r i n g the 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 might 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 o f t h e d e p o s i t . B u t 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 remains 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 ing 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 depth o f i n s t a b i l i t y u n d e r t h e g i v e n wave l o a d i n g .  i s essentially  long  unidirectional.  Scope In t h i s t h e s i s an a n a l y s i s of t h e response of a buried p i p e l i n e  t o wave l o a d i n g tally  i spresented.  layered deposit.  The seabed i s c o n s i d e r e d  Further,  t o be a h o r i z o n -  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 p r o b l e m s i s made, v i z . t h e p r e s e n c e o f t h e p i p e l i n e i n no way  affects either  the  stress field  or r e s i d u a l pore p r e s s u r e s  induced  d i s t r i b u t i o n or the by  instantaneous  t h e wave l o a d i n g i n t h e  deposit.  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 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 can  be d e d u c e d f r o m t h e r e s p o n s e o f t h e s o i l  deposit alone  to t h i s  wave  loading. It following  has  been mentioned e a r l i e r  e f f e c t s on  the  t h a t t h e s u r f a c e waves have  seabed:  1)  I t induces  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 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  Madsen ( 1 9 7 8 ) .  and  presented  (1941).  a harmonic s t r e s s  harmonic s t r e s s f i e l d  and  an  has  They a s s u m e d a n  field.  been a n a l y s e d  elastically  f o r m and  the i n s t a b i l i t y  finite  quasi-static.  deep.  of t h e  In this  The  soil  t h e s i s an  d i f f e r e n c e method has  The  been p r e s e n t e d  The  to analyse  and  Nataraja  s o i l - w a t e r system. of the  pore pressures  them was  uni-  analysis.  soil  (1978).  They d i d n o t  I t has  taken  They a n a l y s e d consider  p r o p e r t i e s d u r i n g a storm  are generated.  d e p o s i t due was  to  into  Predata.  resiaccount  the s e a f l o o r as  the e f f e c t s  activity  the  T h i s method  stability  storm  the  the response of  conditions.  p o s s i b i l i t y of l i q u e f a c t i o n of the s o i l  by S e e d e t a l ( 1 9 7 7 ) ,  degradation  by  was  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  dual pore pressure generation during a design  an u n c o u p l e d  extreme  computer p r o g r a m based on  be r e f e r r e d t o a s t h e p o r o - e l a s t i c m e t h o d o f  d i c t i o n s made by  equations  wave l o a d i n g  deposit considered  efficient  (1978)  s o i l - w a t e r system  seabed f o r g i v e n  s e a b e d t o wave l o a d i n g u n d e r m o r e g e n e r a l f i e l d will  by Yamamoto  coupled  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 . a s s u m e d t o be  the  e f f e c t i v e s t r e s s a n a l y s i s b a s e d on B i o t ' s  They a n a l y s e d  the  as  been mentioned e a r l i e r  the that  of residual the  4  degradation of  of the  soil  p r o p e r t i e s should  s o i l p r o p e r t i e s leads to h i g h e r  the e f f e c t of  s o f t e n i n g has  be c o n s i d e r e d  as  r e s i d u a l pore pressures.  been i n c l u d e d .  storm u s i n g All  t h e Mohr-Coulomb f a i l u r e researchers,  compute t h e p r e s s u r e not  to date,  l o a d i n g on  nearshore regions.  The  seafloor.  has  instability  during  So,  to  l i n e a r wave t h e o r y  shallow water  appropriate  is  theories  stability  in  r e l a t i v e m e r i t s o f t h e v a r i o u s wave t h e o r i e s  and  discussed.  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  the  presented.  L i t e r a t u r e Review Large amplitude  seabed:  1)  i n s p a c e and cyclic  they  induce  time,  and  s u r f a c e waves h a v e t h e f o l l o w i n g e f f e c t s on pressure 2)  l o a d i n g on  the  the  s e a f l o o r which i s harmonic  r e s i d u a l pore pressures  a r e g e n e r a t e d due  to  shear 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  d e p o s i t s were a n a l y s e d uncoupled  failure  surface.  the  stability  t r e a t e d the  of  soft  clay  s e a f l o o r as  an  an a n a l y t i c a l m o d e l b a s e d o n the assumption of a  F u r t h e r , he a s s u m e d t h e p e r i o d s  e f f e c t s of An  presented  He  e q u i l i b r i u m , employing  l o n g enough t o n e g l e c t the  l o a d i n g on  by H e n k e l ( 1 9 7 0 ) .  s o i l - w a t e r s y s t e m and  p r i n c i p l e of l i m i t i n g  fore,  The  be a p p r o p r i a t e when a s s e s s i n g  proposed methods of a n a l y s i s i s  1.3  thesis  h a v e u s e d t h e l i n e a r wave t h e o r y  the  t h e i r ranges of a p p l i c a b i l i t y a r e An  In this  criterion.  a p p l i c a b l e to shallow water c o n d i t i o n s .  l i k e t h e c n o i d a l t h e o r y may  softening  F u r t h e r , Seed,':s a n a l y s i s  b e e n e x t e n d e d 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 o f a  this  the t r a n s i e n t e f f e c t s of  important  the pressure  t h e h i g h e s t wave l o a d i n g s c a n  e x t e n s i o n o f H e n k e l ' s m e t h o d was  of  circular  waves were  loading.  be a n a l y s e d  includes  There-  statically.  p r o p o s e d by W r i g h t e t a l  ( 1 9 7 2 ) w h i c h e m p l o y s t h e f i n i t e e l e m e n t m e t h o d and  the  nonlinear  5  p r o p e r t i e s of s o i l . of  A hyperbolic  0.495 f o r P o i s s o n ' s r a t i o  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  (to represent undrained condition)  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 .  were  B o t h t h e above methods a r e t o t a l  s t r e s s m e t h o d s 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 a n d a r e s u i t a b l e f o r deposits of very low p e r m e a b i l i t y , c h i e f l y , A n 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 by e s t i m a t i n g deposit  the t o t a l  In total  deposits.  i n two w a y s :  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 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 o r  as a c o u p l e d s o i l - w a t e r  clayey  2) by t r e a t i n g  i n the  the deposit  system.  the uncoupled e f f e c t i v e s t r e s s a n a l y s i s the e s t i m a t i o n of  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  appropriate total  stress soil  properties.  i n t h e b e d c a n be made u s i n g  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  p o r e p r e s s u r e s h a s b e e n p r e d i c t e d b y a number o f i n v e s t i g a t o r s b a s e d different  1)  on  assumptions:  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 n o n 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 the f l u i d m o t i o n i n t h e p o r o u s b e d 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 equation leads to the Laplace equation f o r pore pressure. 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 i n d e p e n d e n t 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)  M o s h a g e n a n d T«5rum ( 1 9 7 5 ) a s s u m e d t h a t t h e w a t e r i s comp 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 pore pressure 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 response i s a f u n c t i o n of p e r m e a b i l i t y of the 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 compress 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 comp r e s s i b i l i t y of the 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 higher than the water 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 , M o s h a g e n a n d 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 ( 1 9 7 8 ) a n d M a d s e n  (1978) p r e s e n t e d t h e c o u p l e d  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 Their analysis  effective  equations (1941).  takes i n t o account the e l a s t i c d e f o r m a t i o n of the porous  6  medium a n d 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 . and,  therefore,  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  pore pressures. failure  This i s a coupled  They i n v e s t i g a t e d  failure  zones u s i n g  analysis  s t r e s s e s and  t h e Mohr-Coulomb  criterion. On t h e o t h e r h a n d , 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 t r e s s e s w e r e c o n s i d e r e d by Seed  e t a l (1977)  shear  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 o f r e s i d u a l p o r e p r e s s u r e , was c o n cerned 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 of t h e seabed. s u r e m o d e l p r o p o s e d by S e e d p r e s s u r e under  1.4  cyclic  e t a l ( 1 9 7 6 ) was u s e d  A simple pore pres-  to generate  porewater  loading.  Organisation of the Thesis 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  discussed  i n C h a p t e r 2.  tions are c r i t i c a l l y  are extensively  The m a i n a s s u m p t i o n s a n d t h e d e s i g n c o n s i d e r a -  reviewed.  Chapter 3 d e a l s w i t h the f o l l o w i n g  topics:  ( 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 o f a v a i l a b l e wave t h e o r i e s a n d t h e i r v a l i d i t y ; E q u i v a l e n t Uniform Storm:  (b)  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 f r o m d e s i g n s t o r m 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 sure methods, r e s p e c t i v e l y .  Assumptions,  theoretical formulation,  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 . example problem a r e d i s c u s s e d . approaches  outlined  of a buried  i n this thesis  S o l u t i o n s f o r an  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  have been compared w i t h a v a i l a b l e f i e l d  Stability  limita-  pipeline  i n C h a p t e r 6.  tant c o n c l u s i o n i s presented i n Chapter  data.  i s checked u s i n g A brief 7.  the procedure  summary o f t h e m o r e  impor-  7  CHAPTER 2  GENERAL ASPECTS OF OFFSHORE P I P E L I N E S T A B I L I T Y  In order  t o p r e v e n t damage o c c u r r i n g d u e t o f i s h i n g  anchors, etc., offshore p i p e l i n e s a r e often buried. the main design during is  Therefore,  requirements i s that t h e p i p e l i n e should  i t s operation.  t o be t r a n s p o r t e d  In general,  gear,  one o f  n o t f l o a t up  t h e s p e c i f i c g r a v i t y of the f l u i d  that  b y t h e p i p e l i n e v a r i e s f r o m 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 o f n a t u r a l g a s . i s being  buoyancy f o r c e s w h i c h  t r y to force the pipeline to the surface.  buoyancy f o r c e i s d e f i n e d  transported,  I n c a s e s where l o w  specific gravity fluid  ing  ships'  there w i l l  be p o s i t i v e n e t Net  a s 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 -  w a t e r a b o v e i t s own w e i g h t . In the case of buried  analysed should  f o r the following design  considerations:  of cover  1) s e a f l o o r  should  be  slides  not occur below t h e b u r i e d depth of t h e p i p e l i n e or i n t h e cover  material; provide  pipelines the s t a b i l i t y  2) i f p o s i t i v e n e t b u o y a n c y f o r c e s a r e p r e s e n t  the cover  a d e q u a t e 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 ;  the 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 r e s i s t a n c e should This  be p r o v i d e d  adequate  3) i n  bearing  by t h e s o i l .  t h e s i s i s concerned w i t h cases where p o s i t i v e n e t buoyancy  forces a r e present. of adequate s t r o n g  B u r i a l of p i p e l i n e s a t adequate depth, t h e p r o v i s i o n c o v e r and a d d i t i o n o f a r t i f i c i a l  methods used t o c o u n t e r a c t  pipeline flotation  w e i g h t a r e t h e common  problems.  MacPherson (1978) and L i u e t a l (1979) p r e s e n t e d approaches t o evaluate pipeline.  should  Their  theoretical  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  a n a l y s i s was b a s e d o n P u t n a m ' s a s s u m p t i o n s .  When t h e  8  c r e s t o f t h e wave i s a l i g n e d w i t h  the c e n t r e l i n e of the pipe,  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 it  deposit  (destabilizing  force).  MacPherson concluded that  t h e wave c r e s t a n d t h e c e n t r e periodic  s e e p a g e f o r c e may b e r e s p o n s i b l e  been r e p o r t e d  thesis  ignores  f o r buried  pipelines.  stress field  a l o n e t o t h i s wave l o a d i n g . with h o r i z o n t a l l y layered  2.1  this  f o r the "jacking e f f e c t " which The a n a l y s i s p r e s e n t e d  i n this  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  We c o u l d  p i p e l i n e t o a wave l o a d i n g  generation-  then analyse the response of a  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  Furthermore, since t h i s t h e s i s i s concerned  deposits,  So, t h e o n l y d e s i g n  stability  between  He f u r t h e r s u g g e s t e d t h a t  d i s t r i b u t i o n or the r e s i d u a l pore pressure  d i s s i p a t i o n c a n be n e g l e c t e d .  occur.  be a s  t h e i n f l u e n c e of t h i s p e r i o d i c seepage f o r c e .  Now l e t u s a s s u m e t h a t  buried  f o r a deep  i s independent of t h e d i s t a n c e  of the pipe.  hand,  the centre of  t h e m a g n i t u d e o f 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  l a r g e as 30% of t h e buoyancy f o r c e ,  the  a n d , on t h e o t h e r  a c t s v e r t i c a l l y u p w a r d s when t h e t r o u g h i s i n l i n e w i t h  the pipe  has  force)  t h e wave  a complete s e a f l o o r s l i d e would n o t  consideration  of the p i p e l i n e against  t h a t h a s t o be a n a l y s e d  i sthe  flotation.  Statement of t h e Problem Figure  lishing 1)  1 shows t h e f o r c e s w h i c h m u s t b e c o n s i d e r e d  a criterion for stability  flotation.  Here,  U  =  where  Wp =  U  b  -  W  estab-  The c o m p o n e n t s a r e  The n e t b u o y a n c y f o r c e o n t h e p i p e l i n e , U.  =  2)  against  when  p  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 contents per u n i t length; b u o y a n c y f o r c e c a u s e d by s u r r o u n d i n g water pressure per u n i t l e n g t h .  The e f f e c t i v e w e i g h t o f t h e mass o f s o i l , W, involved i n breakout together with the p i p e l i n e . g  9  FIG.  