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Problems in nonlinear analysis of movements in soils Wedge, Neil Edward 1977

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PROBLEMS I N NONLINEAR ANALYSIS OP MOVEMENTS I N SOILS  N E I L EDWARD WEDGE B.A.Sc, University  of British  Columbia,  197^  A THESIS SUBMITTED I N PARTIAL 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 Engineering)  We a c c e p t t h i s  thesis  to the required  as c o n f o r m i n g standard  THE UNIVERSITY OF B R I T I S H COLUMBIA M a r c h , 1977 fcT)  N e i l Edward Wedge, 1 9 7 7 .  In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y  of B r i t i s h C o l u m b i a , I agree that  the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s .  It  i s understood that copying o r p u b l i c a t i o n  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my wr i t ten pe rm i ss i o n .  Department o f The U n i v e r s i t y  C i v i l Engineering o f B r i t i s h Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  D  a  t  e  March, 1977  s ABSTRACT  The p r o b l e m s of  the  tures  load-deformation are  analysis  are  soils  cussed.  reviewed  are  this  for  anisotropy  soils  of  yield  is  use  merely  reduce  shown,  with  yielded  soil  yield  value  of  elements  It  is  if  and t h e  the  value  dis-  based  authors  on  act  elastic of  of  b u l k modulus  ii  practice  the is  shear  not  soil  and It  is  results  compressible examples,  elements  modulus  reduced.  Many  yield.  further numerical yielded  neg-  represent  ratio  at  the  plastic  to  unrealistically  of of  this  are  materials.  Young's modulus  with  error,  behaviour  as  Poisson's  that  reviewed  stress-path  common p r a c t i c e  value  elastic  relations .  stress-strain  of  the  and  major sources  soil  being  shown,  limitations  of  predicted behaviour  more r e a l i s t i c at  the  used  of  is  a constant  commonly  incrementally  Although s o i l s  it  dis-  effects  n u m e r i c a l examples,  yield. the  yield,  two  b e h a v i o u r by m o d e l s  researchers  that  and the  are  stress-strain  have  discussed.  after  post-yield  after  soils  the  nonlinear and  relations  relations  struc-  advantages  and t h e i r  isotropic,  The r e p r e s e n t a t i o n  materials  in  are  shows t h a t  commonly u s e d  after  relative  analysis  and s o i l  incremental  Stress-strain  soils  nonlinear  soils  E v i d e n c e r e p o r t e d by o t h e r  study  lected.  and t h e i r  stress-strain  that  materials.  of  c r i t i c a l l y examined  These  with  Methods o f  discussed.  assumption  in  response  investigated.  advantages for  associated  is  is  reduced  TABLE OF  CONTENTS Page  ABSTRACT TABLE OF CONTENTS  i  i  i  L I S T OF FIGURES  v  NOTATION  v i i  ACKNOWLEDGEMENTS  x i i  CHAPTER 1  INTRODUCTION  1  2  METHODS OF INCREMENTAL ANALYSIS FOR NONLINEAR MATERIALS  3  2.1 2.2.  General Discussion Analysis  One I t e r a t i o n p e r L o a d Increment  \7  2.2.2  I t e r a t i o n s U s i n g a Convergence C r i t e r i o n  10  2.2.3  Application Forces  11  2.2.4  Modified Method  of  Correction  Newton-Raphson  ;  12 18  STRESS-STRAIN RELATIONS 3.1  General Discussion Strain Relations  3.2  Experimental Studies  24  3-3  Limitations Relations  33  3.4 4  3  I t e r a t i v e Methods 2.2.1  3.  of Incremental  of Stress-  18  of Stress-Strain  M a t e r i a l Behaviour  After Yield  PROBLEMS I N NONLINEAR F I N I T E ELEMENT ANALYSIS OF DEFORMATIONS I N SOIL BODIES Iii  38 42  iv  CHAPTER  Page 4.1  4.2 5  General D i s c u s s i o n of Nonlinear F i n i t e Element A n a l y s i s of Deformations i n S o i l Bodies Common A s s u m p t i o n s  42  4.1.2  Method o f Analysis  44  Numerical  of  Incremental  Studies  47 66  BIBLIOGRAPHY  APPENDIX 2 :  42  4.1.1  CONCLUSION  APPENDIX 1:  ' .'. .'.  70 DERIVATION -OF THE STIFFNESS MATRIX FOR THE LINEAR STRAIN TRIANGULAR ELEMENT STRESS-STRAIN  SUBROUTINES  72 j .  79  L I S T OF FIGURES FIGURE  Page  1  I n c r e m e n t a l A p p r o x i m a t i o n o f a Nonl i n e a r L o a d - D e f l e c t i o n Curve  5  2  One  9  3  Newton-Raphson M e t h o d f o r t h e d i m e n s i o n a l Case  Iteration  p e r Load Increment Method One-  14  4  H y p e r b o l i c S t r e s s - S t r a i n Curve  25  5  Mohr-Coulomb  27  6  Two S t r e s s - P a t h s w h i c h Y i e l d D i f f e r e n t S t r a i n s f o r t h e Same S t r e s s I n c r e m e n t  35  7  F i n i t e E l e m e n t Mesh  52  8  Load-Settlement Curves f o r Analyses i n w h i c h t h e S h e a r M o d u l u s was R e d u c e d t o One-Hundred Pounds p e r S q u a r e F o o t a t Y i e l d and t h e B u l k M o d u l u s was n o t Reduced a t Y i e l d  5^  Load-Settlement Curves f o r Analyses i n w h i c h t h e S h e a r M o d u l u s was R e d u c e d t o F i v e Pounds p e r S q u a r e F o o t a t Y i e l d and t h e B u l k M o d u l u s was n o t R e d u c e d a t Yield  55  Locations of F a i l e d Elements f o r the A n a l y s i s i n w h i c h t h e S h e a r M o d u l u s was R e d u c e d t o F i v e Pounds p e r S q u a r e F o o t a t Y i e l d , t h e B u l k M o d u l u s was n o t R e d u c e d a t Y i e l d , and Twenty Pound L o a d I n c r e m e n t s Were Used  58  Movements o f Nodes D u r i n g t h e L o a d I n c r e ment f r o m F i v e H u n d r e d and Twenty Pounds t o F i v e H u n d r e d a n d F o r t y Pounds i n t h e A n a l y s i s i n w h i c h t h e S h e a r M o d u l u s was R e d u c e d t o F i v e Pounds p e r S q u a r e F o o t a t Y i e l d , t h e B u l k M o d u l u s was n o t R e d u c e d a t Y i e l d , and Twenty Pound I n c r e m e n t s Were Used  59  9  10  11  Failure  v  Criterion  Load-Settlement Curves f o r Analyses i n w h i c h P o i s s o n ' s R a t i o had a C o n s t a n t V a l u e o f 0.42 and Y o u n g ' s M o d u l u was Reduced t o O n e - H u n d r e d Pounds p e r S q u a r e F o o t at Y i e l d L o c a t i o n s of F a i l e d Elements f o r the A n a l y s i s i n w h i c h P o i s s o n ' s R a t i o had a Constant Value of 0.42, Young's Modulus was Reduced t o F i v e Pounds p e r S q u a r e F o o t at Y i e l d , and F i f t e e n Pound L o a d I n c r e m e n t s Were Used Movements o f Nodes D u r i n g the L o a d I n c r e m e n t f r o m O n e - H u n d r e d and Twenty Pounds t o O n e - H u n d r e d and T h i r t y - f i v e Pounds i n t h e A n a l y s i s i n w h i c h P o i s s o n ' s R a t i o had a C o n s t a n t V a l u e o f 0 . 4 2 , Y o u n g ' s M o d u l u s was Reduced t o O n e - H u n d r e d Pounds p e r S q u a r e F o o t at Y i e l d , and F i f t e e n Pound L o a d I n c r e m e n t s Were Used L i n e a r S t r a i n T r i a n g u l a r Element Areas  for  Natural  Coordinates  NOTATION The f o l l o w i n g this  thesis  and t h e i r  is  a list  of the  symbols  used  in  definitions  the i n v e r s e of asymptotic value resultant deviatoric stress  of  the  B  =  t h e r a t i o o f t h e s h e a r modulus a f t e r to that before y i e l d  c  =  the cohesion i n t e r c e p t f a i l u r e envelope  d  =  the r a t e of i n c r e a s e of the tangent of Poisson's r a t i o with s t r a i n  dV  =  an i n f i n i t e s i m a l  E^  =  the  initial  tangent  =  the  tangent  value  P  =  the decrease i n the ratio for a tenfold principal stress  G  =  t h e t a n g e n t v a l u e ..of P o i s s o n ' s r a t i o f o r an I s o t r o p i c s t r e s s equal to atmospheric p r e s sure  =  the i n i t i a l modulus  tangent  G^  =  the  tangent  shear  h  =  the  depth below  K  =  Y o u n g ' s modulus number  =  the i n i t i a l modulus  tangent  =  the  value  =  the c o e f f i c i e n t at r e s t  G. 1  K. 1  K °  tangent  vii  of  change  of  value  volume of  Y o u n g ' s modulus  Y o u n g ' s modulus tangent value of P o i s s o n ' s i n c r e a s e i n the minor  value  of  the  shear  modulus  the  of  Mohr-Coulomb  in  value  yield  soil  value  of  the  surface  of the bulk  bulk  modulus  l a t e r a l earth  pressure  viii  the slope of s t r e s s - s t r a i n curve f o r the o n e - d i m e n s i o n a l c a s e a t t h e end o f t h e l a s t l o a d Increment t h e r a t e o f change o f t h e t a n g e n t b u l k modulus w i t h t h e mean n o r m a l - s t r e s s the i n i t i a l tangent modulus the  tangent  value  Y o u n g ' s modulus  value  of  the  of  the  constrained  constrained  modulus  exponent  a number t h a t i n d i c a t e s t h e r a t e o f i n c r e a s e o f t h e t a n g e n t b u l k modulus w i t h r e s p e c t t o a normalized volumetric s t r a i n atmospheric  pressure  the d d iissppl a l cement increment  at  the  the  end o f  end o f the  i  h  load  l o a d at  the the  th n approximation of the displacement end o f t h e i t h l o a d i n c r e m e n t  the  average  i ^  t  h  load  t  the  load f o r the  i  the  h  load  the the  failure  yield  ratio  stress-level resultant  deviatoric  stress  deviatoric  strain  the  end o f  the  i  t  h  load  the n approximation of increment  the  i  t  h  load  volume the  resultant  t h e ss tt rraaiin increment t  the  at  h  volumetric  strain  at  increment  t h e r a t i o o f Y o u n g ' s modulus b e f o r e to that a f t e r y i e l d the  increment  ix  e  =  the  characteristic  =  the  normalized volumetric  =  the  unit  =  the shear s t r a i n j direction  Aq .  =  t h e change i n d i s p l a c e m e n t increment  AQ.  =  the  i ^  increment  x,.  =  the  1^  natural  =  the s t r e s s increment  =  the  major p r i n c i p a l  stress  =  the  minor p r i n c i p a l  stress  e  vc vn  y Y.. 1  J  n  1  1  a_.l  ^ j~ 2^ult a  a  volumetric  strain  strain  weight for  the  of  i  plane  in  i n the  the  i ^  coordinate th  =  t  h  e  a  s  y  m  P  t  at  o  t  x  C  the  end o f  deviator  I  load  stress  the  deviator  a|  =  the  major p r i n c i p a l e f f e c t i v e  stress  al,  =  the  minor p r i n c i p a l e f f e c t i v e  stress  the  error in a calculated  =  the  mean n o r m a l  =  the  value  of  Poisson's r a t i o  before  =  the  value  of  Poisson's ratio  after  the  tangent  e m  v2 v  t  stress  the  =  a  load  load  (a^-a^)|.  a  h  at  failure  stress  stress  value  of  Poisson's  T.. ^  =  the shear s t r e s s j direction  <> j  =  the  angle  <j)'  =  the  effective  angle of  =  the  vector  i n i t i a l nodal  ±  of  on t h e  internal  i  plane  yield yield  ratio in  the  friction internal  friction  "Vectors: {q}  of  displacements  X  =  {q}. -  1  =  {q}. 1  (Q}  ,  n  the v e c t o r of nodal displacements of the i t h l o a d increment  =  the  {Q}.  =  the n o d a l f o r c e v e c t o r i t h load increment  {Q  e  }.  = 5  {e}  Q  {e}. {e}.  initial  nodal force  end  vector at  the  end o f  th the n vector of nodal c o r r e c t i o n f o r the i t h load increment  =  the  vector  =  the the  v e c t o r o f element s t r a i n s i load increment  1  the  th the n a p p r o x i m a t i o n of the n o d a l d i s p l a c e ment v e c t o r a t t h e end o f t h e i t h l o a d increment  q  1  at  t  of  initial  element  the  forces  strains at  the  end  of  h  =  the n ^ approximation of the v e c t o r of element s t r a i n s a t t h e end o f t h e i t h l o a d increment  6{q}  =  a vector  of  <5{e}  =  a vector  of v i r t u a l  =  t h e change i n t h e v e c t o r o f n o d a l d i s p l a c e ments d u r i n g t h e i t h l o a d i n c r e m e n t  1  '  {Aq}. 1  =  {Aq}. 1  ,  n  _ =  {Aq}!  virtual  nodal  displacements  element  strains  the n a p p r o x i m a t i o n o f t h e change i n v e c t o r o f n o d a l d i s p l a c e m e n t s f o r the i t h load increment t  h  the n ^ c o r r e c t i o n to the v e c t o r of d i s p l a c e m e n t s at t h e end o f t h e l increment z n  {AQ}. I  =  the  {Ae}.  =  t h e change i n t h e v e c t o r o f f o r the i t h l o a d increment  1  =  {Ae}. l  {Ae}!  j  n  = '  i  t  h  increment  of  nodal  the  nodal load  forces element  strains  th the n a p p r o x i m a t i o n o f t h e change i n t h e v e c t o r o f element s t r a i n s f o r t h e i ^ h l o a d increment th the n c o r r e c t i o n t o t h e v e c t o r o f element s t r a i n s a t t h e end o f t h e i t h l o a d I n c r e m e n t  xi  {Aa}.  =  the for  1  {Aa}.  =  l  {a}  j  n  {a}.  =  the v e c t o r o f element s t r e s s e s o f the i ^ h l o a d increment the for  ._^= ^  }.  =  vector  vector the i  t  of  h  initial  element  o f average element l o a d increment  at  vector of errors t h e end o f t h e i  t  [D]  =  the  stress-strain  [D]  =  the  element  [D]^..  =  the element s t r e s s - s t r a i n m a t r i x t o the i l o a d increment  [K ]. T  m  T  e  [Krp] 1  e  i  [K,p]  stress-strain  matrix appropriate  the  matrix  =  the g l o b a l tangent s t i f f n e s s m a t r i x p r i a t e to the i ^ h l o a d increment  =  the  . = 6  matrix  =  t  h  global  element  of load  matrix  element the i ^  1  [K ]  strain-displacement  the n a p p r o x i m a t i o n . o f the s t r a i n matrix appropriate to increment  5 1  [Krp]  h  h  element  =  t  i n element s t r e s s e s load increment  the  .  end  stresses  =  ei  the  the a p p r o x i m a t i o n of the v e c t o r average element s t r e s s e s f o r the i increment the for  h  stresses  [B]  [D]  stresses  th the n a p p r o x i m a t i o n o f t h e change i n t h e v e c t o r o f n o d a l d i s p l a c e m e n t s f o r the i load increment the  ! = """'^  {a  element  h  =  1  {a}..  t  t  Q  {a}.  change i n t h e v e c t o r o f the i load increment  tangent  tangent  stiffness  stiffness  n  stressload  appro-  matrix  the element tangent s t i f f n e s s m a t r i x p r i a t e to the i ^ l o a d increment  appro-  h  .  = the n a p p r o x i m a t i o n o f the s t i f f n e s s matrix appropriate load increment t  '  n  h  element tangent to the i ^ h  ACKNOWLEDGEMENTS  The advice  writer  and g u i d a n c e  Dr.  W.D. Liam F i n n ,  the  advice  Special  t y p i n g the  g i v e n by h i s the  and g e n e r o u s  thanks  gratefully  advice  acknowledges supervisor.  o f D r . N . D . N a t h a n , and  assistance  a r e due t o  g i v e n by D r . K . L e e .  Miss Desiree  manuscript.  