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A simple shear machine for soil Pickering, D. J. 1969

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A  SIMPLE  SHEAR  MACHINE  FOR  SOIL  by  B.Sc.  (Tech.),  A THESIS THE  D.J. PICKERING Manchester U n i v e r s i t y ,  SUBMITTED  IN P A R T I A L  REQUIREMENTS DOCTOR  in  FOR  OF  1943  FULFILMENT  THE DEGREE  OF  OF  PHILOSOPHY  the Department of  Civil  We  accept  required  THE  this  Engineering  thesis  as c o n f o r m i n g  to the  standard  UNIVERSITY  OF  April,  BRITISH 1969  COLUMBIA  In  presenting  an  advanced  the  Library  I further for  this  thesis  degree shall  agree  scholarly  at the University make  that  by  h i s representatives.  of  this  written  e  8,  be g r a n t e d  gain  of British  April  30,  1969  Columbia  Columbia,  f o r reference copying  b y t h e Head  It i s understood  for financial  Canada  of British  f o r extensive  C i v i l Engineering  University  Vancouver  D a t  may  f u l f i l m e n t of the requirements f o r  available  permission.  Department o f The  i tfreely  permission  purposes  thes.is  in partial  shall  that  n o t be a l l o w e d  and  that  Study.  of this  o f my  copying  I agree  thesis  Department o r or  publication  without  my  ABSTRACT  The ations pore  than  new  shear  machine  i  enforces  the conventional t r i a x i a l  pressure  measuring  device  i s an  more  test.  uniform  A  integral  low  deform-  compliance  part  of the  mach i n e . The normal  stress  shear  stress  taken  50  machine up  up  disturbed high. ing,  be  of  problem  and  any  length  This  was  the  conventional strengths  sand. ditions  The  solution  also  it  more  loads  and  cut from  cyclic  c a n be  an o r d i n a r y un-  2 i n s . square  the e n t i r e  elastic  shows  and  1 1/8 i n .  during  sample  is  test-  sub-  f o r the boundary  sample  i n the t e s t s  the v a r i a t i o n the sample.  machine  examined  designs,  between  of the  For the  the  to  stress benefit  theoretical  the r a t i o  the u n i f o r m i t y of s t r e s s e s  of  sample  and  displace-  sample. test  tests,  could  difference  shear  alters  susceptible  results i t was  Some  found  t h e new that  of the r e s u l t s  simple  o f more  than  of undrained  i n t h e new  machine  shear  suggest  and  was  re-liquefaction.  undrained con-  interest.  readily  I t was  of a sand  that  triaxial  theoretical sand  with  the measured  the s t r e n g t h of  machine.  the s t r u c t u r e to  from  over-estimate  between  i s ,therefore,  liquefaction  Static  i s presented,  throughout  shear  triaxial  alternating  alternating  i n . i s permitted  zone;  anisotropic  comparing  test  being  solution  Liquefaction by  cyclic  shear.  are d i f f e r e n t .  triaxial  c a n be  o f ± 1/8  "dead"  simple  In  the  i s no  to height within  l b . per sq. i n .  sample,  deformations  relationship  ments  hole  o f an  future  l b . p e r s q . i n . and  specimen  analytic  described.  field  test  to the a p p l i e d An  of a p p l y i n g  higher.  variation  but there  value  t o ± 500  drill  Height  jected  t o 1000  percent The  described i s capable  found  sample,  induced that  rendering  CONTENTS  ii Page  Chapter  Chapter  Chapter  Chapter  Chapter  Chapter  I  II  III  IV  V  VI  INTRODUCTION Objectives Other types of apparatus H i s t o r y of simple shear t e s t i n g D i s a d v a n t a g e s o f common s o i l testing procedures Intended c h a r a c t e r i s t i c s o f the machine to be d e s c r i b e d D E S C R I P T I O N OF THE APPARATUS Size H o r i z o n t a l motion V e r t i c a l motion The b o d y f r a m e Shear box ends Pore p r e s s u r e measurement Details A s s o c i a t e d equipment T H E O R E T I C A L CONDITIONS OF T E S T D e f i n i t i o n and e x p l a n a t i o n Boundary c o n d i t i o n s A theoretical elastic solution E f f e c t of p r o p o r t i o n of length to height The c o m p l e t e s t r e s s f i e l d R e l a t i o n s h i p between e x p e r i m e n t a l m e a s u r e m e n t s and t h e o r y E x p e r i m e n t a l c o m p a r i s o n o f maximum a n d h o r i z o n t a l shear. Manner o f p r e s e n t a t i o n o f e x p e r i m e n t a l results  1 1 2 3 7  10 10 10 12 12 15 21 22  26 26 31 49 49 50 52 54  E X P E R I M E N T A L PROCEDURE I n s t a l l i n g samples In s i t u samples (sand) U n d i s t u r b e d samples Loading  56 56 59 59  SOME E X P E R I M E N T A L R E S U L T S The t e s t p r o g r a m m e U n d r a i n e d s t a t i c t e s t s on O t t a w a sand C y c l i c s h e a r t e s t s on O t t a w a s a n d Re-liquefaction  63 67 78 97  SUMMARY  AND  CONCLUSIONS  104  CONTENTS  iii  Continued  Page Appendix  A  THE  E F F E C T OF C O M P L I A N C E P R E S S U R E RECORDED OF  PORE 111  B  DESCRIPTION  Appendix  C  A N A L Y T I C E L A S T I C S O L U T I O N FOR THE BEHAVIOUR OF A SAMPLE The M a t e r i a l The G o v e r n i n g D i f f e r e n t i a l E q u a t i o n S o l u t i o n of the e q u a t i o n E v a l u a t i o n of a r b i t r a r y constants  117 119 121 123 133  SUPPLY  D  LIMITING  Appendix  E  E X P E R I M E N T A L PROCEDURE I n s t a l l i n g d i s t u r b e d sand samples I n s t a l l i n g u n d i s t u r b e d samples R e m o u l d e d c l a y and s i l t s a m p l e s Checking the sample A p p l i c a t i o n of normal l o a d  140 149 150 150 151  CYCLIC  153  List  of  F  SHEAR  TESTS  ELASTIC  114  Appendix  Appendix  OF  WATER  THE  Appendix  VALUES  THE  ON  ON  HANEY  CONSTANTS  CLAY  Symbols  156  References  Symbols are and a l s o i n  158  d e f i n e d and r e f e r e n c e s t h e l i s t s on p a g e s 156  g i v e n where and 158.  they  first  occur  LIST  OF  ILLUSTRATIONS  iv  Page Fig.  1  Type  of d e f o r m a t i o n .  Fig.  2 a-d  Machine  Fig.  3  Horizontal  carriage,  Fig.  4  Installing  vertical  Fig.  5  Schematic  Fig.  6  Pore  sub-assemblies.  diagram  pressure  through  7  upper  Fig.  7 a & b  Photographs  Fig.  8  End p l a t e  11  i n body  frame.  motion.  13  of complete  measurement. loading  machine.  head.  15 head.  17  for retaining  membrane. Fig.  9  Fig.  10  Fig.  11  Fig.  12  Fig.  13  Fig.  14  Fig. Fig. Fig. Fig. Fig. Fig.  15 16 17 18 19 20  Shear a & b  21  load  General  piston  and v a l v e s .  26  D e f o r m a t i o n o f dummy p l a s t i c i n e samples. Computer s o l u t i o n . Isotropic material. Length equals height. Computer s o l u t i o n . Isotropic Length twice height.  material.  Computer s o l u t i o n . Length three times  Isotropic height.  material.  Computer s o l u t i o n . Isotropic Length s i x times height.  material.  Computer s amp 1e .  calculated  29 37 38 39 40  Computer s o l u t i o n . Anisotropic Length twice height. deformation  material. 41 of 44  Variation of stress c o e f f i c i e n t proportions of sample.  with 48  C o m p a r i s o n o f maximum s h e a r s t r e s s v s . s t r a i n with h o r i z o n t a l shear s t r e s s (after Cole).  53 57  21  Membrane  ready  Fig.  22  Membrane s ample.  i n machine  23  24  system.  Fig.  Fig.  23  arrangement.  Co-ordinate a - f  14  Section  of the loading  and g r o o v e  12  on m o u n t i n g . ready  Average s t r e s s e s used experimental results.  to  receive 57  in  presenting 66  LIST  OF  ILLUSTRATIONS  v  Cont i n u e d  Fig.  Fig.  Fig. Fig.  24  25  a  a  & b  & b  26  S t a t i c shear Simple shear friction.  loose sand. without adequate 68-69  S t a t i c s h e a r t e s t s on l o o s e s a n d . Simple shear with adequate friction compared w i t h t r i a x i a l . Stress simple  27  t e s t s on w i t h and  paths shear  Oscillograph  compared tests. record  for triaxial  72-73 and 75  of  a  cyclic  shear  failure.  79  Fig.  28  Linearity  Fig.  29  High  Fig.  30  Medium  Fig.  31  Fig.  Fig..  Fig.  32  33  a a  a  & b & b  & b  34  of  speed  speed  by  compare  Fig.  36  Jump  drained  Peacock with  i n pore of  Seed  to  summarized  shear  92  test.  94  load  96  Ottawa  sand.  Re-liquefaction  of  Ottawa  sand  Fig.  40  Fig.  41  Removing  Fig.  42  Loading  hopper  Fig.  43  Closing  membrane  Fig.  44  Top  Fig.  45  of  Cyclic  99 with  strain.  supply  Energy  as  increases.  of  Water  & b  and  pressure increase  39  a  83  F i g . 33.  Fig.  & b  B.  pressure rise  Re-liquefaction  restricted  a  Form  85  cyclic  frequency  38  pore  81  91  A  Fig.  of  Load  A.  S h e a r s t r e s s t o c a u s e 20 p e r c e n t p o r e p r e s s u r e r i s e i n 1000 c y c l e s a n d i n 10 c y c l e s a s a f u n c t i o n o f v o i d ratio and n o r m a l p r e s s u r e .  35  & b  record.  Form  87  Results  a  Load  80  S h e a r s t r e s s t o c a u s e 20 p e r c e n t l o s s o f s t r e n g t h as a f u n c t i o n o f v o i d ratio and number o f s t r e s s r e p e t i t i o n s .  Fig.  37  pressure recording.  record.  T y p i c a l forms liquefaction.  to  Fig.  pore  and  bounds  back  p r e s s u r e system.  for elastic  extension  machine, shear  101  of  end  constants.  136  plate.  141  in place. oyer  tests  on  142  spreader  assembled  116  over  Haney  plate.  148  sample.  152  clay.  154  vi  ACKNOWLEDGEMENTS  In  presenting  this  record  h i s indebtedness  Canada  and t h e U n i v e r s i t y  support. the  work  K.  H.  He w o u l d  Roscoe  The  of B r i t i s h  like  author  wishes to  Research Council  Columbia,  to express  investigators  a n d E . R.  the author  to the National  also  of earlier  thesis,  of  for financial  h i s appreciation  in his field,  of  in particular,  Cole. would  also  like  to thank  the following  individuals : -  His supervisor,  their the  encouragement,  generous K. H.  W.D.  Finn,  a n d R.G. C a m p a n e l l a , f o r  guidance, suggestions  allocation  Roscoe  Liam  of funds  f o r some  very  they  and a d v i c e  provided  welcome  and f o r  f o r equipment.  personal  interest  and  encouragement. The for  much His  criticism  staff  of the C i v i l  meticulous fellow and  work  students  discussion.  on  Engineering  Department  workshop  equipment.  i n graduate school  for friendly  1 CHAPTER  I  INTRODUCTION  Objectives.  A  engineering plifying  few  years  problems  could  assumptions. were  engineering  solution.  reduced  probably  the of  the  prime  of  error  full  a)  of  the  To  build  To  greatest  source  Recent  advances  in  a  modern  design  are  computer herein of  a  series  sim-  of  error  so  in  an  techniques  much  are  no  now  needed,  that  longer  the  of  to  techniques.  The  were:  simple  i n o b t a i n i n g improved  conduct  broad  computer  treatment  soil  engineering.  described new  most  necessary  simplification,  soils  the  work  assist b)  of  the  the  in  of  without  problems,  laboratory results  advantage  objectives  solved  mathematical  Improved take  mathematics  be  many  n e c e s s i t y of  inaccuracies source  the  not  For  assumptions  have  ago,  shear  machine,  laboratory  to  results.  preliminary tests  with  the  machine. Other is  Types  most  shear  tests  chamber  London  apparatus  machine  1  Laboratory triaxial  employed also  the  i s an  of  stresses  also  for  apparatus  ( i . e . load  modified  to  developed  example.  Other  developed. three  described  at  types For  separately by  Kjellman  of  soils  with  direct  and  speed.  testing  apparatus  been  testing  simplicity  appeared  applying was  shear  apparatus,  for  p r e s s u r e ) , but  have  capable  principal  by  triaxial  conditions;  College,  2  have  basically  strain  done  sometimes  designs  confining  ing  Apparatus.  commonly  Various using  of  equipment plunger  give  and  plane  Imperial of  shear  example,  testa  controlled and  a  more  C o r n f o r t h , D.H., "Some E x p e r i m e n t s on t h e I n f l u e n c e o f S t r a i n C o n d i t i o n s on t h e S t r e n g t h o f S a n d " . Geotechnique V o l . X I V , No. 2, p . 1 4 3 , J u n e 1964. K j e l l m a n , W., " R e p o r t on a n A p p a r a t u s f o r Consummate I n v e s t i g a t i o n of the M e c h a n i c a l P r o p e r t i e s of Soils". P r o c e e d i n g s I n t e r n a t i o n a l C o n f . on S o i l Mechanics, Cambridge, Mass., 1936.  recent of  version  shear  are  and  (e.g.  ratio  stress-strain History shear  of  apparatus  now  same  of  procedure the  since of  his  In  3  dynamic  compression to  waves  obtain  procedures  a  do  testing, of  shear not  small  velocity amplitude  modulus  yield  a  and  complete  relationship. Shear  has to  been be  seeks  Testing.  An  d e s c r i b e d by  described but  to  apparatus K.H.  eliminate  or  The  5  to  do  increased  one  simple  Roscoe .  i s intended  i t allows  for  two  basically  the  flexibility  of  disadvantages  in  design. Various  design  advances  have  been  original  model.  Details  have  not  a l l these  latest  Scott .  test  testing,  and  earlier  and  Hardin"*)  but  Simple  testing  type  Ko  longitudinal  measured  Poisson's  by  (Mark  later 6)  machines,  model  has  but  some  made been  by  Roscoe  published  i n f o r m a t i o n on  been  given  by  Roscoe,  simple  shear  have  the  Bassett  &  Cole . 6  Machines in  Scandinavia, ' 7  ages is, as  inherent  4  5  6  7  8  in a  c o n d i t i o n s are the  vertical  deformations  3  8  for but  these  test  specimen  neither  boundaries  around  them  suffer of  plane are  are  from  also  certain  circular  stress  been  nor  flexible,  f u n c t i o n s of  developed  disadvant-  section. plane  the the  That  strain  shapes  and,  of  material  as  K o , H.-Y., a n d S c o t t , R . F . , "A New Soil Testing Apparatus". Geotechnique V o l . X V I I , No. 1, M a r c h 1967. H a r d i n , B.O., a n d R i c h a r t , J r . , F . E . , " E l a s t i c Wave V e l o c i t i e s i n G r a n u l a r S o i l s " , J o u r n a l of the S o i l Mechanics 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 . 89, No. S M l , F e b . 1963. R o s c o e , K.H., "An A p p a r a t u s f o r the A p p l i c a t i o n of Simple Shear to S o i l Samples", P r o c e e d i n g s , 3rd I n t e r n a t i o n a l C o n f e r e n c e on 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 , V o l . I , Zurich 1953. R o s c o e , K.H., B a s s e t t , R.H. and C o l e , E.R.L., " P r i n c i p a l Axes Observed D u r i n g Simple Shear of a Sand". Proceedings, C o n f e r e n c e on S h e a r S t r e n g t h P r o p e r t i e s o f N a t u r a l S o i l s & R o c k s , V o l . I , p. 231-237, O s l o 1967. K j e l l m a n , W., " T e s t i n g the S h e a r S t r e n g t h o f C l a y i n Sweden". Geotechnique V o l . I I , No. 3, p . 2 2 5 , J u n e 1951. B j e r r u m , L . , a n d L a n d v a , A., " D i r e c t S i m p l e - S h e a r T e s t s on a N o r w e g i a n Q u i c k C l a y " . Geo t e c h n i q u e V o l . X V I , No. 1, p . 1, March 1966.  well  as t h e a p p l i e d  stress  field  become  very  i s very f a r from  A the  University  operation but  simple  load.  The r e s u l t  hard  to analyse.  shear  are b a s i c a l l y means  Also  the  actual  deformations  can  uniform. machine  of C a l i f o r n i a  the actual  i s that  has a l s o  at Berkeley  t h e same  employed  been  developed  by  . The p r i n c i p l e s  of  as R o s c o e ' s  t o p u t them  third  into  machine,  effect  are  different. The examine  the tendency  stress. more  machine  at Berkeley  o f sand  to l i q u e f y  The v a r i o u s m a c h i n e s  c o n v e n t i o n a l types  but meticulous  variations  of c o n d i t i o n s from  example,  boundaries force, also X-ray  with  pattern  this  sidering  standard  procedures.  i n which  point  to point  Testing  a model  gives  only.  The o t h e r  type  Engineering  around of  normal were  samples  and  strain  of the  of t e s t ,  complete  for,  con-  or the  is essentially  materials,  If scaling  con-  over  testing  of a model  A model  answer,  In  i t appears  a r e hoped  and p o s s i b l y  a complete  the  Experiments  laboratory  of material.  approach  i n the sample.  Procedures.  the behaviour  i n space.  determine  the sample.  advantages  conditions,  to  the d i s t r i b u t i o n  reconstruction  Soil  either  made  i n sand  used f o r  on d r y g r a n u l a r  cells  force.  to  shear  been  at successive small  within  what  of a specimen  rectly  9  taken  o f Common  load  distributed  permitted  to e x p l a i n  behaviour  from  were  of observing  test  shot  t h e u s e o f a n uncommon  desirable  sists  to measure  of deformations  Disadvantages  several  have  to point  largely  cyclic  mostly  were  point  and f r i c t i o n  lead  photographs  increments;  a  used  eccentricity  done  endeavours  at Cambridge,  were  test,  used  under  at Cambridge  of shear  materials,  For  has been  has been  b u t t o one  t o an e n g i n e e r i n g p r o b l e m  vary done  cor-  problem i s to  P e a c o c k , W.H. a n d S e e d , H.B., " L i q u e f a c t i o n o f S a t u r a t e d Sand Under C y c l i c L o a d i n g S i m p l e Shear C o n d i t i o n s " . Journal of t h e S o i l Mechanics & Foundations D i v i s i o n A.S.C.E., V o l . 94 , No. SM 3, May 19 6 8 .  calculate at  stresses  t h e same  laboratory to  supply  point  at c e r t a i n points,  points,  and hence  the r e l a t i o n s h i p between  behaviour ditions  that  of the material  within  a specimen  t o assume t h a t  represents  the behaviour  material.  testing  into  are uniform  at a point  approach  affect  account.  at a the  When  point,  of the  of conditions  A  and s t r a i n s which  within  but i n p r a c t i c e  to s t r a i n s  second  at every  the behaviour  Uniformity  a specimen,  f o r this  stresses  are taken  these  displacements.  a l l the v a r i a b l e s  reasonable  same  determine  investigation i s required  and t o e n s u r e  relate  con-  i t  is  specimen  a mass  i s thus  o f t-h"e the goal i n  i t i s prob'ably  unattain-  able . If the  measured  when  conditions quantities  fails  stresses  rise  towards  material  will  lead  shedding  increases  load,  zones. a  thin  load  the by  right  involve  the apparatus  through  and weak  from from zone  further  progresses across  as  while  strength  undergoing  will  test,  strains  then  be  will  also  i n narrow  i n t h e manner d e s c r i b e d has passed  until  i t s peak  "fails". laboratory  conditions  and t h e s h e a r i n g  develop  the f r i c t i o n a l the non-uniform  effect  the samples. are  In imposed  i s propagated  In the t r i a x i a l proceeds.  a t t h e ends  deformations  tests for  deformations  as t h e t e s t  restraint  outwards.  shear  within  i n a n u n k n o w n way.  and s p r e a d  peak  v a r i a t i o n of  triaxial  box t h e n o n - u n i f o r m  conditions  with  a test,  zones  large  specimen,  be t h e c a s e  i t s peak  zones  the sample,  o f t h e common  the sample  non-uniform  exceeding  The n e i g h b o u r i n g  non-uniform  shear  material  slight  neighbouring  and t h e sample  direct  stem  onto  the  would  In such  some  and, i n the standard  Both soil  failure,  and c o n t r i b u t e  zone,  test.  within  This  or other  t o one zone  The f a i l u r e  strength  clay  in a triaxial  of s t r a i n .  overstressed shed  are averages.  an o v e r c o n s o l i d a t e d  strength  and  are not uniform  which  They  o f the sample  can s t a r t  The f r i c t i o n a l  test,  end  in a  restraint  5 in  the  Barden  triaxial by  1 0  although be  test  the  this  use  of  due  been  to  the  e l i m i n a t e d by  designed  improves  end  Rowe  platens,  c o n d i t i o n s they  progressive spread  may  of  and  but  still  large  already described.  Non-uniformity lowing  largely  specially  treatment  non-uniform,  deformations  has  specific  of  deformations  difficulties  in triaxial  leads and  to  the  direct  fol-  shear  tests. a)  Measurement  of  deformations tion  of  the  volumetric  are  non-uniform,  total  sample  volume  change.  of  expression for  the  the  direct  This  shear  difficulty  strain.  box  an  soon  unknown  that  strain the  to  the  denominator  i s not  error the  the  as  propor-  is contributing  means  i n measuring  As  known.  In  i s compounded  volume  by  change  accurately. b)  Measurement  of  ment  two  of  the  apparatus with  average  related of the  test,  to  shear  of  destroyed. triaxial The on  the  is  that  toward  a  direct  but  how  shear  this  i s not  which  strain.  compares  known.  deformation  strain,  move-  can  In  gives be  a  the value  directly  However,  the  high  degree  conditions within  the  sample  at  is  progressively  Cone-shaped  zones  at  the  of  are  these axis  axial  between the  relative  test,  central the  the  strain  axial  The  triaxial  sample,  points  least  0  the  axial  of  measured,  shear  u n i f o r m i t y of start  strain.  halves  be  "average"  triaxial for  can  shear  the  restrained cones of  point  the  cylindrical  of  the  of  each  The  s u r f a c e of  the  and  other  result  i s deforming cones  a  shearing.  towards  specimen.  l e n g t h which points  from  ends  is  increases  sample.  The  Rowe, P.W. and B a r d e n , L . , " I m p o r t a n c e o f F r e e Ends i n Triaxial Testing". J o u r n a l of the S o i l Mechanics and Foundations Division, A . S . C . E . V o l . 90, No. S M l , January 1964 .  6 total  axial  movement,  form,  hence,  denominator, centre  and  strains  i s probable  strain  zone  in a than  average but  vary the  of  on  a horizontal  above,  direct  there  usual  box,  In  the  direct  ticable  to  run  undrained  of  useful the  the  sample  lines  possible  trouble A  that  may  even  and  through  more  from  use  pressure plane  In the a  a  of  leads  measured  to  non-  stresses  values.  normal  are  test,  at a l l  effective  and  known,  shear  but  lead  non-uniform  other shear  to  tests.  produced  back  at  enclosed  pressure  direct  the by  are  on  ends  a memimpracthe  Also,  shear  which  For  becomes  measurements.  in a  deforma-  disadvantages  i t thus  triaxial  test  weaker  pore pre-  means than  test,  the  necessity  c o m p l i c a t i o n s of  the  necessary  sealed  leakages  serious  triaxial  i s known  and  readily  contain layers  strength.  drainage  is  failure  material  measured  pore  shear  stresses.  testing,  tests,  already  is far  c o n d i t i o n s may  discontinuity  shear  the  average  various  be  true  strain  the  laboratory  cannot  brane.  take  of  results  are  soil  step-shaped shear  the  the  In  plane  of  shear  deformation  plane  test,  to  the  sealing  shear  non-uniformity  addition  the  any  pressure  In  example,  the  pore  An  strain,  non-uniformity  effective  in  the  on  of  failure  sample.  non-uniform  inherent  that  the  surface.  the  the p r o g r e s s i v e  that  c o n s i d e r a b l y from  stresses  listed  large  measured  stress  direct  again,  of  is  approaching  any  i s g r e a t e s t at the  sample  It  In  selection  toward  uni-  in i t s  non-uniformity  triaxial  Stresses within  or  of  strain  described.  may  fluid  axial  zones  uniformity  a  the  is generally  variation  of  times,  of  the  development  the  tions,  to  source  in a  greater c)  due  diminishes  additional  however,  pressure and  lead  to  trapped a i r .  disadvantage  i f c o n d i t i o n s were  chamber  of  of  perfectly  the  triaxial  uniform  test  within  the  sample, ground are  they  because,  usually  stresses.  properly  way  a point  i n which  ing  shear  The  result  Intended  them  is  designed  45  that  degrees)  planes  occur  is likely  field  i s , simple  with  no  lateral  followed  planes.  in practical  This  c a n be  listed  shear  developchange.  of the  (maximum shear  involves  b u t some It  t o the sample,  of the type  The s e q u e n c e  strain  It which  described.  load  Deformation  engineering.  that  developed  above,  t o be  strain.  corresponds  degrees  t o be D e s c r i b e d .  of  and v e r t i c a l )  a t 45  a rotation  of normal  by h o r i z o n t a l  real  o f maximum  likely  f o r deformation  Type  most  develops.  equipment  shear  Another  as c o n d i t i o n s  of the Machine  and  vertically  f o l l o w e d by  t o be  i n the apparatus  strain,  horizontal  principal  planes  f o r the a p p l i c a t i o n  lateral  load  i t i s very  c a n be  ground.  the planes  be  to  are points  directions,  will  test  that  to d u p l i c a t e  sample  a l l the disadvantages  Fig normal  fails  conditions  points  in level  i n t h e same  any t e s t i n g  are minimized  F i g . 1.  test  i n the  i s supposed  test,  load  as t h e s t r e s s  that  those  to t r i a x i a l  the only  on d i f f e r e n t there  than  a triaxial  deposition  Characteristics  of  in  ground  from  specimen  In the ground  planes  overcome  no  by  a r e always  i s that  unlikely  with  i f the test  in a triaxial  stresses  principal  strain  i n the ground,  the v e r t i c a l .  will  to plane  of a c i r c u l a r  i s that  during  be d i f f e r e n t  e n g i n e e r i n g problems,  the t r i a x i a l  stress  strain  is  closer  represented  problems  to  i n actual  the centre  shear  usually  In f a c t ,  represent  below  would  shear  shown of  planes  (maximum  shear  a rotation  of the  to c o n d i t i o n s which  The c o r r e s p o n d e n c e  at  is  often  8 increased plane ing  because  strain.  shear  with  ever,  It will  the stress  stresses  interior  the shearing  close  place  i n c o n d i t i o n s of  in a later  within  section  the sample  to the d e s i r e d  the boundaries t o be  shown  field  but v a r y i n g  deformations  be  takes  values  that  i s complicated,  through  the  c o n s i d e r a b l y near  the boundaries.  are approximately  rigid,  almost  uniform.  forcing  The v a r i a t i o n  represents  a departure  from  the desired  importance  i s believed  t o be m i n i m i z e d  of  conditions, by  dur-  the  Howthe  stress but  i t s  following  considerations: a)  The u n i f o r m i t y o f d e f o r m a t i o n strains  i n narrow  through  the i n t e r i o r  to  the measured  tion  will  zones  and a c t u a l  Thus  strain  excessive  strains,  of the sample,  strain.  represent  prevents  will  measured  behaviour  be  close  deforma-  reasonably  closely. b)  The  stresses  do  not vary  of  the sample  which the c)  through  ple  stresses that  problems. that  details  to maximize  cut from  samples.  The s p e c i m e n  solidated  with  layering,  as w o u l d  to  enclose  application  be  the sample,  Thus by  that  closely  behaviour  stresses i s , close  a good  to  should  of the machine  dimensions  t o be  expected  obtained.  described, are engineering  drill  i n the machine  orientation,  permitting drained  pressure  be  of the specimen  i n nature.  governed  of the true  to p r a c t i c a l  ordinary undisturbed  stress  sam-  stresses, i t  approximation  relationship  represents  i s largely  to the applied  i s placed  t h e same  of back  strain  i t sapplication  F o r example,  i t c a n be  governed  and t h e b e h a v i o u r  close  stress-strain  intended  the average.  to the average,  the measured  follows  of the sample  stresses.  behaviour  Some  from  i s largely  are close  applied  Since  by  much  the i n t e r i o r  a r e such hole  and  con-  relative  t o any  A membrane  i s used  or undrained  and measurement  of pore  tests,  pressure.  9 The  machine  per  sq.  cyclic  i s designed  in. cyclic shear  applied.  to  normal  stress.  work  with  pressure  Steady  loads  loads  and 50  500  up  to  1000  l b . per  percent  lb.  sq . i n .  higher  can  be  10 CHAPTER I I DESCRIPTION  Size. 1000  The a p p a r a t u s  cyclic  initially would  flatter chosen mum  i n . high.  proportions.  from  by d r i v i n g The  various  Motion.  carriage  with  carriage  sides  shear  spindles, shear  and a s s o c i a t e d  extend  horizontal  linear  hinge  from  were  the miniuncon-  horizontal  f o r trimming)  or from  a  to  sample  a s s e m b l i e s and  equipment.  roller  into  points load  acting  main  to form  Pressed  The s h e a r  force,  however,  a 2 i n . square  o f the sample  upward,  box s i d e s .  box ends.  III,  a sample of  results  sample  has t h r e e  The base  form  i n Chapter  t o be a b o u t  (allowing  and  casing.  horizontal  which  size  or shelby  apparatus  accessories  Horizontal  metal  maximum  a 3 i n . piston  obtained  to have  dimensions,  while  up t o  2 i n . square  be d e s c r i b e d  representative  loads  t o 500 l b . p e r s q . i n . ,  i s considered  deposits,  normal  t o a sample  The a c t u a l  i s a reasonable  up  preferable  the height  alluvial  loads  As w i l l  be t h e o r e t i c a l l y  because  to apply  conditions,  to give meaningful,  area  a  i s designed  dynamic  1-1/8  solidated  cut  THE A P P A R A T U S  l b . p e r s q . i n . and s h e a r  under  it  OF  i s supported  on a  b e a r i n g motion. the lower  halves  the c a r r i a g e  f o r the lower  i s applied  The of the  are bearing  edges  of the  t o the sample  on t h e end o f t h e  by  carriage,  (Fig. 2a.). Vertical bearing line, are  Motion. cam  with  well  upper  edges  guide  frame  frame  so t h a t  rotation  separated,  This  bear  guides, no  ing.  removable  A  i t c a n move  a n d no l a t e r a l  to give  a good  carries  spindles,  of the shear  box ends.  loading  head,  by w h i c h  on t h e t o p o f t h e sample.  the  upper  box  ends  i s constrained  loading  head  are carried  and h i n g e s  as a s i n g l e  freely  in a  movement.  resisting forming I t also  this  moment  load  f o r the upper floating  vertical  to  for  carries  means,  roller  The b e a r i n g s  hinges  the normal By  between  tiltthe  the  i s made t o  the whole of edges  assembly.  of the Dilation  A = Linear roller motion. B = Hinge spindles, lower edges of box ends. C = Sample A = Roller bearing guides. B = Hinge spindles, upper edges of box ends. C = Removable loading head (shaded). D = Sample. E = Normal load.  D = Applied shear load.  FIG. 2a. HORIZONTAL MOTION .  FIG. 2b. VERTICAL MOTION  A = Removable frame holding upper half of box sides.(shaded). B = Roller bearing guides for vertical motion. (Six more inside underneath). C= Bed for horizontal motion. FIG. 2c. BODY FRAME. CUT AWAY VIEW. Fig.  2.  A = Slotted and circular holes for hinge bearings. B = Sample.  FIG.2d. END PIECE Machine  Sub-Assemblies  12 or  c o n t r a c t i o n of  the  "dead"  shear, The  i s  upper and  zone,  Frame.  