FIG.  2  1  Forces  General  Slip  on a B u r i e d  Surface  Still  Water  Sea  Bottom  Pipeline  f o r Shallow  Anchors  Level  10  E f f e c t i v e w e i g h t o f t h e i n v o l v e d s o i l mass c a n 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 w e i g h t , y'» volume of that m a s s a r e known. Should t h e r e be steady v e r t i c a l upward seepage o f 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 a p p a r e n t s o i l u n i t w e i g h t h a s t o be changed t o a  Y" 3)  =  Y' - Y  n  Q  t  n  e  i  w  (2.2)  T h e v e r t i c a l c o m p o n e n t , R^, o f t h e f o r c e s o f s h e a r i n g r e s i s t a n c e , R , on t h e overburden s o i l along the s l i p surface separating the part of the s o i l i n v o l v e d i n breakout from the r e s t of the s o i l mass. g  4)  The v e r t i c a l  component,  C , a  due t o 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 5)  The s o i l s u c t i o n f o r c e s , P , r e s u l t i n g from d i f f e r e n c e s i n p o r e w a t e r s t r e s s e s above and b e l o w t h e p i p e l i n e c a u s e d by a t t e m p t e d u p w a r d movement. F o r i n s t a b i l i t y o r u p w a r d movement one s h o u l d h a v e w  U > p+ s w  W  + R  b  2.2  Estimation  2.2.1  evaluated  buoyancy  per u n i t  precisely.  estimated  v  + C  a  + p  w  (2.3)  of the Forces  Effective  Weight  be  soil.  using  force U  l e n g t h o f p i p e l i n e , and i t s c o n t e n t s  The a v e r a g e buoyancy f o r c e p e r u n i t  the  (Wp) c a n b e  length,  U^, c a n  formula,  V i u  U.  b  where UjV  This the U2  =  TTB  B U2  equation  = =  i s linear.  the diameter of t h e pipe; the t o t a l pore pressures, i . e . , s t a 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 t t h e t o p and bottom o f t h e p i p e , respectively.  i s obtained  circumference  (2.4)  by i n t e g r a t i n g t h e t o t a l  of the pipes  Note that  assuming  the values  of  that  pore pressures  the variation  corresponding  between  around and  t o t h e buoyancy  J  11  f o r c e on a p i p e by  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  represented  t h i s equation as, 2 water  w  I lLi.q u e ff i•e da  and  =  4  Y s "~7~ 4  < 2  4 b  )  soil where  Y  w  Y  S  t h e u n i t w e i g h t o f w a t e r , and  =  =  the u n i t weight of saturated s o i l surrounding the pipe.  Now k n o w i n g Wp a n d U^, e q u a t i o n  2.2.2  and r e l a t i v e d e n s i t y , D .  b  experimental the general  data, Vesic slip  (1971) r e p o r t e d  r e l a t i v e depth, shallow anchors,  D^/B,  fail  along  a failure pattern similar  the general  slip  1971) s u g g e s t s  that this l i m i t i n g  depth,  clay  soft bentonite clay the l i m i t  i t i s approximately  D^/B,  t o punching  The  critical  behave as and, p o s s i b l y ,  A v a i l a b l e experimental  from perhaps 2 f o r a v e r y l o o s e d e p o s i t t o over very  surface.  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 factors.  along  Deep a n c h o r s move v e r t i c a l l y  a b o v e w h i c h t h e embedded o b j e c t s s h o u l d  some o t h e r a s y e t u n c l a r i f i e d  In  s  that shallow anchors f a i l  s u r f a c e a s shown i n F i g . 2.  f a i l u r e and t h e n  (Vesic,  W  Based on e x t e n s i v e  r  a considerable distance producing  shear  involved:  U.  v o l u m e o f s o i l mass i n v o l v e d d e p e n d s , i n g e n e r a l , o n t h e  d e p t h o f embedment, D ,  for  (2.1) c a n be used t o e s t i m a t e  E f f e c t i v e w e i g h t o f t h e mass o f s o i l The  deposit  evidence  i n s a n d may  increase  10 i n a v e r y d e n s e d e p o s i t .  i s a b o u t D^/B  = 2 while i n a  stiff  10.  A f t e r assuming t h e volume i n v o l v e d , an average e f f e c t i v e 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  weight  evaluated.  E f f e c t i v e u n i t w e i g h t o f 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  2.2.3  c a n be  calculated using  Shear r e s i s t a n c e component, R ,  along  v  The  v e r t i c a l component of  unless  criterion  t h e assumed s l i p  can  f o r any  Even a  trial  slip  was' p r o p o s e d by V e s i c  surface.  s u c h as Mohr-Coulomb l e a d s t o r i g o r o u s  the s l i p  surface.  (1971),  who  An  breakout  a n a l y t i c a l approach to t h i s  an  to W  s  p l u s t h e component, R . v  extensive experimental  s a n d , and  concluded  2.2.4  and  s o l i d mass. Phillips  program to study  that Vesic's theory  r e s i s t a n c e because the theory pressures  d i d not  the corresponding  C o n t r i b u t i o n of  and  wood p i l e s  equals  s o l i d mass.  and  required  the s t a b i l i t y  depth i s equiout  of p i p e l i n e s i n  the  stable  soil  consider wave-induced excess  pore  reduction in effective stresses.  s o i l adhesion,  Ca  r e q u i r e s some t i m e .  seem t o i n d i c a t e , .at l e a s t  or exceeds the undrained  The to  e t a l (1979) c a r r i e d  overestimated  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 . nature  expansion  This pressure  Cohesive s o i l s containing a c t i v e minerals w i l l  chemical  problem  o r s p h e r i c a l c a v i t y o f a g i v e n r a d i u s and  o f embedment b e l o w t h e s u r f a c e o f t h e valent  of  compared t h i s p r o b l e m w i t h t h e  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  a cylindrical  be  computations  the d i s t r i b u t i o n  of c a v i t i e s near the 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 solution i s given  surface  the shearing r e s i s t a n c e , R ,  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  stresses along  (2.2).  v  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 simple y i e l d  equation  shear  The  process  Experiences  develop  adhesion  i s of a  physico-  with steel,  i n soft clays,  concrete,  that the  s t r e n g t h a f t e r a p e r i o d of a  days t o , p e r h a p s , a few months of b u r i a l .  M o s t o f t h e known  adhesion few  adhesion  s t u d i e s were concerned w i t h measurements of r e s i s t a n c e to shear.  However,  i n t h i s breakout  object  problem r e s i s t a n c e to t e n s i o n between the b u r i e d  13  and C  a  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 .  i s equal to zero leads to a conservative design, C  zero  i n this  2.2.5  a  C o n t r i b u t i o n of s o i l the process  overburden s o i l  adjacent  underlying  i s relieved  soil  suction force, P  of attempted  w  u p w a r d p i p e l i n e movement, t h e  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 from s t r e s s e s .  This w i l l  porewater p r e s s u r e s above t h e o b j e c t and a decrease  sures below t h e o b j e c t .  I f the s o i l  sures w i l l v a n i s h as they appear.  i n porewater  pres-  i s very porous these porewater  pres-  i n a suction force.  suggested  a p o s s i b l e way o f a n a l y s i n g t h i s  undrained  t e s t s of s o i l  during breakout  also w i l l  Vesic  s u c t i o n f o r c e by  t o sandy o r s i l t y  i t i s assumed t h a t P  pressures (1971)  performing  The e f f e c t i v e s t r e s s e s a n d p o r e  have been s t u d i e d by B y r n e and F i n n (1978) .  i s restricted  permeability,  samples.  l e a d t o an i n c r e a s e  The d i f f e r e n c e i n p o r e w a t e r  above and below t h e p i p e l i n e r e s u l t s  analysis  i s assumed t o be  analysis.  During  in  Since the assumption that  w  lead to a conservative design.  Since the  s o i l s , which have f a i r l y  i s very nearly zero. So, f i n a l l y ,  pressures  This  equation  high  assumption (2.3) can  be now w r i t t e n a s U >W +W +R b p s v  - (2.4)  -  for  t h e u p w a r d movement.  2.3  Proposed A n a l y s i s f o r t h e P i p e l i n e During  storms,  d e p o s i t s and t h e s e w i l l buried  r e s i d u a l pore pressures a r e induced 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  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 b o u y a n c y f o r c e , U i j . T h i s i s because t h e pore pressures around t h e pipe increase.  i n some  soil  of a p i p e l i n e  1  2)  I t w i l l reduce the e f f e c t i v e weight of t h e s o i l mass ( W ) . T h i s happens b e c a u s e o f upward seepage o f p o r e w a t e r c r e a t e d by r e s i dual pore pressures. s  3)  I t w i l l reduce t h e shear s t r e n g t h along an assumed s l i p s u r f a c e a n d , 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  first  pore pressure  two e f f e c t s n o t e d a b o v e c a n b e a n a l y s e d  distribution  i s known i n t h e d e p o s i t .  using r e s i d u a l pore pressure The  shearing  analysis explained  t h e wave l o a d i n g a r e g r e a t e r  than or equal  (Df)  by t h e  I f the stresses  t o t h e shear  induced  strength  i n this  failure region w i l l  The Mohr-Coulomb c r i t e r i o n c a n be u s e d t o d e t e r m i n e f a i l u r e i f induced  stability Highest  s t r e s s e s a r e known i n t h e d e p o s i t .  analysis explained  Df w i l l  occur  a s t h e e x t r e m e wave p a s s e s o v e r ,  analysis  should  be u s e d t o f i n d  pressure  i n the deposit during  This  this.  stability  D f , k n o w i n g t h e maximum r e s i d u a l p o r e t h e storm  activity. problem w i l l  be  considered.  i s concerned w i t h p o s s i b l e f l o t a t i o n during  surface conditions.  depth  when t h e b e d h a s  So, t h e p o r o - e l a s t i c method o f  Three cases of p i p e l i n e f l o t a t i o n Case I  be  P o r o - e l a s t i c method o  i n Chapter 4 c a n be u s e d t o d e t e r m i n e  maximum r e s i d u a l p o r e p r e s s u r e .  1968)  i n the deposit  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 o f t h e p i p e l i n e b u r i e d zero.  i n C h a p t e r 5.  r e s i s t a n c e o f f e r e d b y t h e d e p o s i t a g a i n s t u p w a r d movement  wave l o a d i n g a n d s h e a r s t r e n g t h o f t h e d e p o s i t .  (failure)  T h i s c a n be done  i n detail  .of t h e p i p e l i n e w i l l d e p e n d o n t h e s t r e s s e s i n d u c e d  by  i fr e s i d u a l  Here, t h e theory  calm sea  o f V e s i c o r Reese (Reese e t a l ,  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 o f t h e d e p t h o f b u r i a l .  Phillips  e t a l ( 1 9 7 9 ) 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  by V e s i c  i s s l i g h t l y higher  Vesic's approach, explained  predicted  t h a n t h a t p r e d i c t e d by R e e s e e t a l . earlier  Using  15  R  +W B  where F  c  c >  c  Fq  b 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 o f t h e c a v i t y , as w e l l a s oh t h e a n g l e o f i n t e r n a l f r i c t i o n , and  =  the e f f e c t i v e cohesion of the s o i l d e p o s i t .  1  These f a c t o r s c a n be u s e d zontal cylinders.  directly  f o r embedded s p h e r e s o r embedded  Therefore, f o r s t a b i l i t y  hori-  o r no u p w a r d movement o f t h e  pipeline U, < W + R + W b ~ p v s i.e.,rB y <W + B ( c ' F + y ' V . F )'. 4 w p c b q  (2.6)  2  -  (Factors F  c  and F  Alternatively,  q  are tabulated i n Table 1).  t h i s a n a l y s i s c a n be p e r f o r m e d  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 Case I I  This case i s concerned  a storm c o n d i t i o n  a c c o r d i n g to recommendations  C o u n c i l (ASCE, R e p o r t ,  with the s t a b i l i t y  such that the depth of b u r i a l ,  depth of i n s t a b i l i t y , r e a s o n a b l e t o assume  .  D^,  1966).  of a p i p e l i n e during i s l e s s than the  Under t h e s e c i r c u m s t a n c e s , where  < D^,  i ti s  that  1)  the s l i p vertical  surface i s e s s e n t i a l l y along p l a n e s , and  two  2)  R , w h i c h i s t h e v e r t i c a l component o f 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 . v  U s i n g e q u a t i o n ( 2 . 4 ) , f o r no u p w a r d movement u, < w b p  + w s  Here, U  b  =  f  B O J ^ )  (2.7)  TABLE 1  CAVITY BREAKTHROUGH FACTORS ( A F T E R V E S I C , 1 9 7 1 )  F i r s t number i s F <J), i n d e g r e e s  0  10  20  30  40  50  c  S e c o n d number i s F  c  0.5  1.0  1.5  2.5  5.0  0.81  1.61  2.42  4.04  8.07  0.21  0.61  0.74  0.84  0.92  0.84  1.68  2.52  4.22  8.43  0.30  0.77  0.99  1.26  1.75  0.84  1.67  2.52  4.19  8.37  0.38  0.94  1.23  1.67  2.57  0.79  1.58  2.37  3.99  7.89  0.45  1.08  1.45  2.03  3.30  0.70  1.40  2.11  3.51  7.02  0.51  1.19  1.61  2.30  3.83  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  W  D^, to  t n e  =  g  T h i s c a s e has Case I I I  maximum t o t a l p o r e p r e s s u r e s o n t h e t o p and b o t t o m o f t h e p i p e a t a n y 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 m e t h o d , and  =  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 .  been c o n s i d e r e d  i n the  This case deals w i t h the  sample problem i n Chapter  s i t u a t i o n where the d e p t h of  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 , be c a r r i e d a)  out  to analyse  this  6.  Df.  The  f o l l o w i n g steps  case:  A s l i p s u r f a c e h a s t o be a s s u m e d b e t w e e n l e v e l s Df and D . Guidelines outlined i n Section 2.2.2 c a n be u s e d 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 the i n s t a b i l i t y r e g i o n does not o f f e r a n y 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  b)  The v e r t i c a l c o m p o n e n t o f t h e s h e a r 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 h a s t o be e s t i m a t e d . T h i s c a n be d o n e i n an a p p r o x i m a t e way by c o m p u t i n g 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 the shear r e s i s t a n c e m o b i l i s e d i n the r e g i o n b e t w e e n Df and D^.  c)  An e s t i m a t i o n o f W 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 effects. Now, e q u a t i o n ( 2 . 4 ) c a n be u s e d t o c h e c k w h e t h e r t h e u p w a r d movement i s p o s s i b l e . s  burial, are  18  CHAPTER 3  EVALUATION OF DESIGN STORM WAVE  The  designer  of an o f f s h o r e f a c i l i t y  problem of e s t i m a t i n g design during the future l i f e  s t o r m wave d a t a f o r a s t o r m  of t h i s f a c i l i t y .  loading conditions.  to withstand  i t i s often  t h e worst  small p r o b a b i l i t y of occurrence.  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  possi-  A  numerical  b a s e d o n a number o f f a c t o r s  of the structure, r e l a t i v e cost of 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 a r e of t e n r e p o r t e d (i.e.,  occur  The d e s i g n , t h e r e f o r e , i s a l w a y s b a s e d o n c o n d i -  t i o n s t h a t have a s u i t a b l y  such as t h e l i f e  that w i l l  I n any l o c a t i o n ,  uneconomical 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 ble  i s always faced w i t h the  storm  i n terms of e i t h e r  wave d a t a f o r a p a r t i c u l a r  location  e x t r e m e wave h e i g h t o r s p e c t r a  wave h e i g h t s a n d wave p e r i o d s ) .  