xii  the  Cheung f o r  CHAPTER 1 INTRODUCTION  The  finite  used e x t e n s i v e l y in  soil  kinds of  most  convenient  matical also of  and movements  applied loadings. means  allows  the  use  of  of  soil  adequacy w i t h which the conditions  for The  prior  to  of  stress-path  soils  The established of  not  varthe kine-  structures.  It  i n the  form  method i n p r e d i c t i n g is  l i m i t e d m a i n l y by relations  the  and f a i l u r e  represented. soil  are n o n l i n e a r ,  stress-strain  stress  The  behaviour stress-strain  dependent  and  (9,10,13,17,18). criteria  post-failure  for  soils  are not  behaviour is  firmly  open t o  a  interpretations.  has b e e n a d o p t e d ,  linear  under  data.  of the  Once a model f o r t h e soil  been  geometrical,  relations  a simple problem.  failure  and t h e  soil  stress-strain  can be  dependent  of  structures  modelling of  failure is  relations  variety  soils  has  deformations  foundations  stress-strain  effectiveness  deformations  of  predict  o f r e p r e s e n t i n g the  a n a l y t i c a l or d i s c r e t e The  to  analysis  T h i s method p r o v i d e s  and e q u i l i b r i u m r e l a t i o n s  either  the  method o f  (5,8,9,11,12,18)  structures  ious  element  analysis  one  of  stress-strain  behaviour of  two g e n e r a l methods  may be u s e d t o  1  p r e d i c t the  of non-  deformations  of  2 soil as  structures  iterative  entire  under a p p l i e d l o a d i n g s .  elastic  analysis,  l o a d i n g i n one  strain properties compatible  model o f t h e incremental  soil.  and t h e  assumed t o  be  analysis  of  stress-strain which at  the  applied  loads  i n order to  In the  method,  the  second  loading is  stress-strain  load increment  properties end o f  the  and t h e  used  it  allows  history rather  of  the  i n order to  stress-strain  than j u s t  loading  as  i n the  the  body up t o  case o f  relations  associated i n the  deformations.  the  deformations  The p u r p o s e problems  of  of  with the  the  use  soil In  on  obtain  model o f has  the  the  of  the  with  the  soil.  load-displacement  elastic is  the  strains  interest,  corresponding to  study  is  advantage  loading of  iterative this  called  compatible  the  are  suitable  one may i t e r a t e  determination  the. s o i l  which i s  load increment.  The i n c r e m e n t a l t e c h n i q u e that  which  stress-strain  applied in  are  the  stress-  strains  behaviour of  increment,  known  applying  on t h e  obtain and t h e  l i n e a r w i t h i n each  each  of  a p p l i e d loads  analysis,  increments  consists  s t e p and I t e r a t i n g  used  w i t h the  One m e t h o d ,  to  existing  the  final  analysis. investigate  the  stress-strain  incremental nonlinear analysis  of  soil  CHAPTER 2  METHODS OF INCREMENTAL  ANALYSIS  FOR NONLINEAR MATERIALS  2.1  General  Discussion" o f Incremental A n a l y s i s  A nonlinear material is tionship strains  between  the a p p l i e d stresses  instantaneous  tangent  slopes  be e x p r e s s e d  point  and  moduli  at t h e  t  =  E ({a})  or E ({e}) t  (2-1)  v  fc  =  v ({o})  or v ({e})  (2-2)  t  t  value  stress,  and {e}  t  o f Young's modulus,  ratio,  i s the state  Since  of  the s t i f f n e s s  upon t h e v a l u e s of stress  the tangent  upon t h e t a n g e n t  or s t r a i n s  E  value of Poisson's  Hence,  c u r v e s and t h e y  F o r example,  tangent  the s t a t e  f o r such a m a t e r i a l a r e the  i n terms o f t h e s t r e s s e s  where E ^ i s t h e t a n g e n t  of  and t h e r e s u l t i n g  o f the s t r e s s - s t r a i n  of interest.  depends  rela-  is nonlinear. The  may  one f o r w h i c h t h e  e  strain. matrix of a f i n i t e it  or strain existing  moduli  T  i s the state of  of the moduli,  stiffness  [K ]  {a}  is =  i s the  is  also  element a function  i n that  element.  m a t r i x o f t h e element  based  [K ] , T  g  [K ({0})] T  3  e  or  [K ({e})] T  e  (2-3)  The s t a t e s  of  the  element  finite  tangent  stress  stiffness  and s t r a i n  and,  as  change  a result,  matrix is  d u r i n g the  no s i n g l e  applicable  loading  value  d u r i n g the  of  of  the  entire  analysis. One method o f dependency structure  of  the  d e a l i n g w i t h the  tangent  i n increments  stiffness  and t o  behaves l i n e a r l y w i t h i n each mation to Figure  the  true  1 f o r the For  of  finite  tial  into  response  general  n load increments  {Q},  {AQK .  on t h e  is  strain  load  the  material This  shown  divided into  approach i s  load vector,  acting  the  or  approxiin  case.  a body w h i c h has been  load vector  to  load increment.  one-dimensional  The t o t a l divided  matrix is  assume t h a t  stress-strain  elements the  stress  body,  to  as  a number  follows:  be a p p l i e d  "If  {Q}  then  the  is  Q  load  is  the  ini-  vector,  th {QK,  acting  at  the  end o f  the  i  load increment  {Q}  i E {AQ}. j=l  is  given  by {Q}.  =  °  1  For the i [ K ] {Aq}. T  where  [K^] i s  +  the  th =  global  load increment, {AQ}. stiffness  gent m o d u l i a p p r o p r i a t e t o load  increment  ment  vector  and {Aq}^  resulting  (2-4)  3  is  the  from the  The i n c r e m e n t s  of  (2-5)  matrix based  element  the  {AQK,  stresses  on t h e  i n the i ^ *  incremental nodal a p p l i c a t i o n of  element  stress  tan1  displace-  {AQ}^. resulting  from  Piecewise Linear Approximation  -Correct LoadDisplacement Curve  qi - l  q  i+i  Displacement, q  FIGURE 1  INCREMENTAL APPROXIMATION OF A NONLINEAR LOAD-DEFLECTION CURVE  the  a p p l i c a t i o n o f the  one b e g i n s  the  i  analysis  t  of  the  m o d u l i arid t h e  global  the  load increment  are  of  the  increment.  order to  find  l o a d increment  h  the  are not  increment.  stiffness  Therefore,  both  matrix appropriate  unknown when one b e g i n s  An i t e r a t i v e  known when  the  to  analysis  p r o c e d u r e must be u s e d  in  these.  The t o t a l  nodal displacement  vector,  { q K , at  t h the  end o f '  where  {q}  Q  the  i  l o a d increment i  {qK  Is  =  the  {q}  +  Q  vector  is  i ^  vector,  of  (2-6)  initial  { A e K , resulting  load increment {Ae}.  where  [B]  is  w h i c h depends if  geometrical  the  nodal  is  given  =  [3]  at  the  from the  nonlinearity  end o f t h e  i ^  the  incremental  a p p l i c a t i o n of  the  by (2-7)  {AqK  e  upon t h e  The t o t a l  displacements.  element,  strain-displacement  only  by  {AqK  For a p a r t i c u l a r f i n i t e strain  given  initial is  m a t r i x f o r the  geometry  of  the  element element  negligible.  strain vector,  {eK,  l o a d increment  is  f o r the  given  element,  by  i {eK  =  {e}  +  where  {e}  o  is  the  £  initial  3  strain  The i n c r e m e n t a l  (2-8)  {Ae}.  J-1  °  vector,  stress  vector,  {Aa}^,  for  the  th element,  f o r the {Aa}.  i  load increment =  [D] '{Ae} e  1  is  given  by (2-9)  where  [D]  is  the  element  stress-strain  matrix appropriate  th to  the  i  load increment,  stresses  In the  w h i c h depends  upon t h e  element  i n c r e m e n t and must be f o u n d by an  Iterative  procedure. The end  o f the  I  t  total  {a}. where  {a}  2.2  Iterative  2.2.1  o  the  is  { a K , at  the  g i v e n by  i  = {a}  + E {Aa} .  ° initial  J=l element  One I t e r a t i o n p e r L o a d  improve the  vector,  (2-10)  3  stress  vector,  Methods  The to  stress  load increment  h  is  element  first  Increment  method t o  estimates  of  be  the  considered is  stress-strain  an  attempt  matrix,  [D] ^ ,  th and  the  stiffness  ment w i t h o u t  matrix,  Each time,  calculated ment.  g i  the  tangent  from t h e  These moduli  for  [  K  e  l  j  i  criterion.  The method was  ]  a particular  .  is  load i n c r e -  analysed twice.  f o r the  stress  are used t o  calculate  and t h e  = =  element  [D({a}._ )] 1  C  V  element.  {  0  }  finite  i - l  }  The  elements  c o n d i t i o n f o r the  load increment.  l  T ei,l  moduli  initial  [D] th i  ^ , f o r the [D]  and  used i n the  (8).  load increment  strain matrices, [K]  T  u s i n g a convergence  s u g g e s t e d by Duncan  first  [K ]^,  ]  That  the  stress-  matrices,  is  (2-11)  e  (2-12)  e  The e l e m e n t  incre-  element  stiffness  are  stiffness  matrices  are used t o c a l c u l a t e t h e g l o b a l s t i f f n e s s m a t r i x , [ K ] ^ T  which i sthen flections,  used t o c a l c u l a t e t h e i n c r e m e n t a l nodal de-  ( A q K -j^, f r o m w h i c h t h e i n c r e m e n t a l  element  strains,  {Ae}. ,, a r e d e t e r m i n e d . The i n c r e m e n t a l e l e m e n t i Ji s t r e s s e s , {Aa}. , a r e t h e n c o m p u t e d u s i n g [D] . , i n i, l e 1, i Equation  (2-9).  Since the behaviour  of themateriali s  assumed t o b e l i n e a r w i t h i n e a c h l o a d i n c r e m e n t , age  the aver-  s t a t e o f s t r e s s i n a g i v e n element d u r i n g t h e l o a d  increment,  {a}.  1  i s g i v e n by  S  {a}.  ,  1 , 'S  =  {a}.  1  1 — 1  + Js{Aa},  (2-13)  ,  1)1  i n which {a}. , i s t h e element s t r e s s v e c t o r a t t h e end o f i-l the preceding moduli  load increment.  Improved e s t i m a t e s o f t h e  a r e o b t a i n e d by u s i n g t h e average s t r e s s s t a t e de(2-13).  f i n e d by E q u a t i o n  The c o r r e s p o n d i n g  s t r a i n and s t i f f n e s s m a t r i c e s a r e then [D] . e  and The  CK ] T  response  j 2  e ± j 2  =  [D((c}. J]  =  [ K  t othe i ^ *  j }  1  T  l  j  3  5  ) ]  The p r o c e s s  (2-14) (2-15)  e  i sanalysed  iteration  accuracy  o n c e more  o f t h e s t r e s s - s t r a i n and s t i f f i sillustrated  S i n c e no c o n v e r g e n c e c r i t e r i o n one  stress-  computed a s  e  l o a d increment  u s i n g t h e improved estimates ness m a t r i c e s .  U a }  element  i n F i g u r e 2. i s used and o n l y  i s performed, t h e f a c t o r c o n t r o l l i n g t h e  w i t h which t h e t r u e s t r e s s - s t r a i n curve  material i sfollowed i s the size of the load  of the  increments.  Displacement,  FIGURE  2  q  ONE ITERATION PER LOAD INCREMENT METHOD  10 2.2.2.  Iterations The  U s i n g a Convergence  a c c u r a c y of the  method d e s c r i b e d above  be i n c r e a s e d by c o n t i n u i n g i t e r a t i o n s criterion,  s u c h as  between two  a sufficiently  iterations,is From t h e  {a}. and  . ,  f o r the  [  K  =  (j+l)  [D]  e  l  >  .  +  and  {Aa}  for  j  j  J  +  1  1 ) 3 + 1  analysis  of the  analysis  o f the  L B ( i a } .  [ V  +  some  can  convergence  i n the moduli  satisfied.  + %{Aa}.  T ] i , j l =  ( A e ) .  «  {  =  [B]  =  [D]  }  . ^ n  t  i . j V  (Aq)  e  e  l  j  j  +  i  th  load  Increment  ,  (2-16)  increment (2-17)  e  ]  e  (  2  1  8  )  (2-20)  l j ] + 1  <^>i,j l  1  "  < " > 2  +  21  >_ 1. The  f r o m one  gence c r i t e r i o n . load increment, i^}. {e}. and  t h  th  until  s m a l l change  {a} . ,  -  1  e  moduli  j  Criterion  {a}.  I 1  process  is  continued u n t i l the  i t e r a t i o n to If  this  the  next  occurs  satisfies  on t h e  change the  n^* a n a l y s i s 1  in  converof  the  then  =  {q}.^  =  {e>.  =  {a}.  +  l-l n  Uq}.  + {Ae}.  )  n  i,n n  . + {Aa}. ^ i - l i,n  (2-22) (2-23) (2-24)  11 2.2.3  Application  of Correction  Forces  I n b o t h o f t h e methods attempts  to achieve  agreement  discussed  above one  between t h e p r e d i c t e d  stresses  th and s t r a i n s  a t t h e end o f t h e i  stress-strain  relation  stress-strain  properties  l o a d increment  and t h e t r u e  o f t h e m a t e r i a l by a d j u s t i n g t h e used  i n the a n a l y s i s  of the i n c r e -  ment . An a l t e r n a t i v e  method o f o b t a i n i n g  between t h e p r e d i c t e d s t r e s s e s stress-strain in  and s t r a i n s  agreement  and t h e t r u e  r e l a t i o n o f t h e m a t e r i a l which might  c o n j u n c t i o n w i t h t h e methods  discussed  above  is  be u s e d given  here. th At incremental {aK,  element,  relation  strain vector,  {eK,  o f the m a t e r i a l .  i n any  stress  vector,  for a particular  may n o t be c o m p a t i b l e w i t h t h e I f the s t r e s s - s t r a i n  stress-strain r e l a t i o n of  m a t e r i a l i s e x p r e s s e d by {a}  then, in  load increment,  loading procedure, the calculated  and c a l c u l a t e d  finite  the  t h e end o f t h e i  i f the s t r a i n s  the c a l c u l a t e d  {a K where  {a  f({eK)  =  }^ i s t h e e r r o r i s the true  f({e})  (2-25)  a r e assumed t o be c o r r e c t ,  stress e  =  vector, {ah  -  {aK,  i s g i v e n by  f({ } ) £  (2-26)  ±  i n the c a l c u l a t e d  stress.  This  applying nodal correction forces,  the e r r o r  stress  v e c t o r and  e r r o r may be b a l a n c e d by {Q },  t o t h e element  at  12 the  beginning  of  the  By t h e  (i+l)  p r i n c i p l e of  6{e} {a } dV T  =  e  1  where  T  6{e}  vector  of  is  a vector  virtual  (2-27)  6{q}  of  nodal  increment.  virtual  work  (2-27)  {Q }  T  6  virtual  strains  and 6{q}  T  is  a  displacements.  Introducing Equation  load  t h  the  relation  T  6{e}  = 6{q}  T  E  B  T ^  i  n  e  yields  6{q} [B]^{a } dV T  =  e  1  (2-28)  6{q} {Q } T  e  T and s i n c e  <5{q}  is  {Q }  =  Furthermore,  if  arbitrary  {a  one  }^  is  [B]  position  (2-29)  T  e  not  a function  dV  {a >.  of  position  (2-30)  e  Newton-Raphson Method  In the (2.2.2)  of  e  e  Modified  a function  [B]^{a }.dV  {Q }  2.2.4  and n o t  methods  attempts to  described  make t h e  in  sections  element  (2.2.1)  stresses,  (aK,  and and  th element  strains,  compatible  {e}^,  w i t h the the  lysis  response  element  the  the  end o f  stress-strain  by a d j u s t i n g of  at  stress-strain  and g l o b a l  to  the  stiffness  the  I  relation  properties  load of  the  used  increment material  i n the  ana-  increment/by: r e b u i l d i n g  the  matrices  the  and r e p e a t i n g  analysis  of  the  response  to  the  increment.  The  analyses  o f the  response  to  the  load increment  subsequent yield  new  13 estimates strains, of  the  o f the {e}.,  and t h e  is  and t h e  strain  possible  element  relation  i^* load  element at  the  end  first  the  analysis  do n o t  material.  make t h e  i n the  {e}.,  element  stresses,  agree with  the  adjusting  analysis  of  one-dimensional  The e q u i l i b r i u m s t r e s s the  to  the  stress-  the  stress-  response  to  increment.  