in  a  the  roller  Fig. Shear  can  (Fig.  for in  The  sides,  and  Ends.  the  between  the  vertical  enclose  corners  constant  The  distance  the  sample, end  to  also or  piece  as has  at  varies a  end  the  one  volume  which  guides  motion,  on  and  of  of  pair  any  of  are  attached the  the  horizontal Fig.  mounted  on  3  i t s  the  box  volume  bearings  motions  The  main are  in  distance  varies  or  Frame  These  when  perpendicular  dilation of  Body  three  box.  horizontal ends.  in  the  the  ( i . e . constant for  for  a d d i t i o n to  sample  end  are  i t s Rollers,  pieces  combination one  not  frame.  In  are  the at  ( F i g . 2d.)  there  as  and  but  assemblies  stationary parts.  body  Box  way  to  horizontal  Horizontal Carriage,  a  allowed  compression  various  the  other  the  to  be  2b.).  mounting,  box  motions  therefore  subjected  3.  between  Each  the  assemblies,  connected  occurs  sample  steel  carriage  assemblies,  such  of  vertical  sample  ( F i g . 2c.)  heavy  halves  shows  of  eliminated  Body  united  the  shear height)  contraction  change  providing  and  of  shear.  rotation  13 only  and another  rotation is  made  pair,  riding  and v a r i a t i o n  of distance.  f o r t h e change  i n dimension  The  end p i e c e s  one  end has i t s s l o t t e d  the  bottom.  and  t h e b o x e n d , i s made  the  two ends,  this and  are interchangeable  By  this  of the other  swinging pieces  holes  means,  retaining  arrangement  p a r a l l e l ,  puts  a r c .  horizontal  carriage  in  with  t h e body  shear frame  motion  testing,  i s an added box ends  i n Fig.  4.  during  between  edge  shear.  so  that at  the sample  directions  conditions.  at  Also,  o f one end a t t h e t o p  and, since  upward,  they  are  always  the other  must  be  o f t h e two end  and t h e machine's  i s the f r i c t i o n of course,  only  of the  the inertia  complication.  attached,  both  allowance  and a r e arranged  The w e i g h t s  therefore counterbalance  I n dynamic  allowing  means,  o f t h e ends  to act i n opposite  i f one end i s s w i n g i n g  bearings.  this  any f r i c t i o n ,  the pinned  to horizontal  holes,  at t h e top and the other  end a t t h e bottom  on a downward  By  cross-symmetry  resistance  motion,  i n slotted  i s shown  The  oiled  of t h e  ;  vertical  being  installed  A  ! f  A B C D E  = Normal load, to loading head. = Shear load, to carriage on rollers. Shown at limit of travel. = Vertical motion, between roller guides. = End plates. = Membrane.  Fig.  5. Schematic Diagram of Complete Machine. Section along Plane of Centre Line.  14  15 A described  schematic  so  f a r , i s given  Pore  Pressure  pore  water  loading lium  copper diaphragm  Fixed  edge  conditions  integrally 0.2 0  with  a  In  i s measured  ( F i g . 6).  order by  the  apparatus,  0.22  to minimize  a  It consists  are  of  as  i n F i g . 5.  Measurement.  pressure  head  illustration  cell, of  obtained  supporting  ring  by of  mounted  a heat  i n . diameter  compliance,  and  0.566  treated  0.015  forming  inside  the  the  beryl-  i n . thick. diaphragm  i n . diameter  and  i n . thickness.  A = Foil strain gauge . B = Diaphragm (beryllium copper) and C = Integral mounting ring. D = Diaphragm retainer. E = 0-ring seal. F = De-airing passages. G = Teflon lined non-displacement valves. H = Recess for upper membrane spreader plate. J = Socket for vertical loading plunger. Fig.  A  6.  diaphragm  is  of  type  cemented  water.  An  Pore Pressure Through Upper  to  foil  the  strain  side  additional,  the diaphragm, would  all time  tests to  have  rise  been  of  gauge,  providing  the diaphragm  waterproofed provide  short  noticeably  Measurement. L o a d i n g Head.  a  duration,  and  a half  a half  remote  gauge,  full  Section  on  bridge  from  has  the  the other but,  t e m p e r a t u r e has bridge  bridge,  been  pore side  so f a r , not  had  adequate.  16 The is  cell sealed  groove when is  i s held against  f o r the  the  cell  obtained,  sidered After  the  O-ring  hence  body  a  tube of  metrically  the  opposite  Removal  of  for  stainless  wide  also  support  for  The  shown  Figs.  7a  the  just  and  7b  the  the  seal  and  a l l water  ing  pressure head.  cell The  deflection  of  a maximum  leads,  the  O-ring  pressure  upper  by  i s con-  is  of  having  to  face  of  supplied valve,  valve  flushing  cell,  contact  pressure.  and  the  a i r cannot  threads  the water and  i n the  be  right  remove a i r .  ring  diaphragm,  in  dia-  water  interrupted  c o n n e c t e d to  the  that  tube  and  filled  to m e t a l  of water  permits  passages  by  1/32  in.  beryllium  trapped  in  this  the n o n - d i s p l a c e m e n t v a l v e s  the  the the  head,  to  head  steel  the  ring  the water  negligible  retaining passages  system  is  is  them of  and  the  practically are  the  solid  the  The  a volume  i n volume  39  of  0.49  of  x  10 x  ^  is  i n . under  10~6  cu.  O-ring and  stainless  diaphragm.  diaphragm  built  sample  expansion i s that  of  dia-  either  appreciable  deflection  handles  F i g . 7b  the  on  compliance of  between out  The  while  non-displacement valves  are machined only  assembly.  i n F i g . 7a,  measuring  passages  involving change  loading  are prominent  because  loading  the  Metal  similar  ring,  entrances  the  words,  A  stainless  into  pressure,  completely  for saturation  the  so  show  valves  shows  eliminated  have  The  water  first,  arrangement  Expansion of  to  in i t s seating.  change  below  ring,  side.  the  the  the  and  steel  ring  under  head.  steel  phragm  the  in.  steel  i n F i g . 6.  two  clearly  fully  screw  t o be  compliance of  over  over  holes,  hollow.  O-ring  air is facilitated  slots  diameter copper  head,  an  stainless  a non-displacement shut-off  load  the  by  the  with  a  i s designed  sample,  through  the  by  negligible  mounting a  leaks  i s screwed  t o be  through  i n place.by  in.  due  to  calculated 100 In  p.s.i. other  the measuring system, i n c l u d -9 e t c . , i s a p p r o x i m a t e l y 4.9 x 10 cu. i n . per l b .  Fig.  Underside  of  Loading  Head  18 per  sq.  and  between  so,  taking  the  per  lb., a  change  water  in.  to  The the  flow  volume  of  shut-off  the  passageways  valves  out  pressure  of  the  the  i s approximately  c o m p r e s s i b i l i t y of of  below  of  water 1  sample,  as  into  the  .066  cu.  in.  10 ^  sq.  in.  3.4  x  sq.  in.  head,  as  l b . per  cell  -  causes follows  —6  Total  .0049  x  10  .224  x  10 —6 10  .229  x  —6  Compliance apparatus ible  usually  metal  tube,  approximately  that  sample If  can  be  infinite  there  is  equation  is  a  the  is  Q  new  the  pressure  for  Vy  is  water. =  length sample  to  x  10  to  ^  cu.  for  a  the  in.  the a  pressure  piece  typical  Ap  First  at  time  value  time  i n v e s t i g a t e the  k  flex-  l b . per  response  permeability  of  transducer.  i n . per  compliance. and  water  pore  of  i s minimized,  pressure  0.167  the  the  at  a  disc  x  =  10^.  "  1  time  sq. of  in.  a  lag  in  consider, a i n . per:  t  =  at  through the  cell  2.75d,  for  a  value  t  t, F  time  of  A  « P  0,  sec.  the  ^  existed before  occurred  later  porous  F  which  which  compliance  Taking kt  -  pressure  allowing  of  the  Ap  recorded  V  to  l b . per  laboratory  the  equation  of  -  x  pressure  pass,  on  0.5  volume  change  P  0  is  sq.  modern  change  P - P  Pp  i n . per  due  P - P  where  cu.  adapted  recording,  of  of  this  Hvorslev's  pressure  i n . compression  connecting  would  1 1  cu.  depends  assuming  piezometer  in. vol.  of  However , be  cu.  is  which and  flat  o f yr~  Vy  =  =  0  time  and  the  disc 4.605  -  is  the  p  factor,  pressure  is  0,  the  intake  the y^  p  t  must  density  filter is  enough 6  to  H v o r s l e v , M.J., " T i m e L a g and S o i l P e r m e a b i l i t y i n G r o u n d Water Observations". B u l l e t i n No. 36, U.S. Waterways 1951. Experimental Station, Vicksburg  19  P ~P make the of  = 99 p e r c e n t .  1  P  0 ~  P  value  the change  i s correct within  10 , 9  1 percent  o f t h e amount  when  k.t f o r the range  required  4.605 = 0.167fct'x  i  recorded  So,  So, p u t t i n g  forp  =  2 7 . 6 x 10  of t y p i c a l  to equal  p  ±  i n .  9  soils,  the time  theoretically  .01 Ap  i s approximately  as  f oHows : Clay Permeability (in. p e r s e c . ) 0.4  x 10  Time  69  (sec.) This  greater  stiffness  cut  down  for  waterways,  volume  taking  retaining with of  could  ring.  with  these  to about  passages  they  means  kt  x 10  1 2 , 12 x 10  , 12 x 10  be  allowed  reduced  o r by the  diameter  This  of  and  hollow enclosed  would  ^ cu. i n . per  ^ f o r the  little  hole  i n the  the volume  l b .  existing  i n recorded i s only  pressures being -9 4.7 x 10 i n . So, -9 -6 x 10 , 0.4 x 10 ,  s a m p l e s w i t h p e r m e a b i l i t i e s o f 0.4 _3 x 10 a n d 0.4 i n . p e r s e c . , t h e t i m e s -3 -6  become  If  a  Furthermore,  of the space  9  c a n be  from  tubes  a smaller  x 10"  were  could  in liner  t o .039 x 10  result  69  6  of water  .01 c u . i n .  t o 0.23  0.4  on s a n d .  t o make  finer.  Sand  3  x 10~  the volume  up m o s t  o f Ap w h e n  x 10  69  diameter  them  Clean  f o r tests  However,  By  would  0.4  are simpler  compliance  This  1 percent  3  by d r i v i n g  s q . i n . , as compared  within  x 10"  b e made  be r e d u c e d  arrangement.  0.4  could  the t o t a l  69  of view.  to d r i l l  the diaphragm  reduce  for  they  considerably  ring  6  Ample  a p r o j e c t i o n to take  water  per  point  the time  x 10"  i s required,  since  Sand  0.4  i s adequate  considerably.  machinist's in  result  9  Silty  required -9  a n d 12 x 10  would  sec. respec-  tively. The drawal the  effect  of the volume  pressure  being  of a n o n - i n f i n i t e required  recorded.  to r e c o r d (Hence  sample  i s that  the pressure,  the time  withreduces  to record the  20 pressure  is  tity  tends  the  and  affected,  pressure  bility  of  actually  to  but  reduce  water  and  o c c u r r i n g , to  compliance  the  is affected  the  of  the  this  is a  delay.)  depends the  the  second The  on  soil.  the The  pressure  apparatus  were  order amount  by  relative ratio  which  zero,  small  is  of  quan-  which  compressithe  pressure  would  occur  given  by  i f  the  1 1 K  Where  is  the  K.B  +  compliance 1  B  V  '  C  .  of 1  .n  the  +  V  =  Volume  C  w  =  C o m p r e s s i b i l i t y of  water  C  g  =  C o m p r e s s i b i l i t y of  soil  n  =  Porosity  Appendix The  x  2  s  which  (See  2  and  C  w In  apparatus  x  1.125  A  for proof  volume =  of  4.5  of  cu.  the  sample  of  sample C  in.  this  statement.)  is  is  skeleton  approximately  approximately  3.4  x  10  ^  w s q.  i n . p er  lb. It  about lb.  .25  and  0.229  x  or  a  soil  figures  10  the  ^ ,  percent.  if  K  were  as  the  0,12  with  extremely  less  than  0.5  p o r o s i t y of than  0.12 =  x  10^.  i +  or  ^  sq.  With  stiff  in.  per  equal o  soil.  resulting  99.5  K  than  0—T2—x~~0"—229  exceedingly the  10  less  to  r  Also,  pressure  ratio  percent.  .039  the  effect  being  stiff,  percent  =  ratio an  a  stiffer  described,  x  pressure  even  described.  B  - — • — - — r - i  Therefore, on  make  i s with  modified  1 +  sample  imagine  pressure  This  become  to  skeleton  these  97  would  i s hard  with  of  the  recorded,  dense the  soil  water  finite is  and  less would  passages  size  of  than  3  be  the percent  reduced  modified,  as  to  21 Details. reduce  Some  details  friction  between  the  box, a l u b r i c a t e d  the  form  thick  t o be d e s c r i b e d .  the sample  rubber  of a box-shaped  latex  dimensions  i t encloses.  The s m a l l  enough  stretch  are 5 percent dimensions  side  membrane  has a h a l f - i n c h  diameter  the recess  measurements in  i n the loading  a r e made.  t h e t o p and b o t t o m  size,  prevent  sample  during  tension  and a n c h o r  hole  head,  Hardened  a  o f t h e membrane, at the corners  t h e t o p and bottom  on  brane  to s t r e t c h  away  from  the corners  by b a c k  to  pressure  pro-  communicate pressure inserted  i t to  corners  full  the  o f t h e membrane  vertical  the  top of the  deforming  four  overcome  than  are  to spread  The  c a n be  The  plates  mountings.  tendency  less  pore  respective tend  t o 0.014 i n .  i n i t , to  from  takes  the s h o r t e r  test.  where  steel  It  a r e needed  to e l i m i n a t e b u c k l i n g  of the deformed  with  0.012  to  and ends o f  i s employed.  envelope,  diagonal  In order  and t h e s i d e s  membrane  and i t s l i n e a r  sample vide  remain  to  their  o f t h e mem-  of the box.  i n the pore  F i g . 8. End P l a t e s . A Groove, f o r r e t a i n i n g a membrane c o r n e r s h o w s t h r o u g h s l o t t e d hole of p l a t e to r i g h t .  This  fluid,  22 but  during  cannot  be  samples are  placement applied.  formed  retained  long, tucked  ners  where  t h e membrane  from  torn  be  membrane  membrane  the  1/2  will  through  be h i g h ,  rounded  through  during  shear  described  without  saves  onto  Equipment.  The method  and rub  frictional  t h e box  samples,  The e l a s t i c i t y  dealing  sides a  of  are inserted  to approximately  i n the s e c t i o n  thick  recurrent  i t , the  i n the top and, f o r the  described  deform-  grains,  or clay  as t h e s a m p l e s  stretched  and  are  t h e membrane,  probably  adequate.  cor-  the s t r e t c h i n g  f o r sand  For s i l t  prove  hole  and have  t h e membrane,  excessive.  h a s t o be  be  of padding  i n grooves,  (see F i g . 8.).  parts  membranes, a l s o ,  i n . diameter  hole  ends  action  of  7/16 i n .  inserted  The membranes  padding  might  must  and t h e n  these  placement  moulding,  Otherwise  to press  sample  probably  the  through  purpose,  3 i n . square,  with  preparation  samples.  Associated course,  more  cribed,  loads  four  sizes,  suitable piston  or l e s s  of normal of about  was  acting.  controlled revolve  by  two  at equal  leading  allows  loads with  ends  different  to load  f o r any t y p e  of l o a d i n g  speeds,  now  was  cycles,  valves,  b u t c a n be  of  selection  of  maximum  The  piston  controlled  testing.  testing  and, i n o r d e r to  the loading  of the shear  des-  A range  of shear  i n mind  i s , of  being  a  120 l b . p e r s q . i n .  testing  T h e two  loads  of pistons.  to shear  designed  but dynamic  flexibility  double  In the case  b y means  are controlled,  apparatus  allow  of applying  on i n t e r c h a n g e a b l e mounts,  pressure  programme,  optional.  are applied  ratios  pressures The  Such  of a sand  thinner  of  over  to r u p t u r e .  sides.  trouble  as  fits.  pressure  c o r n e r s o f t h e membrane  t o be p o l i s h e d  t o a c t as a measure  box  would  have  any t e n d e n c y  during  rubber  o f t h e box  o f t h e membrane  reducing  this  t h e membrane  grooves  enough  reason,  the v e r t i c a l  the sides  can lead  i n the box, back  i n . diameter  These  ation  drag  1/16  inside  around  working  the  For this  in situ,  by  placed  of a sample  load  pistons  piston  are  s e e F i g . 9.  The  advanced  retarded  and  are  valves  23  Fig.  9.  Shear  other.  relative  to  each  rotating  at  constant  ing  cycles  and  introduced  by  by  the  varying  tons but  were  a i r has  rise  time  house into  a  nected  bank to  vertical swung of  2  supply  of  to  a  body.  small various  of  setting  load of  of  f i l l e d  sec.)  is  passed,  and  is  access  mounted to  the  with  two  stem  load-  to  as  in  gantry,  machine  stiffen  them  and  .02  use.  sec The  regulators,  turn,  required. a  pis-  to  adjustable  and  The  (about  simpler  be  ports  valves.  water,  is  on  ports,  Different  valve  These,  inlets,  stems  response  via  reservoirs. valve  the  with  Valves  a p p l i c a t i o n can  the  rapid  per  piston  allow  valve  shapes  run  and of  sufficiently  cycles  loading  samp1es .  to  in  Piston  consist  forms  the  relative  pressure  the  aside  speed  varying  given  at  They  different  designed  Load  are  con-  The which  can  be  installation  24  Fig.  10b. General Arrangement. Vertical L o a d i n g Frame Swung A s i d e .  25 The rical  shear  and a l o a d  machine  could,  itself,  i f desired,  opposite  end o f t h e h o r i z o n t a l  for  this  purpose,  can  be  added  later,  Loads diaphragm the  load  periphery  calculated  motion.  resistance  the displacement  This  i s capable  another  of shear  could  r u n so f a r ,  pressure  pore  response recorded  copper  s e t around Strains  are  differential  of the h o r i z o n t a l  pressure recording  up  system  the p i s t o n .  bridge.  t o 1000  o n t h e same  i t i s convenient  f o r supplying  i s required.  and  displace-  oscillograph. cycles  per sec.  instrument but,  to record  t h e method  i s , therefore, The  general  the a s s o c i a t e d  de-aired  i t on  at part  of i t i s d e s i r a b l e ,  of i n s t a l l i n g described  arrangement  equipment  water  I t i s n o t an i n t e g r a l  but a d e s c r i p t i o n  understand  apparatus  of  be  gauges  left  machine.  the. m a c h i n e , to  load,  loading  beryllium  of a l i n e a r  to a multi-channel  Apparatus able  strain  the displacement  of l i n e a r  load  the tests  reads  has been  circular  a full  Outputs  normal  with  to give  which  Space  to dismantle  four  are carried  in  are measured  at the  controlled  having  ment  The  having  symmet-  applied  motion.  a strain  without  be  cells,  from  transformer,  so t h a t  is practically  samples.  i n Appendix  i n Figs.  of order  This B.  of the machine  a r e shown  in  adjust-  and  some  10a and 10b.  26 CHAPTER THEORETICAL  Definition  and  is  as  defined  where  u  and  V  are  respectively.  In  shear  shear  deformation  the of  boundary  a  elastic  load  i s described  are  compared  designing  a  considered. effect  of  Boundary tion.  with  The  2h  has  and  i/ axes  A  strained are  resulting  samples  and  sample  11.  and  directions the  The  different  normal  small  load  are  simple  of  sample  1  A  tests  theoret-  under  shear  deformations effect  of  proportions and  the  is  combined  examined. the  sample  transferred  to  deformaa  system  and  height  equal  shear  angle,  OJ .  vertical  Co-ordinate  and  describes  theoretical  load  apparatus  considers  distribution  length a  %x  load  and  plane  3y  conditions.  illustrated  of  - y  chapter  sample  deformation  through  horizontal  Fig.  shear  Fig. 1 this  of  y  x  Uniformity  obtained. of  or  that  This  stress  actually  application  shows  co-ordinates.  the  and  normal  boundary  the  for  load  the  "by  x  soil.  for  those  Conditions.  of  around  i n an  stated  objective.  various  and  the  apply  of  TEST  either  in  to  OF  shear  i t was  sample  stated  machine  normal  been  a  solution  F i g . 11  _  to  under  ical  I  intended  conditions  deformation  i n which  Chapter  discussion is  was  Simple  displacements  under  deformation  CONDITIONS  Explanation. pure  III  respectively.  System.  to  The  x-  Taking  27 the  case  and  botton  on  the  of  end  an  undrained  test  boundaries  cannot  boundaries  must  ar-direction.  It  follows  the  bottom  corners  move  -co. h  top  and  bottom  boundaries  the  top  and  do  not  on  there  i s no  extensional strain.  sions over  box as  sufficiently  a) i n c r e a s e s ,  the  exactly  boundaries  ends. the  This  same  as  to  the  simple  shear  it  does  appear  that  not  uniformly  in this  the  increasing  for  maximum  bottom.  In  directions opposite sample in  strain the of  to  ends.  either  friction. required  So,  Thus,  since  out  simple  a  the  the of  shear  best four  shear,  box  boundary  three  can  ends  not  vary  c o n d i t i o n s would  be  Unfortunately  made to  ends  act  long  at  to  of  for  enough top  the  or  two  direction  stress  aim  extend  their  in a  shear  to  provide  i s designed  on  to  provided  the  deform  for, is  deformation be  the  deformations  contact  condition  and  dimen-  would  sample.  conjugate  slipping,  distance  d e s c r i b e d , f o r one  for  at  changing  can  would  friction  stretch  necessary  slip  of  prevent  same  metal  lengths  to  hence  ends  move ixt.h  the  could  the  the  friction  required  direction,  for  having  having  shearing,  total  point  the  If  their  and  I t becomes  and  both  a l l boundary  metal  arrangement  that  for  throughout  l e n g t h by  the  top  Each  u>.y i n  corners  I f we  shear  the  way.  top  moves  make  simple  uniform,  amount  extensional strains  would  be  an  change.  allow  the  vertically.  i s adequate  point  the  change),  , t h e r e f o r e the  every  of  by  the  boundaries  these  volume  deform  that  and  bottom  (no  zero  conditions and  one  c anno t .  Provided  Desired Top  and  Bottom  Ends  Since patible cannot  with be  one  simple  uniform  u V  = k.h = o  u V  =  of  =  the  shear,  simple  and  u V  -k. h  u  k.y 0  = k. h a n d = o =  k.y  = 0  T  xy  boundary  c o n d i t i o n s i s not  c o n d i t i o n s through  shear  -k.h  and  the  i t i s necessary  com-  sample to  consider  28 how  actual  conditions differ  to  test  of  two c o l o u r s o f p l a s t i c i n e  plasticine  reversing Fig.  sample,  strain  and  bottom  the  plasticine  had  a sample,  i s shown  these  "friction"  had been  12b, i t w i l l  uniform  shear  were  approximately  piano  that  most  there  are areas  but  even  three  cycles,  there  has been  zones  with  to  While  working  with  would  happen  inadequate  to prevent  tests  performed  were  introduced paper  effectively  ticine. will  The r e s u l t  be n o t e d  sample shear  strain  pressible  shear  a small  a test  strain zone  that  to a s o i l  plane  strain  plasticine  could  only  deformations.  each way), or of  the average. i t was  decided  friction  boundaries. bands These  were  Some  of oiled  restraint  paper  strips of on t h e p l a s -  i n F i g . 12c. I t  markedly  the centre  because  Around  through the  approaches  a  deformations.  plasticine  i s incom-  skeleton, the deformations are  The p l a s t i c i n e  under  boundary  near  which  deformation,  plane  than  to the boundary  may b e a r g u e d  compared  shear  i s shown  varies  that  of p r a c t i c a l l y  deformation.  samples,  the f r i c t i o n a l  and  of a magnitude  t o p and bottom.  of such  "friction".  (30 d e g r e e s  narrow  with  Looking at  consist  at these  several  corresponding  comparable.  performed ible  that  and o n l y  It  not  broke  full  i f t h e t o p and b o t t o m  with  top  contact  slipping.  higher  dummy  slipping  a t t h e sample  at the  o f any f a i l u r e  markedly  cycles of  i t was a s s u m e d  of reduced  strains  s e e what  wire;  strains  shear  i n F i g . 12a.  to the p l a s t i c i n e  of s t r a i n  no d e v e l o p m e n t  full  their  to the boundary  boundaries  made  For the t r i a l i n  of t h e sample,  the  after  i s shown  plates  to prevent  be n o t e d  samples  i n direct  adhered  fine  corresponds  placed  In order  i n the machine.  three  to develop  enough  through  testing,  boundary  plates  with  dummy  i n F i g . 12b.  and a l l o w e d  t o b e c u t away  Fig.  desired.  strained  undergoing  metal  o f t h e sample  the test,  were  before  after  12b, s a n d - b l a s t e d  After  those  the uniformity of deformations,  Such A  from  tests  described  and t h e p r a c t i c a l l y  accommodate  As a p o s s i b l e  itself method  so f a r , were incompress-  to the uniform of allowing the  Fig.  12a.  Undeformed  Fig. 12e. Plane Stress, No F r i c t i o n , 1/2 Cycle  F i g . 12b. Plane Strain, Friction, 3 Cycles  Fig. No  12f. Plane Stress , Friction, 1 Cycle  30 plasticine  to  behave,  two  compressible,  three  they  narrower  were  cut  boundary plane,  surfaces  but  lateral if  there  soil,  is,  of  ent  material  plane  course,  sample ary  the  the  It  and  density  of  total  volume  under  changes  whole  sample  Fig. the with  12d same  proper  relieved,  in  This  so  are  the  be  restraint  seen  of  are  shear  to  boundary  importance 12e  of  have  top  colours  with  directly  a  do  undergone  results  to  the  the  and  the  most  soil  of  to same the  strain  cor-  shows  that  boundaries  and  in-  from  adhere  shows  shear  compressible  the  stresses  F i g . 12e  bottom  soil,  almost  in  not  still  F i g . 12b  comparable  on  in  at  differs  joints  com-  compliance  are  simply  the  bound-  changes  Looking  compressive  and  narrow  d e n s i t y of  F i g . 12d  as  It  a  to  due  deformation.  the  due  not  however,  boundary  test  satisfactory  with  different  local  undrained are  con-  context,  the  at  two  i t represent  above  deformations  slipping  total  represents  make  average  plasticine.  differ-  analogous.  constant.  are  (no  given  an  hence  F i g . 12d,  frictional  and  and  a  for  In  the  to  result  the  indicate  the  well.  to  12d  that  of  responding  Fig.  stress;  The  since  to  and  for  compression,  the  x-y  compressed  squeezing  to  The  the  have  sample  represent  small  in  happens  but  box.  sample  conditions  in  tested,  instead.  although  meant  box  would  and  i t were  shear  the  somewhat  that,  layers  still  of  available  i s because,  sample  extra  to  very  slight  distribution  assume  that  friction  favourable can  be  effectively  of  the  what  strain,  skeleton.  F i g . 12b  the  of  were  the  displacements  not  seen  nature  the  other  be  side  narrower  should  are  total  boundary  chiefly  same  are  soil  remains  plasticine.  each  of  in  the  plasticine  changes  compressible 12b  a  with  different  i s emphasized  i t will as  proof  though,  expansion  of  sideways  plane  the  stresses  expansion. pression  of  results  allow  tensile  expand  However,  give  width  each  though  samples  m a t e r i a l which  under  remembered, may  on  real  instead  should  be  a  (soil)  stress  conditions,  Thus  not  the  i n contact  room  could  change).  ditions  must  was  plasticine  than  were  expansion.  i t were  volume  more  d i m e n s i o n a l l y , as  would  skeleton.  whereas  12d  31 shows far  conditions  from  to  the  being  sample  action. been  In  but  can  undergoing  no  ners  the  be  sand  sample  further  of  ing  alternate  the  central  and  f r o , with of  tests  dummies can the  displacements  linear be  of  truly  that  the  once  the  will  not skeleton  i f i t only whether  of  friction  be  the  a  one  experienc-  corners;  sliding  bodily As  i t was  to  a  decided  intended were  aims, adequate  boundary. The  results  helpful  changes  trials  However,  as  an  an  through  to o b t a i n value  a  solution  cannot  i s not  linearly  elastic,  the  such  a  problem  to  either,  solution  example, the  sample  in  solution  is  i n terms  elastic, apply  solution  for  yields  with empirical  analytic  solution  strains  analytic  of  in variables,  analytic and  cor-  undergoing  friction  linearly  elastic  and  arise  the  i s not  such  top  could  samples,  boundary  Soil  the  i n F i g . 12f.  achieve  o b v i o u s way  appropriate  that  as  boundary  as  exactly  boundary  be  the  shear,  even  after  satisfactory.  stresses  The  plasticine  shown  A  does  could  sample  be  further  not  they  sample  has  does  when  loosening  Solution.  effects  elasticity.  objected  ever, be  the  the  strain  cyclic  bottom  sand  sample  bottom  never  and  terms.  solve  a  probably  appear  dimensions.  general  and  plasticine  at  Elastic  investigation  sample  and  slipping  examining  on  shearing  i n F i g . 12f,  corners  question  very  happened  a non-cohesive s o i l  deformation,  would  top  Theoretical  the  of  above,  resistance  Such  of  little  that  plasticine  the  have  any  F i g . 12f  r e - o c c u p y empty  strain.  portion  under  "compress",  hand,  shear  shown  sample  them  shear  offer  mentioned  d e n s i f i c a t i o n and  the  prevent  sand  i n a d e q u a t e top  the machine  provided  stage  forth  sample  other  reduced  large  a As  However,  with  cycle  to  the  them.  exhibit  in  and  would  reversal  cannot  the  shows  original  deformation.  plasticine  would  A  back  expected to  into  to  After  what  one  the  plasticine  slide  On  fell  result  after  the b e h a v i o u r of  re-expand.  that  the  F i g . 12e  shows  of  boundaries slide.  of  would  F i g . 12f  resistance.  represent  ideal,  i n F i g . 12e,  plasticine  bottom  so.  to  F i g . I 2 f much  removed  shearing  close  leads  and  i t  soil. but to  of may  How-  i t will  32 displacements  very  If  the  elastic  of  plasticine,  linear  linear this  close  to  solution  i t implies  elasticity  do  not  that  the  resemblance tion  is  elastic  to  material  a  size  certain  of  solution  for  terms  Appendix  in  tropic axis  It  the  material  given  sample  great  soil  by  and  isotropic  a  elastic  the  with  such  to  a  solu-  solution  for  constants  Ratio  of  for  0.5.  more  about  to  approximate  solution  symmetry in  solution  and  in  from  reasonable  elastic  given  constants  displacements  an  sample,  Poisson's  with  radial  yield  Roscoe ,  samples.  departures  seems  The  is  the  elastic  5  case  together  having  ( v i z . equal  a  pursuing.  was  C,  very  therefore  in  plasticine  predict  solution will  worth  isotropic  the  i n v a l i d a t e the  conditions  therefore  of  can  that  p a r t i c u l a r problem.  assume  those  The  general for  any  aniso-  vertical  a l l horizontal  direc-  tions) . In tions,  there  soils,  of  tion  was  was  the  in  be  no  five  only  pattern could  developing  to  the  representing anisotropy  thought  simple  as  was  to  wide  the  the  a  of  range  of  the  form  by  computer,  and  Each  of  these  stresses  angle  of  varying these  shear  Fig.  has  of  a  the  inten-  figures  gives  a  now  be  added  to  Fig.  order  that  of  the  study  materials  reasonably  possible,  solution, plotted in  in  a  Figs.  sample  after  different  and  varying  deal  of  explained 13  13,  in  14,  a  material.  in  to  assist  into  six  in  unit  represent Because  material,  detail.  16  summary  small  figures  condensed  dimen15,  comprehensive  The  great  deformation  development.  given  will  and  as  are  sample  contain  constants;  behaviour  elastic  deformation.  