S p e c t r a a r e much m o r e u s e f u l b u t  i s u f f e r from t h e disadvantages compute. involved  Complaints  of being  extremely  t e d i o u s and e x p e n s i v e  to  a r e o f t e n made a b o u t t h e two m a j o r u n c e r t a i n t i e s  i n the estimation of a design  s t o r m wave d a t a ,  either  i n terms  o f 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  uncertainties are: 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 studies are available only i n limited locations 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 estimate relevant data 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 f e w y e a r s and t h e y have t o be u s e d t o e s t i m a t e t h e d e s i g n s t o r m w i t h a r e t u r n p e r i o d o f perhaps.. 100 y e a r s . The  response  p o r o - e l a s t i c method o f s t a b i l i t y  of a s o i l  d e p o s i t t o extreme waves.  with a small p r o b a b i l i t y of occurrence  analysis analyses the So, a d e s i g n  i s required.  e x t r e m e wave  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 generated  I n t h e d e p o s i t due t o c y c l i c  activity.  pressures  shear s t r e s s e s d u r i n g a  A u n i f o r m s t o r m wave i n p u t i s r e q u i r e d f o r t h i s This chapter explains, b r i e f l y ,  storm  purpose.  a simple basis f o r obtaining  an  " e q u i v a l e n t u n i f o r m s t o r m w a v e " c o r r e s p o n d i n g t o t h e d e s i g n s t o r m wave . d a t a 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 m o d e l f o r a small specified  probability^of  occurrence.  Here,  certain  an e q u i v a l e n t u n i f o r m  s t o r m wave i s d e f i n e d a s a s i n g l e wave o f c e r t a i n wave h e i g h t ( H q ) a n d e  period  ( T q ) w i t h a c e r t a i n number o f c y c l e s  t i o n of t h e storm, which terms  e q  ) acting  weather  f o r the dura-  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  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  deposit.  shear s t r e s s e s i n t h e  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 maps i s p r e s e n t e d  i n F i g . 3.  I n nearshore l o c a t i o n s ,  s t o r m wave d a t a may a l s o b e a v a i l a b l e . are considered but t h i s approach  O b t a i n i n g Design Storm Storm  Here,  storm-produced  c a n be e x t e n d e d  g e n e r a t e d by, f o r example, earthquakes  3.1  (N  e  Waves  the required  say, a b i g t a s k i n i t s e l f . seemingly u n p r e d i c t a b l e .  s i t e may b e p r e s e n t e d i n t e r m s  So, a s t a t i s t i c a l  The s t a t i s t i c a l  t o r i c a l data a r e used storms. wind  storm data f o r a s i t e  approach  i s , needless to  f o r selecting  of the sea s t a t e i s accounted  design for is  p r o p e r t i e s o f s t o r m o c c u r r e n c e r a t e s and h i s -  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 i g n i f i c a n t wave f i e l d s c a n b e e s t i m a t e d b a s e d  speed,  spectra.  The s e a 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 a n d  conditions i n which the v a r i a b i l i t y important.  waves o n l y  or landslides.  wave d a t a f o r a p a r t i c u l a r  task of c o l l e c t i n g  direct  t o c a s e s where waves a r e  o f 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 ) a n d wave The  from  f e t c h , d e p t h , beach s l o p e and wind  duration.  on c o n d i t i o n s of T h i s method t o  20  Weather  Maps  Storm Data: Wind V e l o c i t y and Duration  P r o b a b i l i t y Model, D e s i g n Storm Data  S t o r m Wave C h a r a c t e r i s t i c s H, T, number o f waves  Wave  Theories to  Calculate  Pressure Loading  Equivalent Storm  Uniform Wave i  FIG.  3  Flow Chart Equivalent  t o C a l c u l a t e G e n e r a l S t o r m Wave D a t a a n d t h e U n i f o r m S t o r m Wave S y s t e m  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 and  for different  shallow water c o n d i t i o n s i s described  by W e i g e l  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 cane c a n be e s t i m a t e d field  (1964).  significant  Harmonic p r e s s u r e  b e now e v a l u a t e d  3.2  Wave  time increments  loading exerted  by t r a v e l l i n g  larity  (Nataraja et a l , s u r f a c e waves  Theories  s e a o f c r e s t s and t r o u g h s  on t h e water s u r f a c e because o f i r r e g u -  o f wave s h a p e s a n d t h e v a r i a b i l i t y  i sparticularly  and c o n s t a n t l y  i n the d i r e c t i o n of  i n t e r a c t i o n between i n d i v i d u a l waves. through slower  transform  such as wind  into  by t h i s  interaction,  turbulence  depths. nature to  and o f t e n c o l l i d e w i t h each  and spray.  Wave e n e r g y d e r i v e d f r o m  on b r e a k i n g  other sources  by i n t e r a c t i o n  with  and a t t h e bottom i n s h a l l o w  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 seemingly  unpredictable  o f s e a 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  formulate.  H o w e v e r , t h e r e a r e a b o u t 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 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 wave t h e o r i e s c a n b e u s e d t o e s t i m a t e due  and pass  Waves s o m e t i m e s r e i n f o r c e o r  i sdissipated internally within the fluid  t h e a i r above, by t u r b u l e n c e  because of t h e  F a s t e r waves o v e r t a k e  ones from v a r i o u s d i r e c t i o n s .  c a n c e l each other  propagation.  t r u e when t h e w a v e s a r e u n d e r 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 s e a s u r f a c e waves i s d i f f i c u l t  and  distri-  u s i n g wave t h e o r i e s .  Waves i n t h e o c e a n o f t e n a p p e a r a s a c o n f u s e d changing  hurri-  wave  a s w e l l a s t h r o u g h t h e a p p l i c a t i o n o f a n assumed R a y l e i g h  1978).  This  The r e l a -  during passage of a  through consideration of t h i s  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  can  c a s e s s u c h a s deep w a t e r  t o s u r f a c e waves.  c u l a r wave t h e o r y  fluid  flow conditions which  purposes.  the pressure  A l l these  eight  l o a d i n g on t h e s e a f l o o r  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  that a  i s m o s t 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  parti-  22  c h a r a c t e r i s t i c s and s t i l l  water c o n d i t i o n s .  t h e o r i e s h a s b e e n p r e s e n t e d b y Dean ( 1 9 7 4 ) . t h e o r i e s and a b r i e f below.  This w i l l  for  evaluating  the  regions  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  the pressure  of v a l i d i t y  loading  f o rvarious  i m p r o p e r wave t h e o r y  on t h e ocean f l o o r . wave t h e o r i e s .  to estimate  r e v i e w o f wave  T h e s e w e l l - k n o w n wave  d e s c r i p t i o n of t h e i r r e l a t i v e merits  c a r r i e d out a t t h e U n i v e r s i t y of B r i t i s h in using  An e x t e n s i v e  are listed wave  theory  Figure  4 shows  Preliminary  studies  Columbia suggest that the pressure  loading  the error on t h e  s e a f l o o r c a n be e v e n a s much a s 3 0 % . Analytical validity Analytical validity  o f t h e wave t h e o r i e s  i s defined  the d i f f e r e n t i a l  below.  as t h e degree to which t h e t h e o r i e s  the d e f i n i n g boundary c o n d i t i o n s . satisfy:  i s analysed  satisfy  There a r e f i v e d e f i n i n g equations to  equation  and f o u r  boundary c o n d i t i o n s .  These  are as f o l l o w s : 1)  Differential  2)  No f l o w o c c u r s a c r o s s  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 which r e q u i r e s t h a t t h e components o f 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 g e o m e t r y and motion of the free surface.  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 pressure immediately 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 pressure.  5)  Motion i s periodic i n x with o f t h e wave l e n g t h , L .  3.2.1  Linear The  It the  wave  order  only.  f o r two-dimensional i d e a l t h e bottom  flow.  boundary.  spatial  periodicity  theory  l i n e a r wave t h e o r y  s a t i s f i e s kinematic first  equation  i s the simplest  and dynamic f r e e s u r f a c e I t has been r e p o r t e d  that  of a l l the theories. boundary c o n d i t i o n s t o this  theory  can  23  0.05  H  0.00005  0.001 0.002  0.005 0.01  0.02  0.05 0.1  o f Wave  Theories  d QT'  Here,  H  =  Wave H e i g h t  T "=  Wave  d  Mean W a t e r  FIG.  =  4  Period Depth  Regions of V a l i d i t y  0.2  24  be u s e d  successfully  reasonable depths.  accuracy,  i n both  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  although  Furthermore,  i t i s best  the linear  theory  suited  for transitional  water  i s readily applied to representa-  t i o n s o f random waves.  3.2.2  Stokes  higher order  theories  Stokes  theories satisfy  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 orders of approximation. approximation observed  Each e x t e n s i o n i n t h e order of  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  wave b e h a v i o u r .  T h e s e e x t e n d e d wave t h e o r i e s c a n e x p l a i n p h e n o -  mena, s u c h a s m a s s t r a n s p o r t , t h a t c a n n o t b e e x p l a i n e d by t h e l i n e a r theory.  Generally, higher order  Stokes  t h e o r i e s do n o t i m p r o v e 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 computations applicability  i s limited  previously defined  3.2.3  t o 1/25 < d / L .  c n o i d a l wave t h e o r y  function u s u a l l y designated  rather well  H e r e , d and L a r e a s  i s a nonlinear theory.  The t e r m ,  i s g i v e n by t h e J a c o b i a n  elliptical  by c n .  There a r e higher order  t h e y t o o do n o t i m p r o v e t h e a c c u r a c y  computations.  The r a n g e o f  theory  i s u s e d s i n c e t h e wave p r o f i l e  and  involved.  (Sen, 1971).  C n o i d a l wave The  cnoidal, cosine  cnoidal theories  commensurate w i t h t h e i n c r e a s e d  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 t h e waveform and a s s o c i a t e d m o t i o n s .  H o w e v e r , 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 due  wave  to the d i f f i c u l t i e s  i n making t h e necessary  computations  problems  (Shore  P r o t e c t i o n M a n u a l , 1977) . The  w o r k 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 h a s b e e n  s t a n t i a l l y reduced  sub-  by t h e i n t r o d u c t i o n o f g r a p h i c a l and t a b u l a r f o r m s o f  25  functions theory  ( M a s c h and W i e g e l ,  is still  quite involved.  manent f o r m p r o p a g a t i n g d/L  < 1/8  3.2.4  However, a p p l i c a t i o n of  Long, f i n i t e amplitude  wave  move b a c k w a r d s and  a d i s t i n c t wave c r e s t and  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 the pure sense, level.  t h e s o l i t a r y wave l i e s  Long waves, such as tsunamis  p l a c e m e n t s o f w a t e r c a u s e d by  or n e a r l y  forwards  Stream-function  entirely  and  above the s t i l l  waves r e s u l t i n g  s u c h phenomena a s  Numerical  approximations  This theory  to s o l u t i o n s of  i n terms of  and  The  similar  b y Dean ( 1 9 6 5 ) .  to the higher order  Stokes'  mine the c o e f f i c i e n t  of  least  i s obtained  squares  conditions.  sense,  water  earthquakes,  i s broadly  ''  the hydrodynamic  equa-  f u n c t i o n s were proposed  term  The  theory  i s used to d e t e r -  so t h a t a b e s t f i t , i n t h e  t o the dynamic f r e e - s u r f a c e boundary  b e e n shown t o p r o v i d e a b e t t e r f i t t o  number o f t a b u l a t e d p a r a m e t e r s and The  stream  theories.  r a t o r y measurements of p a r t i c l e v e l o c i t y  l a r g e memory.  In  nonlinear stream-function theory i s  each h i g h e r order  T h i s t h e o r y has  A  theory  t i o n s o f wave m o t i o n e x p r e s s e d developed  passage  from l a r g e d i s -  l a n d s l i d e s and  * .  n u m e r i c a l wave  w i t h the  does i t e x h i b i t a t r o u g h .  for. r e l a t i v e l y shallow water.  3.2.5  oscillatory  wave:.trough i s e v i d e n t .  sometimes behave s i m i l a r l y t o s o l i t a r y waves. valid  applicability  theory.  so f a r a r e o s c i l l a t o r y  water p a r t i c l e s  o f e a c h wave and  per-  theory  Waves c o n s i d e r e d The  this  this  waves of  i n s h a l l o w water w i t h a range of  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  Solitary  waves.  1961).  than o t h e r s .  a digital  I t requires a  computer w i t h a  reasonably  stream-function theory provides best f i t over  range i n c l u d i n g a l l of the t r a n s i t i o n a l  and  labo-  a wide  d e e p w a t e r wave r e g i o n s  and  26  also a significant expect,  this  3.2.6  p a r t of t h e s h a l l o w water  i n this  thesis  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  showing t h e e f f e c t s o f a deformable  amplitudes.  Their analyses concluded  soft cohesive s o i l s .  a useful  s e a f l o o r o n wave p r e s s u r e  t h a t p r e s s u r e s on a deformable  meable s e a f l o o r a r e h i g h e r than those on a r i g i d  In  be r e a d i l y a p p l i e d .  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 . analysis  B u t , a s one c a n  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  Theory used All  range.  b a s e by u p t o 1 5 % f o r v e r y  B u t t h i s e f f e c t may b e i g n o r e d f o r m o s t  this thesis,  t h e l i n e a r wave t h e o r y  imper-  sands.  i s used because of i t s  s i m p l i c i t y a n d r e l i a b i l i t y o v e r a l a r g e s e g m e n t o f t h e w h o l e wave  regime.  S o l u t i o n s o f 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 o n t h e s e a f l o o r i s g i v e n by Y AP  =  H  W 9  ,  Cos  {2TT(^ - £ ) }  2Cosh(^)  =  p  Cos  n  {2Tr(f - f ) }  (3.1)  °  where  L  3.3  =  2 - f ^ - tanh ( ^ ) ZTT L  Equivalent Uniform  (3.2)  S t o r m Wave  To u s e t h e r e s i d u a l p o r e equivalent uniform  p r e s s u r e a n a l y s i s , we h a v e t o f i n d  an  s t o r m wave 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 of pore pressure 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 w a v e s e a c h 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 a n d number o f c y c l e s . g e n e r a t i o n model proposed According  to this  This t h e s i s uses a simple pore  by Seed e t a l ( 1 9 7 6 ) ,  (see equation  equation, r e s i d u a l pore p r e s s u r e generated  pressure (5.6)). due t o t h e  27  cyclic  e f f e c t of t h e shear s t r e s s e s  t h e number o f a p p l i e d cause i n i t i a l obtained T  c/°vo  c y c l e s and  a n d N^.  Here, T / a ^ c  defined  stress conditions.  