1  ment,  the  {a}.,  m a t e r i a l without  used  Consider  in  stresses,  strains,  o f the  strain properties the  element  {qK,  increment. It  {a}.,  nodal displacements,  , of  agree w i t h  If  one  and t h e  the the  case  strain  response  to  the  ^,  the  stress-strain  assumes t h a t  (Figure  1^  3)•  obtained load  relation  incre-  of  the  equilibrium stress,  ^,  th is  the  one  correct  stress  may o b t a i n b e t t e r  brium,  by a d j u s t i n g  f o r the  agreement,  the  load  curve  of  the  increment,  and  correct  strain a  strain  f(e^  n  n+  e.  material is  n  at the  j  )  relation  estimate,  K is  is  the  o f the  ]_>  stress  ^  i,n+l  =  i,n  +  Increment,  maintaining  equili-  s t r a i n using  the  slope  the  beginning  of  the of  and a  is  n  the  K  n  , is  then  i the  d e f i n e d by (2-31)  p r e s c r i b e d by t h e for  the  stressth  1  correct strain e  e.  load  n^* a p p r o x i m a t i o n o f  material  ° ? the  i  of the  corresponding to = a.^ - f U . )  6  i n which  If  the  while  estimate  Newton-Raphson m e t h o d . strain  end o f  stress-  an i m p r o v e d  given  by  (2-32)  14  FIGURE  3  NEWTON-RAPHSON METHOD FOR THE ONE-DIMENSIONAL CASE  15 e The  process  is  continued  until  a  is  small  enough  to  be  acceptable. The for  use  in nonlinear  applied nodal  N e w t o n - R a p h s o n method  loadings.  analyses  The i n i t i a l  displacements, ^  {Aq}. , , i , l  5  and  incremental  section ments,  ,,  by  {a}.  '  l , 1  the  of  incremental  stresses,  The f i r s t  element  at  response  estimates  strains,  end o f  {Aa}.  estimates  1,1  ses,  the  readily  modified  of  the  bodies  strains,  the  i  -, ,  of  {e}.  ,,  are  the  {Ae}. .. i , l  f o u n d as  nodal  and  <° i,i  and e l e m e n t  stres-  i , i  th  the  { ° } ^ T. is  i ^  load  a  load increment  are  the  difference,  =  '  by t h e  strains,  &  {  A  £  }  i , l  {  A  0  }  i , l  (2-34) (2-35)  vector  for  the  end  of  increment, {a}.  (2-36)  1  specified  given (2-33)  equilibrium stress  n  {a}. The  +  x - l  l  -  }  Since  E  in  displace-  + {Aq} (  to  incremental  M  element  (2.2.1). {q}.  of  is  1 ± 5  {a  }. , , i , l  stress-strain  {e}.  ,,  nodal  correction  found  from e q u a t i o n  nodal  displacement [K ]. T  between  is  forces  I  and t h e  relation  found u s i n g  of  vector, x  t^ }^  A correction, {q}. =  -, i s  {Q }. e  then _  ]  stresses  the  equation  c o r r e s p o n d i n g to  (2-30).  {Aq}!_  {a},  6  material  (2-26). j_  {Aq}!  a  r  n  ,  f o u n d by  e  for  The then to  the  solving (2-37)  16  in  which  obtain  [K,p]^ i s  the  (AqK ^.  displacement  same g l o b a l  An i m p r o v e d e s t i m a t e ,  vector (q}  l j 2  is  g i v e n by  =  {q} {EK  An i m p r o v e d e s t i m a t e , given  =  0  i n which { A E } | ^ i s ,.  specified  material  f o r the  repeated  enough t o  {E},  the  ,  ° f the  ,  +'{AE}1  be  {eK  on t h e  n  th  stiffness  iterative effort.  equilibrium  process.  (19)  e  n  o  d  a  ±  strains  is  (2-39)  strains  corresponding  ,  is  {a},  and  the  r e l a t i o n of  the  f o u n d and t h e  i t e r a t i o n , {a '  e  }.  i,n  to  process  is  small  and t h e  stresses,  ted  element  is  specify  stress-strain  stress  "residual stress  two a d v a n t a g e s .  This  saves  since  (cK,  have t o  be  First, changed  considerable  method"  the  and t h e  difference stresses  evaluated explicitly  relation  of  the  during the  each  the during  computabetween  specified  r e l a t i o n o f the m a t e r i a l f o r the  strains to  "initial  (14).  Secondly,  stress-strain  possible  2  m a t r i x does n o t  the  the  h  ,  c a l l e d the  T h i s method has  is  t  acceptable.  by Nyak and Z i e n k i e w i c z  tional  element  stress-strain  strains,  method" by Z i e n k i e w i c z  the  °^  to  (2-38)  „ , between  6  by t h e  until, '  of  {a }.  T h i s method i s  global  ,  ±al  vector  The d i f f e r e n c e ,  stresses  is  2  +W  ± j l  2  {qK  m a t r i x used  by {E},  {Aq}!  stiffness  the by  calcula-  Iteration,  it  accuracy w i t h which  material is  followed.  17 While the  m o d i f i e d Newton-Raphson method  easily  a p p l i e d i n problems  strain  curve  problems  is  used,  of  the  strain  behaviour  and t h e  stress-strain  to  different  to  curves  stages  compatible  allow  to is  used  follows of  a  loading.  regarding  the  w i t h both the  strains.  stress-  apply  make a s s u m p t i o n s  i n order to  computed  known  body b e i n g a n a l y s e d ,  c u r v e at  s t r e s s e s which are curves  of  one may have  stress-strain  more d i f f i c u l t  w i t h i n the  stress-strain  In such c a s e s , the  is  i n which a f a m i l y  and a g i v e n p o i n t , different  it  In which a s i n g l e  is  calculation stress-  CHAPTER 3 STRESS-STRAIN RELATIONS  3.1  General Discussion In  structures or  order to  of  predict  model f o r  soils,  behaviour of Therefore,  Stress-Strain the  under a p p l i e d l o a d i n g s ,  stress-strain  behaviour  of  one  soils  to  the  finds  one n e e d s a Because  convenient  be l i n e a r  one r e q u i r e s  deformations  soil. it  Relations  over  soil  constitutive of  to  small load  an i n c r e m e n t a l  of  the  complex  assume  the  increments.  constitutive  rela-  tion . A possible the  generalized  tic  material. Aa  constitutive relation  form o f This  x  D  l l  D _  Aa  2]  Hooke's  is,  D  D  2  D  13 D  2  2  D  l4 D  3  D  15 D  2 i |  D  2  5  D  Ae  2 6  <  Ay  D  or  {Aa}  The  constitutive  and  contains  61 =  D  62 [D]  D  63  D  64  D  5  Ay  °66  xy yz zx  (3-D  { A e }  or s t r e s s - s t r a i n  twenty-one  6  y  Ae  Ay  xy  zx  elas-  Ae  16  Ax  Ax  Is  form,  Aa  Ax  soils  Law f o r a homogeneous  i n incremental  12  for  matrix,  independent  18  [D],  elastic  is  symmetric  moduli.  If  this  19 general model one  form o f Hooke's  for a s o i l ,  different  stress dible  m o d u l i as  or s t r a i n i n the  soil,  the  soil  Aa  D  D  y  Aa Ax Ax Ax  the to  functions  constitutive  evaluate of  w h i c h w o u l d be  the  twenty-  state  a rather  of formi-  task. is  assumed  homogeneous and e l a s t i c ,  Aa  as  w o u l d be n e c e s s a r y  elastic  If as  it  Law were u s e d  l l  D  21  D  12  D  well  0  0  Ae  0  0  0  Ae  0  0  0  Ae  0  0  AY  0  Ay  2 3  D  zx  as  0  13  °31 32 33 0 0 0  yz  isotropic  (3-1)  D  xy  be  equation  D  22  to  0  0  0  0  0  0  0  0  0  D  becomes  X  <  y Z  'xy  A  66  yz  ' zx  Y  (3-2) i n which the  constitutive D  and  D  The c o n s t i t u t i v e tic  material  is  =  l l  12  defined i  s  a  by a s s u m i n g t h a t  D  2  2  =  D  33  D  55  =  D  66  =  °13  =  D  23  the  isotropic  by o n l y linear a soil  is  under s m a l l  of  m o d u l i needed  model i t s  is  reduced from twenty-one  to  two  and  homogeneous  independent  combination of  geneous and e l a s t i c to  symmetric  =  matrix for  moduli '"sinceThus,  matrix is  isotropic  as  elastic  D . ^ and well  load Increments,  stress-strain only  two.  elas-  as  the  homonumber  characteristics  For this  reason  20  soils  are  usually  assumed  While the able  effort  soil,  it  is  to  of  material, the  stress-strain  not  some  accurate  for  the  e ,  a  i n any m a t e r i a l  is  strains, '  reveals  e  . e x  =  stresses  £  + e  that  the  of  i n terms  a  of  (3-4) z  constitutive  normal  Shear  relation,  homogeneous  elastic  strains  independent  stresses,  in isotropic  are  therefore,  materials.  do n o t  of  cause  The same i s  true  materials. sands  and o v e r c o n s o l i d a t e d  and Reese  s t r e s s e s may,  (12)  strains  sands  isotropy  found t h a t ,  not  however,  strains.  f o r a dense  b-e more e f f e c t i v e  than are h y d r o s t a t i c  and o v e r c o n s o l i d a t e d is  clays,  cause v o l u m e t r i c  i n some c a s e s ,  producing volumetric For  element  + e  of the  the  V , of  as  y  a r e known t o  Girijavallabhan  of  volume,  z  f o r an i s o t r o p i c  strains  For  and e y  5  x  stresses.  orthotropic  assumption  i n the  s t r a i n may be w r i t t e n  material  volumetric  change  volumetric  (3-2),  shear  (3-3)  F o r an i n f i n i t e s i m a l  equation  shear  behaviour of  material.  Inspection  shear  consider-  f  v  for  saves  soils.  strain,  an i n f i n i t e s i m a l  the  normal  -  e  the  isotropy  as  dV i s  element  of  i n m o d e l l i n g the  % where  isotropic.  assumption  The v o l u m e t r i c defined  be  correct.  clays  sand,  in  stresses. then,  As a r e s u l t ,  the  the  21 d e f o r m a t i o n s p r e d i c t e d when one u s e s t h e these will ses  soils be  are i s o t r o p i c  somewhat  are  in error  involved.  metric  take  the  strains  pendent  of  sands  any s o i l  and  moduli.  elastic  when l a r g e  shear  stres-  stress-  and o v e r c o n s o l i d a t e d c l a y s  account.  and o v e r c o n s o l i d a t e d usually  especially  that  homogeneous  c o u p l i n g between s h e a r  into  elastic  as  A more a c c u r a t e model f o r t h e  strain properties have t o  as w e l l  assumption  and  volu-  T h i s w o u l d r e q u i r e more  Despite  its  inaccuracy for  inde-  sands  the  assumption of  isotropy  made when m o d e l l i n g t h e  stress-strain  behaviour of  since  it  clays,  stresses  would  greatly  reduces  the  effort  necessary  is  to  produce a model. If geneous e l a s t i c independent  a soil  is  assumed t o  material for  moduli  be an i s o t r o p i c  small load increments,  a r e needed  to  define  its  homotwo  stress-strain  properties. The m o d u l i most Young's modulus, E  f c  , the  tangent  bulk modulus,  and t h e  tangent  as  commonly u s e d a r e t h e  tangent , the  tangent  shear modulus, G ^ , the  tangent  Poisson's  c o n s t r a i n e d m o d u l u s , ]YL_.  ratio,  These are  defined  follows: 6a.  E  G  1  t  (3-5)  Se. l  (3-6)  t  K. t  m v  where a  m  ,  (3-7)  22 6e .  and The  v, t  =  M, t  =  bulk modulus,  J  the  6a. -r-^ 5e.  to  (3-9)  and  two  with  of  a l l be  the  r a t i o by t h e  t  following  E  if  d  M  the  soil  is  t  ratio  may n o t  a negative  matrices,  be g r e a t e r bulk  terms  of  bulk  Y o u n g ' s modulus and  equations:  ( 3  (3  -  "  1 0 )  11}  (1-v) ( 3  ~  1 2 )  than 0.5  and A b e l  [D],  and  (3-11),  it  follows  since  this  that  Poisson's  would  result  modulus. ( 4 ) give  f o r an i s o t r o p i c  material: in  (3-5)  isotropic.  Desai tive  to  The s h e a r ,  (l+v)(l-2v)  =  From e q u a t i o n  in  independent.  by e q u a t i o n s  I(Ao  =  the  positive.  = 271+77  t  K  (3-9)  Y o u n g ' s modulus and  moduli defined  may be r e g a r d e d as  G  n  = e. = 0 k  j  constrained moduli are r e l a t e d  Poisson's  a  e.  shear modulus,  c o n s t r a i n e d modulus must Any  (3-8)  oe .  v , f c  the  following  homogeneous  constituelastic  23 t t  v  D  12  "  D  44  "  D  13  "  D  55  "  D  ( l + v  23  E  ) ( l - 2 v  t  )  t  t 2(l+v ) E  D  and i n t e r m s  Drnevich terms  of  of  and  D  ll  D  12  =  D  13  D  44  =  D  55  (7)  = 22 D  gives  =  =  =  K  t  °23  =  K  t  66  =  D  =  °22  =  D  12  =  D  D  44  =  D  a soil's  convenience  the  degree  G  +  4  "  /  < t>  3  G  2 / 3 (  V  t  following  13 55  =  D 3  =  D  D  =  of  constitutive  =  23  =  66  constitutive  M  matrix  in  M  =  of  the  t  G  -  matrix,  Poisson's  ratio  is  one-third  times  the  2 G  t  t  test  properties  necessary  i n the  uncertainties [D]  .  value ratio  of is  value  b u l k modulus  of  them and  determined. Poisson's  elements of  if  the  is  Young's modulus. the  to  depend upon  may be  i n the  one-half,  uses  determine  For example,  0 . 4 8 , the  one  will  to  w i t h which they  uncertainties  large  Poisson's  t  o f w h i c h two m o d u l i  certainty  may c a u s e  3  stress-strain  Small  of  23  D  the  l l  the  value  t  ,  The c h o i c e  ratio  "  and G ^ :  D  define  66  value  eight  the of  and  But i f  b u l k modulus  the is  24  infinite. less  Uncertainties  i m p o r t a n t when i t Since  stress-strain  the  3.2  is  soils  relations  stress  are  generally  the  to  define  be . d e t e r m i n e d as  their their  functions  Experimental Studies  tutive  matrix,  tangent  value  Poisson's  drained  [D], of  ratio  a soil  the  stress-strain  constant  for  sands  relations  <r )  (  -  a^)  d i r e c t i o n of °2^ it u  4).  be e x p r e s s e d  "*"  is  the  the  E^ is  the  of  tangent  They  t r i a x i a l tests  1  deviator  are  a^,  loading of  if  conduc-  stress,  for primary  the  of form  o-^) 3 ult  stress,  major p r i n c i p a l  initial  value  (3-13) ( c? -  ±  the  found that  —  asymptotic  s  consti-  d a t a on Y o u n g ' s modulus  and c l a y s .  observed  3 E  In which  , and t h e  by h y p e r b o l i c e q u a t i o n s  =  3  the  i n terms  of the minor p r i n c i p a l  may be r e p r e s e n t e d  (Figure  be w r i t t e n  and p r e s e n t e d  values  (a-L -  (.a^ -  proposed that  or undrained compressive  at  the  (9)  Young's modulus,  ted  soils  for  ratio,  and P o i s s o n ' s  in  however,  state.  Duncan and Chang  of  are,  nonlinear in  moduli used to  have  or s t r a i n  ratio  smaller.  behaviour,  constitutive of  in Poisson's  is  stress,  the a^,  strain and  value  of the  deviator  value  of  modulus and can  the  stress  as E. 1  =  K p a  a  (^) a p  n  (3-14)  25  FIGURE 4  HYPERBOLIC STRESS-STRAIN  CURVE  26 in  which K i s  a dimensionless  p_ i s a  number,  atmospheric  number c a l l e d (CT^-  the  ^^^ult  failure,  a  ^-  y  w a  ^f>  Coulomb f a i l u r e  Sweater  s  criterion  in the  i - "s'r  which c i s angle  of  cohesion  v^ult  (a  ±  where R  f  R^, i s  range  defined  tests  n  (9)  deviator  = the  0.6  tangent  stress  by t h e  at  Mohr-  sincj> ( 3  of  the  R  f  (a  the  x  -  t  a ) 3  h  u  the  e  l  failure ratio. to  (Figure  expressed e  Q  u  "  1 5 >  and <b i s  soil  soil  ~ ° 3 ^ f ky  d  f  of  5)•  relationship a  i  t  o  n  (3-16)  t  Typical  values  of  0.95-  value  c o n d u c t e d at  of Young's modulus,  a fixed  value  of  ,  for  may be  as  E  d(a t  =  Using (3-17),  d  a ) 3  £  (  l  equations  Duncan and Chang  Y o u n g ' s modulus as equation  3  f r o m about  7  and  a )  known as  The triaxial  -  a  dimensionless  The v a l u e  intercept  i n t e r n a l f r i c t i o n of  (?