13  been  the  displacements  dimensions  figures  first, tion  and  real  In  soils  equa-  for  field  machine.  anisotropic  evaluation,  elastic  stress  in  the  actual  include  sionless  of  of  shear  included  Results  17.  solving  anisotropic  study  extended  and  Some  following  the  informathe  explanation. Fig. right  hand  one  13  is  gives  divided details  of  the  panels;  example.  the In  bottom  this  bottom  33 panel  the f i r s t  constants example under  four  which  figures  adequately  illustrated.  discussion,  stants.  They  Young's  there  c a n be  ratio  Poisson's  define  are five  of the e l a s t i c  the material  f o rthe  material  of the type  independent e l a s t i c  i n horizontal direction, E (EX on t h e c o m p u t e r  solution)  in vertical direction, E (EY on t h e c o m p u t e r  solution)  in horizontal y (MUXX  ratio  plane, on t h e computer  solution)  in vertical plane, U (MUXY o n t h e c o m p u t e r  solution)  xy  Shear  modulus  in vertical plane, G (GXY o n t h e c o m p u t e r xy  The  shear  ent,  modulus  i t i s given  i n a horizontal  plane  solution)  i s not independ-  by: G  x  =  X  E  2(1 + y  X  *xx  Any  con-  taken as:  modulus  Poisson's  the values  For anisotropic  modulus  Young's  give  material  of the type  ) under  discussion  i s iso-  tropic i f E  V  x  and  E =  xy  x  E  /  /  and  y  = y  xy  xx  t h e example  xy  i n F i g . 13  the bottom  right  hand  that: y  xy  5  a  "  n  3  Also for K  therefore  = 1  the m a t e r i a l  values  G  y  =  °G  E  G xx  shows E  =  and  In panel  x  , K  d  V  =  xx  ' ° °  =  2  (  °1  therefore -  5  +  i s isotropic i n the bottom , K  a n d A.  therefore  - - -  ^ >  with  -  E  X  =  E  y  y  = y  G xyxy  =  -xx XX  -xy  G -xx  y = 0.5.  right  hand  panel  o f F i g . 13 a r e  These  are functions  of the  34 elastic poses, value  constants, but  L/HT  sample. ation L/HT  of  no  is  the  It  for  a  interest ratio  should  height;  is unity,  is  p r i n t e d by  be  coefficient  to  of  noted  sample  of  the  the  length HT  equal  is  stress  computer present  that  i t i s not  the  the  height  the  In.  to  square.  of  pur-  Fig.  value  which  The  the  computer  In  The  uniformity,  checking  discussion.  to  is  for  abrevi-  13, of  will  since  A  =  be  .776  defined  presently. The in  a  row  ferent  of  remaining  three  stress,  bottom,  where  stresses  are  along  and two  a  two  five the  in  panels  top,  the  normal  each  Fig.  of  remaining  displacements  (the  of  are  stress  13  arranged  which  plots  a  spaces  along  the  plotted. in  are  the  The  dif-  three  x-direction),  a  * (the  normal  y  stress  in  the  ^ - d i r e c t i o n ) and  T  (the  shear  xy s t r e s s o n a n x-y p l a n e ) and t h e y h a v e a l l b e e n r e d u c e d to dimensionless f o r m by d i v i d i n g t h e m by T where T i s the J  applied  shear  tion  of  the  have  been  force  divided  sample.  labelled  In SIG  Starting  with  ation  dimensionless  SIG  of  X.  The  The  distance  from  Fig.  x  creases, 1.0.  upper  an  of  at  the  left  values  are  cal  axes  of  SIG  X  two  ends  of  the  sample.  part  of  X/L.  the  Each  panel of  a  different  y  =  0.8/z,  and  y  =  h  on  and  symbols  Y  =  at  there  these  sample y the H  =  =  appear.  Fig.  five  At at  = the  the  These  a  smaller,  lower  as  a  function  of  varies  from  0.0  =  curves  top  =  x  two  verti-  represent  that  in  the SIG  variation 0.4/z, y  hand  opposite  of  into  1.0,  boundary  left  symbols  The  showing  the y  upper  to  as  seen  0. 2h,  vari-  and  and,  X/L  be  the  re-direction,  abscissae.  represents (y  shows  sample,  13)  and  stresses  R e f e r r i n g back  the  as  It w i l l  height  panel,  in  0.0  curves  l.Oh).  of  the  sec-  respectively.  the  X  sample.  horizontal  form,  XY  in  SIG  end  are  a  this  portion  written  X/L  panel,  shows  the  X/L  of  TAU  stress  upper  end  area  and  top  portion  (labelled  These  Y  normal  the  => 0  x/l  has  the  av  dimensionless  SIG  hand  av  J  by  this X,  left  panel  portion.  11,  the  6  the  upper X  vs.  of  =  at  0.6h, the  sample,  corner, the the  value  of  35 SIG  X  =  1.8,  the  upper  SIG  X  boundary,  actually  able  scale  this  is  the  indicate  in  the  top  goes  reason  negative  of  information. top On  upper  portion  "Y  =  0"  of  SIG  will X  at  The  at  the  be  seen  of be  a  the  sixth  line,  remaining  at  through  i t increases  with  y.  mate  value  The  polation.  the  SIG  function  to  the  vertical  axis  set  out  X  horizontally.  in  ten  equal  tours  of  SIG  X  these  segments  .20L,  .25L  contours portion  The  of  X X  = =  repeat, the  =  f o r Y/H half  increments  from  Y/H  Y/H  at  0.5L of  panel,  the at  end  with  SIG  to  the  a  SIG  It  sign, mid-  above  mid-  also  by  increases  the  approxi-  intercomprehen-  left  the top  axes  of  the  and the  of  hand  and  height,  SIG  X  The  pattern at  conof  .10L,  actual given  are  was  eleven  .05L,  a  bottom  boundaries  through  0  as  mid-height  sample  the  X  will  at  f u n c t i o n of  the  =  of  reverses  upper  at  X  plot  h.  give  -1,  the  a  i t is plotted  mid-length.  course,  =  It  at  at  of  p i c t u r e as  length  the  actually  but  0,  and  of  zero  =  as  i t s values 0  is  the  length  of  X  X  in  this  y  time  is plotted  represent  to  from  in  0  SIG  loss  and  a l l values  and  on  legend  sample,  plotted of  off  i t  ends.  the  This  and  i s used  taken  upper  SIG  1  in  X  represent  enough  portion  variation =  the  SIG  without  start  sample,  are  the  =  and  graph  SIG  the  point  lower  Y/H  y  1.8  -0.6  the  for  reason-  at  side  and  sample,  near  re-plotted.  height  etc.  of  described  make  sample,  the  0  this  mid-length.  to  of  =  between  in  However,  Since  the  X  any  boundaries.  on  SIG  at  X  the  with  X  SIG  panel,  at  skew-symmetric  at  hand  sample  gradually  p o s s i b l e , the  left  on  a  zero  cut  X  corner,  of  are  off  become  upper  curves  plot  to  cut  the  length  contours  of  shows  the  at  closer  of  depth  zero  the  represents  heights  any  height  of  four  strains  the  the  introduced  and  are  SIG  maintain  of  curves  hand  This  was  off  two  to  for  In  top  values  mid-height  throughout  panel  cut  left  seen.  the  that  as  be  panel,  height  sive  flat  curve  X/L.  but  cut-off  the  can  intermediate  passing  a  f u r t h e r i n s p e c t i o n of  represents X/L.  infinity,  the  f u n c t i o n of  Negative  hand  way.  to  of  A l l stresses  values  left  a  for  boundary.  start  as  plotting,  so  the  the  in  mid-height  .15L, values the for  36 all  lengths,  values again  just been  increasing inside  cut  reasonable  off  in  vertical  the  for  normal  are  cut  These  curves  X,  Y  and,  of  of  upper  are  cut  off  assist  in  Y  at  .8h  (one-tenth  off  just  1.8  It will  SIG  Y  the  the  maximum  and  be  same  SIG  as on  of  boundaries  than  X  The dimensionless  SIG right  shear  SIG  and  X  TAU  (X/L be  =  the  omitted  and  to  portions  i s easy  of  approximately so  plot  within  to  the  see  as  a  values  of  SIG  this  hand  of  Y  leads of  curve  at  X,  the  except  curves  that  tend  representing  i s cut  general  each  boundary  the  that  curves  panel;  from  boundary)  .6h  is  off  cut  lower  distribution i t is  to  of  greater  occur  them)  and  panel  of  F i g . 13  in  the  In  F i g . 13  Y/H  the  more  the  top  of  near  to  heights The  for  function  corner  them.  plotted  X/L  top  top  TAU  the  of  not  at  further  and  the  end  from  the  shows  the  the  upper  =  curve  in  too The  right  has  top  the  (see  the  different  and  Y  =  end,  the  0 like  following  together were  to  panel. with  same  plot  For  in  must  to  example,  a maximum  value  be  therefore  i t i s necessary  hand  upper  as  mid-sample  hand  labels  curve the  H  close  contours,  the  at  =  left  cases  are  highest 1.1  for Y  the  i n most XY  same way  separated  at  computer.  the  XY  well  contours  TAU  the  that  XY',  stresses.  fairly  the  maximum  been  the  However,  TAU  that,  dimension-  below  hand  of  has  function  different  (although  by  the  HT  distinguish  it  plots,  have  maxima.  contours  labelled  both  maximum  X  curve  and  Y).  legibly  use  the  shows  This a  and  left  stress,  .500)  SIG  X  inside  XYare  labelled  figures)  keep  SIG  the  previously described of  The  =  the  (instead  could  of  with  but  SIG  boundaries  length  and  1.8,  seen  points  below  at  Y  Y. as  identifying  SIG  below  SIG  SIG  the  o f f at  contours  to  F i g . 13  first  at  cut  two  -0.6  of  X.  than  is  is  and  panel  curve  s t i l l .  Values  stress,  values  off  to  boundaries.  SIG  greater  being  other,  as  values  different  corners  corners  1.8  second  same way  different  at  the  bounds. The  less  the  toward  value  both represent  37 SIG X as a function of X / L  T A U XY as a function of X / L  Y = H  SIG Y as a function of X / L for Y = 0, Y = . 2H, Y = . 4H . Y = H  SIG X as a function of Y/ H  SIG Y as a function of Y / H  T A U XY as a function of Y / H  for Y = 0, Y = . 2H, Y = .4H  for Y = 0, Y = .2H,  for X = 0, X = . 05L, X = . 10L  X = . 5L  DIMENSIONLE S S  U / H as a function of X / L  STRESS  V / H as a function of X / L  8  S  EX/ET = 1.00  "  a  8  L/HT = 1.000  MUXT = .50 MUXX = .50 EX/GXY - 3.00  ' U/H  -.n  -.n  1*°  — I  -.*»  .i n  I .SB  1 .SB U/H  I.CDS  —Tm— —  1.I0O  i  U / H as a function of Y / H  DIMENSIONLESS Fig.  13.  -.RB  -.•on  Y/H  -.BO  -.an  Kj = .750 V/H ao .an  1 ,  Era  rf  1 .an  t .an V ' H  l.ODD •  i.«n  i -  K, = .750 fl = .000  K. = .750  l.WD  u  V / H as a function of Y / H  CONSTANTS FOR CASE IN THIS FIGURE  DISPLACEMENTS Isotropic Equal  \ = .776  Material. to Height.  Sample  Length  38  DIMENSIONLES S  Fig.  DISPLACEMENTS  Sample L e n g t h Isotropic Material. T w i c e H e i g h t . Equal to  39  DISPLACEMENTS  Fig.  15.  Isotropic Material. Sample E q u a l to Three Times H e i g h t .  CONSTANTS  Length  s  s  8  8  Yi(  I  I  8  40  S  8  8  STRESSES  Fig.  16.  Isotropic Material. Sample E q u a l to S i x Times H e i g h t .  Length  I 8  41  Fig.  Anisotropic Material. Sample Length Equal to Twice Height.  42 the  mid-height  (Y = 0) c o n t o u r  Y = 0 on t h e l o w e r this the  contour  the  plot  be  Y = 0 contour  decreases  from  Therefore, in  the contour  must  successive closer  axis.  contours  the  eleven  boundary  conditions zero  a l lthe stress  elastic  would  plots  included  must the  sections  zero  may  and t h e  plot  all  lying  along  to shear  in a  forces  with  I f the boundary instead of  throughout  t h e sample  eleven  o f X a n d Y.  o f SIG X would the axis. vertical  Therefore  contain  The lower  lines,  s i x lines, plot  would  a l lc o i n c i d i n g  with  axis.  SIG Y = 0 f o r a l l v a l u e s  o f X a n d Y.  of  be as d e s c r i b e d f o r  SIG Y would  therefore  The  plots  X.  TAU XY  = 1.0  f o r a l l values  o f TAU XY w o u l d superimposed  of  TAU XY w o u l d  at  TAU XY  The  lower  l i e along the  displacement  stresses  shows s i x  distribution  described.  vertical  plot  the axis,  subjected  already  upper  all  be  as f o l l o w s :  plot  how  maximum.  Therefore  represent  the stress  when  the  SIG c)  that i t  no o t h e r  the upper  SIG X = 0 f o r a l l v a l u e s  the  identified  the contours  one of w h i c h  show  at the ends,  contain  b)  plots,  l i ealong  contours,  sample,  be i d e a l , a)  may  conditions  friction  i n the  be s e e n  magnitude. plot  (half  Having  to the end, with  i n the lower  .500  maximum  i twill  plot,  at  t o t h e end of t h e sample.  The s t r e s s  linearly  plot,  of decreasing  one o f which  shows  a t X/L =  this  occurs  t h e maximum o f  t h e one f o r m i d - l e n g t h .  the centre  order  In  plot  containing  value  since  occurs  i n the upper  and c l o s e r  contours,  plot  at Y = 0 i n the lower  a steady  this  Similarly,  on t h e u p p e r  length),  lower  plot.  since  plots  f a r the r e a l  o f X a n d Y.  contain  a t TAU XY contain  The  sixhorizontal  = 1.0.  eleven  The lower  vertical  upper lines, plot  lines, a l l  = 1.0. of s t r e s s e s  conditions  a r e thus  a r e from  being  an i n d i c a t i o n ideal.  of  43 In cribed.  F i g . 13,  They  bottom  left  ments,  u,  are  hand  which  by  the  ment  has  been  Contours plotted  of  of  of  would U/H  the  plot  U/H  1  through  i n the  U/H  would  not  vary  showing  (since U/H  axis  vertical  =  of  on  lines  the  x.  X/L  i n the  eleven  at  displacethem  labelled  U/H  displace-  stresses. have  upper  values  been  plot  of x  a  to  the  of  and  as  i n the  of  the  lower  their  final  shape  should  lower  plot  some  scale)  vertical.  of  the  Similarly  should  should  show  be  started the  displaced  a  the  any as  s e t of  a  as  show u  U/H  the  picture  and  of  the from  be  lines  zero along  during  a  lines  so  a  which  therefore  V/H  lines  at  shapes  a  drawings  started  upper  be  In  horizontal  deformed  can  verti-  o f y.  s e t of  the  final  the  were  deformation  function be  lines  it,  o f x,  calculated  relevant  a  should  of  plot  assumed)  vertical  function  A  and  would  d i s p l a c e m e n t of  drawings  upper  superimposed  deforming  of  h,  plot.  and  as  shapes  V  the =  lines  vertical  horizontal.  machine,  replotting  V  final  before  U/H  the  straight  horizontally  of  to y  lines  of  Y/H.  ideal, 0  to  eleven  horizontal  and  the  value  axis  be  words  The  and  would  =  and  d i s p l a c e m e n t was  V/H  lines  dimension-  X/L  straight  of  sample  y  horizontal  plot  a  of  lead  superimposed  of  the  made  from  would  shear  be  s i x superimposed  network  side  could  steadily  mid-height.  show  functions  spaced  angular  upper on  as  This  cal  from  of  increase  unit 0  If  which  functions  the  t o make  horizontal the  different  h,  been  as  heights  six equally  the  V/H  have  way  conditions  with  the  lines  height,  desIn  horizontal  displacement plots  boundary  leading  (to  by  same  d i s p l a c e m e n t V/H  everywhere,  other  of  dimensionless  for eleven  other  showing  lower  drawn  plots  divided  t o be  displacements.  d i s p l a c e m e n t axes  and  Y/H  vertical  =  two  remain  plot.  values  of  been  still  showing  d i s p l a c e m e n t as  If  but  are  plotted  The less  panel  for six different  functions lower  plots  The  computer,  panels  the  have  dimensionless.  two  plot of  lines  sample  drawn,  scale  of  by  which  44 will  make  unit  The  lines  its  proper  deformation  must,  of  equal  course,  be  position  along  "sample".  This  not  out  values  of  U/H  hand.  It  i s shown  of  the  picture  by  material  and  Fig.  size son  in  diagram  for  lines  y  y  x  at x  -l.Oh,  =  It  =  0  =  (left  V/H  with  L/HT  elastic  =  0  l.Oh  (top  hand  end),  be  boundary) x  placements  that  which  plasticine  are  same  13  isotropic  (Fig.  13),  to  (Fig.  16).  a  Comparing  been  16  are  sample  length  displacements having  elastic  is  the  compari-  forms  y  =  vertical  solution the  a  with  =-0..5h,  (right  by  as  for  0.286?;, x  =  1.0Z  resembled  material.  different  this  closely  to  represents  the  linear  same  0.5h,  lines =  0.429Z-,  hand  end).  yields  results  dis-  of  the  trials. Figs.  the  the  =  isotropic  started  with  0.143Z., x  =  a  Calculated  that  and  print  such  F i g . 18  (centre line) , y  at  the  computer  F i g . 18  sample  of  (the  As  shear).  plotted  plot  for  samples.  x seen  to  1.778  Thus  plasticine  y  at  but  used  =  each  height  i n F i g . 18  0.5711, will  and  was  degrees  and  computer,  described).  imaginary  horizontal =  (30  Deformed E l a s t i c Sample By C o m p u t e r .  deformed  an  length by  and  7z//3  separated  the  done  sample  machine  the of  a  18.  the  with  was  to  are  equal these almost  obtained  a l l examples  of  They  that  vary  proportions to  in  from  conditions in  a  times  the  figures  shows  that,  with  in  non-ideal  figure  square  six  uniform  each  height as  a l l cases boundary  hoped, and  that,  conditions,  45 the  stresses  specimen. compared more  are  not  However, to  and  Fig.  figure cent  length  of  the  stress  displacements represents  Examination  of  the  the  the  greater  than  the  vertical,  modulus  since  '  modulus  in  a  /G  in  may  described  of  interesting  to  note  the  anisotropy  under  out  In  in  that  uniaxial vertical  anisotropy,  dilation  elastic  Poisson's  ratios  although  only  Appendix  D, 17  It  u  for  Fig.  to  height  is  therefore  is  and  be  of  the  is  x  IG  times  u  exist  ratio  in  Fig.  be  compared  were D.  D  50  per-  are =  xy  the  6.00, '  shear  aniso-  material,  the  bound-  examined.  This  passing,  i t is  shows  that  show  as  that  with  material  net  be  the  be  shear  to  14,  be for  assume that  than  Checking  0.5,  in  represented  anisotropic.  Fig.  to  greater  this may  note  material  seen  i f  possessing  to  with  volume  Thus  necessary  material.  will  a  triaxial  interesting both  this; the  compression.  may  of  that  In  can  is mildly 17  ensure  i t i s not  found  and  to  during  also  can  to  E  material  regarded  compressible  i t will  in  are  behaviour;  energy.  uniform.  ratios  and  0.4  possible  described,  type  of  order  Appendix  clays  release  whether  Appendix  consolidated  represent  is  becomes  anisotropic  modulus  2.4  plane  constants  heavily of  =  soon  part  Poisson's )  xx  increases  more  an  the  plane.  elastic  set  increase  u  represent  is  of  +  vertical  exist.  should the  2(1  questioned  examination  type  =  xx  a  be  can  results  values  x  horizontal  It  tropic  £  the  in hand  h o r i z o n t a l Young's  and,  even  conditions right  of  specimen  become  lower  centre  distribution  the  shear  ing  17  near  that  n  type  the  except  shows  equal the  as  i t s height,  favourable  sample.  ideal  The  2.000  l e n g t h •• and  i t  isotropic 1  material. that  1 2  a  It  was  noted  considerable  by  degree  Zienkiewicz, of  anisotropy  Cheung was  and  Stagg  2. ,  unimportant  Z i e n k i e w i c z , O.C., C h e u n g , Y.K., and S t a g g , K.G., "Stresses i n A n i s o t r o p i c Media w i t h P a r t i c u l a r R e f e r e n c e to P r o b l e m s of Rock M e c h a n i c s " . J o u r n a l o f S t r a i n A n a l y s i s V o l . 1, No. 2, 1966.  46 because where It  i t had l i t t l e  the boundary  some  mostly  effect  effect  on i n t e r n a l  anisotropy have  a considerable  this  suggestion,  where  i t may  of the type  displacements  region  sample  of zero  increases  i n size  the  sample.  whole  normal  stresses  desired  unit  the  desired  ideal  proportion ditions  one  of the sample.  case.  The v a r i o u s  easily  by l o o k i n g  indicate  it  appears  will  stress.  medium  i s a shear  uniformity. it  examples  This  b e made  The p o i n t  closest  t h e sample  centre,  proportion  as t h e r e g i o n  that  on  panels,  since  the test  referred  the d i s t r i b u t i o n  to i d e a l  o f the sample,  and w r i t i n g  stress  at this  true  conditions  uniformity  equal would  con-  little  distribution complete most  s e e m s an being  of d i f f e r e n t  a coefficient be  large  which  This  concases  of stress t o as X and of  shear  i s at the  to the shear give  16,  ideal  compared  stress.  will  a  vary  the  zero  where  of ideal of  16,  of  in Fig.  throughout  a region  hand  of  i s a region  the region  For comparison  centre  point,  to F i g .  are therefore  coefficient  to depend  Fig.  as i n d i c a t i v e o f a  to define  on  13  region  with  of comparison,  appropriate  ends,  effect  and, f o r example  at the top r i g h t  test.  near  i t follows  shear  of  s t r e s s , the  and t h e d i s p l a c e m e n t s  taken  the dimensionless  appropriate sidered  coincides  c a n be  from  are achieved  to another,  stress  acts  Because  f o r the others,  the sample  i s a known  that  t h e same  conditions  i t may  In support  i t spercentage  seen  stress  though  distribution,  near  increases,  be  even  where  stresses.  stresses,  It will  that  has i t s g r e a t e s t  on  and i n c r e a s e s  shear  example  any one  effect  length  f o r one s t r e s s  conditions from  of material  increases,  the  that  had  larger  t o be  on d i s p l a c e m e n t s .  conditions  normal  b u t a much  stresses,  tunnel,  conditions  anisotropy  on s t r e s s  be n o t e d  and l e a s t  As the  effect  a  stresses.  boundary  appears  o f known  effect  one o f t h e b o u n d a r y  variation  with  implication  around  o f known  displacements,  consist  has l i t t l e  that  displacements,  The  conditions  the stresses  consisted  here  o f known  on s t r e s s e s .  boundary  on  conditions  i s i n t e r e s t i n g to note  consisting  of  effect  T  =  T C  throughout  the  uniformity, average Since  we  them  the  can  absolute  the  result  through  be  + 2hl  a  of  to  desirable  these  be  -  T  to  compare  to d i v i d e  by  T  +  2hl  but  shear  stress  Figs.  13  ,  as  one be  ; to  T  below  added  T  c  ,  and  dy  )  dx  be  to as  17  However,  results  T  1,  the  to  have  Figs.  the  plots,  the  comparison. value  of  X  This  the  is  depends  trend  plot  of  X  as  a  f u n c t i o n of  elastic  plot  X  as  a  f u n c t i o n of  sample  19,  This  has  i t should  would  not  yield  mixed  boundary  the  is  both  been  be  same  of  them; show  they  to  a  constants.  of  19.  somewhat  In  the  present a  quali-  graphs  of  and  meaningful It  is  for  better  various  looking  i s because implying  different  and  with  a l l isotropic  u,  would  proportions  make  in  detail.  plotting  This  the  stresses  obtaining  sample  exceed  shown  variation  by  that  c o n d i t i o n s ; changes  X  of  proportions,  results.  to  17  in Fig.  remembered  compressibility, lead  patterns.  done  to  time  difficult  stress) is  uniform  d e s i r a b l e to  same  constants.  more  considerable  achieved oh  shear  values  to  in  elastic  materials.  It  13  of  the  added  i t seems  c o n c i s e l y , at  dx  condition will  The  sample  summarized  computer  unity,  unity  dy  -i)  actual  sample.  here.  i t seemed  a  uniformity  any  closer  through  more  The  =  should  having  of  X  the  defined  ( -EX  (complete  by  cases,  leaving  -h  0  through  displacements  of  result  must  different  '+h  condition  value,  Fig.  c  xy  -h  able  this  to  serious  from  as (T  represented  X.  as  the  non-  +h  then  tative  of  m  0  1  ideal  series  adding  departures  expressed  'I  X. -  then  is  o  degree  •  order  The  T  the  a l l departures  sample,  above  'I  1  T  indicate  together  the  values  can  To  add  variation  the  In  sample.  at  materials of  the  variations  stress  48 It reverse such a  a  and  case, the  i n one  curve  sample  will  seems  several  the  which  noted  case  i n F i g . 19  A actually  unlikely,  times  boundary  ends,  be  as  conditions  would  be  extremities  most  uniform  shear  condition, tend zero,  19.  towards and  so  of  i t would  unity the  demand only  the  no  on  In  sample  case  case  with  such  a  friction,  the  be  In  short  on  top  and  confined  to  zones  would  at  a  be  V a r i a t i o n s of Stress Coefficient w i t h P r o p o r t i o n s of Sample.  stress  but  of  curvatures  extreme  i t i s long.  therefore  near  Fig.  an  but  would  the  a maximum.  tall,  A l l deformation and  as  would  bottom. the  has  consider  high  that  as  zero. be  the  curves  This  very  nearly  length  must  would  have  not  be  uniform.  to height maxima.  ratio  the  desired  Thus tends  A  would towards  49 Effect the  of Proportion  proportions  able  t o have  have  passed  the  larger  a length  approach  four.  too  There  long.  although applied bottom  are very  I f there  and b o t t o m  plates,  non-uniform,  shear.  The p r o p o r t i o n s  = 1.778,  described.  were  the entire  most  However,  of the sample  i t i s believed  choice,  as p r o d u c i n g  of  with  reduced  liability  Complete  ment  dealt  soil  tests,  and  with  load,  tions  stress  the sample  before  shear It  need  must  follows  i t is  resist  already  Figs.  12  through of  displacements.  theoretical shear  load.  commencement  shear  developMost  normal  i t se f f e c t .  lateral  load, The  ver-  of shear, i s  strain.  and t h a t  It  the d i r e c -  and h o r i z o n t a l .  the x - d i r e c t i o n  from  c a n no  When  longer  and t h e d i r e c t i o n s o f  not.  t h e above  shear  that  and t h a t  19  uniformity  rotate.  n o t b e t h e maximum test,  the  reasons,  a vertical  are v e r t i c a l  a d i r e c t i o n of p r i n c i p a l stress  maximum  to  a compromise  b u t no  undergoes  i s applied,  less  having  that  includes  strain  of the  against  to non-uniform  involve  field  of p r i n c i p a l stress shear  most  described,  o f an a p p l i e d  applied  by v e r t i c a l  that  horizontal  the  will  sample,  a t t h e t o p and  The f o r e g o i n g  the e f f e c t  usually  accompanied follows  Field.  however,  the complete  tical  be  Stress  the sample  low  ceasing  for-practical  this  The  long  three  proportionately  of the machine  chosen  point i n  d i s t r i b u t i o n c h a n g e s and  vindicate stress  i n making  on f r i c t i o n  change  about  to uniform,  have  of  little  than  fact,  theoretical  i s any s l i p p i n g o f t h e sample  becomes  L/HT  seems  f o r a very  ends  In  the rate  greater  close  depends  the r i g i d  material.  f o r caution  that  desirf o r X to  enough  t h e more  and t h e r e  need  be s e e n  field  surfaces;  effect. top  i s also  considering  i t appears  However,  ratio  In  large  ratio,  the i d e a l .  conditions stress  ratio  to height  to height  It will  apparatus,  f o r any p o s s i b l e  to get l e s s  the length  to Height.  to height  i t s maximum  X c a n be s e e n  making or  of a possible  the length  conditions of  of Length  the x - y  shear  at the beginning  of  50 Relationship During  the  England,  theoretical were  times  uniform  Experimental  development  placements three  Between  the  in  F i g . 15  boundary  i t s height.  sample  simple  calculated  conditions  of  of  length.  confirm  for a  earlier  and  whose  i t was  complete theory.  Theory.  at  centre  throughout  more  and  machines  sample  these,  occur  The  the  shear  stresses  From  should  Measurements  Cambridge, line  length  diswas  concluded the  middle  results Based  that third  presented  on  the  1 3  theoretical set  out  to  measure  stresses. third and  The  there  hence  third So,  results,  i f we  shear  were  the  the  stresses  as  idealized  0  and  T  sand;  third. he  did  volume)  tests  machine  was  normal  on  the  sample Cole  middle also  on  medium he  two  ends  and  of  each  From  the  use  statics  stresses  and  their  of  shear,  the the  T  uniform  stress,  of  the  soil  normal the  middle  being and  tested.  horizontal  stresses  in  extensive testing  and  could  to  third  under  are  included  i t s eccentricity  boundaries.  middle  vertical  " d r a i n e d " and  that  third  determine  zontal  of  made  the  able  work  dense,  and  middle  was  minimum  so  force  (friction)  on  boundary  the  xy  Cole's series  and  Cambridge,  f o r the  f o r the  , these  at  that  b e h a v i o ur  sought,  y  middle  showed  stress-strain  the  workers  displacements  displacements  relationship  define  earlier  results  uniform  the  and  actual  Cambridge  that  was  Cole  the end  "undrained"  loose  separately  and  surface  of  both  detailed to  , f o r the  top  determine  length.  vertical  and,  middle  shear  also and  the  force  separately, bottom  maximum  f o r the  It  normal  the  information obtained,  directions  sample  His  measure  sample  dry  (constant  samples.  the  of  (uniform)  follows  stress,  a  and  that  , and  he  could  the  hori-  portion.  xy  In  the  apparatus  normal  stresses  cannot  argued  that  average  the  be  now  being  measured.  measured  described, However,  stresses  are  horizontal  i t will close  to  C o l e , E . R . L . , "The B e h a v i o u r o f S o i l s i n t h e S i m p l e Apparatus". D i s s e r t a t i o n f o r t h e D e g r e e o f Ph.D., Cambridge, August, 1967.  now  be  the Shear  51 stresses  actually  formity,  that  operating  i s , close  in  to  the  a  and  zone  normal  stress  normal  stress.  tests.  That  measured average  showed  at  the  is  to  average stress  sample  area.  stress  was  bottom  the  average  is  middle  was  close  say,  the  Cole  top  measured  than  to  friction; that  each  was  in  d i r e c t i o n , thus  sample  at  the  boundary  apparatus  being  mid-height.  friction  on  for  achieved, were third  being shear  shear  sides  that  uniformity,  to  unity.  the  .  is  with  this the  in  practice.  T  , is xy  Fig. &  15  shows  dimensionless to  the  give  be  say,  T  x  is  -  force  In  bottom on  to  the  the  about  10  that  net  frictop  height  and  ratio  percent after  approximately  from  the  y  The  longitudinally  and  that  to  in  largely  one.  forecast  force  the  negligible  fact  that  top  apparatus,  by the  the  the  shear  theory, middle  average  c a u s e s no d i f f i c u l t y b e c a u s e , i n a c c o r d a n c e w i t h t h e o r y , we shearing  shear  stationary  split  r e s u l t was  different  the  fixed.  moving  showed  The  on  and  the  top  stresses  by  frictional  length  Cole  measured  third  friction  should  third.  that  equal  average  sample  applied  the  That  the  hence,  friction,  the xy  a  middle  hence,  T  i t on  apparatus),  stress,  of  and,  for  achieved  '  on  to  side  stress,  force  net  assumed  the  from  a  is  result  boundary shear s t r e s s a c t u a l c o n d i t i o n s are straight  relieving  a  ascribed  piece  The  showed  and,  exerted  increasing  (Cambridge  allowing  one  the  divided  was  were  the  for  sides  in  to  d i f f e r e n t from  sides  boundaries,  than  be  vertical  undrained  Cambridge  difference  was  equal  stress  the  bottom  less  The  the to  for  middle  described,  Theory  three  in  average  load,  the  vertical  context,  shear  found  side  two  and  tion.  of  was  as  this  that  average  the  the  close  taken In  However,  friction.  