by  i s the cyclic  0  f o r a given  T /O^ C  Q  t h e excess pore pressure  Now,  by  i s t h e number o f c y c l e s r e q u i r e d t o  l i q u e f a c t i o n under given  a p p l i e d c y c l e s and  the  Here, N i s  L  N  from a l i q u e f a c t i o n p o t e n t i a l curve which i s a p l o t  is uniquely that  i s a f u n c t i o n o f N/N .  T /O C  ratio  shear soil  i s uniquely  between  stress ratio.  deposit,  Since  C  we c a n c o n c l u d e  Q  L e e and Chan ( 1 9 7 2 ) .  uniform  s t o r m wave c a n be e s t i m a t e d  It will  by  reducing  t h e method proposed  be seen i n Chapter 4 t h a t  the soil-water  s y s t e m c a n be u n c o u p l e d and t h e s h e a r 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  II.  simple  s o l i d mechanical p r i n c i p l e s as explained  L e e and Chan s u g g e s t e d t h a t N „ , t h e e q u i v a l e n t n  given  reference  wave, c a n be g i v e n w  N  I  - ^ N Lr  n  N  = ec  where  *  r=l ^Leq  =  ^Lr  =  L  T /O^ .  an equivalent  accuracy using  N  g o v e r n e d by t h e number o f  o f a l l w a v e c o m p o n e n t s t o z -> 0 a n d u s i n g  v o  c a n be  L  good  i n Appendix  number o f c y c l e s o f a  by  (3.3)  r r  of c y c l e s required to 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 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;  n  u  m  D  e  r  number o f 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 - r wave c o m p o n e n t a t z -> 0; t n  N. n  w  =  number o f w a v e s o f t h e r  =  t o t a l number o f wave c o m p o n e n t s .  t  n  wave c o m p o n e n t ;  To c a l c u l a t e t h e number o f c y c l e s t o c a u s e i n i t i a l s t r a i n f o r a given the  shear s t r e s s r a t i o ,  l i q u e f a c t i o n or limited  one has t o e x p e r i m e n t a l l y  c u r v e known a s 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 .  performing c y c l i c  t e s t s on ' u n d i s t u r b e d '  samples.  This  establish  c a n b e d o n e by  The t e s t s c a n b e  28  carried  out  i n either  triaxial  or simple shear  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 , of  a v e r a g e v a l u e , 0.60  for  (Seed,  the r e s u l t s of c y c l i c  shear tion  1979;  triaxial  c o n d i t i o n s or simple shear (Seed,  apparatus.  S i n c e ocean  a correction factor,  N a t a r a j a e t a l , 1979) t e s t s to reduce  c a n be  used  to p r e v a i l i n g  simple  t e s t s c a n be u s e d w i t h o u t a n y c o r r e c -  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 c a n a l s o be o b t a i n e d f r o m penetration tests. Calculated f i e l d  T h i s method  i s based on d a t a p r e s e n t e d  v a l u e s of c y c l i c  stress ratio  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 one  t o n per =  N P  square V  (  N  f o o t and  standard  by S e e d  c a n be p r e s e n t e d  f u n c t i o n o f 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.  of  Cy,  as  a  H e r e , Np  e f f e c t i v e overburden  i s g i v e n by t h e f o l l o w i n g  (1979).  is  pressure  equation.  f  where  C  where  N  =  1-1.25  =  e f f e c t i v e overburden p r e s s u r e i n tons per square f o o t a t the 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 o f N j ;  a\  =  one  Nf  =  standard 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 per f o o t .  ton per  F i g . 5(a)  r e s i s t a n c e , Np,  ing  and  square  interpolated 5(a)  the corresponding  earthquake and  (b)  cyclic  limited  s t r e s s r a t i o which strain  i s shown.  i t i s p o s s i b l e to get an a v e r a g e  i n c r e a s i n g by  causes Correspond-  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 ) .  p o t e n t i a l curve f o r various N a l suggest  foot;  the 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  l i q u e f a c t i o n or a s p e c i f i e d  Using Figs.  et  (3.4)  5Q  In  initial  l o g (|^)  p  values  10 p e r c e n t  liquefaction  ( N a t a r a j a et a l , 1979).  Nataraja  the. l i q u e f a c t i o n p o t e n t i a l  curve,  (a)  0.6 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.  w h i c h was b a s e d strength of  on f i e l d  of the s o i l  under  o c e a n wave l o a d i n g  dimensional  data  conditions  during  earthquakes.  two-dimensional  i s somewhat h i g h e r of earthquake  plane  than  loading.  This strain  that under  i s because the conditions the three-  31  CHAPTER 4  PORO-ELASTIC  4.1  ANALYSIS  Introduction The e f f e c t s o f w a v e s o n f o u n d a t i o n  been important In the design it  considerations of offshore  i n the design  s o i l s and s t r u c t u r e s have of offshore  installations.  s t r u c t u r e s such as p i p e l i n e s , sea w a l l s , e t c . ,  i s common t o i d e a l i s e t h e s u r f a c e w a v e s a s p l a n e t r a v e l l i n g w a v e s .  When t h e s e w a v e s p r o p a g a t e o v e r a p o r o u s s e a b e d ,  fluid  flow  i s induced  in  t o deform.  Thus, t h e  t h e b e d a n d t h e p o r o u s medium  itself  i sforced  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 mechanical effects.  bed  t o wave l o a d i n g  deformation presented  should  t h e response a n a l y s i s of a  a coupled  Governing  sea-  i n c l u d e p o r e w a t e r f l o w , volume change and  c h a r a c t e r i s t i c s o f t h e b e d . Yamamoto ( 1 9 7 8 ) a n d M a d s e n 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  (1941) f o r a deep s o i l  4.2  Therefore,  (1978)  equation  deposit.  Equations  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 theory: 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 that t r a n s i e n t response of t h e pressure loading caused by s u r f a c e waves c a n be n e g l e c t e d . Therefore, t h e p r e s s u r e l o a d i n g c a n be a n a l y s e d quasi-statically.  2)  Theory o f l i n e a r  3)  Plane s t r a i n conditions  4)  Darcy's law i s a p p l i c a b l e . Figure  elasticity  6 shows a s o i l  i sapplicable.  prevail.  deposit  of constant  t h i c k n e s s D.  The  32  Free Surface y  s  = y Cos (Xx -a)t) Mean Water Level  Pressure Loading Ap = p Sea-bed  Porous Bed  A  Element A  ^Rigid Bottom  FIG.  6  Wave P r e s s u r e s  on Ocean  Floor  Q  Cos (Xx-ajt)  33  x-axis  i s t a k e n on t h e bed s u r f a c e ;  b e i n g v e r t i c a l l y downward shown i n F i g . 6.  the p o s i t i v e z-axis  f r o m t h e bed s u r f a c e .  to the 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) u n d e r p l a n e s t r a i n s a " ^ - T n e | £ z . 2 w 3t 3z  . 2 3x  where  ^x»^z  =  t  *  conditions:  3(£  2  x  Consider element A  D a r c i a n f l o w of c o m p r e s s i b l e porewater i n a compres-  s i b l e p o r o u s medium l e a d s  k  i s shown a s  =  Y  +e,) * 3t  principal permeabilities  i e  (4.D  Z  w  i n t h e x and  z directions, respectively; p Y  W  n e  and e  x  z  t  =  the excess pore  pressure;  =  the u n i t weight of porewater;  =  t h e p o r o s i t y o f t h e bed;  =  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 p o s i t i v e a s e l o n g a t i o n s , and  =  time.  The c o m p r e s s i b i l i t y o f . p o r e w a t e r , 3, i s g i v e n  If where  -  p  t o be  by  >»H  =  the density  of porewater.  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 a n d z d i r e c t i o n s c a n be w r i t t e n as  where  3o' x ~Z— 3x  3T  .  xz 5 — ~ 3,z a , a and T —  +  x  Here, i n e r t i a quasi-static  2  =  z  x z  3a' 3T z zx 3p "a— ~^— ~ 3z 3x 3z the incremental e f f e c t i v e and s h e a r s t r e s s e s . ,  a  n  loading  +  d  terms a s s o c i a t e d  ,, ~ (4.3)  =  with accelerations  stresses  are neglected  and a  i s assumed.  Assuming t h a t material  3p •» 3x =  the s o i l  skeleton  then, under the plane s t r a i 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  condition  N  34  _ £  x  £  z  "  _  _3v 3x  (1-y) , , 2G x  _  ( C J  v ,N 1-v V  (  4  4  N  ^ ' ^  and  where  3w ~3z"  =  (1-v) , , ~ ^ G °z "  =  -  =  displacement  G and  =  shear modulus  v  °x  A  =  a' z and a l s o ,  shear  and P o i s s o n ' s  ratio.  )  G  C  3x  +  1-v  aV  (4.5)  stress  _,3v , 3w. 3z 3x  equations  4.3  vectors;  - 2 ( t - ) G (^- + ^ -5-) l-2v 3z l ^ v 3x  substituting  is,  N  )  ( 4 . 4 ) c a n be r e w r i t t e n a s  l - 2 v  xz  third  , °x  v and w  Equation  By  v  (  equation  (4.5) i n e q u i l i b r i u m  i n t e r m s o f t h e unknowns v,w  equation  Boundary To  and p.  i n t h e b a s i c v a r i a b l e s v,w  in principle,  equations Equation  and p .  ( 4 . 3 ) we  get  two  (4.1) g i v e s a  Therefore, a  solution  possible.  Conditions  s o l v e t h e a b o v e b o u n d a r y v a l u e p r o b l e m , we  dent  boundary c o n d i t i o n s per boundary.  dary  conditions are:  At the s e a f l o o r  need  three  indepen-  surface the  boun-  (z=0)  1)  the v e r t i c a l  effective  2)  shear  x  3)  p o r e p r e s s u r e p i s g i v e n by p  stress  x z  stress,  i s negligible Q  a  z  =  0;  small or x  x z  =  0;  Cos(Ax-iot) .  Therefore,  zx  =  0,  r  p = p o r  C o s ( Ax-cot)  at  z = 0  (4.6)  35  H e r e , A a n d to a r e d e f i n e d a s 2TT/L a n d 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 a n d T b e i n g t h e wave p e r i o d ; p pressure l o a d i n g which The  Q  i s the amplitude  c a n 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  of the  wave t h e o r y .  boundary c o n d i t i o n s a t t h e bottom boundary of t h e seabed  are 1)  displacements  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 w o r d s , i t i s assumed t h a t t h e b o t t o m boundary i s r i g i d and impermeable, i . e . , v  4.4  =  w  =  - |  £  =  3z  at  0  z = D  (4.7)  S o l u t i o n Technique Since the third  i n b o t h t i m e and space, also periodic  boundary c o n d i t i o n i n e q u a t i o n i t i s reasonable  i n t i m e and space.  t o a s s u m e t h a t v,w a n d p a r e  Since d i f f e r e n t i a t i o n  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 problem  i n terms of complex v a r i a b l e s .  r e a l p a r t o f t h e complex s o l u t i o n . , a' z  =  x  = 0 ,  xz  Equation  c  p  =  Now,  p e o  (4.6) i s p e r i o d i c  with respect to  t o s o l v e t h i s boundary  The s o l u t i o n equation  value  i s t a k e n t o be t h e  ( 4 . 6 ) becomes  i ( Ax-cot)  r  ( 4 . 1 ) c a n now b e w r i t t e n a s 2  - K A p + K 2  X  From e q u a t i o n  Z  M r + i Y ngcop _ 2 W 3z  =  - ly t o ( i A v + -^) W 3z  (4.8)  ( 4 . 3 ) we o b t a i n 3x  iAa' + x and  3z  =  - piA  (4.9)  36  3a' z , + iAx 3z  - I  =  xz  introducing  (4-10)  2  3z  the .expression f o r p i n equations  2 3 3a' 3 x K ( 1 + 1^) - ( K A z -2 iA „ 3 x 3z 3z = i y co(iAv + |^) 9  W  - i  'w Y  (4.8) and  (4.10)  , 8T n0u>) (a'' + - ^ ) x i A 3z (4.11)  oZ  and 3a'  3a' =  + iAx xz  Z  3z  .  3z  3 x 2  P . 2  iA  (4.12)  3z Substituting  s t r e s s e s i n terms of displacements  from equation  ( 4 . 1 2 ) a n d e l i m i n a t i n g w, we o b t a i n 6 „ K 2 4 . K v . 2 J c _ J i i v 4 J i . K ,2 .4 K 3z z A 3z z  (4.5) i n  (4.11), 3  A  - A  {  K {-£ z  6  K  2  +  }  2 - \} A  v  =  +  A  { 1  +  2  2  J i  2 l z ,2 . 2 A 3z a  }  J  0  (4.13)  where 2  .  {nB+(l-2v)/(2G(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 c i e n t s c a n be o b t a i n e d v  by  coeffi-  substituting  . r A z i(Ax-cot) Ae e  =  equation with constant  . . . i n equation  (4.13) 2  This leads to the c h a r a c t e r i s t i c  equation which i s cubic i n r  and has  roots r  2  Realizing  =  K,. 2 1,1, ^ z A  that r  2  = 1 i s a repeated  (4.14) r o o t , we o b t a i n t h e g e n e r a l  solution  37  v  {(A  =  1 +  1  J and  A^  2)e  +  (A,+A z)e3 4  X z  A  +  A,e  T l A z +  D  A,eo  r i A 2  } «**- t) e  z  A  I t s h o u l d be n o t e d  be c o m p l e x and  t h a t c o n s t a n t s A^  using the  t h r o u g h Ag,  o n l y the r e a l p a r t of the complete  c o n s t i t u t e s t h e s o l u t i o n to our  4.4.1  (4.15)  a  2 "72  K  ^  A z  o Z  ->- 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  conditions. will  A  boundary i n general,  complex  solution  problem.  Horizontally layered deposits In  t h e case of h o r i z o n t a l l y  type  (4.15) e x i s t s f o r every  will  t h e n be  s o l v e d by  f o l l o w i n g manner.  s e t t i n g up  (4.7).  tions are v  n  6 x N simultaneous  constants  These a r b i t r a r y equations,  con-  i n the  the t h r e e boundary  b o t t o m o f t h e d e p o s i t d e f i n e d by  equations  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  = v  w  n-1  n  = w  a' = cr , zn z(n-l)  r  =p  n  ., *n-l  T  1  n-1 dp  K  dp = K ,  condi-  zn  9z  _  p o r e p r e s s u r e s and An  equations.  z(n-l)  , , v , xz(n-l)  1  (4.16) 9z  of e q u a t i o n  T h e r e f o r e , t h e t o t a l number o f solving  i n displacements,  (4.16) w i l l  simultaneous  stresses,  adjacent  l e a d to 6 x equations  these equations, i n c r e m e n t a l s t r e s s e s , pore  s u r e s , e t c . c a n be o b t a i n e d s t r e s s e s and  = x  f l o w n o r m a l t o t h e i n t e r f a c e b e t w e e n two  e q u a t i o n of the form  After  xzn  . v  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  6 x N.  the  satisfied,  p  layers.  e q u a t i o n of  number o f a r b i t r a r y  S i x equations a r e o b t a i n e d from  c o n d i t i o n s a t t h e t o p and and  The  6 x N, w h e r e N i s t h e number o f l a y e r s .  s t a n t s c a n be  (4.6)  layer.  l a y e r e d d e p o s i t s , an  i n any  layer.  (N-1)  are pres-  These i n c r e m e n t a l v a l u e s f o r  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, 1 9 7 8 ) ' . -  a'  = a ' = p Aze ^ z o  and  —Az = p Aze o  xz  p  = p e o  Analysis An  state.  =  a'  =  a'  T. xzt  =  T  a'  n  a  T  y.z b  xzt  =  z  Q  t  *  i e  and  a'  xo  The  =  K  o  stresses  a t any p o i n t ;  a'  of t h e s o i l ;  =  the c o e f f i c i e n t of earth g i v e n b y (1-Sin<j>').  to find  a'  zo  (4.19)  zo  the depth of the point  f o r each  as  f o r a uniform deposit are  =  c a n be e x t e n d e d  weight  c a n be w r i t t e n  the i n i t i a l effective stresses (i.e., during t h e calm p e r i o d ) .  cb' i s t h e a n g l e o f i n t e r n a l  Here,  effective  t h e buoyant u n i t weight  =  a t any p o i n t  equi-  (4.18)  =  a' where  idea  a  stresses  =  the i n i t i a l  z  and a '  zo  stresses  from  induced  + a'  initial  K  effective  t h e wave  xz  °zt  x  to evaluate  X  zo  ° t>  These  Instability  + a'  XO  a'  where  (4.17)  i n s t r e s s e s and pore p r e s s u r e s  The g r o s s  xt zt  Cos(Ax-cot)  of t h e Depth of  changes  a'  Sin(Ax-cot)  a n a l y s i s has been p r e s e n t e d  incremental librium  —Az  conditions,  Cos(Ax-tot)  Z  r  X  x  4.5  Under t h e s e  considered; pressure  friction. by u s i n g  assumed  For layered  t o be  deposits  the appropriate  buoyant  t h e same unit  layer.  