^ -  a  modulus  as  Duncan and Chang between  and n i s  than the  cosij) + 2 a , r^-siHi  =  the  c a l l e d the  w h i c h may be d e f i n e d  2 c ( a  pressure,  modulus e x p o n e n t .  a  ( -  quantity  (3-13), (9)  a function of  (3-14),  expressed the  (3-15), the  stresses  3  "  the  7  )  (3-16)  tangent by  1  FIGURE  5  MOHR-COULOMB FAILURE  CRITERION  28  R (l-sin<j>) f  1  -  ( 2_~°2a  a_ n  K p (-^)  2c cos<() + 2a2Sin<))  Q  a  Pa  (3-18)  The t a n g e n t as t h e respect  instantaneous to  rate  of  a constant  ration,  ,  i n which G i s in  same as  G is  of  the  f  1  value  ,  of  v,  fy  i  J  increase  in  the  and d i s  de-  the  w i t h s t r a i n and a l l o t h e r p a r a m e t e r s  [D],  (3-18).  rate are  For undrained conditions zero. proposed that  the  f o r a s o i l be w r i t t e n i n t e r m s o f tangent  f c  presented  of  g i v e n by  when Q = p , F i s j a  s h e a r m o d u l u s , G , and t h e  and t h e y  value  (3-19)  3  Domaschuk and Wade (6)  tangent  is  2c cos(() + 2a2Sin(j)  i n equation  matrix,  tangent  and c l a y s  o n e - h a l f and F and d a r e b o t h  tutive  test.  R (a -a )(l-sin^)_  for a tenfold  increase  the  the  s t r a i n with  (  Pa  a  of  defined  °3/p„) F loi dCo^ - a^)  a  •  ratio is  of r a d i a l  triaxial  value  1 -  of  change  f o r sands  G -  crease  of Poisson's  a x i a l s t r a i n i n the At  Poisson's  value  d a t a on t h e  constithe  bulk modulus,  b u l k and s h e a r m o d u l i  of  sands. In order to in  shear,  investigate  the  b e h a v i o u r o f a sand  t h e y p e r f o r m e d a number o f d r a i n e d t r i a x i a l  pression tests  at  constant  values  o f the  mean n o r m a l  comstress,  29 a  , on sands nr  of various  ;  tests  deviatoric S.  •  stress,  (<°1-V  the r e s u l t a n t  e  in  +  (  ° 2 - % ,  d  -  which G. i s  modulus,  to  o f S^.  initial  +  (  3 - %  0  e^,  strain.  )  2  ) '  S  (  -  3  2  0  )  g i v e n by  The g e n e r a l  form o f  by t h e h y p e r b o l i c  stress-  &  « -  value  o f the tangent  as t h e i n s t a n t a n e o u s e^,  and b I s  density  t  In  =  shear  o f change  of the  equation  shear  asymptotic  (3-22)  at t h e  with  following  (3-23)  2  d  t h e b u l k modulus o f  Domaschuk and Wade ( 6 ) p e r f o r m e d a number o f tests  on sands  respect  modulus  order to investigate  consolidation  of  was p e r f o r m e d .  G ( l- b S ) ±  »  o f t h e s a n d and t h e mean n o r m a l  f o r the tangent  G  2  f o u n d t o depend upon t h e  Domaschuk and Wade ( 6 ) a r r i v e d  expression  rate  the i n v e r s e  B o t h G^ and b were  relative  2  i  By d i f f e r e n t i a t i n g  pic  resul-  1 + ' £ 0 .  s t r e s s at which the t e s t  sands,  2  was e x p r e s s e d  the I n i t i a l  defined  with respect  e^,  )  strain,  t h e mean n o r m a l  d  to  between t h e  relation S  value  These  S^, g i v e n by  deviatoric  relationships  strain  2  densities.  d  which e is m  these  in  relative  y i e l d e d d a t a on t h e r e l a t i o n s h i p s  tant  and  initial  of various  initial  isotrorelative  30  densities versus  and p l o t t e d  the  volumetric  relationship as e  the  instantaneous  . and t h e v'  K_  is  zero  e  .  tangent  rate  of  mean n o r m a l s t r e s s , They f o u n d t h e  of  a  ,  following  bulk modulus,  change  a  ,  defined  with respect  to  mean n o r m a l . s t r e s s , 0 ' m =  fc  In which K. i s v  o f the  strain,  between t h e  K  e  curves  K. + ma  the  value  and m i s  the to  (3-24)  m  o f the rate  and m depend upon t h e  of  tangent  b u l k modulus  change  when  of  K, w i t h t  a . m  i n i t i a l relative  density  of  Both the  sand. F o r a sand o f  a given  r e l a t i o n s h i p between K, and a Is ^ t m be e x p r e s s e d  by two  (3-24),  of which i s  of  a  m  each  equations  relative  from t h e  Domaschuk and Wade ( 6 ) between t h e  deviatoric  tests  form o f  a p p r o p r i a t e to  equation  a range  of  values  .  relationship the  the  a b i - l i n e a r one w h i c h may °  Domaschuk and V a l l l a p p a n of  density,  was  strain for  o f the  Consequently,  by e q u a t i o n related  to  solidation  resultant  resultant  deviatoric  and t h e  as  of  stress  work the  stress  and  compression  that  between t h e  deviatoric  The v a l u e s  the  They f o u n d t h a t  clays  i n i t i a l void ratio  pressure  extended  drained t r i a x i a l  relationship  (3-23). the  clays.  same form f o r  the  modulus and t h e  to  (5)  for  sands.  tangent was  shear  also  given  and b f o r a c l a y of  the  clay,  mean n o r m a l s t r e s s  its at  are  preconwhich  the  31 test  was  conducted. The t a n g e n t  found to by t h e  be r e l a t e d  to  , of  a clay  a normalized volumetric  was  strain,  ^  v  n  ,  relationship K  i n which and N i s K, t  bulk modulus,  =  t  is  K. (1 + N £  the  E  of  K.  i n i t i a l value  vn  =  l  .  stress  tangent  to  a  mc  and t h e  the  /e  intersect.  )  of  (3-25) the  the  tangent  rate  of  b u l k modulus  increase  the  Is  volumetric  values  s t r a i n at  of  infinite,  the  isoinitial  line  stress-strain  equation  stress-strain  the  which the  c u r v e and a s t r a i g h t  portion of  When n i s  e x p r e s s e s an e l a s t o p l a s t i c  by  (3-26)  characteristic  final  of  given  vc  stress-strain  approximation to  1  The i n i t i a l b u l k modulus  i n which a and e are mc vc  curve  -  n  a number w h i c h i n d i c a t e s  with that  tropic  n y  (3-25)  curve.  According  Domaschuk and V a l l i a p p a n ( 5 ) o , E and N were f o u n d ^ mc' vc by a method o f l e a s t s q u a r e s a d j u s t m e n t d e s c r i b e d by Deming to  5  (3).  The n o r m a l i z e d v o l u m e t r i c  strain,  e  3  £  vn  =  Drnevich modulus, mate  1  |E /e v  (7)  vc  1  the  compression.  constitutive  given °  by "  (3-27)  proposed the  t o g e t h e r w i t h a n o t h e r modulus define  is  3  I  M, i n n o n l i n e a r a n a l y s e s  one-dimensional  , vn  of  of problems  the  [D].  the  constrained  which a p p r o x i -  T h i s w o u l d be  s u c h as  matrix,  use  used  s h e a r modulus  to  He p e r f o r m e d a number  32 of  one-dimensional  the  d a t a as t h e  e^. of  axial  The r e s u l t s the  compression t e s t s stress,  were  on sands  a , versus  represented  the  and  plotted  axial  by h y p e r b o l i c  strain,  equations  form M.  =  a  e  1  (3-28)  1  e  m initial  i n w h i c h M. i s I  the  defined  as  instantaneous  pect  E ^ , and  to  the  is  the  value rate  E , D r n e v i c h (7)  tween t h e  tangent  M,  the in  following terms  of  a  c o n s t r a i n e d modulus,  change value  =  £ 2 _  M.  m  m  into  of  with  o f the  (3-28)  )  ,  with  f o r the  be-  (3-29)  2  (3-29)  for  £ and  e q u a t i o n (3-28) , D r n e v i c h (7)  expression  respect  and  1  equation  res-  strain.  following relationship  c o n s t r a i n e d modulus,  By. s o l v i n g result  the  equation  found the  v  i n g the  of  asymptotic  By d i f f e r e n t i a t i n g to  of  tangent  substitutobtained  c o n s t r a i n e d modulus  33  3.3  Limitations of Stress-Strain The s t r e s s - s t r a i n  section  ( 3 . 2 ) have  b a s e d upon t h e and t h u s  none  these  assumption of  i n the  state  of  like  tests  same  since  the  expressions  which r e l a t e  the  values  ition  is of  of  elastic  stress  satis-  discussed  the  moduli  theory,  in to  they act  the  are  not  more  may l e a d t o  percentage resist  stress-level, of  different  stress-paths  different  values  to of  a  modulus.  m o b i l i z e d to  classification duces  of  be  of  significant  o f y i e l d when s o i l s  for s o i l s ,  Using the that  to  materials.  state  particular  may not  Secondly,  point  due  The use  which develop  a r e b a s e d upon e l a s t i c  all  behaviour  them.  therefore,  soils  are  s h e a r may have  strains.  beyond the  in  they  elastic  shear  stress,  plastic  due t o  establish  relations,  Thirdly, the  used to  First,  due t o  (3.2),  applicable  discussed  f o r volume changes  changes  when d e a l i n g w i t h  volume changes  section  isotropic  o f them a c c o u n t s  stress-strain  factory  relations  certain limitations.  s h e a r e v e n t h o u g h volume occurred  Relations  of  the  of the  an a p p l i e d l o a d i n g as  Lade and Duncan  loadings  strength  into  primary  defin-  ( 1 3 ) have made a l o a d i n g which p r o -  monotonically increasing stress-levels soil,  the  soil  above  those  previously  e x p e r i e n c e d by t h e  u n l o a d i n g which produces  decreasing  stress-levels  and r e l o a d i n g w h i c h p r o d u c e s  increasing  stress-levels  below those p r e v i o u s l y  experienced  34 by t h e  soil. The s t r e s s - s t r a i n  soils  i n primary loading  or r e l o a d i n g  since  the  is  behaviour  different  strains  this  difference  increased.  generally  Duncan and Chang  cohesionless  from t h a t  in  unloading  induced during primary l o a d -  i n g are p a r t i a l l y i r r e c o v e r a b l e . that  of  Lade and Duncan ( 1 3 )  increased  (9)  as  the  found  stress-level  found s i m i l a r r e s u l t s  for  clays. Lade and Duncan ( 1 3 ) published data t h a t , paths the  for  cohesionless  another.  i n going  However,  f r o m one  stress-paths  r e l o a d i n g would produce  only primary loading  stress  another.  to  strains  any two  by p o i n t  Duncan and Chang expressions  (equation  (3-18))  stated that triaxial  their  f o r the  A to  (10),  tangent  and P o i s s o n ' s expressions,  compression  that  f r o m one  minor p r i n c i p a l s t r e s s ,  involves state  of  modulus  (3-19)), obtained  constant to  by p o i n t B.  Young's  with parameters  were u n a b l e  of  discussion  (equation  t e s t s performed at  of  produce  represented  of  or  stress-path  would  in their  ratio  state  (6),  from t h e  values  to  stress-path  P - 2 , which  i n going  stress-  any u n l o a d i n g  in Figure  of  nearly  stress  than a  i n going  than would s t r e s s - p a t h  represented  of  only primary l o a d i n g ,  some u n l o a d i n g and r e l o a d i n g , stress  state  strains  For example,  which i n v o l v e s  larger  the  soils,  involving  smaller  involving  their  from a r e v i e w  i n v o l v i n g o n l y p r i m a r y l o a d i n g would produce  same s t r a i n s  P-l,  concluded  values  predict  from of  accurately  35  ro b  Final  I  Stress-Level  S t r e s s - P a t h P2 Produces Smaller S t r a i n s than S t r e s s P a t h PI  co co cu PH  -p  CO U O  -P cti  •H >  CD Q  Initial  Stress-Level  Minor P r i n c i p a l  FIGURE 6  Stress,  TWO STRESS-PATHS WHICH YIELD DIFFERENT STRAINS FOR THE SAME STRESS INCREMENT  36  the  strains  fine ses  induced i n a t r i a x i a l  sand o f were  the  soil  stress-level  controlled, for  specimen o f a  when t h e the  s t r e n g t h m o b i l i z e d was  Yudhbir  clays  density  i n c r e a s e d i n s u c h a way t h a t  available that  low r e l a t i v e  test  was  percentage  constant  (that  stresof is,  so  constant).  and V a r a d a r a j a n  drained t r i a x i a l  several  principal  uniform  tests  stress-paths.  (18) conducted on n o r m a l l y  stress  consolidated  The s t r e s s - p a t h s  were  as  follows: (a) (b)  increasing with  o-^ i n c r e a s i n g w i t h c o n s t a n t  (c) (d)  constant;  constant  with  stress;  decreasing;  i s o t r o p i c decrease  of normal  They e v a l u a t e d t h e modulus and P o i s s o n ' s  mean n o r m a l  tangent  r a t i o u s i n g the  stresses. values  of Young's  a p p r o p r i a t e Hooke's  Law e x p r e s s i o n  for isotropic materials  f o r each  stress-path.  In the  s t r e s s - p a t h (b),however,  o n l y the  tangent  case o f  s h e a r modulus c o u l d be e v a l u a t e d d i r e c t l y and an of  Poisson's  r a t i o was u s e d t o  estimate  the  estimate  tangent  value  of  Young's modulus. Y u d h b i r and V a r a d a r a j a n variation  of the  initial  tangent  with consolidation pressure of the  the  same f o r m as  equation  f o r each o f the  value  (3-l4)  but t h a t  by  the  modulus e x p o n e n t ,  different  the  o f Y o u n g ' s modulus  c o u l d be e x p r e s s e d  modulus number, K , and t h e  different  ( 1 8 ) found that  stress-paths;  equations values n,  of  were  K varied  37  from s i x t e e n  for stress-path  stress-path  (d)  while  stress-path  (c)  to  The t a n g e n t  value  test  about n i n e - t e n t h s  i n which E ^ i s defined  upon t h e tenths  stress-level,  as  (1 -  =  f e  the  R S)  i n equation  S,  of  stress-path  on t h e  tangent  clays.  stress  value is  to  to  very fine  and B , a m i x t u r e o f  (3-3D Y o u n g ' s modulus and R^ The v a l u e  just  one.  sixfor  the  between t h e  (18) concluded that  stress-strain  significant  Parzin  a  particular  for normally c o n s o l i -  and K r i z e k  on two  soils,  (2) conducted A , a mixture  sand,  The t e s t s  of  of  percent  w i t h da^/da-^ v a r y i n g  d a ^ / d a ^ was u s e d  stress-path  state  and c l a y .  stress  the  silt,  were  from  i n each  used a f f e c t e d  tri-  of  five  silt  the  r e l a t i o n s h i p and  sand c o n t a i n i n g a b o u t  O n l y one v a l u e  They f o u n d t h a t ships  o f R^ d e p e n d e d  over n i n e - t e n t h s  o f Y o u n g ' s modulus at  conducted using s t r e s s - p a t h s to  as  strength  ±  (3-16).  on t h e  compression t e s t s  coarse  expressed  a  form  tangent  (a)  very  Corotis, axial  was  of  (c).  effect  dated  any s t a g e  ( f r a c t i o n of  Y u d h b i r and V a r a d a r a j a n  of  (a).  