result one  the  to  applied  showed  friction  bottom  be  stress.  also  tests,  especially  can  boundaries.  average side  a l l his  normal  higher  and  in  were  uni-  xy  that  Results  approximate  T  y  Cole  of  since can go  middle  central stress  approximately  zone  is  equal  third  close to  the  xy  applied  shear  strains  i t was  force shown  divided by  both  by  the  theory  sample and  area.  For  shear  p l a s t i c i n e models  52 that,  i n t h e zone  approximate placement  of uniform  uniform  shear  stress-strain  directly,  using  and  the applied  using  area,  and  was  the average  -  y  to the boundary  reached  above, the  c o n d i t i o n s c a n be  now  divided  being  by  of measuring  are .necessarily  accomplished  horizontal  given  shear  stress  could  being  described cannot  t h e Mohr consider  i n terms  of a  -  y  because,  stress  maximum.  tests  as s t a t e d  earlier,  and so t h e r e  circle.  However,  t h e measured  horizontal 2 degrees samples  initial  shear  shown  stresses maximum)  and s u b s e q e n t l y place  stress  by C o l e  becomes  reverse.  magnitude.  than  However, results  the v e r t i c a l ,  to the i n i t i a l  presented  tendency  now  being  by C o l e ,  hence  shear.  f o r undrained i s presented  tests  No  a shear tests noted  n o t be p u r s u e d complete  This  of 1 or  for reversal  The  and h a v i n g  a more  to the  f o r dense  shows  of  dilates,  horizontal  be a p a s s i v e  drained  will  i n F i g . 20.  f o r drained  as t h e sample  causing  described,  the matter  will  with  f o r l o o s e , and  The r e a s o n  but not l a t e r a l l y .  to  interest  at strains  vertically  to the d i l a t a n t  data  ) compares  The r e v e r s a l  i s that  normal  a t 45 d e g r e e s  zero  strains  now  y  that  can expand  t h e work  Maximum  be o f  X  tests  opposite  and xy  horizontal  (T  i n drained  than  , T  Shear.  i t may  shear  greater  stresses  a r e n o t enough  on p l a n e s  at smaller  to a greater  reaction  in  shear  (initial  takes  proceeds  it  I t has been  on s a n d ,  Cole.  n o t be a s c e r t a i n e d i n t h e a p p a r a t u s  be m e a s u r e d  how  and x  o f Maximum  capable  by  normal  13 the  not  y*  Comparison  to  for strain  the h o r i z o n t a l  d e s c r i b e d , was  measurements  Experimental  fix-  obtained  deformation  shearstrain.  stress  dis-  c o - o r d i n a t e s , f o rthe  measured  loads,  stress  no m e t h o d  results  in x  uniform  machine  the d e t a i l e d  There  closely  for stresses.  The of  equal  the conclusions  relationship  of approximately  sample  strain  displacements  of the sample. Summarizing  zone  stress,  pressure,  stress are reported the r e s u l t s further.  resume  of  a series  Cole's of  tests  53 in  which  strain. this a  both  T^^  Also  i n F i g . 20  makes  test.  and  for easier  F i g . 20  dissertation,  as  results  present  are the  were  discussion.  plotted  ratio  comparison  is a plot  Cole's his  T•'  of  plotted  It w i l l  X  y l ^  after  results  i t i s not not  T  be  a  m  functions a  the  of  i n the  first  one  form  that  of  shear  is plotted,  x  taken  copy  seen  as  stages  from of  figures his  desired  of in  figures, for  for undrained  F i g . 20. A f t e r Cole (1967). Comparison o f Maximum S h e a r a n d H o r i z o n t a l Shear.  as  the  54 tests,  the i n i t i a l  strain  o f 0.1  small.  degrees  The r e s u l t  undrained the  test  maximum  Manner  i s that,  largely  and a f t e r  after  shear  tests.  the applied  shear  methods  Shear  In selected,  stress  could  very  i n an  i s approximately  the r e s u l t s  in practical  same  plane  simple of  plane.  the second  method  where  shear  behaviour  i s of  e n g i n e e r i n g problems,  involves  and v a r i a t i o n  a normal of shear  load load  the  on a on t h e  plane.  I t appears shear  as l o g i c a l  stress  a plane  changes  o f maximum  planes).  strains  shear  associated. associated during  with  on l o g i c a l  rapidly  as t o  method  grounds;  trace  i s ,  perhaps,  the measured  i n the d i r e c t i o n  shear,  the other  with  which  hand,  t h e maximum  application  of  (on a s u c c e s s i o n of  are strains  On  which  shear,  the f i r s t  horizontal-vertical  the development  plane,  o f maximum  In fact,  be p r e f e r r e d  shear  to trace  on a g i v e n  becomes  to  of  reasons:  that  usually  are at  o r i t c o u l d be  on a h o r i z o n t a l  t o be p r e s e n t e d ,  It i s considered  given  There  as a f u n c t i o n  stresses)  stress  f o r the following  situation  Results.  be g i v e n  (or principal of shear  interest  b)  i t remains  strain  shear  (at a  of presenting the results  strain  as a f u n c t i o n  a)  that  a small  of Presentation of Experimental  maximum  destroyed,  shear.  two p o s s i b l e  is  i s soon  or l e s s )  on s a n d ,  least  given  shear  some  shear  of normal  load  of the  they a r e of the s t r a i n s  stress,  occur  and a r e u s u a l l y  ignored. c)  The a p p a r a t u s ment In  load  i s greatly  of h o r i z o n t a l  simplified  normal  summary, t h e v e r t i c a l  a r e measured;  hence  stress load  the average  when  i s not  measurerequired.  and h o r i z o n t a l  values  of O  and T  y known. are  The h e i g h t  and h o r i z o n t a l  known and so t h e a v e r a g e  shear  deflection strain  shear are  y  x  of the samples  i s obtained.  55 The shear change  strain  2  of  ±  30  i n volume.  triaxial —  apparatus  degrees.  If  specimen  is e  (l-e^)  .  tan  1  resulting  shear  described  the  the  simple  triaxial  values  of  ±  allow  11  strain  can  be  axial  a  percent of  a  shown  to  strain,  shear  apparatus  20%  be  the  believed  covers  30%  18°50'  9°1"  i s , therefore,  tests.  strain  10%  4°12'  It  allows  compressive  shear  four  also  to  strain i s :  5% 0)  axial  , the For  It  i s designed  the  that range  ±  29°17  30°  strain  included  in  1  in most  56 CHAPTER EXPERIMENTAL  Installing  Samples.  developing  a  sand  set  out  cepts of  to  placed  evident  Disturbed  the  are  any  samples  for  .005"  without space  undue  of  length. when  It  than  the  a  thus  possible  seems  and  this  brief are  either must  can  be  to  very  a  the  of  is  the  con-  two  types  box  and  or  in  the  be  of  further box.  It  type ( i ) .  dimension  box  get  (after  them  about close  0.25 to  to. f o r m  method  been  approxiallowing  into  expansion  preferable has  date,  of  the  be  type.  Subsequent strain  to  strain  must  have  of  may  basically  in situ  spent  especially  review  of  sample  was  gained  outside  minimum  width  time  developed  formed  i n order  linear  samples  A  formed  of  disturbance.  such  a  samples  thickness),  represents  length  be  as  There  undisturbed  could  less  membrane  with  ( i i ) samples  that  experience  E.  ( i ) samples  of  installation,  procedure  here.  in position and  the  i n Appendix  Pre-formed mately  that  i s given  they  disturbance is  others,  involved  f o r sample  order  in detail  sample;  then  In  PROCEDURE  c o n s i d e r a b l e amount  technique  samples.  available  A  IV  place  to  f i l l  the  percent.  the  actual  samples  adopted  in  for  The box  situ  sand  samples. In  situ  Samples  (for  screws)  with  the  little the  water by  punched  bottom  assembly  drainage  wide  open,  and  means  under  hole  and  in  full  The  space  of  a  and  i s then  then  tube.  the  is  the  then  top of  for  mounted  the  the  correct  top  of  height,  allowed  to  i t s base  in  membrane  sample  sample  the  on  i t ( F i g . 21). the is  block, This  machine  and  stretched  with  i s placed  the the  holes  ( F i g . 22).  is filled  s a t u r a t e d sand  With  a l l necessary  in position  of  the  has  inside  placed  size  membrane  membrane  plate  the  de-aired  at  The  friction  the  water,  located  (Sand).  sand, top  close  de-aired  under  level  friction  over  i t .  water  and plate The  cap  is  Fig.  Fig.  and  loading  de-airing,  head  21.  22.  Membrane  on  Membrane i n Machine Receive Sample.  are then  loading,  Ready,  assembled  e t c . , can  over  commence.  Mounting  Ready  To  the sample  and t h e  58 In ficulty a)  were  the above as  The  s t r e t c h i n g o f t h e membrane  from  tends their  retained  placed  to p u l l  points  of  dif-  position.  inserted  in special  Locating  adds  in  samples.  situ  inside  lengths  described  of neoprene  placing  of care  the sand.  as p o s s i b l e ,  must  be u s e d  leave  a fairly  preparing  in as  amount o f  and t h e  surface.  adjustment  c a n be  but  must  a minimum,  to avoid  uni-  material  density)  level  then  in their  I t h a s t o be  surface be  and  i n Chapter I I .  of  pouring, this  rteoprene  i s required  the c o r r e c t  are  of  moulding  ( t o get the d e s i r e d  must  away  corners  t h e membrane  grooves,  degree  form  process  short  to the complications  considerable  actually  corners  The v e r t i c a l  the lengths  grooves  i n the sample  the v e r t i c a l  i n p o s i t i o n by  moulding,  A  the chief  follows:  space  b)  procedure,  After  carried out, sample  dis-  turbance. c)  After  the sample  matter  to place  face  and c l o s e  must  be done  to  avoid  place. the is  plate  likely  above  able  puncturing  exercise  plate.  i t and hence  proper  corners  o f t h e membrane  of the upper will  be  grains  plate. likely  into by  on  the  top of the  of the  loading  and  prob-  It is essential t h e membrane  that  actually  when  the moulded  seat  I f they to form  snap  onto  to l e a k i n g ,  to ensure  also  t h e sample but  grains  over  and  as s n a p p i n g  seating  lead  control  i t , i n order  membrane  sand  o f t h e membrane.  closing  ners  such  sur-  Everything  disturbance  Sand  proper  i t .  disturb  t o wash  prevent  head  to  not only  the sand  o f membrane  movement,  will  of the s t e e l  steel  to avoid  i t i s a delicate  on  over  any p o r t i o n  sudden  extremely  plate  t h e membrane  letting Any  placed,  the upper  gently  membrane,  top  has been  on  the  cor-  do n o t , t h e  a fold  and, a l s o ,  59 the  plate  corners  a r e more  likely  to cause  a  puncture. Undisturbed form  even  remoulded  described After in  Samples.  now  Vacuum  to  stretch  to  the correct  and  place  size,  inside  i s then  corner  membrane hold  because  The  would  material  ensuring machine  i s now  have  on i t s b a s e  of t h e sample part that  soft.  This  this  i s now care  released  T h e memover  one  corners  of the  essential  for in situ des-  i s n o t s o much as b e c a u s e  volume  and y i e l d i n g at the  type  of  at the  mid-sides.  deformation.  lubricated  and  not to scrape  are placed  into  the the box.  in position  i s not pinched. ready  lowered  i t was  as i t i s l o w e r e d  be p u t t o g e t h e r  i s cut  of the type  and y i e l d i n g  of t h e box s i d e s  can then  Although  constant  taking  sides  lifting  the moulded  are strong, held  be  put i n place.  f o r samples  block,  t h e membrane  o f t h e membrane  i s then  o f t h e membrane  resisted  i n the machine,  upper  that  involve bulging  so f a r t e s t e d  sample,  corners  corners  s i d e s and  The sample  The vacuum  corners.  are very  t h e membrane  installed  The  they  porous  the top p l a t e ,  i s not necessary  the sample  corners Clays  this  the machine.  a j i g and c a n e a s i l y  and e n s u r i n g  the v e r t i c a l  unless  inside  over  procedure  i t on i t s b l o c k ,  to the sides  stone  to  ( F i g . 21) t h e s u b -  i n . square.  and p o r o u s  f i t to the p l a t e  samples, cribed  2-1/2  outside  box w i t h  applied  using  closed  a t a time  samples  t h e membrane.  the top plate  brane  to  i t to about  formed  difficult  and t h e  and m o u n t i n g  i n a square  c a n be  be v e r y  in situ  a l l samples  for in situ  i s mounted  top.  into  as  probably  samples  t h e membrane,  t h e same way  open  would  clay  covers  preparing  assembly  It  The r e s t  t o commence  while  of the testing  procedure. Loading. the  Application  l o a d i n g head  operations sand while  which  samples,  of normal  load,  and a p p l i c a t i o n must  flushing  of back  pressure  be  combined  in a logical  the height  locating  p i n may  a i r i s flushed  away  and b a c k  a i r out of  be  pressure  are three  order.  left  With  i n place  applied.  Then,  60 if  normal  load  balanced, normal It not  the  load  is  is pin  may  may  be  important  be  added  resting  be  with  the  some  the  removed  added  that on  until  and  the  loading  odd  back  pressure  the  water head  grain  of  is  intended supply  just effective  valve  open.  and  upper  plate  sand,  above  the  should general  level. Chapter  III  discussed  sufficient  friction  at  to  the  s l i p p i n g on  prevent  ing  on  top  and  bottom  surfaces  sand  grains  to  or  the  soil  soil  described,  one  being  the  adequate  because  during  steel  became  strain  was  most of  boundary  used  for  after  on  friction  was  steel  plates  fairly  face,  over  after  clay.  strength and  the  was ribs  under off.  because  after  many  The  away  from  in  with  plates  With  this  of  the  there the  was  of  with  were  clay.  sought;  .05  The  there cycles  was of  a  For  mm.  were  plates no  sign  strain.  polishof  to  of  adequate  sands, of  at  on  their  30  degrees  centres, to  develop  polishing found  some-  hardened  ground  mm.  was  plates  that  at  appeared  of  deformed  pair  ribs  and  of  testing  2  the  ends.  absence  sides  It  the  the  full  sand-  intensity  with  of  of  samples  second  high,  the  re-sandblasted  suggest  transverse  ridges  These  to  evident,  mid-half  s a t i s f a c t o r y appearance  a l l combined  not  III.  material,  adhesion  paper  hardened  load,  to  the  machine  i n which  pattern  Chapter  did  the  tests  at  the  became  cyclic  Depend-  emery  In but  of  several  of  made w i t h  edges.  surface  coincides  pair  This  approximately  dying  pattern  vertical.  sharp  friction  the  develop  p o s i t i v e was  The  the  and  developed  very  from  sands.  sand,  polished  removal,  thing  faces.  to  same  the  testing  samples  on  for  f r i c t i o n , "calculated  polishing,  gluing  sandblasted  friction  testing  by  was  plates,  The  provided  boundaries,  friction  plates  was  This  be  necessary  developing  plates.  of  intense  the  the  boundary  sandblasting.  allowed  ing  the  of  sample  or  scratched  surface  length  may  bottom  plates  tests  blasted was  tested,  and  necessity  the  set  develop  top  the  of  with adequate  their  that  sur-  with  61 light  normal  supported contact  loads,  on  its ribs,  with  required.  the  This  plate  the  order  In  of  2  to  With the  available  fore,  than  locating  the  pin  would  drive  start  swelling.  with  water  pressure desired valve  at  and  be  closed  to  used  is  head  is  and  to  reached. full  and  permits  a  may  be  no  further  opened  to  better  check  change  height  back  pressure,  Shear  load  could  of  the of  0.1  height pressure clay  air  rapidly  both  the  drainage  allowing  This  When  dial  the  the  consolidation  events.  and  to  back  until  applied.  vertical  in.  the  apply  this,  a l l  There-  back  the  before on  of  exhaust  about  other,  load  the  to  out  then  occur  grain  loads  could  allow  After  complete, in  shape  vertical  each  normal  is  used.  and  and  balance  starts  indicate  the  into  and  seating  cutting  flush  pressure  load  and  application  to  load  consolidating.  upward,  better  strain  are  size  be  surface  density  samples  Thus  and  lateral  strains  size  size.  slight  lateral  the  were  require  causes  so  the  travel while  be  pressure  sq.  standard  minimal  normal  back  can  a  on  on  might  seating  cm.  per  loading It  and  a  kg.  nominal  the  reason  depends  plate  i t s main  described,  likely  cannot  upper  experiments  clay,  are  load  of  bringing  this  sand  vertical  samples  higher  For  the  4  type  without  seating of  ribs.  ribbed  sand.  characteristics  the  the  the  gauge  will  drainage  valve  consolidation  to  start.  method has  has  been  applied  to  load  be  in  either  direction.  pulling run.  A  the  of  a  cycle  ends  of  on  loads  desired  round  three  the  are  ten  and  this  adopted  loads  to  d i r e c t i o n may i f a  cycles  per  by One  other  test  cyclic  from  about  second.  setting tank two  be be  static for  variation  tanks. the  shear  provided  produced  supply  but  device  hand,  speed  about  piston  by  box  are  The  allowing  The  gear  to  controlled  applied.  continuous  desired  pressures  both  and  of  strain  piston,  valves  motor  capable  The  to  been  acting  one-twentieth  mon  yet,  double  tests,  required  as  a  selected, is  not,  be  the is  tanks  comsupply  62 one  to  tank  each  and  equal  end.  n i l in  the  alternating  unequal  loads.  ton  be  can  ference  other  loads,  of  two, while  three  the  any  between  pattern  putting a  With  either  between  interposed cyclic  By  the  tanks. piston  loading.  pressure  in  i t i s p o s s i b l e to back  tanks,  pressure  two  given  pressure  the  i n any The and  total one  tanks,  load or  common  guarantee  i n one  tank  rotating  the  on  makes the  the  pis-  dif-  valves,  allow  almost  any  63 CHAPTER SOME  The  Test  Programme.  programme whether usual  of  the  test  programme  (a  I t seems  obtained  should lead  Seed .  should  other  to a p p r e c i a b l e  has  For  9  shear  undrained  already  been  loading  the present  with  triaxial  saturated Because  was  machine,  are d i f f e r e n t words,  the  initial  establish  from  those  initial  the d i f f e r e n c e s  differences compared  on  the  saturated  test  in  in soil  with  of  test  behaviour.  triaxial  sands  by  testing  Peacock  3  simple  to  that  d i s c o v e r whether  = o* ) f o r c y c l i c  and  In  RESULTS  reasonable  f o r a novel  results  shear  2  EXPERIMENTAL  procedures.  conditions Simple  tests  V  be  capable  designed  of to  work,  i t was  testing  decided  under  static  to  compare  loading  of  sand. the machine  dynamic include  described  testing, tests  of  the  h e r e i n was  initial  intended  test  liquefaction  programme  under  cyclic  loading. During ary  conditions  conditions  testing,  assumed  No  slipping  boundary b)  Zero  c)  Rigid The  investigated was  poses  and  to  the  discussed  i n Chapter  III.  on  and  boundThese  this  of  on  sample  the  sample  than  friction  initial  the  top  bottom  ends.  test  on  the  that  had  of metal  sides  of water  or  negligible  achieved boundary  within  sample  programme,  considerable without  rigidity  were  the  boundaries.  already  machine  boundaries  of  friction  i n the  far higher  finished  given  plates.  effect  The is  was  included:  a)  testing  and  attention  a  as  this  and soil  limits  c o n d i t i o n was  not  extra  not  Thus  bearings The  rigid  practical  examined  of  variable.  skeleton.  of  was  the volume  needle  "backlash". the  ends  pur-  further.  64 With planned  t h e above  points  i n mind,  a programme  was  to:  a)  Establish  test  procedures  f o r the simple  shear  machine. b)  Perform (i)  undrained  Compare and  (ii)  Run  Compare  the r e s u l t s  for  material  of a d e s c r i p t i v e t h e same  The fall  allowing  de-aired  height.  Both  density under  = 93  no  cu-  comparable should  be  load, ratio  this  with  was  remarked  applying a large  under  drained  testing,  specifically were  with  by  others  by  letting  dry  a low  from  ratio  volume  by  o f 0.78 ( D r y  was was  vibrated reached.  = 107.5 void  The  l b .per  ratio,  t o be  I n p a s s i n g , however, i t densities  and a d e q u a t e  In f a c t , 0.47  to achieve  controlled  of  widely  water  t h e minimum  greater  of about  attempting  performed  strain.  ratio  being  checked  void  sand,  was  the r e s u l t  (Dry density  load  Ottawa  here.  A d r y sample  as  much  normal  using  a minimum  be u s e d  found  and  a steady  0.54  conditions.  a void  was  through  practice.  that  lique-  material  require  presented  a maximum  taken  normal  general  to  was  This  can r e a d i l y  ratio  until was  lead  plates,  a  of r e s u l t s  Furthermore,  to f a l l gave  would  i t would  l b . per cu. f t . ) .  void  f t . ) and  void  sand  to o b t a i n  C109.  a low h e i g h t  methods  normal  resulting  known,  the r e s u l t s  i n a i r from  machine  triaxial  ribbed  of the tests  Standard  nature.  maximum  shear  plates.  material  with  using  a comparison  f o r most  well  of top  undrained  tests  and  comparison  sand  shear  faction  as, being  available,  shear  of c o n d i t i o n s which  and r i b b e d  types  plates),  from  picture  give  two  and p l a i n  simple  simple  d e s c r i b e d i n A.S.T.M.  tests  by  with  from  i n the simple  plates  cyclic  The  selected  plates  (Ribbed  plain  as  to  the r e s u l t s  bottom  tests c)  tests  load  c a n be  cyclic  i n the course was  reached,  high  density.  rather  than  of  achieved  shear other  without A l l tests controlled  65 In tests as  were  also  a trial  they  a d d i t i o n to the tests performed  o f methods  are described All  on t h e p o r e  test  chosen  anticipated is  high  little under  enough  cases, from  In  i s not the angle  adopted for  with  this  used  could  obtain  stress  obliquity,  tests  tests,  should  without simple  will  tests  from  different  symbol,  usually the  applied  ratio  stress  9.  an  apparatus  that  simple  shear  the simple  of primes  stress /  (a  -  and  shear quantities,  angle  that  for  i t  be d e n o t e d  t o G.  of  the  as t h e s t r e s s  To e m p h a s i z e will  To  assumption.  be d e f i n e d  shear  i s t a n 0 a n d t a n 0' = x  reason  the angle  The f r i c t i o n  be a p p l i e d  simple  <j), u s u a l l y  of the measured  The n o t a t i o n s  of applied  angle  The b a s i c  and hence  <j) , t h e a n g l e  t o <f> w i l l  pressure).  I I I , the  making  i n terms  different  change  obliquity,  t o compare  therefore  the usual  back  In a l l  on a h o r i z o n t a l p l a n e .  planes,  on a h o r i z o n t a l p l a n e .  tests  has been  represent  i n Chapter  obliquity  of  i s of  readings.  testing.  relationship. will  the  pressure  the f r i c t i o n  stress  but i t i s considered  any assumed shear  results,  b e made  given  over  pressure  against  involves  be p r e s e n t e d  and  back  f o r any  comparison  pressure  stresses  on o t h e r  done  run with  the back  pressure  to t r i a x i a l  measure  an a s s u m p t i o n  triaxial  test  as e x p l a i n e d  the s t r e s s e s  maximum Such  only  pore  o f maximum  reference  i s that,  easy  (drainage  presenting  were  i t s actual value  the back  of pore  conditions  were  Providing  pressures,  values  a few  and p r o c e d u r e ,  pressure  cavitation,  a l l reported  the given  These  use of the transducer  So, t o a l l o w  back  from  initial  used  to prevent  different  The back  of pressures.  importance.  subtracted  fluid.  sand,  F.  t o be d e s c r i b e d  to optimize  range  clay.  preparation  i n Appendix  pressure was  on a l o c a l  of test  the tests  on O t t a w a  is  by a  and s u b s c r i p t s  In other  to a p p l i e d  words normal  u) .  y  To the than  assumption that  compare will  simple  b e made  accompanying  6^ .) aa  shear that  and t r i a x i a l  after  i n simple  a small shear  test  results,  strain  x ^  =  (less  T ^ .  66 This  assumption  Chapter is  <b = t a n 6.  beginning 9.  with  simple  max  maximum shear,  test  shear  tion  ( i . e . planes  i s shown  triaxial  obliquity  sloping  i n this  n o t be  true  are given  circle  case  at the i n terms of  t h e r e f o r e , f o l l o w the  on one p l a n e ,  tests  stress  since,  results  stress.  To make will  that them  on p l a n e s  comparable i n terms  o f maximum This  i n F i g . 23.  SIMPLE  i s the plane  be r e p o r t e d  a t 45 d e g r e e s ) .  diagrammatically  TRIAXIAL  would  results,  obliquity  s t r e s s e s and s t r e s s  stress  this  , described i n  Mohr  *  and hence  shear  by C o l e  , the complete  be d e t e r m i n e d ,  However,  and s t r e s s  ultimate  on t e s t i n g  = T xy  of a test  The s i m p l e  stresses  of  If T  III.  k n o w n a n d (b c o u l d  sin  of  i s based  SHEAR  Fig. 23. A v e r a g e S t r e s s e s Used I n Presenting Experimental Results.  shear  representa-  67 Undrained with  and  Static without  controlled middle in  Chapter  of  IV  the  comparison  and  4.5  ranges in  kg. of  terms  common  functions  of  one  that  or  this  increase  cm.,  proper  friction  to  are  developed  degrees  lower  when  and  plain  more  at  or  which  and  are  upper  presented  angle  24b.  plates  0'  2.0  and  as  24b.).  and  less  for  were  middle  Fig.  the  plain  used  results  24a.  °  and  chosen  friction  in  described  were  the  The  ( F i g . 24a.  value  in  testing.  in Figs.  the  slip,  results  sand  plates,  noted  beyond  Ottawa  pressures  these  remains  on  the  stress-  Ribbed  be  difference  compare  friction,  done  liable  and  strain  will  were  normal  as  To  bottom  develop  The  stress  shear  two  to  laboratory  shear  It is  sq.  Sand.  density.  found  tests.  per  of  used been  and  tests  relative  had  Ottawa  top  shear  were  which  on  proper  simple  range  plates,  Tests  are  that  9'  used  constant  and  as  i s reached.  max  strains Below  max  strains  about  appears  to  2  decrease On  be  noted  tion  occurs  at  at  There  extending  over  shear  resulting shear  A  of  A  of  1  or  stress,  may  test,  shear  causes  with  shear  stress  ing  shear  strains,  the  s t r e n g t h reduces  as  strain, shear  exceeds pore  stress  the  further,  a  result,  deformation  considerable  strain,  the  pore  10  an  is  follows.  Soon from  pore  of  an  under to  out  accelerates begins  the  strain  pressure  stage  continues  The  reduces  resistance  this  after  the  increase.  stress  the  pressure  or  gap  the  magnitude  leaving  deforma-  of  s t r e n g t h and,  pressure  8  will  plotted,  shear at  large  i t  points  normal  but  a  resulting  given  plate  stress,  f a r as  pressure  where  of  i n the  as  strain  a  i s reached  of  shear  cause  pore  type  increases.  2 degrees  develop  m o b i l i z e d by  increment  of  going  The  the  stress  curves  which  point  increasing  As  the  range.  the  force.  normal  of  c o n s i d e r a b l e gap  stress,  resistance  small  a  effect  reduction in effective  increment. still  this  condition  beginning  applied  strains  is a  the  the  constant  degrees.  the  as  examining  that  unstable  degrees,  is  not.  means  that  the  result-  rise. of  Thus  balance  until, to  is  at  drop  shear a  again;  68  Fig.  24a.  S t a t i c Undrained Simple Shear. Ottawa Sand. N o r m a l l o a d 2.0 k g . p e r s q . cm. C o m p a r i s o n o f Two T y p e s o f T o p a n d B o t t o m P l a t e s .  69  Fig.  24b.  S t a t i c Undrained simple Shear. Ottawa Sand. N o r m a l L o a d 4.5 k g . p e r s q . cm. C o m p a r i s o n o f Two T y p e s o f T o p a n d B o t t o m Plates.  70 shear  resistance  established.  then  increases  The w h o l e  a f r a c t i o n of a second.  tan  remained  stress  approximately  increased  and p o r e  deformation  just  and  termed  will  be  In and  ribbed  slightly 2.0  In  plates,  ately  after  strength  strain  that  after  higher  test,  was  19  general,  of  were  by  ribbed  absence However,  ideal;  sample  surface  ultimately  shear  plain  when  tests  with  this  of  large  that  t o be  the top p l a t e type  now  undue  shear  plates  as h a v e  friction strength  types  of  the p l a i n developed by t h e of t e s t s .  have  into  prove  been of the place,  and  preferable.  between  static  t h e same  already  in  s t r a i n s , the  disturbance  tests,  show  the true  judging  may  noted  that  a series  to the comparison  plates  show  adequate  i s lowered  of surface  and s i m p l e  ribbed  after  strength be  the f r i c t i o n  ribbed  cause  of  t a n 9'.  at small  adequate,  that  yield  again  p a r t i c u l a r case,  of the plates  other  tests  immedi-  re-mobilize  f o r t h e two  In a l l cases,  may  to  as h i g h e r  stress  i n this  strength  plates  to i n d i c a t e  t h e same  the t e s t .  and i t w i l l  d i d not mobilize  normal  at  s t r e s s , the  of increase  these  plain  In the t e s t at  began  the ribbed as w e l l  occurs  i s because  plates  failed  appears  triaxial  shear  later  r e s i s t e d by  greater  the rate  strain  that  Turning undrained  with  the ribs  some  The r e g i o n  throughout  and t h e sample  i t i s not claimed  proved  the applied  "yield"  plates.  show  high  of abrasion  that  plates  adequate.  plates  seen  plain  approximately  plates  occurs  d e f o r m a t i o n s , '.  to again  of s t r e s s  plates  plates  With  i t appears  referred  ribbed  results  plate;  be  e f f e c t i v e normal  However,  the p l a i n  were  increment  decreased.  with  i s considered  the sand.  while  plain  resistance,  the recorded  results  with  degrees  It  be  initial  less  shearing  constant,  the values  smaller  was  the large  but apparently  earlier.  with  and  stress  yield,  deformation  After  will  i twill  i s greater  the other  strain  i sre-  "yield".  k g . p e r s q . cm.  strength  large  pressure  described  comparing  lower  stability  of t h i s  within 9'  until  been  simple quoted,  71 were  used.  tests  were  confining the  In order done  angle  with  pressures  simple  shear  8 has  been  and  a r e shown  they  equal  max  terms  will  be  shear  and  used  were  checked,  tan  8',  sq.  cm.  more  shown  the  but  range cent  o f 33  does  stress  values.  tests,  varies degrees  not  density. appear  on  The  strength  strength. transducers  pore The  of  pressure shear  oscillograph,  manometer,  reading  i n terms  functions of  o f 2.0  and  4.5  of shear  kg. per  respectively. 8'  i s higher  46  8'  after  f o r , as w i l l  the simple  1/2°  be  On  shear shear,  t o 39  on  test  density  percent.  