stress  s t a t e a t any p o i n t  i s given  by c r , x t  Oy  t  and T  x v t  .  Let  39  Uo' Sin*(x,z,t)  - o ' J +4x , ,  =  zt If  } (4.20)  xt  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  occur bility  i f <j)(x,z,t) > (j) . 1  w i t h i n which  bility  will  F o r t h e d e s i g n e x t r e m e wave, a r e g i o n o f i n s t a -  <j>(x,z,t) > <j>' c a n b e d e f i n e d .  STAB-MAX, h a s b e e n d e v e l o p e d compute i n d u c e d  instability  based  A computer  on t h i s method.  T h i s program c a n  s t r e s s e s , pore p r e s s u r e s and p r e d i c t  d u e 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  program,  the r e g i o n of i n s t a -  layered deposit of f i n i t e  depth. 4.6  Values f o r E l a s t i c Except  priate  Constants  f o r some s p e c i a l  values f o r the l i n e a r  s t r e s s e s u s i n g t h i s method.  cases mentioned  has been found  We r e q u i r e 2 l i n e a r  i s i n e l a s t i c and h i g h l y  e l a s t i c c o n s t a n t s and I n general, the stress-  strain-dependent.  Based on c y c l i c t e s t s of average  i n l a b o r a t o r y , Seed a n d I d r i s s secant  modulus a t l o w s t r a i n l e v e l The  (10  - 4  o r i nthe laboratory.  t u r b a n c e o n t h e measurement o f G of  (1970) p r e s e n t e d t h e  percent) w i t h shear m a x  purpose.  m a x  i nthe f i e l d .  of t h e  (Fig. 7). either i n  Because o f t h e i n f l u e n c e o f sample m a x  dis-  i n the 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 i t i s desirable to  S e i s m i c methods a r e e x t e n s i v e l y used  B a s e d o n t h e number o f r e s o n a n t  column t e s t s  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 i n d i m e n s i o n l e s s form a s  strain  , c a n be determined  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 ,  measure G  elastic constants.  shear modulus f o r sands a s a f r a c t i o n  shear modulus a t l o w s t r a i n l e v e l , G  the f i e l d  But, i t  t h a t s t r e s s e s c a n 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 secant m o d u l i a s the l i n e a r  variation  appro-  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  they 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 . strain relationship  i n S e c t i o n 4.9,  carried  for  this  out i n the  a simple equation f o r G  m a x  10  4  10  3  SHEAR  FIG.  7  10  10  2  STRAIN  , 7-  1  PERCENT  V a r i a t i o n of Secant Shear Modulus w i t h Shear ( A f t e r S e e d a n d I d r i s s , 1970)  Strain  41  G  320.8 P  e  =  void  OCR  =  overconsolidation ratio  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 zero p l a s t i c index r has a v a l u e of z e r o . ( H e r e , r i s a s s u m e d t o be' z e r o ) ;  cjjjj  =  mean e f f e c t i v e  where  (  ; ? ! i " (1+e)  2  a  a  e  )  (OCR)  2  S) P a  (4.21)  =  max  r  h  ratio;  normal s t r e s s  given  by  1+2K  3 K  So,  vo  Q  =  c o e f f i c i e n t of earth pressure at r e s t be e q u a l t o ( l - s i n c b ' ) ) ;  <j>'  =  effective  P  =  atmospheric  a  knowing G  m a x  ,  we  angle  of i n t e r n a l  friction;  pressure.  r e q u i r e shear  strain  by t h e wave l o a d i n g t o  the secant  by  a s s u m i n g a v a l u e f o r s e c a n t m o d u l u s and t h e n m o d i f y i n g  strain  developed  ciable  difference  iterative was  before  and p e r f o r m i n g  analysis  assumed  modulus.  induced  estimate first  shear  (assumed t o  this  i s not performed.  t o be g i v e n by e q u a t i o n  B  =  m  where m  by Duncan  K,P b a  shear  The l i n e a r (4.21),  i n an i t e r a t i v e  there  elastic  i t f o r the  i s no  moduli.  appre-  In this shear  and c a l c u l a t e d  way  thesis,  modulus  using  stresses  o f t h e wave l o a d i n g .  B u l k modulus v a l u e s were only,  analysis until  i n t h e i n p u t and m o d i f i e d  the application  stress  T h i s c a n be done  found  e t a l (1978) .  t o be a f u n c t i o n o f m i n o r n o r m a l  They  proposed  (4.22)  (^) P a  =  b u l k modulus  constant;  =  b u l k modulus  exponent c o n s t a n t ;  42  =  The  average e f f e c t i v e minor ( a s s u m e d t o be-K-b^).-.  same i t e r a t i v e a p p r o a c h  principal-stress  u s e d f o r s h e a r m o d u l u s c a n be u s e d f o r b u l k  modulus to get c o m p a t i b l e b u l k modulus. was  a s s u m e d t o be c o n s t a n t and  b a s e d on  4.7  was  In this  thesis,  the b u l k modulus  calculated using equation  (4.22)  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 .  Example The  Problem s i m p l e e x a m p l e shown i n F i g . 8 w i l l  t h e a p p l i c a t i o n o f t h e m e t h o d and obtained.  The  depth  i s 12 f t .  to  illustrate  the k i n d s of r e s u l t s t h a t can  deposit i s a uniform  The mean w a t e r  be a n a l y s e d  be  sand of r e l a t i v e d e n s i t y , D  Other  r  =  54%.  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 o f t h e d e s i g n maximum wave a r e a wave h e i g h t = 9.0  and  a wave p e r i o d = 7.0  0.1953 x 1 0 "  7  ft /lb 2  sec.  The  c o m p r e s s i b i l i t y of water  (Madsen, 1978).  The  u s e d f o r t h i s d e p o s i t i s shown i n F i g . 9. induced  p o r e p r e s s u r e s and  d e p o s i t of i n f i n i t e  out.  element  i s taken to  M a d s e n ( 1 9 7 8 ) s o l v e d t h e wave  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  t h i c k n e s s and  constant p r o p e r t i e s .  He  2)  the s o l u t i o n s were v e r y s e n s i t i v e to r e l a t i v e c o m p r e s s i b i l i t y o f t h e s o i l s k e l e t o n and p o r e w a t e r ; -  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 a n y i n f l u e n c e o n t h e v a l u e o f s h e a r stresses.  Two  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  soil  permeability K  z  = 0.002 cm/s  and  0.2  cm/s  the v a l u e of  which  correspond  that  thickness  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  t y p e s w e r e a s s u m e d by c h o o s i n g  stiff  concluded  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 coarser than s i l t ;  s t r e s s dependent e l a s t i c  be  discretization  1)  To and  finite  ft  carried  vertical to  typical  43 Wave Height = 9.0' Wave P e r i o d = 7 . 0 5  = 64.0 l b f / f t  3  V o i d r a t i o =0.7 _ E f f e c t i v e u n i t weight = 47.6 l b / f t " * Compressibility of porewater = 0 . 1 9 5 3 x l 0 f t / l b Bulk, modulus c o n s t a n t s K = 950, n = 0.03 = 33.0 - /  60  FIG. 8  '  b  L  S o i l D e p o s i t used i n Example Problem  H = Wave Height  Depth - Ft  Node No.  o  I  2  2  4  3  6  4  8  5  10  6  rElements  7  8  15  20  Nodes  9  25 30  10  36  11  46  12  60  13  FIG. 9  F i n i t e Element  Discretization  44  values  f o r f i n e and  of K ,  s o l u t i o n s were o b t a i n e d  z  1,2  and  5.  The  coarse  sandy d e p o s i t s , r e s p e c t i v e l y .  f o r h y d r a u l i c a n i s o t r o p y K /K K  c a l c u l a t e d maximum i n c r e m e n t a l  non-dimensional form i n F i g s . 10(a),(b),(c) labelled  by a two  permeability:  2 for K  z  = 0.002 and  second d i g i t  i n d i c a t e s the anisotropy r a t i o :  tively.  a t t e n u a t i o n of pore p r e s s u r e  Fig.  The  10(a).  The  solutions.  curves  S i n c e one  The  attenuation i s unaffected  c a s e of f i n e sand, but  vertical  10(b)  cm/s, K /K x  the  pore pressure by  and  curves  This  z  = 1,2  and  b u t e t o an  increase i n the v e r t i c a l  t h e wave.  I t can  pressure  be  inferred  same b e h a v i o u r  vertical  0,  in  the  permeability  pressure. incremental  t h a t , as  under  into  the  the contri-  i n the case of  increment i n v e r t i c a l  Increases  increase i n a , z  effective  the other  pore  effective  f o r f i n e s a n d s and  sand d e p o s i t s .  seabed t h e v e r t i c a l  s h e a r s t r e s s , on  Madsen's  seepage f o r c e s w h i c h  is a  in permeability  and  also close  stress gradient  increases w i t h increase i n the a n i s o t r o p i c permeability The  as  in  e f f e c t i v e s t r e s s under the c r e s t of  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 s u r f a c e of the  respec-  b e e n 9p/9z =  sand, i n c r e a s e d  s t r e s s i s independent of the a n i s o t r o p y r a t i o i n coarse  The  shear s t r e s s e s were observed.  f r o m F i g . 10(b)  a t t e n u a t i o n w i t h depth, the  f u n c t i o n of the r a t i o  5,  e x p e c t e d as water i s f o r c e d  s o i l d e p o s i t under the c r e s t causing  in  are  respectively.  maximum e f f e c t i v e s t r e s s o c c u r s  i s t o be  of  indicates vertical  shows t h e v a r i a t i o n o f t h e maximum The  values  Z  the h y d r a u l i c anisotropy r a t i o  i n the case of coarse  effective stress.  c r e s t o f t h e wave.  the  The  the h o r i z o n t a l d i r e c t i o n l e d to a f a s t e r decay i n pore Figure  in  (d).  o f t h e b o t t o m b o u n d a r y c o n d i t i o n s had  ' k i n k ' i n wave i n d u c e d  value  with depth i s i l l u s t r a t e d  exhibit essentially  the  in  and  0.2  each  stresses are presented  d i g i t number w h e r e t h e f i r s t d i g i t  1 and  For  ratio.  h a n d , a p p e a r s t o be  fairly  to  FIG.  10(a)  Porewater  Pressure  i n a Deposit  of F i n i t e  Depth  46  0.1  0  0.2  0.3  0.4  0.5  •  FIG.  10(b)  Induced of  Vertical  Finite  Depth  Effective  0.6  0.7  25  Stresses  in a  Deposit  47  i n s e n s i t i v e to both p e r m e a b i l i t y  and  anisotropic  (Fig.  bed  is elastic  10(c)).  Assuming t h a t  shear s t r e s s e s principles. that  a l s o be  T h i s can  be  calculated using  d o n e by  T h i s method w i l l  both permeability  stresses  and  o b t a i n e d by  be u s e d w i t h o u t much e r r o r  pore pressure a n a l y s i s to c a l c u l a t e induced  The  increase  s e p a r a t e mechanisms. effective stress. tive stress.  The  seepage f o r c e s the  shear s t r e s s e s  One  The the  i s the  maximum v a l u e o f t r o u g h of  increase,  other contribution the  flow.  which decreases with  d i r e c t i o n away f r o m t h e w a v e c r e s t and  tropic  increase  in a  permeability The  horizontal  the  G3  the  solutions  stress  soil  two  in vertical effec-  response  i s a function increase  under the  i s g o v e r n e d by  by  act  in  of  aniso-  on  the  troughs. crest.  to  soil  This  There-  both the  aniso-  permeability.  Yamamoto ( 1 9 7 8 ) had  -»• 0,  method  the v e r t i c a l  towards the  result i n tension  i s c o m p l e x and  r a t i o and  residual  shear  increase  seepage f o r c e s  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  these solutions. depth, with  x  The  method  wave.  the  This contribution  in  seepage f l o w w i l l  The  Since  is affected  to the  i s g o v e r n e d by  permeability  f o r e , net  the  i s i n phase w i t h  tropic  horizontal  ratios.  due  shear  simple e l a s t i c  i n Chapter 5 uses t h i s e l a s t i c  This contribution  i n d u c e d by  cyclic  i n F i g . 10(c).  in horizontal effective stress  horizontal pressure gradient  the  00  in  independent  P l o t of  other curves, the  c r e s t and  such  This procedure i s explained  cyclic  shear s t r e s s e s .  o c c u r s h a l f w a y between the  mechanical  f o r a l l p r a c t i c a l purposes.  explained  induced  a stress function,  t h i s method i s l a b e l l e d as the  ratios  impermeable,  c o m p r e s s i b i l i t y of w a t e r .  t h i s c u r v e compares w e l l w i t h can  give  and  simple s o l i d  constructing  i t s a t i s f i e s boundary c o n d i t i o n s .  Appendix I I . of  can  the  permeability  shown t h a t ,  a great effect  f o r a seabed of  f o r the wave-induced  stresses  on infinite  reduce  FIG.  10(c)  Induced Shear F i n i t e Depth  Stresses  i n a Deposit  of  O-x'/Po  FIG.  10(d)  Induced H o r i z o n t a l Deposit  of F i n i t e  Effective Depth  Stresses  in a  50  to  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 , a s shown i n e q u a t i o n  partially  saturated  f l u i d s may  the e f f e c t i v e compressibility  exceed t h e c o m p r e s s i b i l i t y of pure water  considerable analysis  soils,  (4.17).  of t h e p o r e -  ( a i r free)  amount d e p e n d i n g on t h e d e g r e e o f s a t u r a t i o n .  by V e r r u i t  (1969) g i v e s  of a w a t e r - g a s m i x t u r e  For  a n u p p e r bound f o r t h e  by a  A  simple  compressibility  as  1-S 3  =  3„ + — 5  ;  1 - S  <<  1  (4.23)  t i n which S  g  pressure,  3 and  saturated  porewater, r e s p e c t i v e l y . F o r 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 by y 3 i s 10 cm c o m p a r e d t o 4 x 10 cm f o r pure water.  lity The  i s the degree of s a t u r a t i o n , P 3  defined  Q  saturation  further analysis  t h e l a y e r s and  is printed are  out.  i s well-documented  i s not considered  STAB-MAX, s t r e s s e s  Based  on t h i s , v a r i o u s  f o u n d t o be  by M a d s e n ( 1 9 7 8 )  in this  z  and,  thesis.  a r e computed a t t h e m i d d l e for  stability  c o n t o u r s o f cf>' c a n be d r a w n a n d = 0.002 cm/s  and K / K x  z  =  1.  problem considered  ( $ ' = 33°)  failure  and  was  12.5'.  The r e a l u n d e r w a t e r  here.  and  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 o f  c a n be e s t i m a t e d f o r t h e e x a m p l e  cated  pore  saturated  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  shown i n F i g . 11 f o r t h e c a s e o f K  Assuming  static  w  I n t h e program, of  i s the t o t a l  a r e the c o m p r e s s i b i l i t y of p a r t i a l l y  e f f e c t of p a r t i a l  therefore,  t  seafloor  stability  problem i s more c o m p l i -  t h a n t h e one w h i c h h a s b e e n 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 u s e d There i s a l w a y s energy t r a n s f e r between  s o i l because o f work done a g a i n s t  the shearing  This contributes  t o t h e damping of t h e waves.  method,predicted  tensile horizontal  stresses  t h e wave and resistance Further,  under  the  of the  the  moving soil.  poro-elastic  t h e wave c r e s t .  Since  51  F I G . 11  C o n t o u r s o f <j) D e v e l o p e d  52  soil  cannot c a r r y t e n s i o n , the  i s not dict  accurate.  However, t h i s  the response of the  4.8  stress field simplified  p r e d i c t e d by a p p r o a c h can  seafloor with reasonable  Comparison of the Theory w i t h F i e l d  this  be u s e d t o  Data  California deals with c o l l e c t i n g ,  b o t h s a n i t a r y and and  dispersed  Ocean.  The  approximately existing  storm wastewater f l o w s .  t o be  The  by  embedded t h r o u g h o u t m o s t o f a s much a s  25  Pacifica,  California,  of water.  