f o l l o w e d and r a n g e d f r o m a b o u t  for stress-path  state  stress-path , at  for  for  E  2  f  initial  stress-path  stress-path  for  one-half  stress-path,  by an e q u a t i o n o f t h e E  o n e - h u n d r e d and f i v e  n v a r i e d from about  conducted with a s i n g l e  mobilized)  to  of Young's modulus,  a f u n c t i o n of the  is  (a)  zero  test.  relation-  and b o t h Y o u n g ' s modulus  38  and  Poisson's  ratio  It stress-path effect  to  apparent  followed  on t h e  soils. the  is  f o r both of the  correct  relations  f o r the  relations  only  tested. studies that  d u r i n g l o a d i n g may have  The e x p r e s s i o n s  be  from a l l these  stress-strain  stress-strain  soils  of  f o r the  significant  and e l a s t i c  elastic soils  a  moduli of  may,  stress-paths  the  moduli of soils  therefore,  and  be  f o r which they  said were  obtained.  3.4  Material  Behaviour After  Although the generally that tic  that  Yield  behaviour of  of  a plastic  material,  many a u t h o r s  deal with  soils  materials.  strain  This  simplifies  soils  (3-18)  g i v e n by e q u a t i o n Poisson's  y i e l d as  modelling  ratio,  together the  sand i n an a n a l y s i s  of the  load-deformation  strip  footing.  Y o u n g ' s modulus t o As simple  may be  soil  stress-  small  the  value  of  behaviour of response  of  a  a  value.  value  of  (3-10)  b u l k modulus.  correspondence),  the  and  Y o u n g ' s modulus  i n a p r o p o r t i o n a t e r e d u c t i o n of  (personal  as  y i e l d by r e d u c i n g  seen from e q u a t i o n s  reduction of  results  stress-strain  They s i m u l a t e d  a very  s h e a r modulus and t h e Duncan  elas-  with a constant  define  yield  of  soft  ( 9 ) used Young's modulus,  to  the  is  behaviour. Duncan and Chang  buried  yield  Duncan ( 8 ) r e p o r t s  after  the  after  However,  both  (3-11), at  the  according  b u l k modulus  to  39  immediately that  after  yield  Immediately  prior  should not to  yield.  r e d u c i n g Y o u n g ' s modulus t o yield the  is,  therefore,  ability  stress  is  of  the  greatly It  soil  of both Poisson's  to  ratio  the  immediately before  yield  is  yield, at  v^.  v , 2  that  resist  however,  value  of  value  R  of  of  1 +  one-thousandth  lus  to to  is  value  of  the  ratio  be i n c r e a s e d t o  in  reduced.  immediately  ratio  is  after  immediately  value  yield  before  immediately  of  the  given  bulk  after  modulus  by  2v.,  of that  Poisson's  of  values  yield  greatly  Y o u n g ' s modulus i m m e d i a t e l y  value  Poisson's 0.4995  after  immediately  ratio prior ratio  after  to  yield  is  prior  to  yield  was  y i e l d would  i n o r d e r f o r the  b u l k modu-  remain unchanged. If  value  that  hydrostatic  change  R times that  to  set  have  implies  i n w h i c h Y o u n g ' s modulus  maintain the  -  to  Poisson's  if  then the  it  at  b u l k modulus r e m a i n s u n -  Poisson's  For example,  0.45,  the  case  immediately p r i o r  and t h e  possible  yield is  which w i l l  yield  simply  small value  changes i n  s h e a r modulus  The v a l u e  equal to  since  of  and Y o u n g ' s modulus at  C o n s i d e r the  and t h e  some a r b i t r a r y  acceptable  changed even t h o u g h t h e  yield,  The p r a c t i c e  from  reduced.  is,  s u c h a way t h a t  not  be v e r y d i f f e r e n t  of  the  v  2  is  chosen  using equation  s h e a r modulus i m m e d i a t e l y  (3-32),  after  yield  the is  then  40  B times that  immediately  prior 1  B  ~  =  +  R -  tion  above  (3-32)  yield  then,  would g i v e  equal to  limit  Poisson's  choosing  to  avoid computational  equation  (3-32)  shows,  r a t i o were z e r o  Poisson's  ratio  Poisson's  of  that  prior  to  y i e l d must  o n e - h u n d r e d at  ratio  to  a reduction of  r e d u c e d at (3-32) yield.  to  values the  yield. select  However,  It  of  doing t h i s the  Poisson's  ratio  approaches  occur  for  elastic  to  t h a n about  in  0.49  Inspection  even i f  the  the  0.495  of  value  value  of  of  i n order to  leave  r e d u c i n g Y o u n g ' s modulus  yield.  Poisson's may l e a d  b u l k modulus  shear  Therefore,  limiting will  lead  to  ratio to  use  f o r use  after  computational  tends to  infinity  and b u l k m o d u l i a r e  dif-  as  soils.  u s e d as  these d i f f i c u l t i e s  is  equation  one-half.  constants  compressible  common p r a c t i c e  w o u l d be d e s i r a b l e  since  independent  computer  b u l k modulus when Y o u n g ' s modulus  values  the  when u s i n g a  yield,  be  after  yield.  o f n o t more t h a n 0 . 4 9  ficulties  If  to  difficulties.  b u l k modulus unchanged w h i l e  by a f a c t o r  to  after  is  less  however,  Poisson's  by  equa-  s h e a r modulus  prior  it  values  a c c o r d i n g to  the  Duncan ( 8 ) ,  order to  the  of  analyses,  ratio  given  (3-33)  0.00097 times that  perform nonlinear  B is  -  a value  According to to  where  1 + 2v  =  example  yield,  l  V  R + For the  to  do  One may s i m p l y r e d u c e  the not the  41  s h e a r modulus value  it  at  yield  had p r i o r t o  and m a i n t a i n yield.  the  b u l k modulus  at  the  CHAPTER 4  PROBLEMS IN NONLINEAR F I N I T E ELEMENT ANALYSIS OF DEFORMATIONS IN SOIL BODIES  4.1  G e n e r a l D i s c u s s i o n o f N o n l i n e a r F i n i t e Element A n a l y s i s of Deformations i n S o i l Bodies  4.1.1  Common A s s u m p t i o n s In o r d e r to  element  analyses  of  perform incremental n o n l i n e a r  soil  bodies,  c e r t a i n assumptions  finite are  commonly made. The d i s p l a c e m e n t may be assumed t o order. the  f o r the  finite  element the  constitutive  following 1.  actual  stiffness  model f o r t h e  The d i s p l a c e m e n t  the  linear  stiffness  body i f  the  u s e d and  ( 4 ) ) are  field  met:  within  between  states  allows  finite  displacement terms. 42  finite  body  elements.  f i e l d allows  i n the  rigid  constant  elements.  f i e l d must  contain  of  be an  elements.  The d i s p l a c e m e n t  is,  material is  e l e m e n t s and c o m p a t i b l e  strain  a higher  the  that  are continuous  displacements, of the 3.  of  ( D e s a i and A b e l  The d i s p l a c e m e n t s  adjacent 2.  field,  elements  mesh u s e d t o model a body w i l l  conditions  the  finite  q u a d r a t i c , or of  F o r any assumed d i s p l a c e m e n t  u p p e r bound t o rect  be l i n e a r ,  field  That  the  cor-  43 If of the  the  finite  above  element  body i t  represents  element  mesh  conditions  by u s i n g f i n e r  ( D e s a i and A b e l  strain triangular  order elements  i n the  c a n t i l e v e r beam t o of  the  higher  It  was  higher order  has been t h a t roughly  one  elements  for  to  stress  six  the  higher order  They f o u n d t h a t , constant  stresses  and d e -  ten  elements.  that  constant  results.  their  experience  element strain  is  triangular  They r e p o r t t h a t is  elements  even  less  efficient  in nonlinear analyses,  could prevent  slip  the  conrela-  analyses. the  use  of  deformations  elements.  Another and d i s p l a c e m e n t s T h i s means t h a t tionships  a  f o r meshes  gave b e t t e r  ( 1 1 ) state  in elastic-plastic  s t r a i n elements  along y i e l d e d  elements  element  of  same number o f unknowns,  computation.  strain triangular to  to  response  those which used  linear strain triangular  equivalent  finite  higher  found t h a t ,  found f o r r e c t a n g u l a r  and M i l l e r  the  accuracy of  of  of the  elements,  F o r meshes w i t h t h e  Finn  tive  linear analysis  gave more a c c u r a t e  The same t r e n d s were  stant  with that  order  those with the  of  of the  ( 1 ) compared t h e  finite  elements  that  (4)).  elements  loading.  stiffness  to  subdivisions  same number o f  flections.  the  mesh may be made c l o s e r  B r e b b i a and Connor constant  are met,  common a s s u m p t i o n i s  occurring  strains  the  d u r i n g an a n a l y s i s  o n l y l i n e a r terms  between  that  a r e needed  and d i s p l a c e m e n t s  strains  are  i n the (see  small. rela-  D e s a i and  44  ( 4 ) ) . Also,  Abel  it  means t h a t  the  body may be u s e d t h r o u g h o u t t h e It 12,17,18)  that  means t h a t the  is  also  soils  Chapter T h r e e , correct due t o  for  however,the  soils  appropriate  elastic  materials.  are needed t o  soils.  susceptible  isotropy to  This  determine  As e x p l a i n e d  assumption of  which are  order to  values  gent m o d u l i of  of  soils  stress.  a l l o w the  of the  incremental analysis,  the  of  the  volume  in  is  not  changes  shear. In  state  moduli  equations  of  (2,5,6,7,8,9,10,11,  commonly assumed  elastic  geometry  analysis.  are i s o t r o p i c ,  o n l y two  stress-strain  initial  it are  elastic is  functions  functions  to  to  stresses  at  any s t a g e  4.1.2  relate since of  these  of  values  i n section  of  the  an  tan-  existing  (3-3),  however,  stress-path.  a soil's  that  tangent  it  is  moduli  strains.  Incremental Analysis  was  shown i n C h a p t e r  may be u s e d f o r  strain relations  any s t a g e  only of the  than are the  incremental nonlinear analysis methods  of  the  a r e u s u a l l y more a c c u r a t e l y known  an a n a l y s i s  Method o f As  the  at  ( 9 ) p o i n t e d out  Duncan and Chang desirable  moduli  commonly assumed t h a t  As p o i n t e d out  moduli are a l s o  determination of  soils  2 , s e v e r a l methods  are a v a i l a b l e . if  the  Any o f  appropriate  of these  stress-  a r e known.  In-many n o n l i n e a r p r o b l e m s , d i f f e r e n t  stress-paths  45 are at the  f o l l o w e d by t h e the  same t i m e .  soil  point  of  relations  soils  derived those  In such c a s e s , relations  and one u s e s  from t e s t s  followed  ensure  relations,  a family of  problem of  analysis  of  f r o m one  stress-strain of  families  d e s c r i b e d above makes d i f f i c u l t  forces,  v i n g a unique methods, the  culate  of the  the  value  behaviour of the  curves  stress-strain  (3-13),  curve.  s t r e s s e s .'which a r e  stress-strain if  interest.  one might  i n order to  and t h e curves  assume find  soil  than  the  mass goes  another. curves  o f methods  such  i n problems  In o r d e r to  use  these  with regard to  r a t i o had a  r a t i o o f the  cal-  the  For example,  d e s c r i b e d by  Poisson's  as  invol-  m a t e r i a l i n order to  u s e d were  as  a p p l i c a t i o n of  compatible with both  correct  exist  curves  other  stress-strain use  followed.  do n o t  computed s t r a i n s .  that  the  stress-strain  During  family to  to  possible  stress-paths  one w o u l d have t o make a s s u m p t i o n s  the  different  however,  which are very u s e f u l  stress-strain  stress-strain  at  stress-paths  m o d i f i e d Newton-Raphson method o r t h e  correction  occur w i t h i n  stress-strain  curve i n the  mass  particular  the  p r o b l e m , a p o i n t w i t h i n the  The use  the  a  soil  w o u l d be n e c e s s a r y  that  the  performed with  i n the  the  at  which i n c l u d e a l l  one w i s h e d t o  stress-strain  it  of the  failures  followed  used are a p p r o p r i a t e to  Such g e n e r a l for  local  parts  s o i l mass may be d i f f e r e n t  stress-strain if  if  stress-path  loading.  stress-paths  in different  Moreover,  the  w i t h i n the  stages use  mass,  soil  equation constant  m a j o r and  46  minor p r i n c i p a l  s t r e s s e s and s o l v e equation  In analyses paths followed  (3-13)  f o r them.  o f problems i n which t h e s t r e s s -  a r e n o t those used t o e s t a b l i s h t h e s t r e s s -  s t r a i n r e l a t i o n s a v a i l a b l e , t h e best  that  one may do i s t o  assume t h a t t h e e l a s t i c m o d u l i a r e f u n c t i o n s o n l y o f t h e existing  s t a t e o f s t r e s s , and a r e g i v e n by e q u a t i o n s  from t h e s t r e s s - s t r a i n r e l a t i o n s .  M e t h o d s s u c h a s t h e one  i t e r a t i o n p e r l o a d increment method, d e s c r i b e d (2.2.1),  i n section (2.2.2),  perform incremental the  i n section  o r t h e i t e r a t i o n method u s i n g a c o n v e r g e n c e  t e r i o n , described  analyses.  derived  cri-  may t h e n be u s e d t o  T h e s e m e t h o d s do n o t r e q u i r e  c a l c u l a t i o n of the correct stresses for given  strains.  When u s i n g t h e s e m e t h o d s , h o w e v e r , one may n o t e x e r c i s e direct  c o n t r o l over t h e e r r o r s which a r i s e from t h e f a c t  that the s o i l ' s  s t r e s s - s t r a i n behaviour i s a c t u a l l y nonlinear  over t h e load increments  used.  I n t u i t i v e l y , one f e e l s t h a t b e t t e r between t h e s t r e s s - s t r a i n  agreement  c u r v e s g e n e r a t e d u s i n g t h e one  i t e r a t i o n p e r l o a d i n c r e m e n t method o r i t e r a t i o n s u s i n g a convergence c r i t e r i o n and t h e t r u e of t h e s o i l ,  may be o b t a i n e d  stress-strain relations  by d e c r e a s i n g  load increments used, provided  the size of the  that t h e equations  used t o  r e l a t e t h e tangent moduli t o the s t a t e o f s t r e s s a r e correct.  However, s i n c e t h e e x p r e s s i o n s  tangent moduli o f t h e s o i l take  stress-paths  used t o r e l a t e t h e  t o t h e s t a t e o f s t r e s s do n o t  i n t o a c c o u n t , one d o e s n o t know w h e t h e r  47 or  not  cause  the  r e d u c t i o n o f the  convergence  stress-paths  to  the  size  of  the  load increments  correct result  other than those  used to  will  i n problems with  establish  the  expres-  sions .  4.2  Numerical  Studies (18) performed n o n l i n e a r  Y u d h b i r and V a r a d a r a j a n finite  element  which the The  soil  analyses  w a l l r o t a t e d about was a s e d i m e n t e d Constant  the  finite  element  of the  its  base  system  away f r o m t h e  soil.  clay.  mesh w h i c h r e p r e s e n t e d mesh was a t  w a l l w h i c h was t e n  the  feet  same  high.  the  soil.  level  nodes to  s i m u l a t e d by t h e  as  the  A l l nodes  adjacent  the  to  on the.  to  to  the  assess the  used  T h e s e nodes  tangent state  of  of  assumed t o  values  o f Y o u n g ' s modulus  stress  by e q u a t i o n  importance o f the  i n e v a l u a t i n g the  results  were  Wall the  be  fixed  wall. The  related  it.  h o r i z o n t a l movement  in  The  l o w e r b o u n d a r y o f t h e mesh were assumed t o be f i x e d . r o t a t i o n was  in  s t r a i n t r i a n g u l a r e l e m e n t s were u s e d  lower boundary o f the base  for a retaining wall-soil  o f the  using t r i a x i a l  (3-18).  