difference  to depend  f o r simple  shear  o f 26  percent  stresses overlap  and  the  as  pressures  In simple  The  shear  results,  pressure  show  f o r 46  given  6  i n terms  electronic  a mercury  of the simple  more.  but  stresses  strength  in relative  and  results,  recorders.  the simple  tests  friction  results,  stress  especially  However,  o f 52  by  that  to v a r i a t i o n  conditions  normal  seen  the  o f (b c a n b e  t o as  tests,  recorded  25b  s t r e s s e s of  effective  and  normal  pore  and  densities  relative  shear  tive  25a  densities  8'  shear  piston.  and  triaxial  relative  less  referred  possible,  c a n be  sensitive  relative  be  for initial  "yielded".  The  will  fed to chart  stress  i n Figs.  axial.  t a n 6'  although  initial  results.  or  strain,  were  for triaxial  tests  , values  8'  i n the load  It than  normal  earlier,  the r e l e v a n t  the simple  when  shear  strain,are  of  t o as  increments,  pressure  max'  triaxial  and w i t h  o f <p t o p r e s e n t  i n the t e s t  to measure  the  described  a t 9'  xy  stress  the outputs  stress  As  instead  of the r a t i o  For were  ratios  d i s c u s s i n g the experimental  referred  applied  comparable,  to the i n i t i a l  to x  M  In in  equal  used  T  tests  t h e same v o i d  tests.  assuming 6  to render  within  shear  seems f a r than  and  25  the  tri-  1/4° f o r  the other  hand,  specimens  vary  8'  from  varies  degrees  i n 8'  f o r 50  f o r the  the d i f f e r e n c e seen  shear  presently,  the s t a t e d  in the  ranges  a  per-  simple initial effecof  0'  72  Fig.  25a. S t a t i c Undrained Shear. Ottawa Sand. I n i t i a l P r e s s u r e 2.0 k g . p e r s q . cm. C o m p a r i s o n o f S i m p l e S h e a r and T r i a x i a l Results.  73  0  5  10  Shear  Fig.  15  20  Degrees  Strain  25b. S t a t i c Undrained Shear. Ottawa Sand. I n i t i a l P r e s s u r e 4.5 k g . p e r s q . cm. C o m p a r i s o n o f S i m p l e S h e a r and T r i a x i a l Results.  74 Considering strains  higher  than  axial  specimens  shear  specimens.  lower  9'  axial  test.  are  At  and  ing  values  as  which  shear  higher  and  than  axial  stress 9'  so  in  the  cussion,  this  lower  9'  as  be  is  to  than  shear  the  for  specimen,  lead-  However,  far higher  normal  stress  mobilized,  strain  becomes  the  tri-  were  effective  greater,  and  the  shear.  develops  actual  tri-  simple  pressures  simple  shear  strength  pressure  triaxial  as  at  of  the  higher  pore  the  pore  that far  strength  is  test. that  within  "yield"  shows  pore  resisted,  strengths  that  high  the  noted  minimum  in  simple  can  be  stress  the  strains,  triaxial  will  show  lower  reduce  corresponding  shear  by  nearly  although  stress  degrees,  small  strength  It roughly  2  results  i s reached,  little  less  very  of  applied  considerably higher  The  shear  "yield"  pressures  were  or  accompanied  higher to  1  the  a  in  minimum  slope  the  simple slope.  will  also  tests,  a  be  range shear, For  of the  ease  referred  strains tri-  of  to  dis-  as  "yield". In pressure occurs  yield  that  the  results test in  of  the  much  this  pore  peak  during  i t was  a  concluded  definite  shear range  tests. of  to  stress  to  10  was  such  a  peak  may  not  to  small.  The  gap  in  the  simple  possibility have  been  triaxial  shear of  a  tests  the  degrees  occur the  had  not  peak  at  and  line  test,  "yield"  overlooked.  in  the  of  in  in  a  thus not  triaxial in  yield  triaxial  a  (1  or  likely is  less;  comparison,  shear  2  triaxial  points is  a  Also,  i t is  strain  yield  involved  the  reliable  as  at  shear).  i f i t does  run  the  controlled  test  in  dur-  proved  force  simple  each  values be  strain  pore  i t s peak  From  shear  of  tests,  phenomenon  yield  been  that  yield.  that  yield  measured  test;  the  The  strain  7  i t could  reduction  record  triaxial  balanced,  occurred  shear  in  be  pressure  no  smaller  to  the  and  compared  be  In  clearly  obtained  degrees peak  quite  stage. had  continuous  be  show  smaller  yield  not  obtained,  simple  showing  pressure  could  would  shear  obtained,  during  reading ing  was  simple  stress  i f the might  75 The sudden gives den  are  occurrence a  true  application,  The  to imply  yield  test  i t could would  strains  type  shear imply  have  the test  sud-  given  i n the t r i a x i a l  in a typical  a possible  involves  I f the simple  phenomenon,  failure  and h e n c e  in practice  strains.  the t r i a x i a l  warning.  too small  "yield"  of this  of which  o r no  of  of large  picture  failures  little  importance  test  engineering  of f a i l u r e  could  be  overlooked. A ing in  comparison  i s shown Figs.  paths  i n F i g . 26.  25a and  have  of the simple  been  25b  The  same  a r e shown  shear  tests  again  and  triaxial  test-  as were i l l u s t r a t e d  but e f f e c t i v e  stress  plotted.  T  Fig.  26.  As plane  Effective  before,  o f maximum  shear  Stress  maximum have  Paths  shear  been  for Static  and n o r m a l  plotted  Tests.  stress  on t h e  f o r the t r i a x i a l  and  76 are  compared  plane  for  with  simple It  of  8'  is  ditions tive  a  plane  show  increase  of  a l l is  paring  the  higher  0'.  two  to  exceeded noted the  of  the  be  shear  normal  pressures tion  of  stress that  to  in  the  Figs.  is  offered  in  the  i s , as only  early  tests a)  more  widely.  for  the  difference  in  initial  difference  of  initial  rapid  yet, a  could  will  is  be also  be  directions  of  the  sufficient  possible of  the  the  tests  of  from appears  stress  of  concluded  would  depend been  on  top  be the  shown  and  the  shear the  application  due  to  pore  the  rota-  releasing  the  It  is  considered  to  prove  of  the  not  pre-  this,  it  difference  paths.  undrained  static  that:  tests  material that  be  to  higher  stress  bottom  "adequate  part,  of  test.  explanation  Insufficient  what  then  effective  simple  during  evidence  results  i t was  up  the  in  stress,  important.  has  shear  e  principal  development  shear  part  due,  set  the  it  range  directions  s t i f f n e s s of  F r i c t i o n at  will  slightly  soon  shear  between  possibly  shear  early  as  is  maximum  sand,  say  m  initial  Com-  the  '- P^  It  strain  a  within s :  effec-  densest  has  n  con-  either.  might  simple  The  portions  on  *  difference  pre-strain,  Summarizing shear  that, S^-lt  of  the  one  in  simple  the  is  the  plane  than  different  25)  and  plane  0'  denser  value  this  but  whether  when  two  value  shear.  load.  during  there  high  greater  the  cm.  lower  indication  the  simple  (see  sq.  the  tendency  This  pre-stress  the  tests  varied  to  The  per  a  paths  specimens  kg.  Thus,  with  stress  due  triaxial.  triaxial  a  and  of  density,  is  loading  horizontal  and  s t r a i n , but  be  the  r e l a t i v e density or  each  with  there  to  on  ultimate  general  tests.  in  6'  that  likely  stress  the  3.0  i f densities  triaxial  of  of  is  used,  plane  normal  test  triaxial  The  densities  that  strain  stress  sample  due  appears  function  are  and  shear.  normal  tests  of  shear  boundaries have  been  friction", being  special  as  tested.  attention  is done  to  this However, must  be  77 given cant In  to the p l a t e slipping  terms  shear than 30  and l o s s  of e f f e c t i v e  machine  shows  the t r i a x i a l  t o 37  degrees  ((p'  30  sumably  results  types  of plane  increase  In  terms  simple ness  interesting  shear The  than  sure as  to note  during  shear.  may  be  unlocks  normal  On  the other  this  with  other  the machine  would  be  expected  force,  initial  the  testing.  i s much  tests  i n simple  of Monterey  sand.  greater i n  and t h e h i g h e r  pore  of i n s t a b i l i t y ,  of the p r i n c i p a l  pore  Itis  cyclic  strength  samples  "yield"  stiff-  i n c r e a s e to  i n undrained lower  the  axes  introduced  of  during  pres-  stress  application  load. hand  i t i s possible that  an i n h e r e n t l y u n s t a b l e  increasing In  in triaxial  this  How-  i s because  rapidly  t h e p r e - s t r e s s s e t up  of  resents  greater This  that  i s pre-  of the specimen.  shear  the r e s u l t s  com-  strain.  because  of  degrees  comparable  and t h e y  shows  found  Both  the r o t a t i o n  degrees),  of plane  tests,  in triaxial  strain  simple  than  and S e e d  9' v a l u e s  of the i n c r e a s e  strength.  9  simple  r e s i s t a n c e to shear  i n simple  values  Peacock  Some  machine  but lower  the  f o r 9 ' o f 26  the s t i f f n e s s  shear  greater  t o 53  restraints,  of total  pressures  37  values  strain  occur. -  shearing resistance  are not r e a l l y  extra  to  ratio  signifi-  may  I t develops  to the e f f e c t  ever,  imposes  greater  test.  degrees). due  stress  (tp =  to t r i a x i a l  Otherwise,  of strength  1  pared =  surfaces.  pore  case  pressures  i t would  seem  vindication  of the simple  to  triaxial  testing  is  reached  several  narrow  remains  only  zones,  small.  condition  at c e r t a i n t o be shear  allows  would  (due t o  densities).  machine when  large  be  compared instability  strains  but the average This  rep-  a considerable  as, presumably  the t r i a x i a l  yield  in  on  strain accordance  78 with new In bered  that  Cyclic ing  Shear  shear  liquefaction.  water sand If  the  the  water  effective  on  was  pore  rises  zero  pressure  confining  may  i t should  show  The  load  shear rises  the  pressure  and  the  reduced  shear  strength  cyclic  shear  stress,  (i.e.  sudden  will  occur.  large  shear  pore  pressure, will  be  even  though,  a  strength.  unlike  In  pore  pressure  some  lower  strain  this  cyclic  cycles  per  run  a higher  stress The  beams  lograph record shown  there  are  the  of  a  i n F i g . 27.  It is a  per  to  the  sudden  onset  is  of  "liquefaction",  not  matter  confining that  rising  a  pore  tests  a l l run  tests  two  paper be  a  intensity  b u i l d - u p of large  to  increase  in  at  the  in  and  the  oscil-  Such  shear  in alternate  strains  the  a  liquefaction,  cycling  pore  to  which  that  reproduced. to  two  intervals.  such  speeds  about required  tests,  are  leading  shows  whether or  was  minute  used,  can  test,  paper  static  at  shear  pressure.  were  f o r the  retain  pressure  sudden  oscillograph  equal  as  i t does  the  figure  the  herein  by  still  at h i g h e r  cyclic  shows  sand  alternating  may  oscillograph  of  large  sand  typically  This  sec.)  also  than  cyclic  typical  the  rising  to  bright  r e c o r d s of  cycles  tions.  of  of  rises  on  equal  were  pressure  i s brought  the  speed  pore  sand  strains)  shear  and  increments  mechanics  light  (2  second  the  the  the  failure  value, provided only  The  If  of  liquid.  where  liquid,  to  the  condition,  definition,  rises  pore  a  referred  is attributable  at  Such  pressure,  strength  v a l u e where  strains  the  equal  to  shear  If  undrained  to  not  applied  an  sufficiently  but  the  to  of  a  reduced,  than  perform-  as  be  less  of  behave  shear  zero.  results.  phenomenon  rise.  will a  to  not  the  the  confining  strength but  the  remem-  different  i s applied  p r e s s u r e may  equal  for  be  intention  investigate  shear  water to  points,  Sand.  to  claimed  I.  sands  Ottawa  cyclic  exhibit  these  denser  tests  As  pressure will  or  advantages,  i n Chapter  considering  Tests  sample,  anticipated  machine  looser  cyclic  sand  the  pressure,  when  the  is  load direcuntil  pore  pressure  79  UNDRAINED  CYCLIC  SHEAR TEST  OTTAWA  SAND  O Normal load 4 5 k g / c m e--0 578  O  ? M  1  Pore pressure  Shear  [«#» 1  Time £  ,  Fig.  27.  reaches  a  for  each  certain  strain  soon  oscillograph  recordings  on  the oscillograph  by  the sample  to  question  specified  paper;  height  angle.  whether  to 1000 c y c l e s  (Fig.  29) t h a t  order  o f 25  These  harmonics  of both  after  27  and  the  except  recorded  f o r some  frequencies  tan-  fluctuareasonable i s  rapid  The  galvanometers presently  lines  strains  rapid  distortion.  and i t w i l l  other  the grid  i t seems  and  i s shown  of the o s c i l l o g r a p h  without  include  pressure  to divide  of the very  fluctuations  per second,  pressure  i n . )to obtain  of the oscillograph per second,  After  strain  with  liquefaction  changes  probably  that  i t i s necessary 1.07  pore  at Fig.  noted  Failure.  load.  of pore  coincide  In view  the fastest  cycles  w i l l  the response  these  response  be  (typically  pressure  to follow  i tw i l l  Shear  the normal  In looking  the scale  of the s t r a i n  enough  up  so t h a t  of pore  than  Cyclic  fluctuations  regular.  inches  tions  less  of a  but the fluctuation  in  gent  |20  are wide  become  1  sec.  Record  value,  there  cycle  1 0 3 kg/cm  ilO  Oscillograph  liquefaction  load  i s linear be  seen  are of the  higher  higher  harmonics. than  80 1000  cycles  shoot  of  per second,  i s small  importance. with  so  the galvanometers,  frequencies  ing  and  The  so  there  but the amplitude  the overshoot  galvanometers  the frequencies experienced. output  was  liquefied  sample  i n which  the cycle  Linearity  of Pore  Fig.  of  be  of  the electric  signal.  at  as  w i l l  be  and  a  Therefore,  fair  that  of  load  n o t be  of  Pressure  cycles The on  major  of  linearity by  over-  the higher  deal-  of the  allowing  fluctuations  a  has  Recording.  with  different  proportionality the record  the sharp  r e p r e s e n t a t i o n of  the shear  noted  The  ( F i g . 28)  the magnitude  the transducer If  it  several  response.  taken  pressure  investigated  to perform  amplification  linearity can  28.  steady,  fications the  should  of  some  are therefore capable  amplified  become  i s probably  true  amplibetween  indicates  peaks  and  changes  valleys of  pore  diaphragm. i s compared  t h e wave  takes  i n Figs.  a different  27  and  form.  28, The  81 form  i n F i g . 27  will  in  F i g . 28,  as  be  a  wave  square  0.02  sec.  Load  Load In  detail, shows sure  a  for a  but  order  few  to  load  CYCLIC  was  that  intended  to  approximately  liquefaction  ( s t r a i n ) and A  and  i n F i g . 29.  on  a  sample  AFTER  OTTAWA SANI \  SHEAR  A  A  hereafter.  i s shown  Form  Form  was  after  movement  N O R M A L LOAD 4 5 k g / c m - 4 Ci -  i n more This  pore  pres-  which  had  LIQUIFACT10N  e = 0 !J60  z  ft  • N  Form  time  events  Load  Load  described  recording  of  as  Load  rise  be  show  relative  to  The  B will  cycles  UNDRAINED  B.  the  speed  load,  referred  Form  Form  high  shear  be  l  E - 3 <) o_ « • w -2< ) 3  in 11 , 0o1' Q. o>. w  fl  : !1  ! 11:, |r; 11 ft - 4- M *  a  a - + 0 3 kg/cr o » in  TiTITT  •—* 1  ;  1111' 1 [• p"'  --0-3  c • + 0 3 in  IBILII  • h  .—  55  --0-3  i  Fig.  just in  the  the  29.  I  High  under  pressure  0.02  sec.,  on  record  sec.,  the  record pore  will  as  be  of  shear  the  rise  also  are  relative  of  the  a  to  occupy time  stated.  The  were  of  and  1  of  time 0.04  high  troughs  of  approxi-  sec.  or  frequency ripple  on  to  electrical  noise.  In  load  is clearly  seen,  the  the  a  marked  smaller  due  shear  portions  The  cycle  load  i  Liquefaction.  loads.  seen  already  pressure  figure, the  recorded  ! Time sec i  •0  9  •  i  j  .i...  Speed R e c o r d A f t e r S h e a r L o a d F o r m A. the  this  7-8  •6  5  4 >  representing  per  of  •3 >  -2  i  pore  cycles  ripple  1  i  liquefied  mately 25  •0  i  cycle  f o r the  as  reversal  82 of  shear,  will  be  cycle new  seen  as  the  of  the  hits  the  dilatency  of  marked  the  trough  probably been  due  ance  to  would  is close  to  to  of  the  In  the  beyond  In  from  suddenly.  stops at  steady  in  the  work  had  this  adopting  the  definition  of  shows  the  Fig.  29  stress  also  reversal  ment  and  pore  half  cycle  until  pressure  time  of  p e r m i s s i b l e to  quarter  cycles  1  4  end  which force  of  0.28  alternate  each  the  static  cycle  the  very  movement  strains  that  the  and  time  would  be  resistmagnitude. in  fact  strain  when  pore  pres-  is  Seed *  partial  as  1 1  is  having  situation  this  from  completion  alter  of  parts,  liquefaction  changes  about  sidered  in  the  The  somewhat  described herein a l l cyclic  characteristic  that  through  hit.  d e s c r i b e d by  failures  arrested  machine  actually  the  aspect  but  the  drop  leading  of  This  a  tests  noted  peak.  liquefaction,  cause  until  the  be  at  condition  a)  strain  sure  the  either  the  part  after  certain  recording that  that  to  a  same  during  analogous  pore  the  It w i l l  a  the  of  only a  of  in  sec.  until  was  act  considerable velocity  at  occurred was  the  machine  value  the  few  b)  new  The  0.01  It  a  when  reaches  or  strain  very  travel  equalled a  rise  continues  strain  F i g . 29  the  to  i t s maximum.  and  the  heavy  to  pressure  effects,  fairly  begins  of  shearing material  pressure  inertia  stress  movement.  start  approximately  begins  arresting  evident  at pore  material.  by  go  very  the  motion  were  have  i s also  ceases  shear  i n pore  achieved  expected  begins  stop.  liquefaction  shear  the  pressure  tendencies  travel  taking  pore  When  pressure,  relative  the  i t s maximum  strain  dilatent  that,  and  when  stress.  the  pore  of  reaches  variation  F i g . 29  the  change  shear  value,  It  from  occurring  pressure  to  pressure  direction,  rate  in  pore  was sec.  the  i s the  already  the of  I t was  shear  directions  load  of  associated  0.09  sec.  Form  separated  by  each move-  out  therefore to  for  given.  beginning  the  about  reason  of  a  conB,  making  quarter  S e e d , H.B. and L e e , K.L., " L i q u e f a c t i o n o f S a t u r a t e d Sands During Cyclic Loading". J o u r n a l of the S o i l Mechanics 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 . 92, No. SM6, November 1966 .  83 cycles in  order  ing By  under to  shear this  read  allow  i n one  means,  components strain  zero  shear any  elastic  direction  i t was  of  stress.  strain.  By  r e c o r d , movements  and  record  the  of  a  rebound few  on  cycles  about  of  a  occur  UNDRAINED CYCLIC S H E A R T E S T Normal load 2 0 kg/cm e = 0-660  designed  after  unload-  and  another. elastic  amplification  0.0002  test  was  plastic  the  unloading  such  form  applying i t in  separate  increasing  each  to  before  to  of  wave  rebound  and  hoped  This  i n . can  can  be  OTTAWA  easily  seen.  i s shown  of  the  be  The  in Fig.  30.  SAND  z  Pore pressure "er  2 0  J - . 0 0 5 In.  3 in  c  o .0-0  -10  </> «  .-005  a.  0  5-.  0  0-  •05-  a> »_  o a.  .0  0  Switching strain scale  5  .c CO  +T V0 2£-4  [- 0 r  kg/cm  1  224' Elapsed-time  50  Fig.  30.  Medium  During quantity  showing  load  i s pore  test  comes  sidered quantity leading when  to  that  to  suitably  the  Speed Record of L o a d F o r m B.  course  significant  pressure,  of  pore  strains  a  pressure real  a  was  only  amplified,  were  to  the  becoming  test,  cyclic large  For  this  reason,  the  only  directly  indication  liquefaction.  Liquefaction.  liquefaction  response  i t s completion.  giving up  High Shear  60  sec  The of  of  progress  strains,  only  shear as  the  i t was  con-  measured  and  although  approximately  the  the  events  recordable  constant  84 magnitude was  therefore  ular  test  pressure compare  i n terms  expressed  some  of  o f t a n 8'.  ratio as  was  after sure  was  i t was  which  was  loose  Curve  No.  lower  cyclic  3 shows  noted  Fig.  31a  but a f t e r  that as  50  to l i q u e f y  percent  tively  few  was  To graph,  leads  on  plot  a test  typical  31  the  void  was  two  stress  then  which  a lower  2,  i t also  more  form  cycles  went  At  reached a test  applied to  of the t e s t s  in  to a  liquefaction.  void  No.  pres-  this  ratio  steadily  than  the  of pore  2 shows was  0.14,  levelled off  had been  steadily  had  over  curves,  cycles.  No.  31b,  i n which  increase  state  Curve  in Fig.  course  and  t o 213  in  the  tan 8  rise  tests  show  proceeded  the r e s u l t s  i f U reached  reached,  extra  dividing i t  again  Figs.  steady  off.  many  pore  2.  and  to It  will  i s t h e same i n  31b.  Commenting failed  113  No.  the general  in Fig.  o f some  a r e shown  alternating  than  of  in  recordable  a  which  number  the r i s e  form,  f o r the other  No  a sample  shear  the l o g of the  the applied  about  lower  several  are presented  1 shows  that  after  to  results  a small  switched  sample,  liquefaction be  assumed  a somewhat  fairly  from  In order  o f t a n 8'.  0.10  only  pore  U,  o f t a n 8'.  cycles.  observed  the test  and  showed  a hundred  stage, and  t o 0.12  partic-  the parameter  tests  although  It  to y i e l d  the plots  No.  those  dimensionless  Some  same  Curve  low and  pressure  shows  with  of  cycles.  Accordingly,  pressure,  instead  tests.  opposed  pore  into  o f any  the r i s e  of  to use  results.  The  U i s plotted three  decided  i n terms  31a  reflect  cycles  as a p e r c e n t a g e .  Fig.  but  few  i t was  normal  o f U and  terms  a  converted  the applied  would  of l i q u e f a c t i o n .  the progress  o f t h e number  in plotting was  the onset  to analyse  which  after  cycles  cycles  terms  before  decided  events  pressure by  just  as a f u n c t i o n  thousand of  until  i n general,  a value  liquefaction  o f 30  ensued  no  test  percent within  a  and  ever once  compara-  cycles. the f u l l  to a meaningless  course  of each  jumble  test  of data  and  on one i t was  combined  85  Tan  6'  'i  1 OTTAWA  —  SAND  r  I  . 3  .2  x  X f1 t  DATE  .1  I  Fig.  10  31a.  e  X  29-4-67  .58  4.5 Kg ±  .64 Kg  D  20-5-67  .66  2.0 Per ±  .24 per  0  26-9-67  .615  4.5 1 cm  .45 'cm.  100  S<  1000  C y c l i c Load T e s t s i n Simple T a n 6' v s . L o g C y c l e s .  1  s<  Cycles  Shear.  U %  100  OTTAWA  - L A R G E STRAINS TEST E N D E D  S A N D  ©  50  o  I  I  Fig.  iO  100  31b. Same t e s t s a s 1/ v s . L o g C y c l e s .  :  above.  1000  Cycles  86 considered test the  necessary  results results  logical shear  in  one  of  a  figures  and  to  normal  equipment  after  instead  an  of  using  of  The  pore  pressure  as  the  the  of  the  and  loss  to  number  time,  gave  number  cycles  of  void  of  cycles  leading  tendency  to  applied  the  tests  of  to  lique-  speed more  control  than  results.  liquefaction, U rise  make  the  to  20  So,  use percent.  value  of  U,  represents  which  might  be  as  important  in to  and  this  to  Furthermore,  point  questionable  to  strength  reduce  cycles  of  many  to  ratio,  of  instability  some  summarizing  plottable  be  duration,  of  relative  to  of  desirable  one  the  of  for  rise  liquefaction.  tion  test  i t was  appeared  number  definite  some m e a n s  Thus  because  hour's  made  very  use  running  was  a  plot.  stresses  However,  half  adopt  complete  faction.  about  to  general,  it  is  an  indica-  liquefaction  in  the  various  tests . Of results  were  a)  the  total  discarded  Erratic rise,  b)  earlier  Figs. considered  valid.  out  normal  of  the  pore  a  normal applied  pressure,  as  resent  the  proper  frictional  by  a  to  results  from  triangles  of  shear  required  tests  load  load  cycles  The  Fig.  32a of  4.5  of  boundary  and  cause  a  void  20  and The  ribbed  top are  and  desired  the  to  Fig.  32b  rise  of  number  plates,  are  are  contours  contours  bottom  drawn  of  carried  show  percent  ratio  with  the  was  Both  rise.  contours It  which  cm.  given  of  tests  tests  cm.  are,  etc.  of  sq.  with  methods  results  per  restraint  rejection  changes  the  kg.  done  to  plate,  sq.  of  pressure  leakage.  corners,  per  to  pore  comparable  such  kg.  tests  squares.  of  2.0  the  these  being  vertical  shows  stress  of  leading  not  many  as  rate  summarize  function  which  and  top 32b  cause  as  of  completed,  intermittent  Examples  the and  the  technique,  retention  32a  in  results  installing  with  in  tests  reasons  slight  results.  proper  of  such  tendencies  implying  later  a  for  Improvements of  with  number  of  repgiving  boundaries. represented  compare  results  Void  ratio  \  \  \  O T T A V \/ A  '±.17  (  SAND  N  \  o±.20 \ o  + .|45  '  °±.I9  \  ——  t  S.38  = *To  t  Tests wil h plain plates, T.  o  written  -°besid<  Tests with ribbed plates s? contour f o r t = ± 0.4 Kg/cm &  = ± 0.6 "  •  Cycles for  32a.  Shear  " " = i | . 0  rise  shear  0.4  100  with  kg/cm  kg/cm  i  rj^ =  2.0  OTTAWA  \  2  Cycles  (i.e.U =  2  \  \ V  0.4  10  simple  t o make u  "  I  I  Cyclic  "  pore pressure to reach  i  .1  Fig.  i.  2  "  kg/cm 20%)  SAND  s°±.3  ^  ^  ^  /  ^  t~^±U3  53  "  °±.96  Tests with slain plates, L W ritten  o  Tests with ribbed  beside  plates  v Contour for Z =± 0.9 Kg/cm a  " •  "  "  = +  0  Cycles 4  I  .1  Fig.  32b.  Shear  i  Cyclic t o make  for pore ! L J  I  simple u  rise  |. |  "  "  ± 1.8 "  "  pressure to ^ e a c h  Z  1  0.9  with  kg/cm  0.9 k g / c m  2  1  10  shear  2  100  o  = 4.5  (i.e.U =  Cycles  kg/cm 20%)  2  88 obtained  with  ribbed  representative the  contours  indicated  by  plates  tests  of  with  Figs.  circles  and  plain  plain  32.  plates  The  with  the  plates.  tests  are  with  Accordingly,  superimposed plain  on  plates  applied  shear  written  representing  plain  plate  are  beside  them.  compared plate ance  If  the  points  with  the  contoured  tests, to  2.0,  3.2,  with  plain  contour  show  with and  6.2.  more  than for  would  Seed  is  less  with  the  Lee *.  none a  the  would  the  of  to  For  For of  In Lee " 1  of  a  that  1  cycles  critical the  that  results  definition  rather  there are  in  resistthan  the  2.2,  2.3,  2.3,  only  three  results  a  i t s value plate  the  greater  are  region  where  i s hard  tests  resistance  in  to  the  com-  Fig.  represented  be  that  with  32b by  tests  results  obtained  on  expected  seen  the  those  performed  shear  with  a  the  to  the  reported  by  Peacock  different agree  reported  that  ratio,  initial  initial  by  material.  numerically Seed  results  void  resistance effective  normal  increases  to  load  void  is  ratios  expected  were  and  a l l agree  ratio  is  resistance  decreasing normal  ratio  stress. to  void  ratio.  the  number  load,  i f the  cyclic  increased. the  above  r e l a t i o n s h i p , Seed  r e l a t i o n s h i p No. invalid.  obtained than  normal  stress,  with and  to l i q u e f a c t i o n  l i q u e f a c t i o n decreases  considering  suggested  32a  ribbed  tests.  void  given  shear  Fig.  are  main r e l a t i o n s h i p s .  given  a  the  be  liquefaction c)  the  plain  r e s u l t s be  given  a  in  these  numerically  increases b)  of  representing  plates  that  the  i t will  a  of  steep  cyclic  following For  32b,  tests  triaxial  a)  Fig.  plate  the  However,  1 1  in  third  agree  that  ribbed  two  so  i s not  , as  Much  and  ribbed  It  over  found  plates  In  plates  surface  contours  and.  plain  However,  herein  be  l i q u e f a c t i o n with  resistance  pare.  i t will  surface  tests  the  should  1  rendered  the  It  is  suggested,  lead  to  questioning  existence  of  a  critical  and  concept however, the  void  ratio.  89 The  critical  terminal a  given  void  void  ratio  normal  increases.  void  ratio  to  depend  example) it  must  on  view  tion  tendencies  ratio  by  critical ratio  void  must  possible  of  more  shear  in  this  of  test.  sample  on  the  the  defined some  l i q u e f a c t i o n has  strains  of  the  that  be  a  stress,  involve strain ratio void an the  no of  for ratio  test.  0.5 a  Taylor,  confining  which  the  has  yet  there  would  i t will  D.W.,  In  prove be  are to  to  void  a  the  "safe"  shear very  stages  tests small  so  strain  words,  pressure  should  change  rate  of  the  (or  change  insufficient be  a  true  to  specify  "Fundamentals  of  Soil  should to  the  a  void be  the  pressure  the  start  results  criterion  necessary  cyclic  perhaps  test  per-  suggests  critical  at  the  exist  zero  pore  equality  shear  ratio of  and  in  for of  to  say  the  loading  Mechanics",  A  that  far  net  l i q u e f a c t i o n under  the  the void  of  may  This  void  is  involve  of  critical  of  intention.  0.1  other  examples  side  which  degrees.  If  void  critical  ratio  barrier  valid,  critical  In  different  at  from  be  liquefac-  to  volume  zero  with  (for  1 5  inconsistencies  cyclic  occurred  tendency  test)  As  energy  the  0.05  degrees.  for  two  d e f i n i t i o n of  given  this  meantime,  the  of  at  the  In  the  might  conditions.  its original  critical  criterion  dilatent  undrained  whether  1 5  stages.  for  critical  sand  l i q u e f i e d , the  of  volume,  formed,  shear  having  a  compare  types  these  valid  to  under  normal  Taylor  specified  valid  as  ratio  other  between  to  not  void  shear  liquefaction  desnity;  critical  the  experimental  to  load  ignores  order  than  as  of  a  and  shear  d e f i n i t i o n s of  test;  time  cyclic  of  or  i n which  completely  line  explanation  a  i t is  Lee,  wrongly  concept  susceptibility  for  under  pressure,  took  variables  under  and  It  such  that  ratio  be  various pore  state  load.  drained  decreases  the  accepted,  Seed  ratio  express  that  determined  quoted  the  is  void  regarded  in  original  determined  this  strains  the  insisted be  generally  large  This  shock  show  now  However, to  under  evidence would  was  is  for  load.  load  liquefy  ratio  N  1948.  90 completely  in  the form  it  seems  it  Having  expressed  a l l shear  will  be  longer  be x  drawn.  shows  i s therefore  handled  cannot  32a  stress  and  be  32b w e r e  as a  a 20  (Figs.  33).  cycles  the a c t u a l  while  a t a 1000  asymptotic, to  It will  the values  and  shown  state  higher.  for  U =  thus  yield  plotted would  from  shows  Any  to a  .  This  described  described.  