The  a t 30,  86 and  60,  during  the  explained sure.  120  f t below the  i n t h i s chapter  These authors  of observed  from a m u n i c i p a l 600  was  seafloor.  type were  Soil  type data  2 and  The  obtained analysis  pore pressure  of  g i v e n by  f l u c t u a t i o n s as a w  using  encoun-  the values  to p a r t i c l e s i z e d i s t r i b u t i o n data  presented  placed  s e a f l o o r , medium d e n s e  i n F i g . 12.  c h o s e n t o be  the  pier in  used to p r e d i c t the wave-induced pore was  the  pressures  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 a s p / 2 y H i n F i g .  Results obtained  in  soil materials  sands near the  are presented  which  f t f r o m s h o r e i n 16 f t  The  dense below 8 f t .  t h i s a n a l y s i s , I^/K^  were s e l e c t e d a c c o r d i n g et a l .  loose fine  dense to very  and  of  the  measured i n  p i e z o m e t e r s w h i c h w e r e IRAD v i b r a t i n g w i r e  i n s t a l l a t i o n process  For  installed  located approximately  t e r e d were predominantly b e l o w 2 f t and  f t below  They m e a s u r e d w a v e - i n d u c e d p o r e  piezometer array  of  i t s length  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  C r o s s e t a l (197 9 ) .  San  Pacific  a t t e n u a t i o n of wave-induced pore p r e s s u r e  i n the planning  using a v e r t i c a l  disposing  s e c t i o n s i n the  c o m p u t a t i o n o f f o r c e s o n 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 field  County of  T h e s e f l o w s w e r e t o be c o l l e c t e d  4000 f t i n a t r e n c h - e x c a v a t e d  seafloor.  is required  t r e a t i n g and  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  O u t f a l l was  pre-  accuracy.  W a s t e w a t e r Management M a s t e r P l a n o f t h e C i t y and Francisco,  method  STAB-MAX a r e a l s o shown f o r c o m p a r i s o n .  presK  z  Cross  fraction 13. The  DEPTH BELOW  MATERIAL  SEAFLOOR  DESCRIPTION  (FEET) •SEA FLOOR S A N D (SP) : L o o s e t o medium d e n s e , 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) : V e r y d e n s e , l i g h t brown, medium t o "v^_ coarse-grained, sub-angular t o sub-rounded  10  y  S A N D (SP-SM): V e r y d e n s e , l i g h t b r o w n , grained, sub-angular t o sub-rounded 20 •  fine-  ^GRAVEL ( G P ) : V e r y d e n s e , l i g h t brown, medium t o / coarse-grained, sub-angular t o sub-rounded S A N D (SP) : V e r y d e n s e , 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 s m a l l amount o f s u b - a n g u l a r g r a v e l  30  Piezometer 4 0  S I L T Y S A N D V e r y d e n s e , brown, to sub-rounded  5 0  (SM) fine-grained,  sub-angular  Piezometer 60  •  J Becomes  70  S  80  light  brown  A N D (SP) V e r y d e n s e , brown, f i n e - g r a i n e d , s u b a n g u l a r t o sub-rounded  Piezometer 90  100  S  110  120  —{  —  A N D (SP-SM) V e r y d e n s e , 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 g r a y - g r e e n  130 BOTTOM  OF B O R I N G  §  130.5'  no  FIG.  12  B o r e h o l e Data  (After  Cross e t a l ,  1979)  Pore Pressure F l u c t u a t i o n 0  i  0.1  ,  OJZ  1  —i  FIG.  13  (L3 1  p/(2y H)  OA  1  Pore Pressure D i s t r i b u t i o n  w  (L5 H  0.6 r  p. 7  n  i n Sand D e p o s i t o f 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 pore p r e s s u r e w i t h depth below t h e s e a f l o o r agrees remarkably  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 m e t h o d .  Only  average  p o r e p r e s s u r e s w e r e m e a s u r e d by t h e s e a u t h o r s , a n d n o t i n s t a n -  taneous  v a l u e s , when a g i v e n wave i s p a s s i n g b y .  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  4.9  The e r r o r  i n prediction  to this.  Some P r a c t i c a l S o l u t i o n s The  analysis,  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 n o t h e r words i t assumes t h a t t h e water  to a coupled problem.  This type of a n a l y s i s  the computer f o r s o l v i n g ; .  i s a coupled  and s o i l  constitute  i s t e d i o u s and r e q u i r e s  F i n n , S i d d h a r t h a n a n d M a r t i n ( 1 9 8 0 ) who  o u t a n e x t e n s i v e s t u d y u s i n g t h e p r o g r a m , STAB-MAX, s u g g e s t e d uncoupled  a n a l y s i s c a n be p e r f o r m e d  deposits.  They c o n c l u d e d  and  simple  f o r a number o f u n i f o r m d e e p s a t u r a t e d  that t h e normal  d e p o s i t c a n be o b t a i n e d by c o m p u t i n g  that  carried  e f f e c t i v e s t r e s s e s induced  t o t a l wave-induced normal  in a  stresses  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 o f each o t h e r and u s i n g t h e p r i n c i p l e o f  effective stress, pressure.  i . e . , e f f e c t i v e stress equals t o t a l  The s h e a r  effective  s t r e s s minus  pore  s t r e s s e s a r e t h e same w h e t h e r c o m p u t e d by t o t a l o r  stresses. T o t a l w a v e - i n d u c e d 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 a s a s e m i - i n f i n i t e e l a s t i c medium, a n d s o l u t i o n s a r e g i v e n by Fung ( 1 9 6 5 ) . a  a  n  It  d  T  =  z  °x  xz  T h e a m p l i t u d e o f t h e s e s t r e s s e s a r e g i v e n by  =  P (e"  o  p  =  A z  0  p  o  (  X  A z  )  / ~Az , -Az. ~ zXe ) e  z  e  ~  A  s h o u l d be n o t e d  the e l a s t i c  + zAe"  medium.  Z  (4.24) ,, (4.25)  (4.26)  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  56  Now k n o w i n g t o t a l  induced  t i o n s f o rpore pressure  attenuation  this,  and M a r t i n  Finn, Siddharthan  types w i t h extreme s o i l  K  z  N/m  t o compute e f f e c t i v e s t r e s s e s . (1980) c o n s i d e r e d  properties.  2  Hard  Soft  10  5xl0  9  Coarse  io-  sands were c o n s i d e r e d . K /K x  z  = 1,2, a n d 5; P o i s s o n 2  pressure  The r e s u l t s o b t a i n e d  i n s o f t - c o a r s e and s o f t - f i n e  w e r e somewhat  has l i t t l e sands.  e f f e c t on t h e  A s i n g l e porewater z  ratio,  g i v e n by  -Az  (4.27)  be used f o r t h e s e sands.  pore pressure  were  r a t i o = 1/3; b u l k m o d u l u s o f  x  e  6  s o f t - c o a r s e , and s o f t -  attenuation curve which i s not a function K /K  P*o can  io"  3  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  porewater pressure  Fine  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  s a t u r a t e d w a t e r = 2.45 x 1 0 ^ N/m . surprising.  were  6  , m/sec  a l s o assumed:  To do  4 different soil  The p r o p e r t i e s c o n s i d e r e d  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 , fine  solu-  SAND TYPE  PROPERTY  G,  s t r e s s e s o n e n e e d s t o know s i m p l e  I t was f o u n d t h a t f o r h a r d - c o a r s e s a n d , t h 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  permeability  r a t i o and g i v e n by -A(K /K )^z X  P  The  o  z  (4.28)  6  pore pressure  attenuation  i n hard-fine  pendent o f 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 a n d s was o b s e r v e d t o b e  inde-  s o l u t i o n f o r a t t e n u a t i o n was n o t  possible. This  s i m p l e method o f d e t e r m i n i n g  induced e f f e c t i v e s t r e s s e s i s -  57  restricted uniform. (coupled  to cases  where t h e d e p o s i t concerned  i s d e e p , s a t u r a t e d and  For layered, shallow deposits a numerical solution) explained  i n t h i s chapter  should  s o l u t i o n of the type be u s e d .  58  CHAPTER 5  RESIDUAL PORE PRESSURE ANALYSIS  The The  cyclic  s u r f a c e waves i n d u c e c y c l i c  shear s t r e s s e s on t h e seabed.  shear s t r e s s e s developed d u r i n g  s t o r m a c t i v i t y may b e h i g h  e n o u g h 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 The d e v e l o p m e n t  of t h e r e s i d u a l pore pressure i s t h e r e s u l t  ing  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  processes.  loading causes t h e pore p r e s s u r e t o r i s e , pore p r e s s u r e which i s governed  magnitude.  o f two o p p o s under  cyclic  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  by t h e dynamic  e q u a t i o n , decreases i t ( M a r t i n e t a l , 1975).  form of t h e c o n s o l i d a t i o n 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 the of  loading acts f o r a very short time. r e s i d u a l pore pressure i s ignored.  I n o t h e r words, But, during  the dissipation  storm 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 pore p r e s s u r e s h o u l d be used i s very large.  since  of  t h e wave l o a d i n g  to  v e r y h i g h r e s i d u a l pore pressures i n t h e d e p o s i t .  since the duration  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  lead  The v o l u m e 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)  the r e l a t i v e density of the s o i l , of soil,  2)  t h e induced c y c l i c  3)  t h e e x i s t i n g r e s i d u a l pore pressure.  r  shear s t r e s s r a t i o , and  On t h e o t h e r h a n d , t h e d i s s i p a t i o n w i l l pressibility  D , and t h e t y p e  depend on t h e p e r m e a b i l i t y  com-  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 o f 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 dissipation effects.  Seed  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 -  e t a l (1977)  and N a t a r a j a (1978)  analysed the  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  5.1  Theory A plane  t r a v e l l i n g wave i n d u c e s  tude a l o n g any h o r i z o n t a l p l a n e drainage plane. tially  activity.  a constant  i n a horizontally  shear  stress ampli-  layered deposit.  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  horizontal  U n d e r 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 to a one-dimensional  problem.  Using  Further,  essen-  Darcy's law, the 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 a s  JL  f  _£  \  =  l£  ( 5 1)  w where  u K Y  z  w  e  =  r e s i d u a l pore  =  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 ;  =  u n i t weight  =  increase i n volumetric s t r a i n being positive. quant i t y .  During  an i n t e r v a l ,  pressure;  of water;  For t h i s problem e w i l l  At, the pore pressure  s t r e s s which w i l l  be a n e g a t i v e  i n an element of s o i l  undergo a change, A t , w h i l e t h e element w i l l of shear  considered  a l s o be s u b j e c t e d t o c y c l e s  cause an a d d i t i o n a l  increase i n pore  3ug/3t At where 3 u g / 3 t i s t h e r a t e o f p o r e p r e s s u r e i n c r e a s e . 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 ,  will  pressure,  Considering  t h e v o l u m e c h a n g e , Ae, o f  t h e element i n t i m e , A t , i s g i v e n by  Ae where As  =  m  m^  =  3u (Au - — & V at  3t)  the c o e f f i c i e n t  of volume  compressibility.  A t -> 0 ~ It  „ ~  m  v  (  ^  3u ( 5  '  2 )  60  combining  equations  (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 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 has  been g i v e n by Seed  m  e  =  v  A r  u 0  tj  o p  A  r  2  5(1.5 - D ) ;  B  =  3/2< r>;  =  pore pressure r a t i o  iiiy  =  c o m p r e s s i b i l i t y a t pore pressure r a t i o , r  m  =  c o m p r e s s i b i l i t y a t zero pore pressure  u  v o  r  2D  initial vertical  0  order to solve equation  p r o p e r t i e s s u c h a s m^, ration,  (5.4)  vo  =  o\^ = In  m  A  r  variation  +  u where  equation f o r this  (1976)  .2 2B 1 + Ar  t o i n c r e a s e w i t h an  K , z  d e f i n e d as u/a^ ;  effective  Q  r  ;  ratio;  stress.  (5.3), i t i s necessary D,  u  to establish the s o i l  e t c . and t h e r a t e o f p o r e p r e s s u r e  gene-  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 b e w r i t t e n a s ( S e e d  et a l ,  1976) g_ 9t  =  g. _9N 9N 9 t  (5.5)  w h e r e N i s t h e number o f 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 . 9u /9N c a n b e e i t h e r e v a l u a t e d b y a l a b o r a t o r y t e s t ••conditions o r by an approximate t h a t f o r many s o i l s practical  purposes  mathematical  The v a l u e o f  simulating the f i e l d  formula.  I t has been  found  t h e r e l a t i o n s h i p between U g and N c a n be e x p r e s s e d f o r i n t e r m s o f t h e number o f c y c l e s , N-^,  r e q u i r e d to cause  61  initial form  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  (Seed  et a l ,  i nthe following  1976) J_  g 7 a vo where  =  2  .  . 20.  — arc s i n (x "  Q =  an e m p i r i c a l  x =  cyclic  rs  )  (5.6)  constant;  r a d i o , d e f i n e d a s (N/N-^) .  This relationship  i spresented  G may b e u s e d t o r e p r e s e n t d i f f e r e n t A v a l u e o f 0 = 0.7 h a s b e e n f o u n d many s o i l s .  By d i f f e r e n t i a t i n g  i n F i g . 14.  r a t e s o f pore  D i f f e r e n t values of  pressure generation.  t o represent t h e average  equation  (5.6)  curve f o r  w i t h r e s p e c t t o N, o n e  gets  _g_  _  3N  1_  _VO  GUNT  ^  v  . 26-1,11  sin  ,U  (5.7)  .  ("2 r ) cos{j r ) u  u  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 p r a c t i c a l purposes The  iS 9t  storm  and i s d e s i g n a t e d as T .  3t  o r  (  5 .  8  )  D  (5.7)  and ( 5 . 8 ) , t h e r a t e of pore  a t any time c a n be determined  a  3u _ £ 3t  So,  i s t h e same a s t h a t o f t h e  Then  D  Combining equations 3ug/3t,  wave.  e  _§a T  =  and f o r  i s u s u a l l y r e p r e s e n t e d by N q c y c l e s o f a u n i f o r m  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 s t o r m  irregular  be i r r e g u l a r  =  the equation:  g. . M 3N 3t a N ^ o _ _ e a 1  =  GUTT) N u  now k n o w i n g  dual pore  from  pressure generation,  T  3ug/3t,  1 . 20- L H  sm  .  (-j r )  equation  pressure d i s t r i b u t i o n .  u  (5.3)  J  ,  K  cos(-^ r ) u  c a n be used  t o determine  F i n i t e element method  the resi-  i s used t o s o l v e  '  FIG.  14  Rate of Pore 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  Loading  63  for Fig.  u.  The  15(a).  detail  5.2  d o m a i n and The  f o r m u l a t i o n of f i n i t e element method  A n a l y s i s of the Depth of  the depth of  instability.  s t r e s s e s i n an  sures develop as  o' zt  =  a  a' xt  =  K  T „,..  1  vo o  the  developed  by  be z e r o .  But,  during  t h e new  i n s t a b i l i t y c a n be I t c a n be  - u)  b ' e  t h e bed  horizontal effec-  the storm  and  0  r e s i d u a l pore  i s t h e maximum c y c l i c  shear  o . x o  pres-  p o i n t can  As  be  =  expected  xt 7—j (a^  So,  2  7  up  ; +  amplitude  h  xzt r a' )  . . (5.11)  fc  t  x  t h e u and  x  x z t  ,  t h e w o r s t c a s e w o u l d be  the higher  z  t  ^  > <j) . 