of the  parameters i n equation  analysis, tests  effect  sets  f o r two  In order  stress-path (3-18)  o f p a r a m e t e r s were different  were  i n c r e a s i n g with  constant,  (b)  decreasing with  constant.  the  evaluated  stress-paths;  (a)  on  and  48  These s t r e s s - p a t h s followed  were c h o s e n t o b o u n d t h e  stress-paths  i n the problem. The i n i t i a l  in the clay  major and m i n o r p r i n c i p a l  stresses  at a depth h below t h e t o p o f t h e w a l l  were  assumed t o be =  yh  =  K  (4-1)  and  a'  i n which  i s the major p r i n c i p a l e f f e c t i v e s t r e s s ,  the  3  o  a'  (4-2)  1  minor p r i n c i p a l e f f e c t i v e s t r e s s , y i s the u n i t  is weight  o f t h e c l a y , and K  =  1 - sin<)>'  (4-3)  i n which a ) i s the angle of I n t e r n a l 1  stress  analysis Two  ried  out.  analyses of the s o i l - w a l l  stress-path  to the state  for effective  of the c l a y . s y s t e m were  In.: o n e , t h e d a t a f r o m t h e t r i a x i a l  formed w i t h  triaxial  friction  tests  o f s t r e s s ; i n the o t h e r , the data from the  t e s t s performed with  stress-path  conditions  Y u d h b i r and V a r a d a r a j a n dicted horizontal  B were u s e d .  (18)  plotted  pressures at the s o i l - w a l l  predicted  used.  the pre-  interface  The p r e d i c t e d  z o n t a l p r e s s u r e s a t any d e p t h were f o u n d t o v a r y the stress-path  In  were a s s u m e d .  v e r s u s depth f o r each o f t h e a n a l y s e s .  with  per-  A w e r e u s e d t o r e l a t e Young's m o d u l u s  both analyses plane s t r a i n  cantly  car-  The h o r i z o n t a l  hori-  signifipressures  f o r a w a l l r o t a t i o n o f 0.006 radians using  data  49 from s t r e s s - p a t h predicted was  A were  h i g h e r at  u s i n g d a t a from s t r e s s - p a t h  as much as t w e n t y  deformation response  of  twent.y  of  inches  quadrilateral  (9)  the  (3-18)  tests  (3-19)  up t o  stress.  Poisson's  value  0.35-  reduced to  i n a plane  of  point  of  failure  criterion.  values  r a t i o was  The v a l u e  t e n pounds p e r  test  l o a d i n g was  of the  loads  f o r the  However,  the  with a given than those The  defined  have  a  after  i n the  p r e d i c t e d average  footing  observed  footing  in  fifty to  after  model t e s t  arbitrarily  footing  observed  prior  pressures  i n the  used  constant  under  i n a model failure  identical.  pressures  as much as  model t e s t  by  failure.  s y s t e m were n e a r l y  s e t t l e m e n t were  by  compression  o f Y o u n g ' s modulus was foot  of  minor p r i n c i p a l  assumed t o  p r e d i c t e d average  lower than those  value  The o b s e r v e d and p r e d i c t e d  footing-soil  observed  as  strain  analysis.  expressed  triaxial  compared w i t h t h a t  footing.  Constant  The p a r a m e t e r s  o f the  square  a depth  strain  s t r e s s was  f o u n d from t h r e e  fixed  measuring  tangent  The p r e d i c t e d b e h a v i o u r o f t h e vertical  load-  sand.  between t h e  state  the  were  c o n d u c t e d at  of  used  relationship  Mohr-Coulomb f a i l u r e  equation  difference  a n a l y s e d the  i n Chatahoochee R i v e r  Y o u n g ' s modulus and t h e equation  those  i n p l a n and b u r i e d at  e l e m e n t s were  The  The  a rectangular footing  by 2 . 4 4 i n c h e s  inches  B.  than  percent.  Duncan and Chang  12.44  a l l depths  associated  percent  higher  failure. failure  with  the  were  50  difference  increasing The  linear  several  of  the  behaviour  of  strip  footing  surface  loading.  The f i n i t e  program developed T h i s program i s  compatible  at  the  problems  The  stiffness  to  six  matrix  NONLIN,  strains  subroutines  for  was  this  a  of B r i t i s h Columbia.  small  node l i n e a r  six  vertical  perform incremental  involving  non-  a sand b e n e a t h a  subjected  University  stress-strain  program uses  incremental  element program used  designed to  The  settlement.  performed  i n c h wide  for  increasing  writer  analyses  analyses  with  nonlinear  using  any  s u p p l i e d by t h e  strain triangular  element  Is  derived  user.  elements.  in  Appendix 1 . The  state  of  stress  would n o r m a l l y v a r y a c r o s s element.  The v a l u e s  of  a linear  the  tangent  vary across  this  element  for  NONLIN u s e s  only  one  of  of  stress  at  the The  earlier bolic the  writer  stress-strain  (3-19)  and  Poisson's yield  as  to  relate ratio  defined  strain  centroid,  developed  for  to  the  the  by t h e  state  state  element.  based  upon  the  hyper-  Duncan and Chang ( 9 ) w i t h  These  tangent  use  However,  on t h e  entire  subroutines  subroutines  The s u b r o u t i n e s the  therefore,  material.  (15) i n order to of  loading  triangular  moduli would,  a nonlinear  relations  i n Appendix 2.  f r o m any  two m o d u l i , b a s e d  p r o g r a m NONLIN.  and  of  element  work done by Roy  computer  sented  set  resulting  values of  use of  stress,  are  pre-  equations Young's up t o  Mohr-Coulomb f a i l u r e  (3-l8)  modulus the  point  criterion.  51  The  tangent  are  then  moduli  values  Y o u n g ' s modulus and P o i s s o n ' s  converted to  for  use  with  The (3-19)  of  f o r the  tangent  parameters  used  density  were  o b t a i n e d from d r a i n e d t r i a x i a l of  b u l k and  shear  thirty-eight  the  (3-l8)  percent.  minor p r i n c i p a l  and  those presented  Sacramento R i v e r  tive  values  the  i n equations  p e r f o r m e d were  Wong and Duncan ( 1 7 ) f o r  stant  of  NONLIN.  analyses  of  values  ratio  sand w i t h  These  tests  a  by rela-  parameters  conducted at  stress  and a r e  con-  as  follows: unit  weight,  angle  of  cohesion  y = 8 9 . 5 pounds p e r c u b i c  internal friction,  < > j = 35-0  intercept,  pounds p e r  c = 0.0  foot  degrees cubic  foot  modulus number, k = 430 modulus  exponent,  failure  ratio,  R  n = =  f  0.27  0.84  Poisson's  r a t i o parameter G =  Poisson's  r a t i o parameter F = 0.21  Poisson's  r a t i o parameter d = 2.9  The described  in  convergence  section  system  wide  rigid  taining  (2.2.1)  were  u s i n g one  carried  iteration  out  as  and no  criterion. The  soil  incremental analyses  0.42  is  finite  element  shown i n F i g u r e  surface  strip  mesh u s e d (7).  footing  t o model the  One-half  of  was r e p r e s e n t e d  equal v e r t i c a l displacements  of  the  nine  a six by  footinginch  main-  nodes  FIGURE 7  F I N I T E ELEMENT MESH  53 closest mesh,  to'the  during  left  hand edge on t h e  loading.  rough f o o t i n g base, horizontally. allowed finite tial  In o r d e r to  t h e s e nodes  The nodes  element  mesh were  simulate  were n o t  on t h e  only v e r t i c a l motion.  upper s u r f a c e  sides  mesh  on t h e  be f i x e d  the  perfectly  allowed to  of the  The n o d e s  assumed t o  a  of  move  were  base  of  in their  the  ini-  positions. The  finite (4-1)  element  Initial mesh were  and (4-2)  element rest, '  mesh.  stresses  at  the  assumed t o  be g i v e n by  w i t h h measured from the The c o e f f i c i e n t  K , was t a k e n t o o'  of  centroids  top  of  the  equations  o f the  finite  l a t e r a l earth pressure  be 0.426 f o r t h e  Sacramento  at  River  sand. A set  of  analyses  modulus was r e d u c e d t o of  the  that is  tangent  a small value  b u l k modulus was k e p t  immediately p r i o r  as  to  The f o o t i n g using different  size  are  shown i n F i g u r e s  are  for analyses  those  foot  at  (3-4)  the  by t h e  load-settlement  load increments (8)  and ( 9 ) .  i n which the  the  yield.  (9)  are  same a f t e r  value  yield  writer. curves  in this  predicted,  set  of  analyses,  Those shown i n F i g u r e  for analyses five  as  This approach  s h e a r modulus o f t h e  s a n d was r e d u c e d t o  shear  y i e l d and t h e  o n e - h u n d r e d pounds p e r s q u a r e f o o t  shown i n F i g u r e  modulus o f  at  tangent  y i e l d was p e r f o r m e d .  suggested i n s e c t i o n  reduced to  i n which the  at  sand was  yield  i n which the  pounds p e r  (8)  while shear  square  54  lOOOr-  30 pound  increments  40 pound  2 0 nound  0.05  0.10  Footing  FIGURE  8  increments  increments  0.15  Settlement  I 0.20  (feet)  LOAD-SETTLEMENT CURVES FOR ANALYSES IN. WHICH THE SHEAR MODULUS WAS REDUCED TO ONE-HUNDRED POUNDS PER SQUARE FOOT AT YIELD AND THE BULK MODULUS'WAS NOT REDUCED AT YIELD  55  1000  0.10 Footing Settlement  FIGURE 9  0.20 (feet)  LOAD-SETTLEMENT CURVES FOR ANALYSES IN WHICH THE SHEAR MODULUS WAS REDUCED TO F I V E POUNDS PER SQUARE FOOT AT YIELD AND THE BULK MODULUS WAS NOT REDUCED AT YIELD  56  The p r e d i c t e d u l t i m a t e footing-soil by t h e  s y s t e m may be t a k e n as  predicted load-settlement  s u g g e s t e d by t h e analyses, to  five  ing  duced to not  which the at  size.  as much as  The w r i t e r  ing  the  asymptotic  load increment For  capacity, soil  all  represented  foot  reduced to  curves  at  five  say  pre-  yield,  redid  analyses  in  pounds p e r  that  than s i x  load appeared to  comparison, the  i n the  half  the  square  asymptotes  hundred pounds.  increase  theoretical  finite  of the  is  f i v e - h u n d r e d and t h i r t y - s e v e n  to  fourteen  percent  six  element  bearing capacity  with  increas-  ultimate the  i n c h wide  mesh,  factors  footing-  footing  found w i t h  given  lower than those p r e d i c t e d i n  the  This  s h e a r modulus was r e d u c e d t o at  y i e l d and a t  least  the  by V e s i c ( 1 6 ) , six  foot  pounds.  as  bearing  is  i n which the  pounds p e r s q u a r e  increas-  size.  f o r the  Caquot-Kerisel  analyses  s i x - h u n d r e d and  assuming r i g i d - p l a s t i c b e h a v i o u r of  system,  about  s h e a r modulus was  square  greater  reduced  v a r i e d from  load increased with  can o n l y  the  s a n d was  those p r e d i c t e d i n the  s h e a r modulus was  yield.  about  the  suggested  predicted in  yield,  i n which the  of  asymptotes  The l o a d - s e t t l e m e n t  analyses,  s u g g e s t e d by them were Again,  at  o n e - h u n d r e d pounds p e r  flatten  foot  foot  The a s y m p t o t i c  i n the  The  curves  and s e v e n t y pounds t o  load increment  asymptote  s h e a r modulus o f t h e  pounds p e r s q u a r e  pounds.  dicted  the  curve.  load-settlement  i n which the  five-hundred twenty  bearing capacity  ten  percent  five  percent  57  lower the  than  shear  square  any o f t h o s e p r e d i c t e d modulus  foot  at  was  reduced  abrupt  increases  tening  out  somewhat. failed  i n the  These  had n o t  analysis  pounds load  per  increments  loads  equal  settlement failed  to  at  began  shown  five-hundred pounds,  to  reduced to  five  are  appear  to  at  base o f  the  finally, the  move i n the  smooth  u p w a r d , away  with  further  the  had  elements in-  footing. elements,  shown  any  was  predicted  at  by  to  square  failure  the  shear  foot  at  m o t i o n s r e p o r t e d by V e s i c  and  modulus yield,  footing. (16) for  the  from  forty was  and  in  downward  horizontal  This  the  The nodes  beginning to  of  increment  were u s e d .  and c h a n g i n g  from the  load  of  mode.  vicinity  five-hundred  trajectories  load-  The l o c a t i o n s  clear  five  (10) for  which the  out.  (11) for  reduced to  in Figure  nodes i n the  increments  footing  elements which  failing  i n which the  w h i c h t w e n t y pound l o a d  flat-  to  flatten  per  after  and,  than that  in Figure  pounds  slopes  showed  and i n w h i c h t w e n t y pound  suggest  analysis  curves  yield  and t w e n t y pounds  i n the  per  failure,  failed  The movements o f are  pounds  which  modulus  and g r e a t e r  e l e m e n t s do n o t  footing  in  shear  were u s e d ,  curve  to  of  i n which the foot  of  failed,  load applied  square  in their  were due t o  coming out  The l o c a t i o n s the  one-hundred  load-settlement  and d e c r e a s e s  which p r e v i o u s l y creases  analyses  yield.  The p r e d i c t e d  previously  to  i n the  is  and,  similar  general  shear  to  sis  FIGURE  10  LOCATIONS OF F A I L E D E L E M E N T S FOR THE ANALYSIS I N W H I C H T H E S H E A R M O D U L U S WAS R E D U C E D TO FIVE POUNDS PER SQUARE FOOT AT Y I E L D , THE BULK M O D U L U S WAS NOT R E D U C E D A T Y I E L D , AND TWENTY POUND LOAD INCREMENTS WERE USED  ( r e m a i n d e r o f mesh (Failed  FIGURE  11  omitted)  Elements are  shaded)  MOVEMENTS OF NODES DURING THE LOAD INCREMENT FROM F I V E HUNDRED AND TWENTY POUNDS TO F I V E HUNDRED AND FORTY POUNDS IN THE ANALYSIS IN WHICH THE SHEAR MODULUS WAS REDUCED TO F I V E POUNDS PER SQUARE FOOT AT Y I E L D , THE BULK MODULUS WAS NOT REDUCED AT Y I E L D , AND TWENTY POUND INCREMENTS WERE USED  60  failure  of  a strip  footing  As was practice out  to  use  stated  in section  a constant  a nonlinear analysis body and s i m p l y t o  lus  yield.  tice  on t h e  response second  Poisson's less  of  results  of  six  of  yield.  to  relate  the  of  stress  prior  first  set  second first  of  set set  analyses  of  yield,  analyses analyses.  are presented The f o o t i n g  the  and t h e  was t h e  of t h i s  The f i n i t e was  the  footing,  value  be 0.42  second  set  foot  of  analyses  the  mesh u s e d  of the  state  used i n  used  in  second  the  in  the  the  set  of  below. load-settlement  load increments  curves p r e d i c t e d ,  i n the  analyses  h u n d r e d pounds p e r s q u a r e The u l t i m a t e  foot  at  yield,  are  shown  b e a r i n g c a p a c i t y of the  load-settlement  in  have a c o n s t a n t  and i n w h i c h Y o u n g ' s modulus was r e d u c e d t o  s u g g e s t e d by t h e s e  of  regard-  square  that  same as t h a t  r a t i o was assumed t o  (12).  a  of Young's  element  The r e s u l t s  prac-  load-settlement  strip  same as  of  Y o u n g ' s modu-  o f Y o u n g ' s modulus t o  which P o i s s o n ' s  Figure  effect  value  In t h i s  size  0.42  of  of the  using different  of  value  surface  used  value  analyses.  