little  values  The  seen  close  to the s t a t i c  In  Figs.  33,  a  are  100  close  occur,  i s close of this  to a  plot is  i f a  smaller  the contour  contours This  set of  be  surfaces  which which  "yield"  friction  but  would  could  one, below  undrained  levels  surface  of cycles  to the highest,  the appl ied  1000  i s almost  that  number  or c r i t i c a l  shear  difference,  of a s i m i l a r  whose  at  never  result  of  after  plotted  contours  readily  be  t o be  stress  that  level  /  Instead,  pressure  32  stress  these  up  presently.  of v a r i a b l e s  makes  given  can  proportional  t o F i g . 33.  not occur,  of the  pore  Figs.  surface  similar  stresses,  contours  and n o r m a l  a sketch  curved  the lowest  would  be  normal  and  as t h e c r i t e r i o n ,  cycles.  i n a manner  presumably  by  32b, d i -  i n terms  l i q u e f a c t i o n would  It will  i s taken  i n 10  liquefaction  stress  for liquefaction.  a sloping  vary  the shear  which  F i g . 33b  20%  ratio  load,  the  32a and  total  of pore  from  using  p l o t t i n g contours  of cycles  represented  of cycles  are  seen  the shear  i n F i g . 33a.  number  be  number  below  the surface  critical  of void  cycles  thus  rise  normal  of  i n t h e way by  tests  rise  t o be  combined  required.  f o r a given  t h e number  percent  function  Figs.  coincide  words,  33  that  reduced  them by  expressed  directly  i n Figs.  evident  to cause  cycles,  In other  is  of the c y c l i c  the relevant  the r e s u l t s  i s not u s u a l l y  non-linearity  Figs.  that  by  with  void  f o r each  t o combine  i s tried  levels  found  contours  t a n 8 do n o t , i n g e n e r a l ,  pressure,  It  to attempt If this  of c r i t i c a l  the r e s u l t s  of a s e t of shear  t a n 9.  applied no  consideration  natural  applied viding  i f any  angle,  would  strength. 0  9  f o r a.  ct given of  shear  a line  stress,  normal  to the e - a x i s  stress  from  and v o i d  the point  ratio,  i s the  representing  the  slope given  91 ±2.0  + LO-  O T TAW A SIMPLE  SAND SHEAR  Alternating Shear S t r e s s to cause a Rise i n U of 20% i n 1000 C y c l e s v s . V o i d R a t i o and I n i t i a l Effective Normal  Stress.  For a large region, slope  is Constant for a Constant Value of e.  Hence slope  can be plotted against e.  Slope e„  Fig. 33b. 10 C y c l e s  A l t e r n a t i n g S t r e s s t o c a u s e a R i s e i n U o f 20% i n v s . V o i d R a t i o and I n i t i a l E f f e c t i v e N o r m a l S t r e s s .  Applied Shear Angle is no longer independent of initial normal stress in any Region and so cannot he plotted against e.  92 conditions. the  plotted  surface. by  In  the  Fig.  surface  33a  there  where  Therefore,  in  single variable  such  is a  a  considerable  line T  this  region  .  However,  8  would  as  l i e along  a  and 0  region  could  no  longer  sufficient  number  of  can  defined  be  not  be  cycles  on  the  ing  form  few  sand;  of  8  replaced  increases,  is  This one  this '  course,  not  rather  r e l a t i o n s h i p between discussion  from  33  Fig.  that  MONTEREY  Fig.  of  for  part  reason  accepted  why  to  predict  intended the  critical  any  that  a v a i l a b l e from  claimed  are  the  no  and of  when  surface  32  could  given  Fig.  Fig. the  to  33  is  32.  The  behaviour  show  the  based resultof  general  variables.  Further  to  the  void  i t will  be  seen  stress  the  ratio,  cyclic  shear  SAND  34. After Reference  a  the  the  Figs.  8  plot.  points  preceding  surface  .  i s , of  they  the  ( F i g . 33b)  into  are  define  reduced Si  plotted  contours  Ottawa  by  combined It  is  to  the  be  a is  of  P e a c o c k and S e e d 9 f o r Comparison  (1968). Results w i t h F i g . 33.  in  93 critical  void  be  though,  noted  sible  ratio  to emphasize  and  Seed  have of  that  combinations  der  required  to cause  of void  ratio  will  be  that  a picture In  be  obtained,  the  Ottawa  35. shear  nearly  normal shear  one  load  was was  returned  remained  the  same  at  .666.  normal  load  test,  any  would  ultimately  have  lower  void  would  a  series  loads  samples  and  of drained  and v a r y i n g  recorded, function which  ratio  could of void  would  shear  cause  they  vary,  this  then  o r two,  used  t o make  that  accomplish  or whether  void  which  might  under with  surface arises  the recorded  the r e s u l t s  Fig.  ratio  of  which the  the  cyclic show  that  under  undrained  ratio  than  .666  starting It this  varying  terminal  is  true,  normal void  ratios as a  t o F i g . 33,  indicated.  to whether results  a r e due  at a  contours  similar  a  before,  i n an  already as  in  reached  of s t r e s s  pressure,  on  .666, by i n -  and  sample  liquefied.  a plot  naturally  greater,  d i d not then  load,  any  loads,  the c r i t i c a l  is  void  after  i t had  tests  and n o r m a l  that  cycling  to  value  ratio  shear  shear  shear  and  reduced  at a higher  cyclic  load  i t i s suggested  have  It  performed  a terminal  The v o i d  never  cyclic  question  stresses  normal  l i q u e f i e d and  ratio  define The  lic  be  a r e shown  cyclic  starting  of r e s u l t s  test  to i t s o r i g i n a l  From  func-  to  stresses  of this  to the value  a  sand.  was  was  load  as  test  f o r a minute  re-imposed.  cycles  shear  reached  ratio  shear  scope.  the type  cyclic  test  load  to increase  but  wider  results  i s similar  of normal  Peacock  to the p l o t  f o r Monterey  of the r e s u l t s  f o r a given  The v o i d  i n 10  In o r -  by  their  of the c y c l i c  stress  results  that  the normal  load  tendency  The  obtained  load,  should  i n a l l pos-  pressure.  i n a s i m i l a r way  to discover  the drained  0.668.  cyclic  It  not apply  of r e s u l t s  the range  drained  may  liquefaction  o f somewhat  order  pressure.  and n o r m a l  contours  and n o r m a l  found  load,  creasing  shows  the form  sand.  I t was  ratio  i n F i g . 34  i n F i g . 33 b u t  normal  conclusion  under  initial  tion  obtained  shear  F i g . 34  seen  with  the s i m i l a r i t y  re-plotted  F i g . 33.  this  of. v o i d  f o r simple  been  giving  increases  the  simply  to such  cycbe-  causes  94  OTTAWA  SAND Normal load 2 k g V c m Shear load ± 0 . 4 k g / c m 2  11 1 1 I U I  C  =  2  .DfC  .670  Pause in cycling approx.-g min.  <  a. o >  End I - test st  SECOND  TEST  .665  100  10  1000  Cycles  After first test normal load doubled momentarily. Then second test under same load as first. _  Fig.  as  inertia  after  o f sand  referred The  normal sequent normal  stress  grains  t o as  dynamic  process from  stress  almost  on  the other cyclic  hand,  shear  the s o i l  involves  s k e l e t o n to pore strength.  This  a relaxation The  could  compared  be  waves,  here-  effects.  volume.  stress  of stress  of l i q u e f a c t i o n  i s ,in effect,  constant  Simple Shear. Sample.  and p a s s a g e  r e d u c t i o n i n shear  at  The  35. Drained Cyclic Two T e s t s o n Same  i s thus  drained  able  transfer  water  with  of con-  transference of of  the s o i l  cyclic  shear  to a process to cause  of  skeleton tests, creep.  relaxation  or  95 creep by  of the s o i l  static  stress that  paths  ratios  tested,  tend  rather  effect  of r e l a x a t i o n  to  the c y c l i c  probably which  due  may  effects the  or  they  governing using  o f the change being  the speed  about with  The  a dozen zero  load,  scatter  cycles  change  a l l other  result  portion sary  oscillograph ual to  cycles mark  l e d to a sudden i n the middle  do  shows  f o r a long speed  n o t show.  the paper  regularly  spaced  ratio,  stress level  results  cycling, of the  on  the e f f e c t  inconclusive  as t h e  every marks,  except  increase  of a long  speed  considered  term was  .01  Instead 10 .05  of speed f o r test.  remained  or  normal  significant.  of the  relevant  A reconstruction of t h i s  Thus,  unchanged.  particularly  test  was  nature,  cycles.  counter  In the a c t u a l  i n . apart,  suddenly  neces-  the  i n . p e r s e c . and t h e the c y c l e  as t h e  instability  of grains  a reconstruction record.  of magnitude  mentioned  packing  to  l i q u e f a c t i o n may  threshold  the already  variables  paper  be p r o p o r t i o n a l  H o w e v e r , i n one o f t h e  of o s c i l l o g r a p h  because,  Dynamic  of frequency.  i n void  36  contacts,  order  i s therefore Fig.  effect are  i s o f t h e same  reported,  control  comparable  reversals.  rapid  Published  normal  hand, the  t o be  case  a r e somewhat  9  26  e f f e c t s , the s t r e s s  might  i n which  one-  sufficient  in Fig.  or creep  stress  some  parameter.  found  of p a r t i c l e  f o r more  require  gradually  of about  order  expected  mentioned  rapid  d i f f e r e n t speeds  now  n o t be  appear  In g e n e r a l , f o r  the other  on d y n a m i c  of s t r e s s ,  simply  dynamic  experimental  of  just  not  i n the e f f e c t i v e  I f , on  during  by  been  of t h i s  readjustment  t o be more  might  have  From t h e  i t does  stresses  the r e l a x a t i o n  apart  accomplished  of s t r e s s .  increases  and  26)  be p r o d u c e d  Stresses  would  those  o f change  expected  tests  tests  tests  as  (Fig.  stress  depends  n o t be  magnitude.  shear  decreases.  jarred  such  rate  effect  normal  to s l i g h t  be  be  of  than  of s t a t i c  tests  cyclic  to cause  stress  paths  o f t h e same  and d e c r e m e n t s  liquefaction.  generally  would  e f f e c t would  of the i n i t i a l  cause  which  i n the s t a t i c  increments  void  tenth to  stresses  the r e l a x a t i o n  applied the  shear  skeleton  was  individused  record, show  two  the  96 narrow  gaps  occurred the  within  paper  could  be  speed,  and  cycles  sq.  cm.  in  the  pore  (average  6630  as the  total easily  cycles  rise  4.5  x  for  Fig.  number  OTTAWA  10~  of  within  shear rise be  time  which  of  0.37  36.  6  pore  kg.  cycles  load  4.5  can  be  x  sq.  actual  cm.  occurred,  the  read  quite  during 0.28  at  actual  these kg.  per  3  per  cycle)  6650  events  maximum i s not  Knowing  From  10~  These  had  cycles  rising  .042  per  affected  20  pressure  cycle). The  cycles  interval.  sec.  After  rise  extra  the  strains  read.  per  3  Normal  SAND  where  (short)  (average  reached  the  cycles  approximately  pressure  reconstructed and  time  of  the can  20  abnormal  the  magnitude  accurately  ing  an  ascertained  record,  twenty  representing  cycles  have  been  rate  of  cycl-  known;  i t  has  kg/cm*  e = . 598  .005  -.005  F i g . 36. C y c l i c Simple Shear. Simulated Oscillograph Record Showing Pore P r e s s u r e Increase D u r i n g Temporary Speed Increase. been  estimated  cycles exhaust  occurred noises  by  the  within during  information, 6 the  sees, test  and  from  the  record,  from  the  rhythm  which  changed  for  an  that of  20  the estimated  97 time  of 2 or 3 sees,  sudden  sharp  returned  increase  t o what In  stress  paths  and  i s considered  of  dynamic,  stated  more  of pore  After  the  pressure  been.  t h e jump that  to normal.  of r i s e  of the foregoing i n pore  to just  testing  remarks,  pressure  liquefaction  as o p p o s e d  that  returning  the rate  i t has  view  it  realized  before  about  just  is likely  cyclic  described,  t o be  shear.  is desirable  effective  a  phenomenon  However,  before  this  i tis  can be  definitely. The  following  The  effect  points  summarize  the c y c l i c  test  results: a)  involves  of inadequate  severe  t o p and b o t t o m  lowering  of resistance  friction to  lique-  faction. b)  Confirmation ships  c)  was  obtained  discovered  by  factors  governing  Drained  cyclic  to  a terminal  stable  Peacock  the general and  liquefaction  shear void  and w i l l  of  under  ratio,  never  Seed of  given below  liquefy  9  relation-  of the  sand. stresses  which  under  leads  the sand i s  t h e same  stresses. d)  Liquefaction  i s probably  cyclic  stress.  partly  due  strain  waves  i n the sand  Re-Liquefaction.  Early  during  it  i f a sample  was  found  drained  and  that  subjected  not a r e s u l t  It i s likely  to i n e r t i a  effects  almost  immediately.  An  37a.  This  figure  shows  test,  ratio  o f 0.68  sq.  cm.,  the  shear  was  then  original  and  an  liquefying load  was  opened, back  effective after  36  returned  allowing  pressure.  the pore As  excess  of  i s given  stress  The  o f 1.0  pore  pressure pore  with  After  herein,  then r e -  load,  starting  cycles.  to zero.  and  cycling  example  normal  1/2  least  passage  reported  liquefied  re-liquefy  first  i t i s at  during  the t e s t i n g was  merely  mass.  to the o r i g i n a l  a  that  of  i t would in a  void  kg. per  liquefaction, pressure  to r e t u r n  water  Fig.  escaped,  valve  to the the  98 original ratio  normal  decreased  reclosed second a  and  and  residual plain  mentioned, and  of  on  the  plate  central mass  region  as  densification so,  tion, to  corners,  sliding would  not  be  immediate ribbed  sample  was  sample  did  and  have  in  resistance  increase  in  density.  showed  similar  zero  in  now  be  behaving  that  occurred  to  one  Fig.  intended the  sample If  the  centre  the  expected. used,  (see sample  but  Fig.  and  that  and  the  i t was  could sample  be was  an  the  a  is  tremendous  bottom  still but  boundaries  considered  into  behaving  empha-  accompanying  therefore forming  plate  immediate  plates  and  when  the  re-1iquefaction,  top  were  almost  of  It  exhibiting  immediate  offer-  redraining  of  mass  37b).  bodily  sample  almost  bottom  this  However,  central  bodily,  where  not  and  abrasion  a  liquefac-  suspected, core  the  loosened  the  the  12f  sliding  were  central  that  If  the  of  ends  into  after of  line  The  dilating,  be  top  to  ascertained  deformation.  in  The  width,  centre  thought  capable  the  pre-  its  expansion.  manner  This  first  densified  l i q u e f a c t i o n and  Ribbed  be  and  were  moving  tendency  as  was  the  therefore  shows  decrease  impossible  If  plates  been  37a  were  sand  half  while  been  plate.  deformations  disperse  occur,  sized  could  of  fro  to  still  it  lateral  in  cycle.  across  i t was  becoming  densifying  bottom  Fig.  the  the  direction.  s i g n i f i c a n t d i s t o r t i o n and  re-liquefaction that  of  then  for  has  to  band  was  noticed  transverse  first  cyclic  resistance.  not  cannot  a  by  one  As  p l a s t i c i n e example  large  likely  and  a  void  percent  first  use.  in  length  r e - l i q u e f a c t i o n would  top  surface  the  region  to  18  observed  the  the  alternately  bodily  to  longitudinal  was  At  without  appreciable  a  about  simulated  central  fro  in  plate  the  in  were  unmarked.  the  re-applied  U went  were  plates  half  in  was  under  the  and  ing  of  somewhat  were  plates  either  were  was  the  valve  liquefaction, in  polished  about  load  and  pressure  l i q u e f a c t i o n was  symmetrically  occupying  pore  test  with  the  then  polished  shear  after  lower  e f f e c t i v e again  The  second  percent  and  distributed and  0.595.  the  weakness  scratched  became  cyclic  In 90  upper  viously  to  the  test.  cycle  area  stress  zones  more  or  of less  99  First test Second >  e = .68 .595  Stresses in both tests  (>  I  ,  Normal I.O Kg/cm Shear +0.08 "  1 i SECOND  \  /r  FIRST TEST  TEST  i  ^ .—' 1 J  1— •  1 1  I  V.  I  10  100  Cycles  F i g . 37a. R e - l i q u e f a c t i o n o f Ottawa sand w i t h plain plates. Two t e s t s o n s a m p l e , d r a i n e d b e t w e e n .  2  100 uniformly,  the  to  characteristics;  machine  rapid  re-liquefaction  interparticle  c o n t a c t s , to  the  and,  same way,  in  the  field  It  was  considered  would  in a  consist a)  of  something make  presumably, layer that  them the  subjected the  could  readjustment  same  such  contacts  and  be  the  due  same  would set  of  two  as  to  the  behaviour  in  occur  of  in interparticle  a way  simply  affecting  change  in either  Re-orientation in unstable  must  be  a l l change  to  change  not  events.  contacts  ways: make  t h e r e f o r e more  a l l the  liable  to  liquefaction. b)  Re-arrangement which were  it  appears  ticles is  the  to  imply  to  such  as  to  render  It  is still  and  under  tacts such  a  would  resistance  reach  type  of  on  void  to  the  a  did  not  approximately  the  same  was of  of the  first tive the  rather  the  first  second  a  type,  liquefaction  second  type.  of  as  an  at  happen.  in  to  the  but  i n which had  the  cleared  therefore, To  would  The  second void  first  a  seems  investigate  low to the  of  type  was  cyclic  for  favour matter  high  of On  re-  always This  contacts readjustment  strains  resistance  favours  increase  ratio.  of  con-  read-  and  liquefaction.  expected  large  of  instability  imply  deformation  be  types  "peaks"  would  i n c r e a s e of  ratios  factor  mechanical  transfer  shear  void  some  these  contacts  the  position  percentages  unless  that  through  i t  practically n i l .  different  large  par-  Otherwise,  unstable  coincidence i f readjustment  type  reasoning,  not  imply  amount  reach  occurring,  making  contacts  considering  Such  particle,  neighbour,  were  contacts.  positions  grains.  one  a  liquefaction  a l l the  reasonable  the  be  at  that  of  would  readjustment  fields,  In  on  aside.  would  transferrence  which  of  re-liquefaction  unstable  liquefaction,  would  of  imagine  contact.  ratio,  readjustment  to  stress  i t appears  involve  points  out  points  motion  unstable  percentage  harder  would  of  factor  with  that  different  justment,  in  a high  high  pushed  type  some  positions  mass,  or  first  imagine  that  h i n d e r i n g the  crushed  If  seek  hard  were  so  of  path.  readjustment  the Inducof  experimentally,  101 it  was  decided  t o r u n some  travel,  t o keep  typical  pre-liquefaction  was  did of  ±  .002  a  came  into  allowing  on h o r i z o n t a l  and c o m p a r a b l e A mechanical  deformations ± 0.01  During  the pore  pressure  limited  travel  38.  would  which  would  i n an o r d i n a r y  test,  of  degree).  until  with stop  approxiThis  the l a s t  density,  The r e s u l t s  liquefaction be h e a r d , travel  stop  few c y c l e s  rise  On  the sample This o f such  In the l i m i t e d  was  showed  been  a series  travel  tests  the stop, type  level  out at  expected  switching at the  an i n c r e a s e  contrary  the stop  of the  and t h e n  have  test.  and  tapping  test,  d r a i n i n g and r e - t e s t i n g  to l i q u e f a c t i o n .  quoted.  a limited  pressure  the value  maximum  higher  approached  the carriage could  stroke.  approximately  Fig.  small  liquefaction  t h e sample  play,  described,  ance  a limit  test.  every  ing  very  i n . (approximately with  with  deformations.  devised,  not i n t e r f e r e  As  at  deformation  accordingly  mately  tests  as t h e  off a result-  in resist-  to the e a r l i e r  tests  of tests  a r e shown i n  the f u l l  cyclic  shear  F i g . 38. R e - L i q u e f a c t i o n o f O t t awa S a n d w i t h Ribbed P l a t e s . T h r e e T e s t s o n Same S a m p l e Drained i n Between.Shear S t r a i n s L i m i t e d M e c h a n i c a l l y .  102 load  was  applied  liquefaction, very the  small  far  the shear  fraction  stop.  stopped  to the sample  with  a jerk  the exact  same  i n t h e two  grains  seek  faction them  travel  with  limited  position been  tests  rotation  placements  shear  is  hard  to  cause  large  liquefaction  a  sand  strains,  with  small  deposit  safer  shear  themthe The  b e met  has of  inter-  plane  to the  the contact. f o r only  more  in a  degree  by h a v i n g  the tangent  to  a sand  resistance  the contact  t o any centres Such  small  dis-  strains. liquefaction  once  deposits  a layer  the shear  one  may  strains  stresses  needed  resulting as  liquefaction  occasions.  involve  has l i q u e f i e d , i t  a r e as s m a l l  tests,  f o r future  make  but i t appears  but d e f i n i t e  I f the shear  travel  length,  the grains  conditions because  arranging  arrangement.  arranging  effective  lique-  to r e -  by  i t can propagate  i n limited  makes t h e  by t h e  contacts  forming  be  of n a t u r a l  factor  makes  from  grains  strains.  are not the  liquefaction  would  need  natural  course,  jarring  such  so t h a t  to a l i n e  t o s e e how  obtained  prevent  that  arranged  involved In  small  some  of r o t a t i o n  of  still  i s not conclusive,  o f t h e two  of the con-  of the necessary  an a r r a n g e m e n t  i s normal  centres  would  as  liquefaction  i t should  under  limited,  contacts  that,  of  of contact  over  particle  hoped  the j a r r i n g  that  but i n which  Such  of  during  i s stable  strain.  contact  b u t i f some  considered  to r e - l i q u e f a c t i o n ,  minimized  and  grain  a to  started  However, i f t h e s e n s i t i v i t y  to the b e l i e f  which  transferred  omission  on a sand  path"  evidence  being  It i s realized,  to the p a r t i c l e  a "cleared  experimental  sensitive  stop.  i s due  travel  weight  kind  f o r only  o f an o r d i n a r y  contact  i t was  t h e sample  I t was  the sole  of test,  an u n s t a b l e  liquefaction  add  types  t h e same  mechanical  stroke.  acting  After  deformation  strains.  forces  strains,  form  selves  but with  alternating  that  shear  at every  on  reversal.  before  the conditions  were r e p e a t e d  siderable  acted  of a second  The h o r i z o n t a l  as p o s s i b l e ,  test  load  at each  from  those might  I f the shear  make  strains stop,  are a l i t t l e  one  larger,  liquefaction  more  susceptible  that  t h e phenomenon  investigation.  as  i n tests  of a natural  to future  without  deposit  liquefactions.  of r e - l i q u e f a c t i o n  may  a  travel  make  i tf a r  I t i s considered  would  repay  further  104 CHAPTER V I SUMMARY  The and  ordinary  interest. a  sample  triaxial  equal  The s h e a r ,  experienced  from  zero  the ground.  sample  i s s e t up w i t h  rotate, field  of merely  might  between The  improve test,  but also  ations  results  simple  Hence,  that  of  test, the compresto con-  i s then  applied  the p r i n c i p a l  axes  i f the shear  degree  elastic  by means  analysis  similarity exhibited degree  provided  i n the regions  of elastic  of displacements by p l a s t i c i n e .  near  case,  in a  achieved  i n  These  the  showed vari-  the boundaries. samples  The v a l i d i t y  was d e m o n s t r a t e d  p r e d i c t e d by a n a l y s i s  The e l a s t i c  uniform  of t h e sample,  by p l a s t i c i n e  analysis.  tests.  by t h e t r i a x i a l  samples.  most  dif-  not only to  t o deform  actually  through  shown  results,  shear  was d e s i g n e d  t h e sample  i n the present  to which  and s i m p l e  plasticine  pattern of deformation  confirmed  machine  patterns  of uniformity  test  considerable practical  of stresses  b y dummy  o c c u r r i n g only  However,  of t r i a x i a l  to constrain  checked  t h e c o n s i d e r a t i o n s above,  interest.  shear  The d e f o r m a t i o n  were  high  the  on a s e t shear  Shear  which  removal  but i s similar  that  p r e s e n t l y , show  on t h e sequence  manner.  The  since  do i n t h e g r o u n d ,  be s u p p o s e d  theoretical  be s u m m a r i z e d  test  strain.  the result  shear,  t o a sample  developed,  i n the ground.  usually  up no  of the p r i n c i p a l  In the simple  lateral  with  sets  of  changed.  ferences  a  would  which  and s t r a i n  i n a l l directions,  planes,  as they  It  to  zero  t o be e x p e c t e d  horizontal  are  orientation.  i s not equal  ditions on  fixed  shear  i n v o l v e s compression  o f one o r more  i s then  simple  i s of considerable  stress,  stress  The s h e a r  with  test  strain  therefore, i s applied shear  planes  sion  triaxial  change  plane  f o rs o i l  a l l round  by a s t e a d y  stresses.  CONCLUSIONS  between  testing  The t y p i c a l under  followed  has  difference  AND  analysis  of by t h e  and those  also  conditions are theoretically  was  showed  improved  by  105 increasing  the  caution  required  for  a  is  ratio  of in  machine,  since  sample  i f the  assumed  become  harder  to  sample  dimensions  tion  to  for  ever,  with  even  boundary sand  although  In  the  chosen  of  to  ratio  for  a  long  realized,  described,  length  applica-  might  distribution. realization  considerable  care  in  they  the  maximize  greater  chosen,  height  are  to  stress  However,  improve  machine  and  dimensions needed  length  primarily  considerations  conditions  height.  conditions  engineering,  the  the  to  conditions  boundary  were  How-  of  the  be  proper  case  of  samples. Using  bridge, sample  England would  strains  very  close  Hence soil  the is  and  1 3  results ,  obtained  from  i n d i c a t i n g that  behave  stress-strain and  length  choosing  realize.  practical  preferable  sample  in  a  manner  at to  the the  applied  desired  obtained  central  the  very  r e l a t i o n s h i p , i t was portion stresses  experiments central  close  to  deduced of  the  and  directly  from  the  the  measured  of  a  desired  the  sample  measured  Cam-  portion  that  stress-strain relationship  at  stresses,  would  be  strains.  for  a  sample  average  of  stresses  deformations. The  distribution the  type  soil.  boundary  of  degree  would  this  would  have  but In  not  was  considered  of  the  elastic could  an  of  anisotropic with  not  of  would  have  known an  the  stress  material  horizontally  concluded  effect  on  the  of  layered  that  displacements,  displacements. of  on  known  appreciable  consist  dealing  it  selected  but  anisotropy  placements  cover  associated  consist  conditions  of  be  to  i n v e s t i g a t i o n of  i n v e s t i g a t i o n i t was  stresses,  boundary  elastic  extended  conditions  anisotropy tion  was  which  From  analytic  where mild  distribu-  S i m i l a r l y where  stresses,  appreciable  the  same  effect  on  dis-  stresses. t h e o r e t i c a l l y with  necessary  constants,  in  to  anisotropic  investigate  order  theoretically exist.  to  the  ensure The  material,  bounding  that  bounding  the  values  materials  values  were  106 calculated this  by  energy  considerations  i n v e s t i g a t i o n i t was  noted  a)  a  Conditions the  for  average  changes  y and b)  E  a  If  and  2  increase vice In  0  versa)  other  or  cohesive  soil  analysed  Sand  and  proper  could  be  ments  where  leakage  avoiding  involves of  the  the  sample  only  the  condition  and  If  of  test the  friction  polishing  on  the  avoid  of  far  is  the  of  devised plates  inadequate  plates,  under  any  of  the might  analyses  by  material  to  install  sand. lead  Results to  to  from  follow  lead  to  leakage, experinormal by  them.  distinguish  friction,  so  and  bottom  for  adequate  is  a  to  results,  essential  after  than  Failure  questionable  there  samples  sand.  probably  This  top  pre-  conclusion  e s s e n t i a l to  adequate  (and  as  s e t t i n g up  with  could  tests.  boundary is  for  appear  which  against so  may  to  condition  slips  will  may  ,  dilate  dilatent  difficult  needed  slight  development  direct  is  valid  material  methods.  intermittent.  errors  apparently  The  tests.  is  procedural  from  or  If  principles.  violating  that  a  such  element  samples  more  desirable  Another tests  are  care  slight  i t is  developed  may  This  finite  elastic  precautions  which  and  without  i t shows  disturbed  considerable the  pattern  soil  behaviour.  by  were  samples  compresseive  anisotropic  significance in  be  the  material  energy  elastic  Techniques  observe  an  a  flocculated clay  since  narrow.  zero,  violating  rules  can  i s not  are  anisotropic  under  compression  of  modulus  xy  small,  without  words,  computer,  and  2u  triaxial  have  clay  3  of  =1  i n volume  compressed  of  y  are  course  xy  =  /E  x  the  unless  + y  xx  bulk  stress  i n volume  during  that  zero  normal  and  valid  condition that  no  part  boundaries. friction  several  similar  an  of  area  d i s t r i b u t e d symmetrically  is  abrasion about  a  107 transverse of  plates  their with  centre was  plates.  due  ribs  they  may  their  into  sides.  top  the  prepared  A  as  It  a  for proper  ary  test  the  the  erly.  load  the  with  the  affects  and  the  a)  on  Ottawa The  sand.  angle  In  of a  usual  associated  <J>,  somewhat  criterion. simple angle  shear  tests  higher  the  triaxial  tests.  sure  that,  was  angle, than  simple  I t was  ing  marked triaxial  test.  of  below  maximum  at  the  the  reached  stage  high  where  shear  shear  i t s maximum.  at  higher a  per-  that  the failure in  comparable  was  much  in  undrained pore  pres-  friction  lower  strength  o c c u r r e d which in a  shear,  but  developed the  were  high  constant  load,  In  tempor-  material.  than  the  than  simple  strain  a  tests.  "yield"  In  the  establish  than  the  same  showed  prop-  developed  than  the  onto  tests  concept  of  at  ribs  down  Mohr-Coulomb  triaxial  i n simple  consisted  of  tests  a  its  found  developed  result  in spite  that  occur  to  shear  shear  off  sample.  i t was  the  simple  shear  notable  the  higher  pressure  ribs  i s lowered  friction  on  grains  the  experiments  the  of  was  corresponding  more  with  The  sample,  sand  to  i s necessary  different  i n undrained  the  i s bedded  tests  tests  pore  the  i s lowered,  i n simple  angle  in triaxial  However,  b)  The  probably  d i s c o v e r whether  static  requires  of  plate  undrained  friction  are  is likely  s t r e n g t h of  Stress-controlled, formed  to  from  grains bouncing  plate  research  projecting  placement,  of  plate  the  pair  difficulties  there  sample  top  a  some  positions  boundary  before  seating,  friction,  s u r f a c e of  trouble  the  that  place  motion  ribbed  sand  are  bottom  During  As  is believed  seating  first  serious  sample.  into  adequate  transverse ribs  horizontal  more  surface  dig  at  place.  