1  the value  t h e e x t r e m e waves  Pore Pressure  shear  the pore pressure  that  as  t o a. d e p t h w i t h i n w h i c h <t>^  of c y c l i c  at  b e f o r e , d e f i n i n g a n g l e , cj>,  stresses i n the deposit,  zt  s t r e s s induced  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  the d e p o s i t to use  o^  b  Changes, i n S o i l P r o p e r t i e s and The  and  (see F i g . 15(b))  seen t h a t the h i g h e r  z t)  extended to p r e d i c t  s t r e s s s t a t e a t any  2 V  be  (5.10)  XZu  induced  . , sin<£, . (x,z,t)  can  initial vertical  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 .  5.3  in  - u  (a' vo  Here, T  XZL  Let  i n - S e c t i o n 4.5  i n t h e d e p o s i t and  expressed  over  i s explained  e l e m e n t d u r i n g a c a l m p e r i o d be g i v e n by  will  v  ^(x  shown i n  Instability  p r i n c i p l e explained  Shear s t r e s s T „  and  problem are  i n Appendix I .  The  tive  boundary c o n d i t i o n s of t h i s  i n the  passing deposit.  Generation  s t r e s s e s h a v e t o be c o m p u t e d  generation  of  equation  (5.9).  It  in has  64  -u = 0 at z = 0  Soil Deposit  _3_[l<z du\ dz|_7w  [du du£\  3zJ' v[at-at m  |^  Basic  Equotion  J  = 0 at z - D  and  Solution  Domain (a)  °"vo-  vo  K  Stress  o °"vo  K  Condition at  Calm  Stress  Periods 15  0 CTvo- )  Condition  Storm  Periods  (b) FIG.  u  S t r e s s Conditions Before and During  Storms  (  u  at  65  been observed due  i n S e c t i o n 4.7  t h a t the shear  s t r e s s e s induced  t o wave l o a d i n g c a n be c o m p u t e d a s s u m i n g t h a t t h e bed  A simple procedure shear  using  s o l i d mechanical  stresses i s presented  uniform  d e p o s i t , ^ /o^ c  deposit.  But,  d e p e n d s on  i n Appendix I I .  i n the case of a non-uniform  the e l a s t i c  the e l a s t i c  reduce (equations excess  deposit  i s impermeable.  t o compute  cyclic  I t i s shown t h a t f o r a  i s independent of e l a s t i c  Q  or  c o n s t a n t s of the  deep soil  s h a l l o w d e p o s i t , T /a^. c  0  constants.  When t h e r e s i d u a l p o r e p r e s s u r e deposit,  principles  in a  i n c r e a s e s d u r i n g a storm  c o n s t a n t s w h i c h w e r e a s s u m e d t o be  ( 4 . 2 1 ) and  (4.22)).  During  the presence  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  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  stress  minor  of  in a  dependent residual  principal  as,  1+2K a' m  =  a' 3  =  =-53 K  o  (a' - u) vo  (a' vo  u)  This w i l l vary uniform  the T /a^ c  d e p o s i t s d u r i n g a storm  m o d i f i c a t i o n of e l a s t i c of e f f e c t i v e greater depth  (5.12)  0  induced  activity.  by  t h e waves i n s h a l l o w o r  I t has  c o n s t a n t s t o be c o m p a t i b l e  5.4  that  the  w i t h the current  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 of  instability  i n the seabed.  An  efficient  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 o f e l a s t i c developed  been found  to perform  stability  levels  a  computer  c o n s t a n t s has  non-  program,  been  analysis.  Example Problem The  s i m p l e e x a m p l e shown i n F i g . 8 i s a n a l y s e d  a p p l i c a t i o n of t h i s method. l a y e r s and  the e l a s t i c  soil  s i s t e n t w i t h the l e v e l s of  T h i s u n i f o r m d e p o s i t was c o n s t a n t s , G and effective stress.  B,  to i l l u s t r a t e  divided into  12  w e r e s e l e c t e d t o be  A v a l u e o f 0 = 0.7,  the  con-  which  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 d e n s e s a n d s was u s e d sure generation (equation (5.6)). i n Table I I . tial  curve  T h e T /a^ Q  0  i n pore  The d e s i g n s t o r m w a v e d a t a  a n d N^ v a l u e s c o r r e s p o n d i n g  pres-  i s given  to liquefaction  poten-  i s given i n Table I I I .  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 was a n a l y s e d f o r two  TABLE I I D e s i g n Wave D a t a 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  Wave H e i g h t f t (H)  TABLE I I I Liquefaction Potential  Shear S t r e s s Ratio  Curve  No. o f 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  different permeability values, K  z  = 2 x 10  J  cm/s a n d K  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  z  = 2 x 10  1  cm/s.  o n t h e d e p o s i t , two  67  types for  of a n a l y s i s were c o n d u c t e d .  I n one,  the changes i n r e s i d u a l pore p r e s s u r e  the moduli and,  were not  modified  i n the other, they  were  modif i e d .  5.4.1  Equivalent uniform The  storm  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  "equivalent uniform  storm"  which i s r e p r e s e n t a t i v e of the d e s i g n  wave i n t e r m s o f p o r e p r e s s u r e p r o p o s e d by L e e  and  T h i s p r o c e d u r e has procedure c a r r i e d  Chan ( 1 9 7 2 ) was been e x p l a i n e d  storm  5.4.2  Results  D i s c u s s i o n of F i g u r e 16  of the storm  = 2 x l O ^ cm/s.  The  -  z  instability,  M o d i f i c a t i o n of  sure.  has  soil  f o r two  equation  the  of "uniform  wave  i n moduli  i s taken  sand.  10 ''' cm/s,  z  = 2 x  -  no  z  = 2 x  10 ^  are presented  cm/s  -  r a t i o s and  into account,  higher  depth  the p r e d i c -  constants  permeable  pressures  f o r changes i n  a p p r e c i a b l e i m p a c t on t h e r e s u l t s .  T h i s example c l e a r l y demonstrates t h a t t h e e f f e c t of  dation i n moduli  should  V.  pres-  The  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 o f p r o p e r t i e s  considered.  of  i n Table  i n the l e s s  the r e s i d u a l pore  t h a t the changes i n the e l a s t i c  e f f e c t i v e s t r e s s l e v e l s had  K  in  the predicted r e s i d u a l pore  i s about 30%  I n the case of K  cases,  (5.11),  ted pore pressure d i s t r i b u t i o n  depth of  I V shows  p r o p e r t i e s f o r the increase i n r e s i d u a l  n o t i c e a b l e i m p a c t on  When t h e d e g r a d a t i o n  w e r e so l o w  Table  storm.  i n Table I I .  maximum p o r e p r e s s u r e  calculated using  pore pressure  method  used to o b t a i n the equivalent  i n S e c t i o n 3.3.  given  The  storm  shows t h e d i s t r i b u 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  t h e d e p o s i t a t t h e end K  i n the deposit.  out to c a l c u l a t e the e q u i v a l e n t storm  heights f o r the design  and  generation  an  be c o n s i d e r e d  f o r proper a n a l y s i s .  was  degra-  TABLE I V C a l c u l a t i o n of Equivalent Wave H e i g h t H (ft) ±  Number o f Waves  Wave P e r i o d TJL ( s e e s )  Number o f U n i f o r m  Wave P  Wave L e n g t h L (ft) ±  ±  Pressure (psf)  Nwi  Cycles  Shear Stress Ratio a t z=0  Number o f Cycles to Liquef. N ^ i  N  E q u i v a l e n t No. of C y c l e s eq wi/%i- ref = N  N  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  200  4.0  66.6  37.4  0.059  100,000.  2.9 0.01  TABLE V Results of Residual Pore Pressure K  z  Analysis K  = 0.002 cm/s  without modification  with modif i c a t i o n  z  without modification  = 0. 2 cm/ s with mod i f i c a t i o n  Maximum R e s i d u a l P o r e Pressure Ratio  0.34  0.46  0.012  0.012  Maximum D e p t h o f Instability (ft)  3.0  5.0  0.0  0.0  69  The by  d r a i n a g e c h a r a c t e r i s t i c s of  coefficient consolidation c  the ting  lower the  the  given  by  sands a r e  K /(va^y^)  .  z  higher  be.  a n a l y s i s b a s e d on u n d r a i n e d c o n d i t i o n s  storm a c t i v i t y w i l l ft.  governed  The  r e s i d u a l pore pressure d i s t r i b u t i o n w i l l  t o n o t e t h a t an  during  v  the  a d e p t h of  10  reduce the  pore pressure  indicate that  A c o v e r of a few  f e e t of  dramatically.  the high  deposit c  v  mainly the  c  v  It is interesprevailing  would l i q u e f y to  material  (gravel)  will  70  FIG.  16  Residual Pore Pressure  Distribution  71  CHAPTER 6  ANALYSIS OF P I P E L I N E FLOTATION  A numerical ciples  example problem i s c o n s i d e r e d  i n C h a p t e r 2.  A pre-stressed concrete  p i p e l i n e of 4 f t diameter  i s b u r i e d a t a depth of 8 f t below t h e s e a f l o o r . d e p o s i t a r e shown i n F i g . 8. fluid  of s p e c i f i c g r a v i t y S The  Q  The p r o p e r t i e s o f t h e  The p i p e l i n e i s t o b e u s e d t o t r a n s m i t a = 0.70 a n d i t w e i g h s q = 80 l b f / f t r u n .  response of t h i s deposit using r e s i d u a l pore pressure  sis to a given design  storm  depth of i n s t a b i l i t y ,  i s presented  , will  occur  i n S e c t i o n 5.4.  dual pore pressures  analyse  Therefore,  a n a l y s i s of t h i s p i p e l i n e Then, u s i n g  f o r no u p w a r d movement  where  U  b  From e q u a t i o n U  b  =  + W s =  t h e buoyancy f o r c e p e r u n i t l e n g t h on t h e pipeline. (2.4),  | B  (l^-Up  B u o y a n c y 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 a t t h e bottom of t h e p i p e l i n e ;  with  T h i s a n a l y s i s gave a  t o C a s e I I , s t a t e d i n S e c t i o n 2 . 3 , w h e r e D^ < D f .  IL < W b p  arid  T h e program,- STAB-MAX, was u s e d t o  f o r r e s i d u a l pore pressure.  v a l u e o f 17.5 f t f o r D f .  (2.7)  The p r e s e n c e o f r e s i -  t h e r e s p o n s e o f t h e d e p o s i t f o r t h e e x t r e m e wave l o a d i n g  properties modified  over  reduce 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 deposit  to higher depth of i n s t a b i l i t y .  analy-  T h e maximum  when t h e e x t r e m e w a v e p a s s e s  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 pore p r e s s u r e s .  lead  to explain the p r i n -  and r e s i d u a l )  corresponds equation  72  U-^  U  =  10 x 64 + 0.263 x  =  765.2  =  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 o f pipeline;  =  6 x 64 + 0.370 x  =  489.7  =  b  lbf/ft  x 4 x  = Weight  lfb/ft  o f p i p e l i n e and  «  -  P  Here, Y j  W  P  S  (o^ ) 0  z  1  i  0  =  6  the  ,  -489.7)  run  i t s contents  x 62.4  Q  =  (Wp)  4^  1 +  =  z  2  u n i t weight of  =  o  2  (765.2  865.5 l b f / f t  (o^ )  fluid  lbf/ft  3  •=  80 + T x 0.7 4  x 62.4  =  628.9 l b f / f t  run  C o r r e c t e d b r e a k o u t s o i l mass Using equation  W  x  16  s  (2.2)  n (W ) s ave where  I|  =  (Y'-Y i)  this  r  =  average volume i n v o l v e d  n  =  t o t a l number o f  i  =  hydraulic  gradient =  =  hydraulic  head  h For  - Y i) V w r r  average e f f e c t i v e u n i t weight of r  =  w  V  (Y'  r  s  in r  layers;  =  dh /dz; s  u/Y « W  example problem, from T a b l e VI  (W ) s  a v e  =  789.8 l b f / f t  Therefore, the factor  of s a f e t y  run against  flotation  layer;  layer;  TABLE V I C a l c u l a t i o n of Corrected  Depth'Below Seafloor  Excess Pore Pressure Ratio  Breakout S o i l  a' vo  H y d r a u l i c Head h (ft) = u/y s  w  i=dh /dz s  Mass  (Y'-Y i)  Volume V f t  ( '-Y i)V  w  3  r  Y  w  r  u/-<4 0  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  =789.8  74  stabilizing force disturbing force W + W _E s U  b  628.9 + 789.8 865.5  of  1.82  4.19  =  1.64  It  should  be n o t e d t h a t  i f softening  t h e f a c t o r of s a f e t y would have a  e f f e c t of the deposit  has not been c o n s i d e r e d  i f a n a l y s i s was p e r f o r m e d a s s u m i n g c a l m s e a s u r f a c e  value and  conditions.  CHAPTER 7  CONCLUSIONS  Travelling the It  seafloor. induces  pressures  s u r f a c e waves e x e r t harmonic p r e s s u r e  This pressure  l o a d i n g has  a harmonic s t r e s s f i e l d i n the deposit.  The  and  two  e f f e c t s on  i t may  depth of  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 if  the e f f e c t i v e s t r e s s f i e l d  i n the deposit  condition for instability w i l l r e s i d u a l pore pressures The  and  occur  a n a l y s i s b a s e d on B i o t ' s t h e o r y uncoupled a n a l y s i s .  STAB-MAX, h a s pore pressures  and  can  be  The  most  severe  t h e maximum  by.  of p o r o - e l a s t i c s o l i d  by a  o r by  an  analysis for a A computer  implement the a n a l y s i s .  e f f e c t i v e s t r e s s e s i n the deposit  for  Mohr-Coulomb  i s known.  a coupled  deposit.  estimated  obtained.either  depth i s presented.  been developed to  be  when t h e s e a b e d h a s  In this thesis,  l a y e r e d d e p o s i t of f i n i t e  such as  on  to r e s i d u a l pore  can  t h e e x t r e m e wave p a s s e s  effective stress field  the  give r i s e  instability  loading  coupled appropriate  horizontally  program, STAB-MAX g i v e s  for a given  wave  load ing. For deep, u n i f o r m , stress f i e l d  can  pore pressures The  be  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  c o m p u t e d by  independently  i n t h e d e p o s i t and  then using  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  on d a t a  f r o m an  Siddharthan  and  extensive Martin  using  pore pressure  effective  stress  stress  t o be  of  and  principle.  r e q u i r e a computer.  Based  t h e p r o g r a m , STAB-MAX, F i n n , analyses.  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  the theory  field  evaluating total  (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  T h e y showed t h a t t h e t o t a l be o b t a i n e d  study using  does not  effective  elasticity  (Fung, 1965).  They f o u n d  independent of h y d r a u l i c a n i s o t r o p h y  for  can the  soft  76  c o a r s e and f i n e sands.  p  =  r  In s t i f f of  r  c o a r s e sand, t h e d i s t r i b u t i o n of pore pressure  =  p e  permeability  not  x  r a t i o and c a n be computed  in stiff  ratio,  f i n e sands i s independent o f t h e a n i s o t r o p i c n o t be e x p r e s s e d  i n simple  functional  T h e g e n e r a l i s a t i o n made f o r d e e p , u n i f o r m ,  saturated  deposits can  deposit  one should  stresses  but i t could  or layered  using  i s very  30%  soil  An e x a m p l e c o n s i d e r e d  higher  cation.  should  because of the r a t h e r  properties  i n loose  stability  take i n t o account pore  long  duration  should  properties The  should provide  deposit  of f i n i t e  load-  the modification give  about  without the m o d i f i -  include m o d i f i c a t i o n of  i n r e s i d u a l pore pressures.  p r o g r a m , STAB-W, h a s b e e n d e v e l o p e d t o e v a l u a t e  soil  that  evalua-  pressure  o f most s t o r m wave  than that obtained  analyses  s o i l properties f o r the increase  on  The  f o r the presence of r e s i d u a l pore pressures,  Therefore,  a layered  a c t i o n o f t h e wave  sandy s o i l s .  i n this thesis reveals  r e s i d u a l pore pressures  in  or layered  STAB-MAX.  c a u s e d by c y c l i c  important  t i o n of r e s i d u a l pore pressures  of  So, f o r a s h a l l o w  a program such as  r e s i d u a l pore pressures  shear s t r e s s e s  dissipation  deposits.  