of  r a t i o through-  o n e - h u n d r e d pounds p e r  The e q u a t i o n ,  to  common  load-deformation response  sand was assumed t o  stress,  tangent  is  was p e r f o r m e d i n w h i c h t h e  modulus was r e d u c e d t o at  it  Poisson's  analysis  i n c h wide  f o r the  state  of  assess  the  analyses  ratio  the  of  (3.4),  reduce the  In o r d e r to  o f the  set  value  of the  a soil at  on s a n d .  curves  is  less  value one  in footing than  about  61  FIGURE 12  LOAD-SETTLEMENT CURVES FOR ANALYSES IN WHICH POISSON'S RATIO HAD A CONSTANT VALUE OF 0.42 AND YOUNG'S MODULUS WAS REDUCED TO ONE-HUNDRED POUNDS PER SQUARE FOOT AT YIELD  62  three  hundred pounds.  g e s t e d by t h e set  of  yield  value  the  0.42  pounds p e r  not  foot are  than that  flatten  at  at  out.  are  failed  failure  stant  i n the value  analysis  of  0.42,  h u n d r e d pounds p e r load  increments  that  the  ments d i r e c t l y  failed  beneath  ted at  i n the  This  y i e l d while  the  loads  to  equal  at  vicinity  of  The o n l y  ratio  figure, o f the  it  had a  very d i f f e r e n t  pound  is  apparent ele-  motions  beneath  the  from t h a t  s h e a r modulus was  b u l k modulus was  con-  one  failed  significant  predic-  reduced  not.  between t h e  from  thirty-five  and f i f t e e n  nodes d i r e c t l y  i n which the  The d i f f e r e n c e  do  the  l o a d Increment  yield,  From t h i s  the  to  curve  elements  o n e - h u n d r e d and  by c o m p r e s s i o n  of  incre-  mode.  foot  behaviour is  analyses  one-hundred  pound l o a d  failed  ( 1 4 ) f o r the  it.  at  constant  Y o u n g ' s modulus was r e d u c e d t o  were v e r t i c a l m o t i o n s footing.  of  i n which P o i s s o n ' s  square  first  predicted  reduced to  of nodes i n the  were u s e d .  footing  as  load-settlement  The l o c a t i o n s  shown i n F i g u r e  sug-  i n the  r a t i o had a  in Figure (13) for  o n e - h u n d r e d and t w e n t y pounds pounds  elements,  which the  The movements  that  not.  y i e l d and f i f t e e n  shown  s u g g e s t any c l e a r  footing  predicted  and Y o u n g ' s modulus was  ments were u s e d ,  began t o  of  of  s h e a r modulus was r e d u c e d  i n which P o i s s o n ' s  square  and g r e a t e r  one-half  curves  b u l k modulus was  analysis of  about  i n w h i c h the  The l o c a t i o n s i n the  is  load-settlement  analyses, while  This  load-deformation  63  A t 12 0 pounds  sis  FIGURE 13  LOCATIONS OF F A I L E D ELEMENTS FOR THE ANALYSIS IN.WHICH POISSON'S RATIO HAD A CONSTANT VALUE OF 0.42, YOUNG'S MODULUS WAS REDUCED TO F I V E POUNDS PER SQUARE FOOT AT Y I E L D , AND F I F T E E N POUND LOAD INCREMENTS WERE USED  64  FIGURE 14  MOVEMENTS OF NODES DURING THE LOAD INCREMENT FROM ONE-HUNDRED AND TWENTY POUNDS TO ONEHUNDRED AND T H I R T Y - F I V E POUNDS IN THE ANALYSIS IN WHICH POISSON'S RATIO HAD A CONSTANT VALUE OF 0 . 4 2 , YOUNG'S MODULUS WAS REDUCED TO ONEHUNDRED POUNDS PER SQUARE FOOT AT Y I E L D , AND F I F T E E N POUND LOAD INCREMENTS WERE USED  response set  of  o f the  analyses  analyses  analysis.  In the  as  reduces  it the  was  the  of  set but  the  sand.  s h e a r and t h e  second shear  set  of  allowed the  This  compressed  to  them.  the  easily  not  footing-soil  more r e a l i s t i c  practice  the  normal  be  of  de-  the  easily  l o a d a p p l i e d to  value  reduces yield.  resistance  them.  Poisson's  the  ratio  values  resistance  of  increases  of  a real  r e d u c e d at  c r i t e r i o n , the  p r e d i c t e d i n the  of  Therefore, in elements to  was r e d u c e d a t  by f u r t h e r  than that  practice  analyses,  c o u l d not  i n the the  of  after  o f Y o u n g ' s modulus was r e d u c e d  significantly  system  set  same  deformations  to  yield.  elements d i r e c t l y beneath  Mohr-Coulomb f a i l u r e  analyses.  footing  deformations  Since the is  first  b u l k m o d u l i at  yielded  be  strains  the  value  modulus  This  shear  in  each  shear  yield.  of  way  in  reduce t h e i r r e s i s t a n c e  analyses,  and n o r m a l  to  elements to  of analyses, the  the  first  set  b u l k modulus was t h e  by f u r t h e r i n c r e a s e s  second  account  analyses,  T h e r e f o r e , i n the  constant  yield  the  of the  elements beneath  compressed  of  second  between t h e  into  Immediately p r i o r  does n o t  formations.  In the  set  resistance  y i e l d but  yielded  first  p r e d i c t e d i n the  difference  sand was t a k e n  y i e l d but  was  system  p r e d i c t e d i n the  from t h e  o f the  r e d u c e d at  yield  at  and t h a t  resulted  which y i e l d  was  footing-soil  the  i n the  as  p r e d i c t e d i n the  set  of  second  the both  This to  load applied  normal defined  behaviour of  first  both  footing  sand t o  yield,  at  the  analyses set  by  of  is  CHAPTER 5 CONCLUSION  E x i s t i n g methods load-deformation response based for  on i n c r e m e n t a l  is  computed f o r  form to  stress-strain  the  a prescribed tolerance.  first, the  of the  the  conform to secondly, and t h e ible the  the  successively  stress-strain  computed s t r e s s e s .  by i t e r a t i v e  adjustment  the  of  analysis  second  the  response  are not  the  computed s t r e s s e s the  con-  within  stress-strain  i n two  ways:  correct  and  c o r r e c t e d t o make them of  the  corrected to relations  material;  of  the  to  be  make them the  objective  elastic  to  stresses  and  correct compat-  m a t e r i a l and Is  accomplished  properties  used  load increment;  in  the  elastic  changed.  Methods o f  form t o  material  by a Newton-Raphson a p p r o a c h i n w h i c h t h e  properties  the  o f the  relations  of the  the  load increment  may be c o n s i d e r e d  The f i r s t  are  A solution  if  computed s t r e s s e s may be assumed  strains  with the  end o f t h e  Convergence to  stress-strain  structures  acceptable  successively  nonlinear  analysis.  m a t e r i a l may be a c h i e v e d  stresses  the  elastic  relations  computed s t r a i n s  computed  and s o i l  considered  and s t r a i n s  relations  soils  iterative  a load increment  the  of  o f p r e d i c t i n g the  incremental analysis and computed  stress-strain  relations  66  strains of  the  i n which  both  a r e made t o m a t e r i a l by  con-  67  iterative  adjustment  advantage  of  computational  soil  body must  only  be r e b u i l t  prior  to  reason, per  used,  it load  Also,  the  and s t r a i n s  conform to  the  material  not  that  sideration the  is  computed  the not  each  increment since  the  one  consider-  matrix  successive  and n o t iterates  stress-strain  matrix  of  d u r i n g the  This  of  the  adjust-  to  perform  to  use  a  on t h e  con-  elastic  computed  stres-  relations  Newton-Raphson method has  rebuilt  the  of  controlled.  stiffness  strains.  have  common p r a c t i c e  directly  The m o d i f i e d advantage  stiffness  accuracy with which the  ses  is  is  properties  they r e q u i r e a  the  criterion.  properties  However, since  iteration  vergence  elastic  effort  For t h i s one  the  simplicity.  able  ment.  of  saves  the  the  body u n d e r  iterative  con-  correction  considerable  of  computational  effort. The m o d i f i e d applied curve  to  is  family  problems  followed.  of  i n which a s i n g l e However,  stress-strain  stress-strain  curve  state  path followed  not  As was tion  of  explained the  involving  in  curves  of  that  is  for  at  more  stress-strain  used  soils,  and  point  if  curves  (4.1.2), this  soil the  the  mass  stress-  obtained. applica-  problems  difficult.  The a c c u r a c y w i t h w h i c h t h e  a  the  were  makes  m o d i f i e d Newton-Raphson method t o  soils  readily  w i t h i n the  that  which the  is  involving  usually  by a p o i n t  stress  section  known  i n problems  followed  changes w i t h the is  Newton-Raphson method  load-deformation  68  response  of  soils  and s o i l  l i m i t e d m a i n l y by t h e properties relations soils  o f the  error:  elastic  of  and t h e  anisotropy of  the  Ideally, which take  the  into  of  effect soils  one  Is  stress-strain stress-strain  assumption  that  and have two m a j o r is  neglected]  neglected.  s h o u l d use  stress-strain  dependent  b e h a v i o u r and  While t h i s  t h a n does t h e  relations  is  aniso-  possible,  relations  the  would r e q u i r e  properties  development  and c o n s i d -  of those  cur-  used.  generally  that  after  y i e l d as values  real  the  soil,  whereas  Therefore,  the  o f the  soils  after  yield  m a t e r i a l , many a u t h o r s  elastic elastic to  resistance  while the  If  materials. moduli  shear  at  yield.  stresses  to normal  s h e a r modulus i s  They  is  deal.with simply For a  reduced  stresses r e d u c e d at  is  is  at  not.  yield,  the  not. one d e a l s  with s o i l s  y i e l d one must be  b u l k modulus i s a constant  soft  resistance  b u l k modulus i s  after  behaviour of  of a p l a s t i c  reduce the  ials  is  account.  Although the  yield  The  stress-path  o f more s t r e s s - s t r a i n  e r a b l y more e f f o r t  soils  of  such s t r e s s - s t r a i n  evaluation  rently  materials  stress-path  soils  development the  are r e p r e s e n t e d .  c u r r e n t l y u s e d a r e b a s e d on t h e  sources  of  can be p r e d i c t e d  a c c u r a c y w i t h which the  soil  are i s o t r o p i c  tropy  structures  value  Poisson's  soft  c a r e f u l that  not reduced at of  as  yield.  elastic  the  value  materof  The p r a c t i c e o f  r a t i o and r e d u c i n g t h e  the using  value  69  of  Y o u n g ' s modulus a t  yield  in  a r e d u c t i o n of the  values  the  bulk modulus.  footing  on sand t o  show t h a t  this  elements. affected  footing-soil  of  both the  of the  vertical  loading,  leads to  a great  unacceptable  Analyses  The y i e l d e d to  is  both the  s y s t e m and f a i l u r e  results  of  a  strip  p e r f o r m e d by t h e behaviour  e l e m e n t s were t o o  extent  it  s h e a r modulus and  response  unrealistic  since  in yielded  compressible.  modes o f  loads  writer,  failure  predicted  in  soil This  of  the  the  analyses. Nonlinear ponse o f the  soils  and s o i l  of  the  structures  modulus  relations.  At y i e l d ,  s h o u l d be r e d u c e d and t h e  As f a r  as  possible,  the  strain relations  used  p a t h on t h e  stress-strain  soil  load-deformation  s h o u l d be p e r f o r m e d  s h e a r modulus and b u l k modulus t o  stress-strain  ged.  analyses  define  the  the  value  b u l k modulus  elastic  s h o u l d model t h e  of  effect  properties.  of  using  soil's the  left  m o d u l i and  res-  shear unchan-  stressstress-  BIBLIOGRAPHY  B r e b b i a , C . A . and C o n n o r , J . J . , ' F u n d a m e n t a l s o f F i n i t e Element T e c h n i q u e s , H a l s t e a d P r e s s , John W i l e y and Sons L t d . , New Y o r k - T o r o n t o , 1974, pp. 129-138. C o r o t i s , R . B . , F a r z i n , M . H . and K r i z e k , R . 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Roy, J . , 'Nonlinear A n a l y s i s o f the Undrained Shear Strength of C l a y , ' a r e s e a r c h p r o j e c t submitted i n p a r t i a l f u l f i l l m e n t o f the requirement f o r the degree of Master of E n g i n e e r i n g , Dept. of C i v i l E n g i n e e r i n g , University of B r i t i s h Columbia, 1 9 7 4 , unpublished.  16.  V e s i c , A . S . , " A n a l y s i s o f U l t i m a t e Loads o f Shallow F o u n d a t i o n s , " J o u r n a l o f t h e S o i l M e c h a n i c s and F o u n d a t i o n s D i v i s i o n , A S C E , V o l . 9 9 , No. S M I , J a n u a r y , 1973, pp. 4 5 - 7 3 .  17.  Wong, K . S . and D u n c a n , J . M . , H y p e r b o l i c S t r e s s - S t r a i n Parameters f o r N o n l i n e a r F i n i t e Element A n a l y s e s o f S t r e s s e s and Movements i n S o i l M a s s e s , ' R e p o r t No. T E - 7 4 - 3 t o N a t i o n a l S c i e n c e F o u n d a t i o n , D e p t . o f C i v i l E n g i n e e r i n g , I n s t i t u t e o f T r a n s p o r t a t i o n and T r a f f i c Engineering, University of C a l i f o r n i a , Berkeley, C a l i f o r n i a , 1974.  18.  Y u d h b i r and V a r a d a r a j a n , A . , " S t r e s s - P a t h Dependent Deformation Moduli o f C l a y , " J o u r n a l of the G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , A S C E , V o l . 1 0 1 , No. G T 3 , M a r c h , 1975, pp. 3 1 5 - 3 2 7 .  19.  Z i e n k i e w i c z , O . C . , The F i n i t e E l e m e n t Method i n Engineering Science, M c G r a w - H i l l Book C o . , L t d . , London, E n g l a n d , 1 9 7 1 , pp. 3 6 9 - 3 7 7 .  APPENDIX  1  DERIVATION OF THE STIFFNESS  MATRIX FOR  THE LINEAR STRAIN TRIANGULAR ELEMENT  The nodal the  forces,  linear  element  {Q},  stiffness  and t h e  shown i n F i g u r e  {u},  of  a point,  p,  l i n e a r s t r a i n t r i a n g u l a r element the  nodal displacements,  {u}  {q},  the  for 15  follows:  The d i s p l a c e m e n t s ,  to  nodal displacements,  s t r a i n t r i a n g u l a r element  may be d e r i v e d as  the  m a t r i x which r e l a t e s  =4  where  [A] as  terms  of natural  u u  (q),  [A]  within  are  related  by t h e  equation  {q}  (1)  y  g i v e n by D e s a i and A b e l coordinates,  ( 4 ) , In  is  1) ? (2?  0 ? (2C 2  0 1  -  2  1  1)  3  -  3  [A.]. =  ? (2?  2  -  1)  ? (2?  3  -  1)  1) 3  4  ?  1  ?  2  4 C 4  V  V  1  ?  2  s  4 4  1)  0 2  ? (2C  -  C  2  3  ?  ?  3  l  4  72  ?  1  '(2)  FIGURE 15  LINEAR STRAIN TRIANGULAR ELEMENT  74  in  which A  A, A A  '1  where  A is  and A^ i s  A  ' 22 g  the  area of  the  the  area of the  o p p o s i t e node  I as  {q},  w i t h i n the  for plane  strain  3  A  3  element  p and t h e  nodes  in Figure 16.  and t h e  strains,  nodal  {e},  at  displacements,  is  =  {e}  point  between t h e  element  ?  triangular  illustrated  The r e l a t i o n s h i p any p o i n t  and  [L]  [A]  [B]  {q}  {q}  (3)  xy in  which  0 0  L  5/6  (4)  y 6/6!  y  For n a t u r a l c o o r d i n a t e s , give  the  following  differentiation  6TJ  b  i n d i c i a l notation b  and  n  x  6 r,  2A  6X  in  3.