abnormal  the  must  the  interfering  fall  cause  sharp  In  inhomogeneities  as  achieve  Unfortunately, there  slight to  To  made w i t h  surface. ribbed  line.  was  correspondthe  yield  load,  well  approximately friction  triaxial  angle  tests  a  108 slight  "yield"  strain  as  slight  " f l a t t e n i n g " of  not  a  Use  of  in  large  plates. plates  usually  high  normal  almost  density)  small  s t r a i n s , but  sand  blasted  much  as  13  as  as  a  loads  high  a  gave  a  this  by  using  ribbed  sand  blasted  (for  a  especially  normal  apparent  not  apparent  strength  lower  did  For  the  plates,  under  plates  low  obtained  ribbed  plates  1/2%  bottom  indicated  Under  only  stress.  friction.  that  same  s t r e s s - s t r a i n curve,  and  than  the  i t involved  constant  less  showed  main  top  approximately but  the  boundary  given  The  at  shear,  blasted  they  strength,  zed  simple  adequate  reason  noted  d e f l e c t i o n at  sand  provide  was  at  loads  the  strengths  as  lower.  results  of  cyclic  shear  tests  on  sand  are  below: Confirmation faction  of  Peacock  and  number  of  was  obtained  most  of  the  Seed .  cycles  to  results  Density  of  (ii)  Initial  normal  pressure  cycles  to  if  number the  cyclic  Contrary found  to  that  quencies number If,  of  after  cycles  it)  then  without far to  more the  It  by  that  the with  the  sand. reduced  increased.  Seed's  a  tendency  results , i t  was  9  for to  increased  reduce  fre-  the  liquefaction.  subjected  sensitive  first.  is  sample  e f f e c t i v e load  drainage,  described  showed  on  and  liquefying, a  normal  lique-  and  alternation  to  on  liquefaction is  load  was  stress  initial and  sand  Peacock  of  of  the  shear  there  effect  l i q u e f a c t i o n increases  (i)  The  the  parameters  The  9  of  to  i t was to was  a  the  was  drained  to  its  (thereby  densifying  original  cyclic  found second  concluded  that  the  sample  liquefaction that  the  load was  than  structure  109  of d)  the  sand  was  Liquefaction  altered  with  plates  occurred  1/3  that  of  sand  at  a  by  liquefaction,  blasted  cyclic  required  for  top  shear  and  bottom  often  as  low  l i q u e f a c t i o n with  as  ribbed  plates. Several is, been  perhaps,  to  questions  be  described,  expected  would  raise  immediately.  In  some  questions  of a)  these The  closure,  lines  knowledge of  The  tests  and  i t was  for  type  than  not  sample  has  solve  comment  on  research.  equipped  However,  the  It  which  i t can to  future  is  of  the  the  desirable  stresses.  described  measurement  to  any  measure work  complete  stress  at  strains  large  clays,  plates  may  be  better  The  vertical  by  other  tests  for  seating  between  be  done  field would  ribs  the  the  It  would  of  upper  and  features.  whether  r e s u l t i n g precompression  measured The  shear  tests  investigation.  In  re-liquefaction  appears  have  are not  include  desirable  be  friction  ensure  con-  sample,  valuable  has to  affects  know  the  stress-strain relationship.  cyclic  there  would  bottom  boundary.  to  the  Before  apparatus.  and  be  the  undesirable the  It  pore  measure  the  top  applied  plate  sand  the  developing  s l i p p i n g at load,  of  to  modify  on  methods  on  negligible.  advisable  appreciable.  investigate  and  compliance  i f necessary,  interference  entirely  obviously  i t would  and,  almost  that  was  Sample  tact  were  considered  compliance  to  of  unanswered.  beneficial.  testing  e)  queries  to  water  d)  more  normal  be  c)  work  described  behaviour  left  that  horizontal amplify  been  i t seems  as  apparatus  and  b)  have  many been  the  leave  questions  p a r t i c u l a r , the to  parameters investigated  shape  many  of  need of at  phenomenon  more  cyclic a l l .  d i f f e r e n t wave  for  testing. loading Such forms  of Also,  which  parameters and  the  110 question  of whether  on  stress  shear  more  on  rate  questions  of  have  or whether  change  stress.  simplifying  the  the  frequency,  of  than  These  seismic design earthquake  amplitude might  f r e q u e n c i e s and  critical  rather  of  design  simplification  range  of  factor  were  amplitude  of  and  lead  into  number  numbers  of  rate  change  stress.  last because  suggest  to a very  of  simply  i t depends  considerable importance,  methods  The  depends  amplitude  recommended  mental  liquefaction  i t s fundaof  cycles.  different  cycles, of  i f stress  Ill APPENDIX THE  EFFECT  OF  It pressure the  was  rises,  sample  COMPLIANCE  ON  mentioned  a certain  to record  A  THE  PORE  i n the text  volume  the r i s e ,  PRESSURE  that  of water  thereby  RECORDED  i f the pore  i s withdrawn  reducing  the  from  recorded  pressure. Consider change  i n total  ditions B  a sample  pressure  i f the pressure  to i n d i c a t e  Take  conditions. i n pore  Also  =  change  C  y  =  Compressibility  of  C  g  =  Compressibility  of the s o i l  V  =  Volume  of pressure  S  =  Change  in V  K  =  Compliance  S  +  =  C  grains A,  for  case  the  decrease volume  by  compliance  conand  due  to A a .  water. skeleton.  measuring  to unit  device.  pressure  of the apparatus,  change.  as w o r k e d  out i n  . V  i  porosity Assuming  mineral  A to denote  text.  W  n  pressure  to a  put  Au  due  subjected  subscript  not a f f e c t e d  actual  the  in  A a . were  V,  of volume  of the  that  i s small  the c o m p r e s s i b i l i t y compared  the decrease i n volume  of the water  sample.  with  i n volume  solid  of water,  of the sample,  of the system or  that  of the  i s equal  and  we  have,  also  to the decrease  112  C . h u . . V . n W A  But  A a .  A a - hu .  =  A C  So  W  A  .Au..V.n H  =  C C  hu  In is  equal  it  i s also  skeleton  case  B,  equal  minus  +  1}  -  1  the decrease  to the decrease  V  n  A a { -gs  =  A  hu .) A  ( A a -  c  O  i n volume  i n volume  to the decrease  the increase  of the pore  i n volume  i n volume  . hu .  +  •  Au  n  £3  =  C  e  J  n . V+ A u  M  W D  or  V )  D  C ,.A  or  (n.V  n  W  (C  .n.V  +  D  Aa  So  Au  +  ( C , . 7  . y  S)  C_.A a . V  =  J  K)  =  C .Aa_.7  D  b  D  0  Aa  -A u  Aa  .  Aa  S.Au„  -  D  W  &  B  D  D  c .v s  C„. S  V +  K  +  . n. V w  .  K  1 +  C  z  °s-  C v  77 +  . n w  s  c  But  soil  of the pressure  D  D  W  And  . A a  O  water.  of the  So C  of the system  device  113 and  the  ratio  of  the  recorded  pressure  to  the  is  Au _ A  C  ,  1  , J, C .V  1  S  + _w  +  1  IC  S  n  ] +  1  +  K  '\  [  C .n +  cj  1  .n  ideal  pressure  114 APPENDIX DESCRIPTION  In saturation  order  reservoir.  pheric  a i r , free  back the  and  sample,  water To  forth,  allow  the  on  use  on  ation, sure  back  head  the other  side  These shown  level to  by  the  side  assembly  admitted apparatus side  supply  reservoir  protected  Pressure  B  and  side  by  so  that  i s s e t up  from  head.  to r e l e a s e  purposes,  side  of the  the inward  to  pres-  them.  and  and  For  do  test  drained for satur-  adjustable side  pres-  and  allow  one  water,  under  the set pressure,  the r i g h t  by  of the  or vacuum  C.  the  to the d e s i r e d can then  appropriate  Atmospheric one  usually  For this side.  a i r contamination  being  by  applied  s e t t i n g s of  pressure  i s from  reason,  The  be  of the valves  flow  the l e f t ,  arrangement  B.  i s adjusted  opening  towards  occasions.  E i s on  one  to d i s t u r b  of Appendix  pressure  of valves  rare  the  at w i l l .  other  pressure  o f t h e a p p a r a t u s by  of the diagram,  paratively  is  a t t h e end  to either  above  shut.  r e g u l a t o r A.  either  water  behind  head,  to apply  supply  atmos-  the space  possible  as  objectives are achieved  i n F i g . 39, The  be  f o r leaks  able  small  i t i s not advisable  liable  or expel  flush  a  of the loading  samples,  test  t o be  water  and  at l e a s t  back  a i d to  contain  To  and v a r i o u s  However,  be  an  dissolving  of the l o a d i n g  to supply  to admit  side  to connect  pressure,  must  head  i t must  f o r leaks  would  to the d e - a i r e d  with  side  as  to put pressure  i t to either  i t i s necessary  loading  the l o a d i n g  able  order  supply  avoided.  of water,  t h e membrane  water,  be  possible  either  SUPPLY  supply  f o r de-airing loose  under  WATER  this  must  to a vacuum.  In tests  lead  t o be  head  vacuum  sure  be  the assembly  is useful  loading  prevent  through  and  pressure  the water  surfaces  the passage  checking it  To  i t must  supply  THE  to use d e - a i r e d  of samples,  supply  OF  B  the  c a n be D.  right  reversed  on  the sealed  surface  of t h i s  a flexible  The  comwater  water  diaphragm  115 and  valve  aired  D2  water  interposed  i s normally into  E,  between of  cedure  dismantling  air  rises  cal  top  G.  As  ready  and a  to  and  can  be the  drain.  loading  gauge  tester,  under  an  gauge  tester.  or  three  K2  open.  opened  head  a  slightly  and for  is H  short  closed,  open.  opened which  and  the  periods.  By  pressure  i s maintained  and  sample,  does  start  bubbles.  of  Kl  ing  allows  Kl  air.  allows If  the  valves  B2,  supply  is  so and  as  to  the  open.  J  water time  with for  supply  in  and  may  Kl  dissolved new E  water  becomes be  closed  sucked  through  D2.  When  de-airing  is  receive desired  or back  supply  air to  is  as  water  level,  Dl  to  a  is  a  system  is  pressure  J  closed  monitored  by  the  at  two  side, with  open.  Then  opening  valve  Kl  Kl  is  fully,  air, in  the  Intermittent  opening  flush  Reclos-  away.  i t s quota  during  is  left  leaving  of  de-airing,  moment, w h i l e  E  usually  attaching  necessary.  set,  valve,  dissolved  exhausted for  needle  B2,  dissolve  complete,  water  pressure  any  coni-  valves  right  not  the  and  be  Dl  by  unwanted  appara-  de-aired, to  the  pro-  electrical  the  will  is  tedious  while  After  closed,\ with  forming  below  the  partial  not  small  required,  admitted  the  collected  the  F  de-  inadvertent  i t is  i s being  are  that  require  sucking  tank  Instead,  are  are  pressure,  atmospheres CI  D2  be  sample  so  calibrating  Dl,  may  auxiliary  i s kept  when  valves H  adjustable  Bl,  When C2,  valve  When  up  when  E.  through D2  except  E,  not  where  below  water  CI,  and  refilling  F,  tube  as  Bl,  and of  The  does  released  more  valves the  top  suck a  supply  a i r bubble  the  rule,  employed tus,  to  closed  vacuum.  the  admission of  an  by  kept  a  half Kl  B2,  is J  fresh  full, closed  and  K2  rYessure gauge Air pressure  Vacuum  iTo gauge tester  Fig.  39.  Water  Supply  and  Back  Pressure  System,  117 APPENDIX ANALYTIC  ELASTIC  BEHAVIOUR  1.  The  that  for  constant,  is  not  varying is  OF  A  or  independent  with  FOR  THE  SAMPLE  assumed  for  bedded  It  been  is  is  y-direction  horizontal,  usually  are  time.  material  a l l directions  would  parameters  or  the  the  as  homogeneous  elastic  strain  i s , taking  same  evenly  the  that  3-directions  elastic  material  stress,  anisotropy  has  the  direction,  That  the  of  that  further  and  are  type  geneous  x-  and  parameters a  given  isotropic.  vertical  Such  assumed  any  It versely  SOLUTION  Material It  and  C  in  trans-  as  the  elastic  x-z  the  represent  a  plane. homo-  soil.  shown  that  1 6  constants,  such  which  E^  Young's  modulus  in  any  E  Young's  modulus  in  a  can  a  material  be  taken  horizontal  vertical  has  five  as  direction.  direction.  y  y  y  xy  xx  Poisson's tion  due  ratio to  a  for  strain  horizontal  Poisson's  ratio  direction  due  for  to  a  in  the  vertical  direc-  stress.  strain  in  any  horizontal  horizontal  stress  at  right  angles. G  Modulus  xy  of  shear y  It  can  be  proved  Five  1 6  that  elastic  —  deformation y  xy  x  =  —  in  a  vertical E  yx  y  parameters,  and  x  =  xx  v  which  will  See, f o r e x a m p l e , A . E . H . L o v e "A Treatise m a t i c a l Theory of E l a s t i c i t y " , 1892.  on  plane.  be  of  the  use  in  Mathe-  118 later  discussion,  are introduced E  K  i  x  = —  -  y 'xy  T T ; —  - y  E E  K  =  2  x  2c7  2  x2/  X  3  =  here;  1 -  K  xu  ( i +  y  xx  )  tf  y  +  2  XX  /K  -  Z  2  K  K * 1 3  2  1  X  -  2  /K  -  2  K  K  *  1 3  2  X^  a 2  1 The  question  naturally  arises  whether  a  and  a  1  are  real  or complex. E  F o r any =  x  E  y  = y  G xx  =  ^xx  G  K  K  yields  =  1  therefore  a  K  It  parameters;  follows  that  1  =1  -  ]  S  the  expressions  Z  f o r a l li s o t r o p i c  D)  of the p o s s i b l e shows  but i t i s completely  elastic  =  into  materials  2  Examination  tive,  K  values  3  = a  (See Appendix  G  these  2  1  moduli  E  =  xy  =  2  material  = y  xy  Substituting for  =  y  isotropic  i t also  a material  that  G  xy  independent shows may  9  that  exist  values must  of the  always  of a l lthe E  which  elastic  be  posi-  other  i s always  positive  is isotropic  119 except K  will  2  The  be  square  in  the  is  slightly  Thus  G  that  smaller  than  root  a  a  less  y  negative  for a  1  G  than  may  Governing  strain  are  affects  biharmonic oped  1 -  and ,  K  2  2  G  than  without  2  .  .  I f , on  will  be  this K  affecting  quantity will a  In  the  larger  be  K  and  l be  then  case,  3  included  other  hand  G  K  and  K  than  1  real  or  .  xy  .  3  complex.  2  The  which  larger  XX  a  and  The  of  of  expressions  1 2.  is slightly  Differential  conditions for equilibrium independent only  the  equation  i n the  Equation  for  u s u a l way,  of  the  nature  relationships the as  material  follows,  and  of  of  compatibility  the  material,  Hooke's  can  Law.  The  t h e r e f o r e be  f o r the  case  of  devel-  plane  strain. Using writing E  Hooke's  the Law  , for simplicity  -y  -y  xy  -y  five  parameters  in matrix later,  we  form,  already i f we  listed,  multiply  x  through  by  get  X  xx  E xy  and  z  e  -y  . E x x  y  . E  x  y  •y  xx  -y  e  . E Z  xy  E _x T T  xy  E G  'xy  ^xy' x  •yz  ' yz  x  zx  ' ZX  X  x xy  2(1  +  y  *xx  )  X  E  For  plane e  strain, =o  z  T  =  T  yz  Y  = Y  'yz  Since y  =  o  everywhere.  0  -  x  a  therefore  =  e o  xx  O  zx  1  2  -  =  zx  y  =  x  e x x  +  y  O ix  y  3  E  a  xy  a  +  =  e  =  o  z  -  =  ( l - y  x  v  y  H  a  xy 2  and  y  ) a  E  e  '  7  x  =  -  y  y  a  y a ' x x3  -  y  (1 +  y  xy  ) a y  xx  x  + —  « a  E  y  y  ) a  x  xy  -  x  xx  E  Similarly,  x  y  xy  a  E  o  y  =  z  -  y  a z  *xy  E  =  -  y  (1 +  Xy  XX  + ( _ ^ - y  2  ) c  #y  g  X  .J/  Differentiating, 9 e ff = (1 -  9  2  X  -  y  a  )  2  XX  2  2  ~  9y  (1 +  XX  3 J T  9 I  9 x  =  -  y  (1 +  y  ) 9 x  2  a y  2  3 j/  a f .+  2  2  ~  Y  9 e E  y  2  2  *  y  9 )  3 -  E  2  y  2  ^  2  a  )  ^ 9 x  2  y  8 Y 2  _ x  'xy  3 * 3 2/ For 9 x  B  ^  3*3y  2  T  xy  . 1  9  9x9y  2  9 x  9 Y  £"  xy  2  #  3  equilibrium,  2  So  ^  x  =  ' xy  3x9y  °  x  2 G  xy  neglecting  a  2  x  3  _  . 9 1  2  a  29 y  2  2  9 x  a  x  2  £ 2G  body  y  2  x  xy  3  2  a  dy a  2  y  forces,  121 For  compatibility  of  9 e  strain 9  2  x  2  we m u l t i p l y  terms,  9  2  y  xy  ~  dx  0  =  dxdy  by E  through  , substituting  and  regrouping  yields  T-rtK  K , 1  2  K  + K a 1 + 7 - 7 LK a  a  dx  where  3  y  +  dy  If  e  2  x  i  and X  2  y  have  dy  3  the values  + K a ] - 0  x  2  already  y  assigned  3  Introducing  an A i r e y  a  stress  function  such  that  - !!• V  dx'  9^92/  the  partial  differential  1  3.  Solution  1 •* OX  2  be  the boundary noted  that  boundary, lower the of  some  3  (1)  i f  dy  a  to the co-ordinate discussed along  opposite  a t t h e same x - v a l u e .  conditions of x  could  system  i n the text,  the upper  a r e the exact  Fig.8  i t will  of the  of those  In other  be r e p r e s e n t e d  and an odd f u n c t i o n  half  of  on t h e  words,  by t h e p r o d u c t  o f y.  This  fact  that * (*,!/)  Trying  2  ay  f o r any p o i n t  function  suggests  0  conditions  boundary  boundary  2  ox  back  conditions  half  a  reduces to  of the equation  Referring and  equation  a solution  =  of this  f ( x ) . f i  form  2  we  ( y )  note  that  9 cb  f o r zero  friction  2  on  t h e end b o u n d a r i e s ,  9 a ; By  must  be z e r o  for x  = 0  and  122 x  for  =  I  o f y.  independently  This  suggests  n-nxf  (x)  = cos  I  1  giving  1 = Zx  l  d <b — I  _  n TT 2  dx dy  ~  2  U t.  ~  2  2  cos  ri^x  I  '.>*  .  f  S i  , \ (y)  2  (y)  XV  7  3/  . f  I  2  c o s ttll  -  (y)  I  k  h  2  . f  cos  k  " 2  ft IT Substituting yields  into  the ordinary K  /  i,  linear  of  the type K  Solving  l l l l  -  2K  l  2  constant  f o rm  K  2  ±  +  /K  cos — —  +  K  f 3  i V  (y)  -  with  solutions  to the a u x i l i a r y  K m  =  h  0  2  equation  o  3  gives  2  -  2  2  K  K  y  2  I*  ( y )  i  leading 2  2  i  coefficients,  1  equation 2  f  k  * ! l f m I  by  equation  ^1  = Ae™^  2  n 7T m  with  Z-  this  2K 2  (y)  /  1  -  2  equation  (1) and d i v i d i n g  differential  (y)  h  1  a  Equation  1  3  IT 3  n  u  2  K  2  l K 2  n 7T 2  I  2  ± /Z  1  2  -  K K ' 1 3  2  n TT 2  or  a  2  Z  2  a  2  1  where  a  and a 1  four  have 2  required  values  ± ILL. Z  J L  2  the values  and  / c T ^  2  = 4. e x p — —  Z/ a  ±  TL _ I  Thus t h e  J _  / o T ^ 2  ni\y (y)  /  defined.  of m are  1  and  already  -m\y  niry  -ni\y  + B. e x p — • — + C. e x p — • — + D. exp  '  Z / a  1  Z / a  1  —• l/a  123 For -B n  vertical  antisymmetry,  a n d C must represents  -D.  equal  i t i s evident Also,  a different  since  solution,  every  we  A must  that  integer  finally  equal  value of  have  00  4>  =  Z c o s ^ f ^  + A  sinhl!^.  s i n h i ^ )  1  where  the subscript K  If  n  = K .K  2  1  2  f^(y)  expression, the f  usual =  ,(y)  then  antisymmetry  A  -B  =  = a  = a and i n t h e above  1  2  to a s i n g l e  HHM.  B,exp  +  term.  C.y.exp  Iv^aT  Z/5T  the  a  y  the factor C = D.  cross-symmetric,  In t h i s  ~ I  I  4. •  applies,  renders  Hence,  rmx ^  [A  l e a d i n g to  the last  get, for K  we  two = K  2  first  .K 3  ,o>,  Constants  the case  where  a  ^ a 1  ating  terms 1  s i n h H ^ + 4 y coshH^L) Z/a ^ Zv^a  Evaluation of A r b i t r a r y Taking  by  Z^cT  2  ,  case,  D.y.exp  iJoT  of conditions s t i l l  and, s i n c e  o f n.  we g e t  TH- +  2  2  a function  3  reduces  methods, A.exp  indicates  (2)  ,  differenti-  2  (b y i e l d s 8 (b 2  -  T  xy  3a; dy -  lH-OLel*™  I  L  i  ~  h  A .c o s h S  . .. ^  (4)  Z / a .  t=l  a X  S,,2  3&  S  I L  7  a  cos  2  I-  v  7  I  „  A.  X a •,  sinh — 2 .  ^ n  ^=l  , ,— Z / a .  (5)  ^  = if*.  2/  3x  2  - I ^ L ! c o s ^  Z  2  Z  I  * ^=l  A. i  sinh S n  Zv^T i  (6)  124 It  has  ential  already  been  equation,  shown,  during  development  of the  differ-  that E  'xy  x  "G  x  y  ( 1 - y  e E X  x y  T  xy  v  H  X  ) a - y  2  xx' x  xy  H  v  ( 1 + y) a xx'  y  E  e E y  e B xx  But substituting into  a  the above of u  atives  -  B  i . . x dx ^„  (  =  x  +  and  e F y x  x  =  F  F,  —  ) a  xy' y  L i 8y  x  so  get  1 -  y  \i  mrx 2 - 1 : * A . sinh y , £ a ^n ^  .  )  2  xxx x  L  I — — s si i nn , I  ( 1+ y  xy xy  +  1  ) a+ (  xx'  and T from equations ( 4 ) , ( 5 ) and ( 6 ) y xy e q u a t i o n s and i n t e g r a t i n g the r e s u l t i n g deriv-  a n d u , we  y  ( 1 + y  Xy  , a  x  E u  where  - y  x  2 v  ^  ^  L  Z  i=l  A . smh %n  y a  i  (7)  (y)  F  - S . Zv^oT  and  y  =  niry  Similarly  Z / a  2  1 E V X  i=l  niTx sin — —  ) )  xx  I  -  2  - y  xy  ( 1 + y  rat  ) J—  cos  nTrx  2 -Jg v  A.  cosh y  i=l F  , " +  2 n v nit I —  x  —  (  " V  c  o  nirx 2 % . — Z a, ^ n Z . , ^ ^=l  s  F (x)  , c  o  s  *i  h  (8)  2  Ey x'xy  But  yields E  =  E x  x 'xy  L  dy  so, d i f f e r e n t i a t i n g ' 6  dx  . mrx  2  s m  2  ( 1 - y  v  ^  o  r  E  1  i=l  -h  xx  ~2  v  + F'(v) + 1  K  F'(x)  2  (8)  )a. + y ( 1 + y )a. t xy xx % v  E  + y ( 1 + y )a. + ( — xy xx't y M  (7) &  _i 2  _3  n i\ 2  ( i L + i L )  h  y  ) a . }A. y "22  t  n  cosh y. t  (9)  125 Equation  (4) a l r e a d y  (9)  must  b e made  T  and y  writing K  By  and K  through  factoring  a.  , i twill = F^(y).  - F^(x)  coefficients  that  Hence  to render  remaining  boundary  and The  x  (9),  i n terms  the required  between and of K ,  identity  holds  F  these  equal  u  =  Uiy  V  = o  Equation  Therefore  F  and  FAx)  so  odd  E u  =  be a  The  TITXX  s i n —;— i n l"  FAy)  where  x  a l l = o  to =  E u>y x  = E to  Ay)  x  = - E w x  t h e t o p and b o t t o m  for y  ±h  (7), contains  reduces  2  ditions  =  ( y ) , and, f o r t h e end b o u n d a r i e s ,  l  both  on a l l b o u n d a r i e s  ] = x=o  both  must  a n d a l l y.  for a l l X  for y  Z-, i t t h e r e f o r e  diy are  derivatives  conditions require  x  except =  them  f o r E u,  equation  terms  On  the r e l a t i o n s h i p  Equation  3  constant,  or  found  and  xy  out of Equation  2  elastic  be  forT  an e x p r e s s i o n  to agree,  the remaining  2 if  •  gives  boundaries,  where  u  = toz/, s i n c e u a n d  f u n c t i o n s of y , constants  which  h will  h.  f i tf o r y  also  =  -  f i t t h e con-  Putting y  = h  - I  x  i=l  y=h  x A .  +  s i n h h.  E uih x  where /ST hence,  on  u  the top boundary  =  u>h, a s r e q u i r e d , i f  E { a'. ( 1 - y ' ) + y ( 1 + y )} A . s i n h h. = 0 xx xy i a;ar xu xx m i i=l i 1  2  1  v  which  c a n be r e - w r i t t e n  E \ 1  *  2G xy  - /K  z  2  - K K  l 3  } A  sinh h  i n  i  E +  {_£ 2ff L  + /K  2  2  - K K }A sinh h I 3 zn ;  =  0  (10)  126 obtain V  To  E  x  v y=h  ,  =  for y  o  =  - y  +  E  X  F  But  F  SO  F  2  re-write  Equation  r nir rWx ) ) — cos  2  N  xx  * I  u  \  V  -h a.  v  L  %=1 nwx  nir "'I  2  '2  (8)  as  A . cosh h vn  /.  ,  i  i  (x)  2  =  ±h,  , , . (1 + y  xy y  /  =  0 1  2  2  (x)  =  -F a)  (x)  =  -F wx + K  X  giving oo  2  ) > { y (1 ^ ^ xy 1  K  v  + y xx  )a.+ t .  p  -  k + F  x  F (-^ F^  )a r } A . cosh ft. — •% %n l  - y *xy'  1  cos  %  l  wx  = 0 Thus  the  is  constant,  a  double  summation the  must  to k  converge  -  E  series Pn~ -n'  I  cox.  n=l,3,5 converges The even  required values  boundary of  n  value  yield  zero  v  of  If  P  ntrx  cos  2  x  is  to  p 8  (1  I  therefore obtained  coefficients  (A.  and  m  =  o  2x —)  i f  for n  even)  i-1 =  P  If equation F  x  for V  const  any  i t i s established that  odd P  is  becomes  v  V  7T, ~ 2  )  Pn  -  2  8 0  "IIX  TT 2x  =  =  for  I  7  -  cos  )  -  k +  E  ux X  ,  k +  _  EX  value a  of  n.  constant,  the  the  127 This  results i n Fx  P = hlE w x  so In  a  n  equation  (11)  d  so,  re-writing {( ZCr  ^  = ^  (1 +  E  j 2*lL ^=1 i. or  k = \lE u  and  x  /  E  y  '  » l y '  K  /  2  I  and,  ^ -  ^  i  Equation  ( 1 1 ) , we now g e t ,  - Z ) a ^  + J f ^ h . cosh 7z. =  for  odd v a l u e s o f  n  MEu  ' 2,  2 i=l  x  «? ^  xy  ~ \  +  simplifying  \«i ) in A  C  S  + / K  -  2G  - K K  2  2  1  { _£x_ -  +  h  h  —  terms,  a ^ {_JL 1  O  ) A  3  i  V s  cosh h  n  1  >  \  "  (  1  2  )  xy E q u a t i o n s (10) and (12) and w r i t i n g  combining  E B  t  1  2 f f  £ B  xy  = ^ 2  k  /K  =  2 G  -  2  K K *  2  1  .  - /K  —  -  2  xy  3  K K  2  1  %  3  We g e t in 4 £ £ ' a) x n 7T  / a a ' B  2  3  3  /cP B  2 A  n  s i n h 7z  1 2 2  cosh  1  h  sinh  1  ft  1  2  - /a"" B  2  2  sinh  1 2  ft  c o s h ft  1  2  2  kl E  OJ  2  /a a  X n TT 3  3  B  1  1 2 * /cPB  2  2  1  cosh  ft  1  sinh  s i n h ft  2  ft  2  1  - /oP B  2  1 2  sinh  ft  1  c o s h ft  2  128 Checking  that  P  i s a-constant,  =  UllL. I { ( I  *  i  I _  +  a  (  2  A  ft TT  4 Z 2 ? to  -  2  %  value  of  n  1  2  .  2  cosh  }  2 2"  2 s i n h ft cosh ft  2  2  1  s i n h ft cosh h  2  1 1  2  as  required.  the  results  }  "  s i n h ft cosh ft - oT^B  1  cf^B  n  ) 4 „ cosh h  A  2  cosh  1«  3  + a~%  2  1  3  1  1 J  ) 4  1 3  / K* - K K  cosh  a  2  3  2  1 i n  X  =  f o r any  .  1  - K K  2  -  4 Z £ " to  ^  + J K  xy  1  1  3  X a ? } 4 . cosh  2  2G  { a~%  3  ) at •+  K  2 xc 7y  2  ft TT  -  (-~-  1 5  1  L  ^  have,  xy  { a"  I  now  E  2  P  we  1  2  2  - of^B  2  2  2  1 sinh h  1  5 cosh ft  2  cc  Summarizing oo  ftTTX  I coB = n=l,3,5 Z  1 ', a = E to X X  to  v  to X  1 u  T  xy  2  2  2  ( i -  to  +  y  sinhy  { "M»  sinh y  a  1  ^  1  1  { A  2  }  2  l f t  + a A'  sinh y  1  1  2  2ft  +4'  sinh y  1  1  lft  sinh y  2  }  }  2  2ft  i n =  { * \  [  cosh y  n  x  o f \  +  cosh y  n  2  y  ftTT  —  2  XX  .  sin  -  x x ^^  ftTT 5  z  - 1  .,  , ,  E 2 a.A\ sinh y . ^ ^ft i=l  Z oo V  1  2  ftTTX  Z  1,3,5 N  xy  + A<  i  for K  have,  2  S  1,3,5 Z =  2  JI  =  sinhy  n  £  cos —  1,3,5 Z  f a r , we  2  I  oo  1  —  1,3,5 Z  so  2  cos —  rv — U . —  X  E  2  oo  1 E  oo  U ;  obtained  .  ftTTX  I  2 „  i-1  . , l  "  .  u  ,,  2  }  2  >  K K  1 3  129 1  2  oo  —v  =  -  "  1 + U  (  M  ) J ^ _ - cos _ i  ~ " xiv i y 1,3,5 Z V i  )  c  o  1  In  -  =  in  A  ^  / aa  2  .  1 2  / aB  3  2  2?  •^2tt  2  B  1  1  2  / aa  n TT 3  /a~B  3  2  = K K .  B  This  =  2  includes  out p r e v i o u s l y ,  2  we  m a t e r i a l s and find  that  = a  already  wiry 2  d i s c u s s e d , and w r i t i n g  , and  boundary constants i n the case  1  a l lisotropic  ft  1  Z / a ?  n  s i n h ft cosh ft  1 2  2  the s o l u t i o n  those  1  = a 1  1 2  where  case  a  the  1  As p o i n t e d  y  5  s i n h ft 2  cosh ft s i n h ft - / a ~ B  2  the case  others.  1  ^2n  w  a;  now  2  s i n h ft cosh ft  2  1 2  1 2  Using  1  s i n h ft  cosh ft s i n h ft - / a B  1  4Z  some  * f  h  u) i n  n ir  z  s  •==— J4  3  K  o n  equations  4Z  Consider  c t  x  t h e above A'  a  cosh y .  2  i in Z i=l  s  y  + hi  • , a. ^  ^  just  n  Z / a ?  are obtained completed.  ni\h  =  b y t h e same  methods  The r e s u l t s a r e  CO  <f\  E u x  =  I cos ^ 1,3,5 Z  {4  sinh y  m  n  + A y 2n  cosh y  n  n  }  x  00  =  0  ,  1  I i U L cos — ^  5  2^2  2  { (A m  + U 2  ) s i n h J/ + A n  2  n  2/ cosh w } n  as  130 1 E to  y 2 V  ft  2 7T  2  r  1,3,5  ftTTX  '  r  c o s  {  z  + A  sinh w  A  y  cosh y  }  1  5" to T xy  x  a  V I  =  n Tr 2  1,3,5  Z  nTrx , . s i n { ( A  2  , . + A  r  I  2  m  v ) cosh y  m  . + A  °n  y 2  »  n  ,  sinh y *  n  n  i  \  -±^ -  J  1 to  —u 00  ^"HEAT  =  -,  mr  Z  5  1,3,5  rmx  ,  s i n  £  -  •  L  { W  +  Z  ) sinh w  1A  m  2  n  +A  M  cosh w }  y znrn  a  n'  /a  00  W  +  1  y  " * " ^  I — 1,3,5 Z  —  s i n  {  Z  A  s  i  n  h  m  y •  + w  «  A  y cosh y 2« w n f f  tf  }  J  + 2/  1  •y  to 00 =  " aw y  E  *  x y  - x +  (  1  +  y  aa  I ~ 1 , 3 , 5 Z-  )  . " y„,) x  —  c o s  y  {&*  j  + 4  m  ) c o s h u + 4 . j, s i n h y } _ L 2n " 2?rn fa B  n  oo  r ftTr 2 — °s 1,3,5 Z  ftTrx  c  {(A  Z  + A  ) cosh v  2  n  2  J5Z  3£ In  w h i c h ,  i  f  C  - j f  t?. 4 -T^M 4Z  A  ^  2  E  n Tr ' 3  '  K  x  ^xy  s i n h  U  ft  2  *  +  +  h  n  x  > &  c o s h ft "  n  #  3  C s i n h  x  ft  cosh  ft  n  + n  4Z  s i n h  A  2G  /cT  h n  C s i n h  Eh  In  n  c o s h ft + n  2G  xy  / c T  + 4 "  w n  sinh y } n  JoT  131 If  K  <  2  K  K  The various  fore of  solution  expressions  actual hence  same  stresses  contain  possible  real  many  portions  to w r i t e  variables,  2  quantities.  a r e , of course,  a l l cancel  the equations  which  > K K but the 13  2  complex  and d i s p l a c e m e n t s  the imaginary  as f o r K  holds  use l e s s  out.  computer  real  It i s  for this  case  storage  The and  there-  i n terms than  com-  plex. Z  0 0  <D  =  cosh y ' s i n y ' - Z 2  1  in  s i n h y ' cos y '  I c o s ^ { - ^ - i 12 . } 1,3,5 I r\ s i n h ft' cosh ft' - £ s i nft'cos ft'  which  expression,  n  1  1  2  2  the undefined  -  variables  2  are obtained  follows 2K  a' = 1  2K  -  a '=  3  y/TT +  K  13  y  ,  ™y  =  yt  ™y  =  /ji =  Z ^  2  1  =  m  3  in z  {  2  hK  42£  x y  5  2  ./TT* 1  2  2  1  2?  9  1 2  D = n  z  K  2  X  a ' a '• a 'x y /a'a'  y  = _ 1 1  E  1 2  E  v^T "  o s h ft' s i nft'} i 2  3 1 . 2 a ' a ' / a 'x y /a'a'  2  a  .  x y  .  2  *\  C  Zu  2  x y  obtain  1  2?  1  1  we  y/  / a ' a '  /a'  V  2  3  s i n h ft' cos ft' + r~i—r l z  X  v  n7T?J  ft' cosh ft' s i nft'} 2 £T^r i 2 " •  3  putting  „  22T  E  now,  2  1  2Z  = — — 33 n T\  K  w  h  2  - { - r ^ - s i n h ft* cos ^,3^3 2ff_, i 3„3 26' n TT x y A 7  Z  m  l/aT  1  2?  Z  SKK 13  2  i-Z?  1  -  3  2  1  ^  1  2  s i n h ft' c o s h ft' - 5 s i n ft' c o s ft' 1  1  2  z  z  + Jl  a n d  ,/rr  2  the following  X A  z  =  ~  2n  2  relationships  —  2  as  132 00  —— a  =  x  9  Y  0  ) 1,3,5  Z  Y  t I —^ "  cos  SaT  Z  2  1  +  SaT  x.„ X ( ^ + - ^ )  V  r  = *  n  )  IT  Z  1,3,5  nTTX  cos  i  l  sinhy'  cosy'  }-*-  2  cosh y  Z  I  2  sin y  • /aa ' 1  2  a  y  c o s n  2  // aa '  1  J  s i n ly  n  -  Z  s i n h y' cos y' } -j-j-  zn  ,2„2 v  —  T  3?  =  W  )  sin  1,3,5  Z  { X Z  2  s i n h y' s i n y' 1 M  + X_  cosh y' cos y' }  2n  1  u  =  W  n~n 1  v  I 1,3,5  .  J  l  sin  X.„ X. r r r i" 2 { C [ — • - —  nra:  I  U  / o T 1  + 2  + [  y  =  2  + [  n  f Z  ^ 1  i^ " 7  {  in  cos y ; ] Z  (Z v  /a" " + Z  /a ")  m  1  7  2n  2  }  : s i n 2/2  C 1 + y ^  /a ") 7  C 1 + ^  sinh y '  ]  sin y ' /a'a'  5  ]  2  "2  y  cosh y j  2  cosh y' s i n y'  +  T  1  1  Z ^ s i n h y; cos » ; ]  1 , 3 , 5 "  + x  in  /o7 2  -  C0S ~  Z  ^ , ? j 1 J cosh y s i n y'  s i n h y' cos y' ] [ 1 - u  +  ~  cosh y' cos y' ] 1  2  £  L i - [—*a'  +  a'  w  -  y  2  ^  ]  }  133 APPENDIX LIMITING  VALUES  Considering of  anisotropy  already  of  the various  Investigating  described,  elastic  constants  the behaviour  f o r the constants,  exceed  their  values  that  limiting  i s that  the s t r a i n  x _  y x  y E  •  _  E  y  x _  XX  be  relationship  mentioned  earlier.  m a t e r i a l , with  The u s u a l  always  be  i f the  constants for limit-  o r , to ensure  positive  constants  numerical  criterion  stable  type  i n the  (Love  this, 1 6  ).  form:  X xy r  x  1_  "xy E  X  has been  and  the  XX E  i  y  E  should  the e l a s t i c  -ESL E x  E  t h e number  n o t be v a l i d  the material  y  1_  would  CONSTANTS  c h a r a c t e r i z e d by  of such  values.  energy  Taking  ELASTIC  a material  values  ing  OF  D  E x  x  xy  xy  2(  l + y E  _  x  The  strain  per  unit  finite,  energy  volume.  problem  the  bounding  stress  caused If this  the s t r a i n  The  1 7  ) XX  energy  has been  fields  values and  by  a  set of stresses  quadratic  form  will  be  stated o f y by  by  _  then  considering  the c o n d i t i o n s  l  i s positive  positive  Lempriere  O ls-~-a C  1 7  ,  who  various  necessary  as  O  de-  required.  