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  i n the deposit, The  ings.  using  z  b e made f o r s h a l l o w  induced  i s a function  -A/K /K Z x ^  distribution  form.  using  -Xz  the anisotropic permeability  p The  p e o  I t c a n be e s t i m a t e d  A computer  the r e s i d u a l pore  depth taking the e f f e c t s of pore  pressures pressures  into account.  cover provided  f o r a buried  p i p e l i n e w i t h p o s i t i v e buoyancy  be s u c h 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 adequate 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 during  wave  loading.  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 m a s s (W ) a n d v e r t i c a l  component o f t h e  s  s h e a r f o r c e (RyO  along  the s l i p  s u r f a c e a r e t h e two f o r c e c o m p o n e n t s  r e s i s t u p w a r d movement o f a b u r i e d p i p e l i n e . instability  There c a n be a r e g i o n  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 any  that  than the strength of the deposit.  T h i s means  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  selected f o r p i p e l i n e breakout problem w i t h i n t h i s r e g i o n . region w i l l  b e d e e p e r when t h e d e p o s i t  a s t h e e x t r e m e wave p a s s e s b y .  has higher  a slip  surface  The  instability  r e s i d u a l pore  The r e g i o n o f i n s t a b i l i t y  that  pressures  c a n be  obtained  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 the presence of higher  r e s i d u a l pore pressures  e l a s t i c method o f i n s t a b i l i t y r e s i d u a l pore pressures and  analysis.  s  An a p p r o p r i a t e  the poro-  Furthermore, the presence of  may g i v e r i s e t o a n i n c r e a s e i n p o s i t i v e b o u y a n c y  r e d u c e t h e e f f e c t i v e m a s s (W )  seepage.  and t h e n u s i n g  o f t h e d e p o s i t due t o t h e upward  example has been c o n s i d e r e d  i n detail  to explain  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 u s e d t o a n a l y s e p i p e l i n e flotation  problems. The  deposit. In these  a n a l y s i s presented  here i s r e s t r i c t e d  to horizontally  B u t t h e same a p p r o a c h c a n b e e x t e n d e d t o a n a l y s e cases,  the influence of s t a t i c  c r i t e r i o n and on p o r e p r e s s u r e must be c o n s i d e r e d . d u r i n g and a f t e r  considered  c h a r a c t e r i s t i c s of the deposit  p o s s i b l e displacement  t h e wave l o a d i n g n e e d s t o b e c o n s i d e r e d .  structures distorts  the pore pressure  h e r e a n d may r e s u l t  seabeds.  shear s t r e s s e s on t h e f a i l u r e  generation  I n t h i s case,  sloping  layered  field.  This  of the deposit, The p r e s e n c e o f  e f f e c t has n o t been  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 reported.  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 b y t h e t h e o r y was f o u n d  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  of t h e P i p e l i n e F l o t a t i o n Research Pipeline Division, pp. 2.  Bea, The  3.  ASCE, V o l . 92, No.  PL1, P r o c . Paper  R.G.  ( 1 9 7 1 ) , "How  O i l and  Gas  B e c k m a n , W.J.  B i o t , M.A.  J o u r n a l , No.  29, p p .  88-92.  (1970), "Engineering C o n s i d e r a t i o n s i n the Design  10, O c t o b e r ,  pp.  Byrne,  Control Federation,  (1941), "General Theory of Three-Dimensional C o n s o l i d a t i o n , "  P.M.  and  15, No.  Christian,  F i n n , W.D.  2, pp.  J.T.,  "Large Diameter  Liam  7.  T a y l o r , P.K.,  Ven,  J . K . C . and  D a v i d , R.E.  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Evalua-  1819-1824.  f r o m Waves," P r o c e e d i n g s ,  Conference,  Houston, Texas,  Paper  83  APPENDIX I  FORMULATION OF F I N I T E ELEMENT EQUATIONS FOR ANALYSIS OF RESIDUAL PORE PRESSURES  The  3  governing  K , z 3u>  H 7~ (  equation i s 3u g.  ,3u  v ^"  =  ^  m  (A.I-l)  (  w At any i n s t a n t o f t i m e ,  (3u/3t  function of z only.  say  9u 3t  g_ ) 3t  m „ (•  v  v  Now,  =  then equation (A;I-1) '3  reduces t o  K  , =  Q  g  Q(z)  , z 3vu  ^  - 3 u / 3 t ) may b e c o n s i d e r e d a s b e i n g a  (  z  v )  w A.I-l  Constructing Functional Functional J f o ra differential J  Using  =  e q u a t i o n o f t h e f o r m Au = f i s  (Au,u) - 2 ( f , u )  this,  the functional J f o r this differential  equation i s  D  ^ ( f |w  )  U  '  2  Q  (  Z  )  u  }  d  Z  0 rD r z i° .y 3z •w  4  K  3  The  u  w  o  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 3u 3z  =  0  at  z  =  0  0  at  z  =  D  (No f l o w o c c u r s a t t h e b o t t o m  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, ? - J  2  K  =  A 9z  — Y  0 A.1-2  of  Technique  us 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  a f i n i t e number o f e l e m e n t s . ^  ^total  Then,  n all  To u s e t h i s m e t h o d function  (A.1-2)  w  F i n i t e Element Let  dz + 2 Q(z) udz  (finite  i  ^  ^elements  elements  element) one must s e l e c t an i n t e r p o l a t i o n  f o r u such that t h e f u n c t i o n a l  c a n be computed.  t i o n a l h a s 9u/9z t e r m , t h e i n t e r p o l a t i o n f u n c t i o n o r d e r 1 i n z. in f i n i t e  ( A . I 3)  This c r i t e r i o n i s referred  element l i t e r a t u r e .  Since the func-  f o r u must be a t l e a s t  t o as "completeness c r i t e r i o n "  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 over an element a s u  -  (  =  N  -i7>  1  T  u - l  i  u  +  i  i :  u  i i  +  i  ( A  t  u  i' i+l u  n  1  element,  u. u  =  t  i  +  l  ;  * 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 . i e  =  d i s t a n c e measured from t h e t o p o f t h e element p a r a l l e l to z direction.  =  .{1 -  rn  N  x  4 )  (A.1-5)  6  where d ^ i s t h e t h i c k n e s s of t h e i ^ e u. -x  - -  n/d^  Now, f r o m e q u a t i o n  n/cL] =  (A.1-2)  nodal interpolation  function.  =  element  t h e f u n c t i o n a l f o r element i  K -5- A w  0 =  say  T  2  dn +• 2Q(n)udn  e e + I 1 2  where  z ,3u. ^ ww ^ K  ,  2  "  d. l and  Now  2Q(n)udn  I,  consider  fl7  31  ^ z ^ u ^ dn Y 3n 3n l l j w ^ '  3{u*} Substituting  for  from e q u a t i o n (A.1-5),  K Y  3{u } - l e  =  [K ] = e  1  1 -1  2[K ]  {u?}  m  -m  -m  m  K and  w  K _z J _ Y d. w 1  2  e  where  u . . -u . 1+1 x d.  z  Y d. w 1  ]  d. l _1_ d.  86  Now  consider d. x 2 Q(z)  udn  d. x  3u 2 m  v  (  IE "  1^  ^  0 Simple t r a p e z o i d a l  i n t e g r a t i o n scheme c a n be u s e d  to evaluate  this  integral, i.e.,  I* Here,  =  1 r , 3U g. ) u ] 2m.. {•£[(• v 2 - 3 t - 3t L  subscript  i and  9u  i  " Tt^  +  ) u ]  i+1  }  d  i  i+1 r e f e r t o t h e v a r i a b l e s a t n o d e s i a n d  i+1  i.e.,  31, =  2  m d. . v i 3u  3{u } -x e  f  2  K  v  2[D ]  =  3t  ).  3t ' i _  dt  . V  m  3u 3t '  v  d. x  element 3{u^}  i  Ju _ _j> 3t 3t 'i+1  Now, 3J  3t i + l ;  where  p  3u  e  v  and  \  3u  3u 3t  3{u?} - X  3{u } 6  - X  (A.1-8)  87  from  equations  (A.1-7) and (A.-8)  •in [K ] { u ? } +  I  [D ]  v  3t ^u  | V  Using v a r i a t i o n a l  #3 iTu }  =  {  0  3t ' i _  3 t ±+l  3t  J  principles,  }  3J  ^  x. e.  &>  element  all elements  _  3{u } e  iE) 3 f x -( . \ ._3u. ( v  X [Ke ] { ux } + 6  [D ] e J  _  v  (_§.) 3t i 3u _ , g. ;  ^3t i+l  3u [D]({-^} - { ^ } )  {0}  =  ^3t i + l  ;  Summing u p f o r a l l t h e e l e m e n t s ,  f  |  ;  one w i l l g e t  a  [K]{u} + where m a t r i x function and  {0}  (A.1-9)  [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 ,  of compressibility  m,  matrix  [D] i s a  which w i l l vary w i t h pore pressure  v  ratio,  { 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. {3u  =  F o r 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  / 3 t } i s t a k e n a s t h e maximum o f i t s v a l u e j u s t  materials,  above and below t h e  node. Equation tial  (A.1-9) c a n be r e g a r d e d  e q u a t i o n s a n d may b e a p p r o x i m a t e l y  t+At as  as a s e t of o r d i n a r y d i f f e r e n -  integrated  over  the interval t ,  follows:  [K](3'{u  t + A t  + a'{u })At t  +  [D]({u  t + A t  >  -  3u " ^ >  A  t  )  =  <> 0  (A.I-10)  88  w h e r e a'+B at  = 1 and  1  subscripts  t , t+At i n d i c a t e the v a l u e s of a  t i m e s t , t + A t r e s p e c t i v e l y , and  the bar denotes an average v a l u e  the i n t e r v a l t , t+At ( c a l c u l a t e d from over  that  interval).  approximations; stable.  t + A t  }  Then e q u a t i o n =  {BQ1}  is  i s a s s u m e d t o h a v e a v a l u e 0.5  1  +  (A.I-10)  over  ratio  to d i f f e r e n t  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  N i c h o l s o n method). [AQ]{u  the average pore pressure  D i f f e r e n t v a l u e s o f a' c o r r e s p o n d  I n t h i s program, B  variable  c a n be w r i t t e n  always  (Crankas  {BQ2}  (A.I-11)  where  After  [AQ]  =  [K]B'At +  [BQ1]  =  (-[K]a'At  [BQ2]  =  +  [D]{u }) t  3u [D]{^}At  incorporating  modified  [D]  e s s e n t i a l boundary c o n d i t i o n s ,  equation  t + A t  }  =  {BQ1*} + {BQ2*}  =  {BQ*}  (A.I-12)  Here, the a s t e r i s k denotes v a l u e s of the v a r i a b l e s c a t i o n f o r the boundary The The  above e q u a t i o n i s solved  program has  the s o i l  the option  i n time. matrix  equation  by  the Gaussian  I t i s assumed t h a t  (A.I-12)  [D] becomes v a r i a b l e a n d  [D] and  repeating  elimination  proce-  the c o m p r e s s i b i l i t y  pore pressure r a t i o ,  equation  (A.I-12)  as v a r i a b l e s  has  t o be  until  the process  values  to  converges.  [D]  forward  then  solved  e s t i m a t e of pore p r e s s u r e r a t i o  t h i s procedure  values  the matrix  c a n be u s e d t o m a r c h t h e s o l u t i o n  I f the c o m p r e s s i b i l i t y values are treated  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 late  modifi-  f o r t r e a t i n g the c o m p r e s s i b i l i t y  p r o f i l e does not v a r y w i t h  i s c o n s t a n t and  after suitable  conditions.  e i t h e r constant or varying. of  is  as  [AQ*]{u  dure.  (A.I-11)  the  iteracalcu-  89  APPENDIX I I  CALCULATION  OF C Y C L I C SHEAR STRESSES  I N AN E L A S T I C  SEABED UNDER WAVE LOADING  A horizontally layered s o i l deposit harmonic pressure shear  increment  subjected  i s shown i n F i g . 1 7 .  T h e maximum  stresses i n t h e deposit a r e required i n order  build-up  i n r e s i d u a l pore pressures.  a r e used  i nsolving for thestresses.  to a travelling cyclic  to determine t h e  The f o l l o w i n g b a s i c  assumptions  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 that t h e t r a n s i e n t response to pressure c a n b e n e g l e c t e d a n d t h e wave p r e s s u r e s as s t a t i c l o a d s .  2)  Theory o f l i n e a r  3)  Plane  4)  The s o i l d e p o s i t i s assumed t o be a c o n 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 b e d . These assumptions reduce t h e problem t o a c l a s s i c a l s o l i d mechanical problem.  elasticity  enough loading treated  i s applicable.  strain condition prevails.  Introducing an A i r y  s t r e s s f u n c t i o n , <j>, t h e p r o b l e m o f d e t e r m i n i n g  stres-  ses reduces t o s o l v i n g ,  4  +  2  7^r  3x Let ting  +  ^  3x 3z  -  0  (A  A  4  i n equation f(z) - 2A  The g e n e r a l f(z)  2  =  C  i e  1  f  Substitu-  (A.II-1) U  ( z ) + f  solution of t h i s A z  II  3z  <J> = f ( z ) C o s A x w h e r e A = 2 i r / L a n d f ( z ) i s a f u n c t i o n o f z . this  -->  + C e 2  _ A z  I V  (z)  =  0  equation i s + C ze 3  A z  + ^ze * -  2  (A.II-2)  -«a i 1I  »•  L  V  'H  \  y = 2 Sin 2 7 r ( ) s  r  d  Ap  =  P  0  Sin  2TT(- -) L  1  I  (N-I)  N  t h  t h  Layer  Layer  (N + I)  ^  X  Layer  st  th  Layer  ^^Rigid  Bottom  z FIG.  17  Wave F o r c e s o n a H o r i z o n t a l l y  Layered  Deposit  ^  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  determined  by a p p r o p r i a t e b o u n d a r y c o n d i t i o n s , <j>  =  ( C ^  + C e"  2  + C ze  A z  2  X z  3  + C ze~  A z  4  )  i.e.,  CosAx  (A.II-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 o f 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 o f t h e f o r m :f  v  4>(n)  =  r  Az [ c , ..e (n,l)  , -Az + c. „.e (n,2)  , + c,  „.ze (n,3)  Az  - i z , + c, z e JCosAx (n,4) M  (A.II-4) The a r b i t r a r y c o n s t a n t s c a n be f o u n d  using the following  boundary c o n d i -  tions .  (1)  (2)  A t z=0,  a  and  a  zz xz  =  Y H — C o s A x 2Cosh(Ad)  =  0  =  z=D, v=w=0 w h e r e v,w a r e d i s p l a c e m e n t s directions, respectively. a  XX  7  v a  zz  a  Special  , and a  XX  xz  are given  2 3 <)> =• — , 2 9z  a zz  n  infinite  deposit.  associated w i t h e^ ±  r  <j> = ( c ^ e  i.e.,  2 3 <r = — , „ 2 9x  2 a xz  z  3(j)  = - . ^ 9x3z :  / A  T T  C\  (A.II-6)  I f D -> °° t h e n t h e c o e f f i c i e n t s  terms should  vanish.  , -Az. + C2ze )CosAx  b o u n d a r y 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  shear  stress, =  i n x and z  by  Using  c  (A.II-5)  Case:  Uniform  x  p CosAx o  (a  xz  )  o  Aze  cyclic  max  Y H w 2Cosh(Ad) p  t h a t t h e maximum  -Az  , , -Az Aze  (A.II-7)  92  It  i s interesting  elastic  coupled  to note  analysis,  t h a t Yamamoto ( 1 9 7 8 ) , who showed t h a t when G3 -> 0,  t h i c k uniform deposit reduces s h e a r m o d u l u s o f s o i l and  H o r i z o n t a l l y Layered The . /  <Kn) The  \  Airy  x  f o r an  c  t o t h e same e x p r e s s i o n .  an infinite  Here, G i s the  g i s the c o m p r e s s i b i l i t y of  porewater.  Deposit  s t r e s s f u n c t i o n f o r the n Az  r  =  performed  [ c , .,e (n,l)  , + c,  e (n,2)  —Az  O N  , + c.  layer i s Az  ,ze (n,3)  + c.  ,.ze (n,4)  —Az -, JCosAx  number o f 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  i n the following  by  solving  simultaneous  equations  set  up  manner.  At the s e a f l o o r ,  z=0  Y H W  zz  2Cosh(Ad)  CosAx  and  At the i n t e r f a c e of the n t h 0  x z > w,  v  s h o u l d be c o m p a t i b l e .  i.e.,  zz(n)  a  xz(n)  CT  xz  =  0  (n+l)th layer,  (2 E q n s . ) stresses  o  z z  ,  Here,  zz(n+l) xz(n+l)  (n)  =  W  (n+1)  (n)  =  V  (n 1) +  This leads to 4 equations per At  d  an  a  the bottom, z =  interface,  D, W  (NM)  " °  „ _ , (2 E q n s . )  (NM) T o t a l number o f e q u a t i o n s a n d t i o n s c a n be The  cyclic  u n k n o w n s i s 4ANM.  These simultaneous  solved f o r the a r b i t r a r y constants c ^  shear  stress at a depth  z i s given  by  equa-  j ^ ; i=l,NM; j = l , 4 .  T  c  •  =  =  a  xz  (z)  c, ..A (n,l)  max A  2 Az e  - c, A (n,2)  • . + c. ,\ {A-A^z }e (n,4) Here, n  i s the  number o f  .2  O N  the  e  -Az  + c,  - " Az  layer  involved.  , 2 , , l A z+A>£ (n,3) N  

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