-U  m  m  where  =  Y  =  y  =  y  0  3  x  -  y-  -  y  -  y  x  2  Desai  and A b e l ( 4 )  formulae: (5)  75  x-Axis  FIGURE 1 6  AREAS FOR NATURAL COORDINATES  6U  a  ,6y in  where : y 2  2  a and  similar  =  =  3  3 l X  l  The  y  [B]  2 l b  3  2  3  2  Y  l  other  in  u 3  1  - %)b  1  - %)a  2  - h)a  - %)b  2  0  - %)a  3  - %)b  3  b  ?  2  0  2 l a  - %)a  +  3  2  ?  c  l 2 a  b  c  l 3 a  3  2 l a  c b 2  c a 3  1  2  +  +  ?  l 2  +  ?  l 2  z  3 l a  c  l 3  ?  l 3  +  a  b  a  b  + c +  + 0  3 l  (c  3  0  3  t, <3-  ?  1  0  l 2  + ? b  +  3  then  %)a  2  2  (C  —  «1  0  ?  (3) i s  0  -  +  derivatives.  equation  - %)a  0 ^b  x  0  0 c b  y  [B]  0  ?  2  l 3  x  1  - %)b  = $  3  the  0  ^3  x y  -  3  for  matrix  - %)b  -  3  "  y  x  formulae  « 1  x  6? m  i n d i c i a l notation  a  6U  m 2A  x  +  a 3  b  ?  2  3  ?  3 l  ?  3 l  a  b  77 The s t r e s s e s , elastic point  m a t e r i a l are  by t h e  {a},  related  at to  any p o i n t  the  in a  strains,  linear  {e},  at  that  equation  [D]  (8)  {£}  xy J for  plane  strain  conditions.  F o r an i s o t r o p i c  elastic  material  [D]  =  k +  y G  k -  k  2  / G  k +  3  3  2  / G 3  V G 3  0 (9) i n which k i s  the  For work,  the  the  virtual  virtual  virtual  real  strains,  {a}  T  external  dV  =  {e},  shear  principal  real internal  nodal displacements. {e}  the  nodal displacements,  work- done by t h e  through the  and G i s  e q u i l i b r i u m , by t h e  work done by t h e  through the to  b u l k modulus  of  loads, (q),  modulus. virtual  {Q},  must  stresses,  be  {a},  corresponding to  acting equal acting  the  Therefore, {q}  T  (10)  {Q}  V which,  upon t h e  i n t r o d u c t i o n of  {i} [A] [L] [D][L][A]{q}dV T  V  T  T  equations  =  (3)  {q} {Q) T  and  (8)  becomes  ( I D  and s i n c e  {q}  and  1  {q}  are not  [A] [L] [D][L][A]dV T  T  functions  =  of  position,  '{Q}  V  or where  [K] (q) [K] i s  =  called  {Q> the  element  stiffness  matrix.  (12)  APPENDIX 2  STRESS-STRAIN  The  SUBROUTINES  w r i t e r developed  stress-strain  b a s e d on e a r l i e r work done by Roy ( 1 5 ) t o the  hyperbolic stress-strain  (9)  with the  routines (3-18)  finite  element  c o n t r o l the  and ( 3 - 1 9 )  to  relate  centroids.  a l l o w the  process  the  r a t i o to  The i t e r a t i v e  These  and use  tangent the  values  state process  of  of  used  is  (2.2.1).  The s u b r o u t i n e s  presented  below.  defined are  at  the  the  one  section  i n FORTRAN and a r e  A l l FORTRAN v a r i a b l e s and a r r a y s a r e  where t h e y  e x p l a i n e d at  sub-  Young's  stress  p e r l o a d I n c r e m e n t method d e s c r i b e d i n written  of  equations  iteration  were  use  o f Duncan and Chang  p r o g r a m NONLIN.  iterative  modulus and P o i s s o n ' s element  relations  subroutines  appear.  the  start  The f u n c t i o n s of  each  79  of the  subroutine.  subroutines  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  C C C C  SUBROUTINE  SETUPCX)  T H I S . S U B R O U T I N E * TOGETHER WITH SUBROUTINE E U C A L C U L A T E S THE I N I T I A L BULK AND SHEAR MODULI FOR THE E L E M E N T S . _. I M P L I C I T RE AL* 8 { A-H, D - 2 ) " " " ~ ~ DIMENSION X( 50) C OM MO_N_ / CN S J J T / P A TRN (1 0,10? ,NCONST ? I P L A N E CO MMO N / E U B L K / TsT"E"P7N3TEP , K S T E P , I F G A CATA I S T R T / O / . N S T F P = 0 * ~ ~ KSTEP=1 IFGA=0 I" F ( T S T R T . E Q . O ) I S T E P = 0 ' ISTRT=1 9  :  JC  " 1 8 " " "C 19 C 20 C 21 C 22 C 23 C 24 C 25 C _26 C 27 C 28 C 29 C 30 C 31 C _32 C 33 C 34 C 35 C  THE ARRAY X CO N T AI N S ~ T H E~TQV C 0 WI NG ELEMENT  INFORM AT 10 NT  1 SOIL UNIT WEIGHT 2 COHESION I N T E R C E P T , C 3 ANGLE OF INTERNAL F R I C T I O N I N DEGREES 4 F A I L U R E R A T I O * RF 5 MODULUS NUMBER, K " " " ~ " 6 MODULUS EXPONENT, N 7 MODULU_S_AFTER_F A I LURE 8 PGISSON•S RAT 10 PARAMETER? G 9 T H I C K N E S S ( P L A N E PROBLEMS ONLY) 1 0 - 1 5 I N I T I A L ELEMENT S T R E S S E S : S I G X » S I G Y , S I G Z t T A U X Y , T A U Y Z , T A U Z X 16 I N I T I A L ELEMENT PORE PRESSURE 1 7 - 2 2 ZERO S T R A I N S EPS X,EPSY,EPSZ,GAMXY,GAMYZ,GAMZX 34^ POI SSON* S R A T 10 _PARAMETER,F 35 POISSON'S RATIO" PARAMETER,^ 36 T E N S I L E STRENGTH 37 ATMOSPHERIC PRESSURE, PATM  o  36 C 38 SCRATCH SPACE (NOT USED) 37 C 39 SCRATCH SPACE (NOT USED) _38 C A0 C O E F F I C I E N T OF LATERAL EARTH P R E S S U R E , KO 39 C 40 C ON RETURN, X CONTAINS THE A D D I T I O N A L INFORMATION BELOW. Al C _ 42 C 23 I N I T I A L Y I E L D FUNCTION 43 C 24 I N I T I A L BULK MODULUS 44 C 25 I N I T I A L SHEAR MODULUS 45 C 46 X(23)=0.D0 47 C ._ 4 8 C " ' C A L C U L A T E THE I N I T I A L BULK AND SHEAR MODULI49 C _50 CALL E U ( E T , U T , X ( 1 0 ) , X ( 2 ) , X t 3 4 ) ) 51 C 52 C COMPUTE THE I N I T I A L BULK AND SHEAR MODULI 5 3 C ' __ 54 X(24}=UT 55 X(25)=ET/(3.D0-<ET/(3.D0*UT))> 56 C . 57 C X ( 2 4 ) AND X C 2 5 ) CONTAIN THE I N I T I A L BULK AND SHEAR MODULI* 58 C 59 FFTURN ___ 6 0 " F N D ~ ' " " "" 61 C __6 2 S UBROUT INE ST I F LT ( X , SS S DE PS 1, EPS V EPS A ) 63 C 64 C T H I S SUBROUTINE, TOGETHER WITH SUBROUTINES EU AND EUD, 65 C CCNTROLS THE S T R E S S - S T R A I N BEHAVIOUR OF THE F I N I T E ELEMENTS. "66 C " " " " " 67 C ARRAY X CONTAINS: _6 8 C i S O I L UNIT WEIGHT 69 C 2 COHESION I NTS RC E P T C 70 C 3 ANGLF OF INTERNAL F R I C T I O N IN DEGREES 71 C 4 F A I L U R E _ R A T I O , RF y  t  .  t  '  •  c h  7 2 73 74 75 76 77 70 79 _ 80 81 82 83 84 85 86 87 88 89 9 0 91 92 93 94  C C C C C C C C C C C C C " C C C C C C S S C C C C  95 97 _9 8 99 100 101  5 M O D U L U S NUMBER, K 6 M O D U L U S E X P O N E N T , N. 7 MODULUS AFTER F A I L U R E 8 POISSON'S RATIO PARAMETER,G 9 T H I C K N E S S * P L A N E PROBLEMS ONLY) 1 0 - 1 5 S T R E S S E S A T T H E END OF L A S T S T E P ( U P D A T E D BY STIFLT) 16 P O P E P R E S S U R E AT T H E E N D O F L A S T S T E P ( U P D A T E D BY S T I F L T ) 1 7 - 2 2 Z E R O S T R A I N S E P S X , E P S Y , E P SZ » G A M X Y , G A M Y Z , G A M Z X 23 YIELD FUNCTION 2 4 - 3 3 MODULI U S E D I N P R E V I O U S S T E P , R E T U R N E D M O D U L I FOR N E X T S T E P * 34 POISSON'S RATIO PARAMETERS 35 POISSON'S RATIO PARAMETER,D _ 36 SCRATCH SPACE (NOT U S E D ) " 37 ATMOSPHERIC PRESSURE, PATM 38 S C R A T C H _ S P A C E _ < NOT_USE_DJ .__ 39 SCRATCH SPACE (NOT USED) AO C O E F F I C I E N T , KO X(41)-X(50) MAY B E I N I T I A L I Z E D F O R S U B S E Q U E N T U S E IN S T I F L T . ' S I S TO R E T U R N T H E M O D U L I WHICH A R E T O B E U S E D WHEN THE S T E P IS REITERATED. P E P S IS T H E I N C R E M E N T A L S T R A I N[S F O R 'j_HE_ L A S T S T E P . E P S V IS T H E S T R A I N V ' E L O C I T T E S A T ~ T H E P R E S E N T TIME. E P S A I S T H E S T R A I N A C C E L E R A T I O N S AT T H E P R E S E N T T I M E .  C P E A L * 8 X ( 6 3 ) , S S S ( 1 0 ) , D E P S ( 6 ) , E P S V ( 6 ) , E P S A ( 6 ) ,D ( 1 6 ) , S D { 4 ) , D E P S 1 ( 6 ) I NT E G F R * 2 I F A L ( 5 0 0 ), IAD 0 COMMON / F A L B L K / D T , D E L T A T » D T N E X T , I R E S T P , I E L T , I F G O , I C H N G , I F A L , * IPLANE,IDEBUG,ISTAT,IFDT,IFDIAG,ITHAX,LSTEP,IFERRF COMMON / E U B L K / I S T E P ,_NST E P,_K5 T E P » I F G A _ _ _  __  103 _1 0 4 105 106  NSTEP=1 I FGA- I FGO C C  107  C  /FALBLK/  CONTAINS  GOVERNING  INFORMATION  FOR  THE  PROGRAM  AS  A  WHOLE.  1 0 8 C I P E S T P I S THE I T E R A T I O N NUMBER ( 1 ON THE F I R S T P A S S ) . 109 C IFLT I S THE NUMBER OF THE ELEMENT B E I N G PROCESSED BY S T I F L T . HO C IFGO I S A FLAG SET WHEN THE_ ACCURACY C R I T E R I O N IS V I O L A T E D "111 C F O R A ~ G I VENTSTEP- I"F NOT 0 IT C A U S I E ' S n r H E S T ' £ T ^ p ~ B E 112 C REPEATED ( U N L E S S I R E S T P > I T M A X ) . 113 C ICHNG I S A F L A G TO I N D I C A T E THAT THE ELEMENT MODULI FOR 11* C THE NEXT STEP ARE THE SAME AS FOR T H E P R E V I O U S S T E P . 115 C J16 C OF THESE, ONLY IFGO AND ICHNG ARE CHANGED 8Y S T I F L T • 117 C 118 C 119 C UPDATE THE ELEMENT S T R E S S E S . _ 120 C " 121 I F ( I R E S T P . G T . l ) GO TO 100 122 IFGC=1 123 DO 101 1=1,4 '• 124 101 0 = PS( I ) = D E P S 1 ( I ) / 2 . D 0 125 99 CALL GF T D ( X ( 2 4 ) , 0) . 1 2 6 " C A L L BL0WUP(D,D,4) " "~ 127 CALL DGMATV(D,DEPS,SD,4,4,4) 128 00 1 1 = 1,4 129 1 X'Q"+"9 "j="X"( 1+9) +SDTTJ ; 130 C 131 C C A L C U L A T E NEW MODULI. _ 132 " C ~ ~ " I F ELEMENTHASYIELDED,THE BULK MODULUS I S 133 C NOT CHANGE0 FROM ITS LAST V A L U E . 134 C 135 C ALL Eli ( E T , U T , X f 10) ,X( 2 ) , X ( 3 4 ) ) 136 C 137 I F ( U T .EO.O.ODO) GOTO 5 0 0 _ -  139 J^4() 141 142 143  500 600  X ( 2 5 ) = E T / ( ' 3 . D 0 - { E T / t 3 . D 0 * U T ) J) GO_TO_ 6_0_0_ CONTINUE X(25)=ET CONTINUE  -  144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179  C C  X ( 2 A ) AND  X ( 2 5 ) ARE  THE BULK AND  SHEAR MODULI  RESPECTIVELY.  c  SSS{1)=X(24) SSS(2)=X(25) ICHNG=1 ;  -  c"  RETURN c  100  IFGO=0 DP 102 1=1,4 102 D E P S { I J = D E P S 1 ( I ) GO TO 99  c  END c  SUBROUTINE c c c c c c  EUD(ET , U T , S T R E S S , P R 0 P 1 , P R 0 P 2 , C R , S I , S 3 , I STEP,NEB)  THIS SUBROUTINE CHECK FOR F A I L U R E AND COMPUTES THE VALUES OF YOUNG'S MODULUS AND POISSON'S R A T I O . ( R E F : DUNCAN AND CHANG, ASCE,JOUR. OF S.M.GF.E. D I V . , V 0 L . 9 6 , N0.SM5, SEPTEMBER,1970) IMPLICIT REAL*8(A-H,0-Z) DIMENSION S T R E S S ( 1 4 ) , P R O P i ( 7 ) , P R 0 P 2 ( 7 )  c  60  90  I F ( N E B . G T . l ) W R I T E ( 6 , 6 0 ) CR,S1,S3 FORMA T( • CR = • ,F10.5,'S1='»F10.5» S3 = "»F10.5) FA=PROP1(2)/57.28DO CO=DCnS(FA) S I = DSIN(FA) FFF=2.D0*PR0P1(1)*C0+( S1+S3)*SI I F ( N E B . G T . l ) WRITE(6,90) FFF FORMAT( «FFF=« , F 1 0 . 5 ) R C S = 2 . D 0 * ( P R 0 P 1 ( 1 )*C0 * S 3 * S I ) / ( 1 . D O - S I ) HCS=RCS/PR0P1(3) ,  180 181 182 183 184 185  ~S30PA=S3/PP0P2(4) FS=2.D0*CR/FFF S T P E S S ( 14) =FS  186  187 188 189 190 191 192 193 194  ~'  C C C  TF ST  I 70  _  C C C  :  FOR F A I L U R E USING MOHR-COULOMB  CRITERION.  F (FS.rt7i.b6)GOTO31"'  " "  W R I T E ( 6 , 7 0 ) ISTEP FORMATC ' E L E M E N T ' , I 5 t ' H A S F A I L E D BY MQHR-CQULQMB ) 8  TEST FOR T E N S I L E  FAILURE. ._  31 41  IF{S3.GT.0.DO)GOTO 33 WRITE(6,41) ISTEP FOPMAT('TENSILE FAILURE  .  IN ELEMENT ,15) 9  GOTO3 0  19 5  196 197 198 199 2 00 201 202 203 2 04 205 206 '2 0 7 208 2 09 210 211 212 '213 214 215  "  C C C ~ 33  C C C  C A L C U L A T E TANGENT MODULUS FOR U N F A I LED  ELEMENT .  I F { F S . G E . l . D O G O T O 30 EI=PPCP1(4)*PR0P2(4)«( S30PA**PR0P1 ( 5 ) ) ET=E~l*( ( l . D 0 - ( 2 . D 0 * C R / H C S I ) * * 2 ) IF(ET.LT.PR0P1(6)I ET=PR0Pli6) GO TO 4 0 _ _ _ _ _ ~ ~ ™ C A L C U L A T E MODULUS AND POISSON'S R A T I O FOR F A I L E D _ _ 30 F T = P R 0 P l T 6 l ' . : UT=O.OD0  _ ELEMENT. '  GO T O 5 0  'C C C  " COMPUTE POISSON'S RATIO 40  FOR U N F A I L E C  STRAIN=2.D0*CR/(EI*t1.C0-(2.DO*CR/HCS) PRAT=PR0P117)-PR0P2(1)*DL0G10(S30PA) UT=PRAT/(l.D0-PROP2(2)*STRAIN)**2  ~  "  "  "  "  "  "  "  ELEMENT. ; ))  • ~  oo .  0 1  216 217 _21 8 219 220 221 222  50 C C C  c  IF(UT.GT.0.495D0)UT=0.495DO ET=  YOUNG'S MODULUS AND UT^=  RETURN ENO  :  "  "  "  POISSON'S  ~  RATIO.  223 SUBROUTINE EUCET,UT,STRESS,PROP1,PR0P2) _2 24 C 225 C 226 C T H I S SUBROUTINE, TOGETHER WITH EUD, COMPUTES THE 227 C A P P R O P R I A T E MODULI FOR THE F I N I T E ELEMENTS* 2 2 8 C 229 C PRCP1(1)=COHESION,C 230 C P P 0 P 1 ( 2 ) = T H E ANGLE OF I N T E R N A L _ F R I C T I O N 231 C P P C P 1 ( 3 ) = T H E F A I L U R E R A T I O , RF 232 C P F 0 P 1 ( 4 ) = T H E MODULUS NUMBER, K 233 C P P 0 P 1 ( 5 ) = T H F MODULUS EXPONENT, N 234 C P P O P l { 6 ) - M O D U L U S AFTER F A I L U R E " 235 C P P 0 P 1 ( 7 ) = P C I S S O N * S R A T I O PARAMETER, G 236; C PRCP2 ( 1 )=POI SSON S R A T I O PARAME TER, F 237 C P P 0 P 2 ( 2 ) = P O I S S O N ' S RAT I C P A R A M E T E R D 238 C P R 0 P 2 I 3 ) = S C R A T C H SPACE (NOT USED) 239 C_ P R 0 P 2 ( 4 > = A T M 0 S P H E R I C PRESSURE 240 " C " P P 0 P 2 I 5 J = SCRATCH' SPACE (NOT USED) 241 C P R 0 P 2 ( 6 ) = S C R A T C H SPACE (NOT USED) 2 42 C PR OP 2 ( 7 ) = C 0 E F F I C I E N T OF L A T E R A L EARTH P R E S S U R E , K3 243 C 244 C S T P E S S d - 6 ) CONTAINS THE ELEMENT S T R E S S E S . 245 C STRE S S { 7 ) CONTAINS THE PORE P R E S S U R E . ~ 2 4 6 ~ ~ C " S T R E S S ( 8-13) C O N T A I N T H E ELEMENT STRAINS." 247 C S T R E S S U 4 ) CONTAINS THE Y I E L D F U N C T I O N . 248 C 2 49 " I M P L I C I T R' ETAL * 8 V A ^ H T O - T T 250 DIMENSION S T R E S S ( 1 4 ) , P R C P 1 I 7 ) , P R 0 P 2 ( 7 ) 251 COMMON/EUBLK/ISTEP,NSTEPj^STEP,IFGA 8  t  252 " ' DATA N E N T E R / O / " 253 I F ( N E N T E R . N E . O ) GO TO 6 2 54 NFMTER=1 ; 255 C 256 READ(5,4) NN NEB 257 C ' 25 8 " C N N I S THE NUMBER' G'f^rNTTl '_LE"ME'RTS« " 259 C NEB I S THE DEBUG OUTPUT L E V E L . _2 60 C 261 4 FORMAT(215) 262 6 CONTINUE 263 STPESS ( 14) =0.00 ; 264 C 265 C C A L C U L A T E THE P R I N C I P A L S T R E S S E S , S I AND S 3 . _2 6 6 c 267 STRE S2= — ( S T R E S S ( 2 ) + STRESSC 7 ) J 268 STRESl=-(STRESS(1)+STRESS(7)) 269 CC=(STRES1+STRES2)/2.D0 270 BB={STR E S l - S T R E S 2 ) / 2 . D 0 271 CR = DSCRT( STRESS (4 )*STRESS (4)«-BB*BB) _2 7 2 Sl^CC+CR ; 273 S3=CC-CR 2 74 C 275 C UPDATE THE ELEMENT NUMBER, I S T E P . _ 2 76 C " " " ~ " ~ 277 I S T E P = ISTEP«-1 _?7 8 C^ 279 C GET NEW TANGENT MODULUS AND POISSON'S R A T I O . 280 C 281 29 CALL EUD(ET,UT,STRESS,PROP 1 , P R 0 P 2 , C R , S I , S 3 , 1 S T E P , M E B ) 282 C 283 I F ( N E B . G T . l ) W R I T E ( 6 , 2 0 3 ) I STEP _28<t 2 0 3 FORMAT ( ' ELEMENT » 15) . 285 C 286 C COMPUTE THE BULK MODULUS AND STORE I T I N THE 287 C LOCATION OF POISSONS R A T I O , UNLESS.THE ELEMENT V  '  _._ ,  :  ;  ;  1  .:.  co ^  288 289 290 "2 9 1 292 293 294 295 _2 9 6 297 2 98 299 30 0 301 _30 2  C  HAS  FAILED.  C  IF ( UT . E Q.O.QDO) GOTO 9 1 0 0 U T = E T / ( 3 . 0 0*11-DO-2.DO*UT)) 9 1 0 0 CONTINUE I F ( I S T E P . N E . N N ) GO TO 4 0 _ _ IF ( N E B . G T . l ) W R I T E C 6 , 2 4 } N N ~ ~ " ~ 24 FORMAT(•THE NUMBER OF ELEMENTS I S = , I 4 ) 38 ISTEP = 0 40 CONTINUE :  e  C C C  303 304 305 30 6 " 307 3 08 309 310 END O F F I L E  FT I S YOUNG• S MODULUS. _ UT I S THE BULK MODULUS.""  \ " '  _  ~  C C CR=2.D0*CR I F ( N E B . G T . l ) W R I T E ( 6 , 1 0 0 ) CR 1 0 0 FORMAT (•S1-S3=•,G15„5) I F ( NEB.GT.i >WRITE (6,202 ) E T , U T ~ " ""' 2 0 2 FORMAT!'MODULUS »•,G15.5,1 OX,•BULK MODULUS^ ,G15.5 J 9  C  ;  .  RETURN END  oo oo  

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