then  deduced  possible  to ensure  that  the  Lempriere, B.M., "Poisson's Ratio i n Orthotropic Materials". J o u r n a l , A m e r i c a n I n s t i t u t e o f A e r o n a u t i c s and A s t r o n a u t i c s . V o l . 6, No. 1 1 , N o v e m b e r 1 9 6 8 .  134 resulting  strain  analysis,  which  below. tic  The  form  cipal  minors  of  is positive.  considers  necessary  should  1_ E  energy  be  and  a l l the  C should  be  more  elastic  sufficient  positive  A  rigorous  parameters,  conditions  definite  are,  positive.  So,  that  that  form is  given  the  a l l the  of  quadraprin-  putting  D  x  y  1_ E  V E  E  x  x  xy E  x  y  y E  r  xx  E  x  E  x  -  xy  E  x  y  XX  _  X  D  x  1 i? a:  xy  E  x  y  1_  xy  E  D  E  x  1 E  =  1_  xy  x  0 =  1_  All  D  should  xy  D  G  0  0  1_  0  xy  =  D  1  0  'xy  0  0  1  0  0  Gi 'xy  0  the  Gi  D  1  'xy  0 2 +  2y E  0  x  a: a;  D be  values positive,  135 D  Since will  ^ i — 3  G  ^  "  will  xy  t h e main  it  be p o s i t i v e .  of  i f G  diagonal,  i fE  D  .E  z  2  3  x  by E  minors  'x E  =  l E  G  xy  2(1  + y  x  / E  positive  G  i t s only  y  will  E*  or E  3  E^  must must  2  xy y  be  must  be  -y  i s evidently  2  i f E /E  i s positive,  this  i s sat-  energy  their  reduce to  positive  *  C  l  ^xx^  +  m  u  s  t  b  e  P  o  s  i  t  i  always  i s positive  y means  and g r e a t e r  than  D y  2 2  c a n be .  Since  xy  E  that  negative,  must  be  positive.  y The  tive  and t h i s  last factor  condition  requires  can then  that  be o m i t t e d  1 + y  be  posi-  D , since, i f  from  3  one  factor  i s always p o s i t i v e , In D t h e t e r m E /E  x  3  E 1  x  e  positive  x  be  v  positive  xy  x  i s that  positive  Kyi  2  itself,  the elements  n o t change  strain  be  by  be p o s i t i v e  ,  )  only  appears  multiplying  be  •CSC  t h e rows and  and h e n c e ,  ~ xx>"  { 1  i s equal  requirement  D  that  for positive  - y  y  way  always  xy  must  Since  E  In whatever  The r e q u i r e m e n t  = E /E x  *  5  = gi— t h e r e f o r e  D  D  i s positive,  D , which  and t h e r e f o r e  and thus  so, the c r i t e r i a  D  >0,  i s positive  the other  sign  xy  be p o s i t i v e .  on  i fD  that  3  of C are interchanged,  columns  isfied  i t i s evident  xy  be p o s i t i v e  D  to  = D  *  discussed. /E  i s always  y  - y  both  I t i s evident  must  positive  be p o s i t i v e .  be p o s i t i v e ,  -1 < y  y  t h e o t h e r must be t o o . (1 - y ) - 2y remains  that  - 2 y  s o , t o make Since  xy  2  xy  i s negative  t h e term  1 - y  to  2  xx  arid  positive,  and 1 + y  < 1 and t h e c o n d i t i o n s  must we a r e  136 finally  left E  a)  with  are:  ,  and  E  x  Absolute  c)  E  /E  y  xx  bounding , y  except  The  surface  space.  xy  G  xy  shape  The  , which  of  of the boundary  never  exceed  y  surface.  = -1, where  Fig.  less  2 y  >  2  ^xy  expression  c)  surface  includes  and  =1  and  we  a l lthe  c)  can  x  that  i s part  However,  there  i s a discontinuity  40a.  of the  for Elastic  values  Constants.  y  The could  continuous  F i g . 40b. Bounds  others.  i n F i g . 40a.  possible  ,  parameters  of the  i s such  y  ^  limit  containing  obtain  i n E /E  = -1  xx  i s shown  expression  one.  this  the s o l i d  Energy  y  independent  boundaries  of  than  zero  to zero,  y  xx  than  0  between  the energy  unity  ) -  ^xx  greater  of y  i s completely  shape  bounding  y  equating  each  y  value  (1 -  y  x'  G x  b)  By the  y  at of the  137 constants, G„  is  i t must  cut  be  off  by  a  remembered  vertical E  that  ent  (so  only E  long  the  /E  .  x  as  a l l three  Poisson's The  are  Ratios  actual  shape  In are  y  p o s i t i v e ) and  which of  E  and  x  xy  wall.  are  bounded  and  any  any  vertical  also  independ-  really,  i t is  by  is  by  straight  y  interesting  =  1,  bounding  surface  section with  constant  y  to  lines  y  =  '  xx  is  of  parabo-  note  that  passing  E  constant the  through  is /E  y  line  =  further points G  of  is  and  y  line  represents  y  Hence  will  be  =  seen  - 1 .  is  of  could E  point  interest  emphasized  by  to  that,  have  stresses  strain  energy  the  a  of  a  00  y  would  + be  2  zero, =  4  y_  xy  only  a  parabola.  be  generated  /E x  =0,  y  emerge.  line  given  -  i f  ff  behaving  y  the +  represented  material 2  y„  .  xx  by  2  a  y  under  In plane  to  zero  Figure °  by  a  E  N  y;  shear equal  and  y  off  40b).  this at  y  point .  It  line. = on  is  under  stress.  0.5 this thus equal The  a l l round  a2  }  *xx>  has  E  isotropically  subjected  *xy  = x  cut  r-rr—• 2(1 +  =  xy  The  (Fig.  l i e on  represented  by y . -  4  must  AB E  which  i t is  anisotropically material  x'  1  materials  material  material  and  for  expected,  i f and  normal s t r e s s a i s r ~ { 2 + E /E  is  as  However,  normal  this  a l l materials  isotropic  isotropic  possible  2E  is  y  xy  This  parabola r  x  surface the  a  xx  xy  independence  x  ratio  0.  Some  E / E  the  the  h o r i z o n t a l section with  It  If  to  y  loid,  It  addition  bulk 40,  surface  modulus  such  passing  and  material through  the  138 E  points v  x  /E  E  y  ,E x/  E  This  y  x/  surface  obtained, y xy y = xx line.  is  y  -1.  =  =  0  y  =0.5  y  =  0  y  =0  y  xy  xy  xy  tangential  0  to  i t along 1  to  the  the  a r  y  a  xi2E  there  strain  1  x  is  uniaxial  {  =  xx  0 =1  xx  boundary  from  the  E /E = x' y materials  point  stress  is  the  condition  -  2  H  y xy  d i r e c t i o n of  material  has  volumetric uniaxial portion  y  xy  of  the  and,  hence  /E  when  subjected  that  with  a  by  i s negative in  the  to  the  and  y  when  sidering  boundary possible  rj  and  test. seem it  compressed. behaviour  can  without  indicate be  seen  which  a  violating The  caution to  a  is  needed,  certain  especially  in  This of  amount  It  point  be  usual  the  =  °»  material,  for of  , a  energy  anisotropy layering sands,  xy  upward  in  If  a  xx this  plane,  swells  under  40b  shows  the  this  plane  actually  swell  of  course,  material  may  Z may  be  of  interest  clays  of  kind  possible  considerations  uniform  y  y  outside  which  O  subparallelism of  Fig.  follows,  and  y  sloping  lies  precompressed  i t may  the  same  . a  small  anisotropy  that  E  the  2  material  X  P a r a l l e l i s m and to  the  materials  compressive  large  the  plane,  direction.  y swell  a  =  y  and c o n t a i n i n g t h e a x i s o f xy l a r g e enough to put i t o u t s i d e  energy  shows  to  y  compression  /  x  K  applied  i f E  strain  represented  strain  E  } a y  volumetric  is  point  4, y = 2 , ' xy ' m u s t l i e on this  SB  This  already  is  E /E x' y  no  =0  xx  energy  line  A l l incompressible  volumetric  Thus,  \i  =  xx  If the  y  -2  touching  0,  H  ,E  =  clay under  for  under  in  con-  triaxial  p a r t i c l e s would discussion,  such  soil  to  and  swell  criterion. may in  apply natural  (greater  i t might  to  or  also  sands,  sands  may  smaller) be  due  to  but be  but, a  due  139 statistical anisotropy would  be  discussed  as  orientation the  likely  to  herein.  of  interparticle  second, depending take  a more  on  contacts.  arrangement  c o m p l i c a t e d form  of  than  Such grains that  140 APPENDIX EXPERIMENTAL  Installing tured  Disturbed  membrane  delivered diameter without  must  with hole  any  a  Sand  in  the  be  The  specially  prepared.  tubular  upper  remainder  PROCEDURE  Samples.  first  short  E  neck  face.  (which  The  might  tube  fold  in  seating)  and  without  unduly  enlarging  the  holes  in  the  top  bottom  must  be  screw  holes  bottom  in  plate  bottom  it  The  are  is  membrane fit  i s then  into  the  aluminum  stretched  to  into  membrane  are  adjusted  3/16  to  rubber  7/32  membrane, purpose place  between  of  the  to  from  and  The  exterior  cated  with  the  are  sides  silicone  then  to  and  position,  o i l .  and  fitting  the  than a  finger  membrane  through of  block.  corners  pieces  of  The  the  the  i n . diameter  the  The  An  The  aluminum  place.  and  membrane,  corners  1/16  the  and  which  of  cor-  place  pushed  corners  in  plate  in. less  the  the  sample.  block. be  the  that  stage.  into  The  and  of  the  block.  The  moulding  mouldings  in  are  in  the  corner,  equidistant  located,  the  stretch  of  the  mem-  against  the  block.  mouldings ends  to  in  badly  the  this  the  these  in  of  the  plate  of  plate  top  placed  hold  Once  and  the  out  .005  moulding,  inserted  bottom. these  space,  membrane is  of  the  the  with  the  the  block  f i t to  are  a vertical  retains  the  neoprene  block  they  adjusted  brane  or  to  in  small  plates.  i t so  to  folds  about  proper  Four  downward,  across  shows  removal  sample  i n . long  while  top  the  21  hole  allow  and  of  a  later  hole  Pieces  Fig.  off  made, m a t c h i n g  membrane  through  inch  cut  prevent  spreader  or  are  half  be  hole.  otherwise  seating  the  must and  face  Also,  required,  having for  the  screws  then  inserted  top  by  with  block  and  f i t , as  hold  to  f i t properly  puncture.  block.  can  then  to  mounting is  bottom  is stretched  must  interfere  size,  is  to  secured  and  membrane  enough  sample be  the  liable  then  i n s e r t e d , convex  corners  will  top  membrane  of  sharp  then  steel  then  the  corners  corners. ners  the  is  membrane, a n d  and  membranes  attached  This  manufac-  of  The  in  position,  the  membrane  membrane  are  i s now  then ready.  lubri-  141 The horizontal holding sides Fig.  next  travel  stops  the moving  and u p p e r  step  aluminum  block  the  machine,  ensuring  them using  entered,  i n place  the rubber  ings while then their  that  on o n e  41.  fairly  carefully  Once  must  Using to press  Extension  be  a piece  o f End  grooves,  a t t h e same  time,  and h o l d i n g  inward dowels  are placed  until down  they  into  of  retain  i n place, shim halves  both  them  i n position.  the lower  will  Plate  doing  are secured,  their  are cor-  the upper  their  sides  they  with  into  enter  secured  into  t h e box upper be p r e s s e d  block  box  and membrane,  o f t h e box s i d e s  easy  Removing  mouldings  side  locating  halves  machine,  end p l a t e ( s e e  mouldings  o f t h e end p l a t e s .  and  and t h e upper  lowered  the rubber  i n i t sbase.  i t i s then  Fig. of  pins  i n the shear  block  are then  and t h e m o u n t i n g  the taper  material,  inside,  the lower  the v e r t i c a l  of the p r o j e c t i n g  The m o u n t i n g  i n the sides  that  i n mid-position,  extension  the  rectly  are i n place  parts  41) a r e r e m o v e d .  housings  i s to ensure  by  sides,  mould-  i n place They  must  screwing i f this i s  142 not  done,  Once  the rubber  this  retained  has been  done,  i s now  brane,  ready  design  are inserted  membrane,  of  right  has  with  The  reached.  water,  parent  be  from  water  corners,  b l o c k may  ( F i g . 42).  as t h e sample,  This  hopper  and i t s i n t e r i o r  Fig.  so t h a t  Around  42.  and  and  Loading  retained  now  be  removed  from  i n F i g . 22 with  de-  any a i r b u b b l e s  filled  made  nearly  has p l a i n ,  of  trans-  to the top  vertical  cross-section i s  can f a l l  of the hopper  Hopper  outwards  on t h e  filled  hopper,  t h e sand  of the  c o r n e r s of the top  horizontal  the base  special  and p u l l e d  i s then  A special  installed  of a  illustrated  space  o f t h e mem-  top opening  c a n be  temperature  etc.  i s then  place.  hooks  corners,  handles  The sample  the  into  Four  and t h e s t a g e  faces  same  t h e mouth  the centre  aluminum  at ambient  plastic  to open  i t s upper  interior  down  corners w i l l  provided, at the four  t h e membrane,  removed  t h e membrane  through  i n their  specially  been  aired  grooves.  the sample.  into  the machine.  inside  out of t h e i r  necessary  to form  the holes  screws,  spring  i n place. It  until  mouldings  i n Place.  straight  there  i s an  143 O-ring does  seal,  not  then  so  leak  that  away.  sand  prepare  i s weighed  suitable.  The  void  of  ratio  usually excess  and  the  of  sand,  and  would  three  amounts  It  weight  of  slight grain  and  inner The  sand  glass  but  the  tube  i s placed  below  sand  will  r e p l a c e d by be  moment  kept the  The  sand  and  by  of  is  pour  t i p of  keeping  a  the  tube out  the  during a  silty  into  sand,  two  one  or  supply  that  the  up  the  tube  any  with  to  that  the  t i p of  water  water,  and  in  the  hopper,  i t s volume  water than  and  inverted  stops  long  from  the  a l l flow  i n water  trap  i t s neck  Good  place  to  i t s very  flask.  the  in.  hollow  i t s contents  free  A  i f i t is  that  from  1/4  tendency  into  gently  supply,  conical  fall  is lifted  weighed  a i r . Vacuum  about  water  such,  the  the  sample  will  because  i n one  to  range  d e - a i r e d water.  prevent  into  exact  percentage,  prevalent  slightly  filled  will  small  the  with  a  under  more  tube  is  slight  or  up  remove  glass  s u r f a c e of out,  a  the  reduces  Instant  drawn  is  leaving  i s added  to  condition  situation,  g r a i n s pour  not  The  as  using  and  hollow  in a  make  dry  desired  grams a  course,  to  brim  having  pressure  water, the  enough  ISNthen  the  of  the  somewhat  material  water  d e - a i r e d sand  tube.  to  of  through,  i s now  glass  begin  flask  the  tube  only  graded  and  apportioned  (This  120  of  flask  on  surplus,  present  filled  just  the  size  Before  atmospheric  out,  size  to  or  have  the  ml.  depends  a well  cooling.  the  can  water  p r o p e r l y p r o p o r t i o n e d , the  for long  f i l l  be  of  excess  500  segregates  require,  be  any  half  face,  flask  inverted,  of  to  A  115  to  form  grain  installed,  pushed  The  coming  be  i s then  sand.) tip.  have  boiled  must  diameter, the  would  during  sand  separate  This  A  about  With  of  sizes.  flask  stopper  at  the  supply  slight  required  grains  to  a  flask.  but  the  surplus being  applied  the  sand  ranges  supply  supply,  in a  sample.  is required  material is  in place,  prepared  is desirable  desirable  range. of  of  finest  different  each  placed  because  the  be  sand  specimen,  non-representative  the  previously  a  amount  adequate.  placing  for  A  i t i s secured  required. To  it  when  will  control the surface.  In a i r ,  enough  to  establish loose  any d e n s i t y  current,  i t i s possible  sand.  The d e p o s i t i o n  of the sand  controlled.  I t i s probably  impossible  form,  b u t by p o u r i n g  ensure  that  geneities, silt  by  layers  be  o f one s i z e  supply  towards  by  sizes.  structure what  difference  extent  sample sible  must  this  well  be  jarred  closing  be u s e d  the d e s i r e d  ficult  to p l a c e , at t h e i r would  motion  during  unduly  compress  placed  sample  top  o f the box The  manner.  later  stages  Very  as c l o s e even  sand  making loose  The  surface  has  deposit,  depend  the  to  dense as  pos-  loading  travel  samples  are  They no  of the  may to  dif-  should  excess.  Excess  vertical  and t h i s  would  aim i s to a r r i v e to the d e s i r e d  a t t h e same  dif-  i t impossible  but with  of preparation,  can  of the sand.  vertical  the weight  as p o s s i b l e  of  A very  is critical.  sample.  this  a sample  layered  and p l a c i n g  height,  carrying  a loose  height  at a  density as t h e  sides. corners  of f a l l i n g  necessary  nominal  such  roll-  other  t o g e t i t as dense  the sand,  load.  the  deposit,  desired.  thin  to grains  the sample  the e n t i r e  and t h e h e i g h t  involve  a level,  pattern  normal  full  height  with  t h e membrane  only  On  to the s t r e n g t h  and v i b r a t e d  i n densifying  apply  a natural  Where  approached  definite layers  whether  density  i s n o t done,  sample.  inhomo-  size.  c a n be  due  a natural  f o r completing  the r e l a t i v e  I f this  start  makes  on  before  head.  imitates  of  to  d i f f e r e n t to  and a l l o w i n g  a uniform  I t i s n o t y e t known  Techniques some  flasks, uniformity  to imitate  truly  pattern  The m i x i n g  uni-  i t i s possible  t o t h e sample  d e l i b e r a t e l y pouring  that  to get i t t r u l y  a regular  i n turn  very  be c a r e f u l l y  are s u f f i c i e n t l y  at a time.  tends  to o b t a i n  must  pattern,  compared  flasks  i f i t i s desired  ferent  form  or the s i z e s  the various  attempted  or  are small  o r more  sideways  hand,  a  which  two  using  ing  the v a r i a t i o n s  i s present,  require  i n a regular  144  to bring The u p p e r  cause  sand  and c a r e  the sand edges  unexpected  up  interference  and f r e q u e n t  the  pauses are  i n an a p p r o x i m a t e l y  of the hopper  with  are m i l l e d  uniform level  and  145 by  laying  a  right,  the  can  used  be  into  a  blade  minimize  so  that  amount  density. when  the the  flask with  even  as  inverted weight  of  percent  the  sand  sand  of the  and  i n the  draw  reach  this  will  not  rough  the  void  neck  of  remaining  dried  i n the  In the  cor-  achiev-  enough  can  be  to  obtained  With  scale  bottle,  weighed  in  flask.  the  pouring  and  help  ratio,  pouring out,  by  intended  guide,  the  desirable  that  the  be  excess  place.  ensure  will  A  any  surface,  exactly  and i t  course,  first  to  to  height  sand  sample.  to  of  correct  upper  placed  but  i t s length  I t i s , of the  excess  scale  and  of  been  with  surface  possible  the  weight  a paper  the  i t is desirable  has  one  them  middle.  i s complete  within  to  upper  d i s t u r b ance  course,  correct  the  i n the  sand  sample  fixing  the  sample,  of  Of  indicate by  this  over  reaches  smooth  pile  the  spatula  just  i t i s as  finishing  ing  to  small  to  rect  T-shaped  is  the  marked  for  various  levels . Having sample  space  siphoned which to  enough the  out  was  take  of  care  the  to  the  As the  first  the  plus edges  the  into the  help  be  the  f o r the water  that  and has  to  to  very  siphon  excess  brought  be  removed,  remain  sand  part  gentle.  It i s  properly, by  cause  about  gently,  at  accident.  a j e t of  placed  the  of  i t s level  starts  within  is  close  up  and  within  the  It i s necessary  pick  the  in  water  i s not  flow w i l l  probably upset  may  any  siphon  be  sand  excess  middle.  flow  of  subsequently broken  been  found  the  i s removed  the  i s not  hopper  will  In required.  level  of  the  uncontrolled  hopper  the  the  and  at  f l o w must  ensure  reverse  water  water,  intake  the  amount  together with  a pile  excess  sand,  the  sand,  of  the  correct  i t s surface,  surface  attempt,  Otherwise  Once  the  to  the  hopper,  into  sand  necessary  water  the  that  sample.  also  levelled  scraped  approaches  of  and  deposited  and  sample. a 1/4  in.  the  sur-  up-turned  membrane. closing  Most  of  in achieving  the very  the  membrane,  precautions gentle,  very  delicate  described  are  g r a d u a l movements  handling intended until  the  is to  146 membrane sure  is sealed.  t o wash  Three  rods  holes  i n the  smaller  the  are  This  particles  needed  upper  in. thick,  the  machine  and  two  slots  i n . apart  in  the  long  a  basin  and  place.  This  The  special  the  plate  the  rods.  the  plate  slight let  a  rest  on  above  now the  adjusted  the  be  by  square  and  sides  of  sample.  plate  i s now  This  the  must  be  disturbance  done  plate,  where  grains  i s tedious will  densify  the  and be  be  membrane  The  of  the  support  they  with  are  then  half  wash be  i f any  effected  as,  to  the  sand  grains  tolerated. are  of  by  and  sample  the  plate  water  and  be  spreader  above the  sand  the  rods,  the  surface.  the  resulting  the  top  of  Removing  any  such  onto  means  to  rods  the  then  the  not  a  strips  If  up  with  care  must  overlooked,  except  necks  under  i f i t slips,  water.  the  plate,  lined  into  in  supporting  unscrewing  way  under  in  holes  machine.  i t s edges  gently  taking  strips  water  lifted,  the  the  43.  four  in  supporting  partially  about  under  the  above  of  the  in Fig.  kept  i t s surface just  The  cannot  their which  the  later will  also  sample. Having  must  with  to  are  distance into  carefully  they  not  sample.  members  By  lowered  of  t r a p p i n g a i r , but  the  true,  i s sure  removal  water  that  plate  be  engaged  the  surface. so  are  under  correct  eye,  to  three  strips  located  sand  into  and  end  i n use,  i s placed  rods  into  the  accommodate  plate  strips  frames  mid-point,  ( d e - a i r e d ) i s pushed  screwed  support  end  to  de-aired plug  prevent  dig  screwed  should  the  is slipped  corner  plug  the  of  strips,  their  appear,  is  screw  a neck  of  rods  spreader  has  side  These  plate,  to  each  f i t the  support  across  enough  are  the  The  tilt,  should were  and  reach  to  rod  two  wide  porous  rods  are  movement  undesirable places.  Each  and  upper  allows  one  sudden  threaded  plate.  to  any  into  end  required  having,  the  sand  one  enough  rods.  The  with  Also  1/8  necks  of  spreader  diameter.  1  i s because  allowed are  partially  to  easily  close,  lowered  over  sufficient  i t . to  the  plate,  the  The  forces  in  move  the  plate  membrane the  from  stretched its  147 correct at  position,  once,  letting hand.  them This  brane and  lifting  so i t i s b e s t the hooks  come  forward  the r e s u l t  to snap  clear  and e v e n l y ,  operation, i t hard  i s a disturbance  two o p p o s i t e  of the machine  gently  is a critical  a r e enough  t o do  i n the water  plate.  As  over  the p l a t e ,  to hold  free  and  some  machine the  of the surplus parts  capacity  were the  required necessary  using  i s not serious  sizes.  i t i s necessary  the  porous  through the  held is  been  by  may  through fine  grains  membrane  invisible  that  t o make  frequent In  level  above  t o d r y and water  t h e edges  After  with  the  t h e membrane furtheri t  t o be  any  grains  and t h e t o p o f t h e p l a t e . satisfactory,  surrounded  finger  the area  as s m a l l  by w a t e r  t i pwill  i s somewhat  A  grains  and v i e w e d  detect  check.  of  a l l four with  proceeding  appear  pauses, doing  the excess over  in  even  very  encumbered,  I t i s , however,  i s very  light  and does  t h e membrane  snaps  down  not drag  the  with i t .  mistake,  edge  bearings  not a v a i l a b l e  begin  i t and t h r e e  there  but f o r  water.  over.  Before  reduced  If a l l  be one c o r n e r  a reasonable  the touch  Although i n fact  of the p l a t e ,  movement  A  sand  will  rods.  and, a l t h o u g h  i s possible  edges  over  whether  t h e membrane.  essential  by  there  flow  washed closes  needle  surface  cannot  i t must  t h e membrane  almost  were  surplus  the d i s c  t o wash  removed  to check  away  i n s p e c t i o n i s not very  become  ones  the water  otherwise  smoothly  between  i n the design,  d e s i r a b l e to suck  supporting  advisable  visual  it  It i s also  lying  back  trapped  t o keep  and i s l i a b l e  have  membrane  to suck  so t h a t  the d i s c ,  plate  hooks  disc,  spilled.  I t i s d e s i r a b l e t o make  tube  so,  air.  is  this  a plastic  entrap  water  are rust-proof,  steel  slip  t h e membrane  be  and s t a i n l e s s  to  and sand  will  intended  each  i n t h e mem-  water  of load  then  one w i t h  i f i t i s allowed  the top of the spreader i t scapacity  and  the forces  onto  corners  will  force  of the p l a t e ,  i t does except a small onto  not c l i n g  very  a t t h e corners. quantity  i t supper  hard  closely Thus,  of water  surface.  when  released  tothe any sudden  out, round t h e  This  would  also  148 be  likely  to  use great  top  t o wash care.  of the p l a t e  them  along,  work  them  under with  to  alter  avail the  care,  indicate  while  is difficult  without  dragging which  onto be  upper  screwdriver  grains  must  been  at this  the l e v e l  must  be  only  enough  stage.  have sides  t o be  done  t h e t o p , by suitamounts  disturbance o f any  Reinsertion  of the water  and i t w o u l d  above  of  the sand  accomplished  unevenly  o r i n any  the sand.  plate  c a n now  be  by u n s c r e w i n g  unscrewed,  a little  at a time  Closing  or back  It i s not, usually,  surface,  43.  to ease  any c o n s i d e r a b l e  have  the sand  Fig.  over  on  and e i t h e r  an o p e r a t i o n  grains;  anything  shift  possible  I t i s , of course,  a t t h e membrane  would  The  there  keeping  surface  manner  too f a r .  appreciably.  t o t r y t o undo  hooks  other  odd s i n g l e  the sample  are discovered  i t i s often  Such  not to drag  that  necessary  i n t h e t o p o f t h e membrane  of the p l a t e .  f o r removing  sand  stage,  o f sand  t h e end o f a s m a l l  t h e membrane  of  t h e t o p and i t i s s t i l l  I f any g r a i n s  out of the hole  great  able  over  at this  with  t h e edge  pulling  sand  Membrane  lowered, the rods.  i n turn,  Over  very  The r o d s  to lower  Spreader  gently, must  the p l a t e  Plate.  150 material which  would  would  need  have  Remoulded  Clay  remoulded  clays  cut  an  up  in  the  the  same the  head  the  gauge  thousandths  as  appears  to  sample  Having head right  will  the  outlet  dial  and  the  minute are to  may  steps  of  is  outlet  with  the  can  then ten  be  pounds  pressure  out.  per  during  the  can  then  of  set  which  the  disturb  ruptures.  outlet  into to  test.  to As  be  each  during  to  stand,  most  reached  pressure under  bubbles  mid-range,  i s switched  on  and  both  inlet  flush  in the  reduction for  a  dissolved air  i t is  de-airing process.  for  to  pressure  and  the  pres-  checked  slightly  and  is  above  expected  each  Most  seconds,  mid-range at  flow  pressure  inch,  opened  the  valve.  few  the  must  dial  release  allowed  first  of  locating  complete  or  back  two  reduced  means,  the  or  pressure  through  Having  to  set  weight  one  in  air  Once  the  allowed  this  be  without  the  then  square  are  passages  admitted  while  valve  success  read-out  the  apparatus  is  of  the  the  flushed  By  of  added,  the  motion  be  concern  the  chance  and  closed  set i t  motion.  leaking  during  d i s s o l v e d a i r and  is  so.  flushed  pressure  through  approximately  or  water  them,  d e s i r a b l e to  carry  slack  start  opening  a  decrement  check  gauge,  valve  Water  may  to  for  movements,  slight  increment  test.  membrane  on  remould  water  admitted,  by  anticipated  to  been  tests  water  i t is  later  pressure  highest  steps  the  no  has  of  vertical  pounds  in  water  be  the  vertical  start  the  to  matters  full  the  against  and  purpose.  sample.  that  five  raised,  out  on  machine,  m a t e r i a l , and  valves,  i t will  slight  least,  and  truly  any  will  flushed  The  be  pressure  the  be  leaks  the  accomplish  remoulded  mid-travel  there  through  leaks.  principal  inch  about  air  sure  The  of  at  set  at  undisturbed  no  into  i t i s necessary  an  opening  cause  To  as  lifting  once  the  of  be  an  for  Samples.  head,  Thus,  gauge,  the  at  built  block  i s admitted,  loading  dowel.  modified  should  Before  pressure  be  Sample.  loading  samples  silts,  way  should  dial  have  Silt  and  there  bubbles. a  and  appropriate  Checking that  to  to  necessary  The  pore  and  outlet  151 valves with  are  the  solved,  closed  valves or  a  slow,  but  inlet  valve  flushed few  in  the  closed,  leak.  is  i t is  The  i t becomes  head.  If an  indication  pressure  readily  re-opened.  out  quite  readily  seconds  every  minute  drop  may  apparent  Any by or  the.pressure  be  dis-  imperceptibly  the  jump  entrapped  air  is  until  off  a i r being  by  opening so  of  drops  the  outlet  the  air  when  the  usually valve  for  been  re-  has  a  moved . Application or  flushed  be  opened  valve  is  of  Normal  out,  the  again left  open.  to  the  back  minimum  of  one  and  expanded  to  The  a  the  gantry  maintained  with  an  dition,  the  connection  into  been  the  made,  must  is  that  the of  the  water  is  desired  shear  the  piston  upward  pressure  which  just  the  be  pushed  loading  forces  on  the  test,  kept  a is  fully  When  vertically  is  usually  balances In  hand,  this  piston  is  machine.  down b y  head.  then  centimetre  loading  The  must  is  box.  normal  the  not  inlet  the  always  secured.  of  the  for  and  motion  will  system  square is  dissolved  and  but  in  per  been  closed  membrane  the  carrying  be  has  finished,  kilograms  that  vertical  piston  a l l air  pressure  walls  place  the  valve test  half  into  of  When  pressure  swung  weight  the  ensure  against The  then  outlet  until  reduced  recommended,  Load.  this  to  the con-  screw  connection  moving  parts  by  pre-set  its has  are  as  follows: a)  Weight  b)  in  The  back  pressure  the  loading  The  to can  force  in  the  and  pore  balanced  pin,  acting  water,  acting  by  vertical  in  the  sample.  connecting  plugged  into  and the  set  the  lead  After to  the  i t s diaphragm, to  back  a  suitable  pressure  and  normal  and zero the  this and  up  on  shear.  on  the  upward  piston.  exerting  be  on  the  i s , therefore,  head,  depending  head  balanced  piston  loading  on  the  locating  the  switched  down,  pressure  motion  downward  acting  a  load  zero the  intended  piston  read-out  read-out scale  net  is  factor, normal  load.  152 The  top  Fig. ton  of  4 4 .  the  machine,  Pressure  downward,  gradually  from  the  ton  just  balances  gauge  will  move,  (about  .002  found,  of  It  will  its  hole  is  now  or  a  in.).  found  and  giving  new  zero  Fig.  zero set  4 4 . After  normal the  load  test  whatever  when  i t may  i s the  the  Top  the  of  time  the is  most  load  piston.  to  the  to  When  the  normal the  dial  so  far  balanced. freely  load  value  pis-  setting  method,  moves  pisback  vertical  is just  p i n now The  the  its original  sensitive  and  force  in  in  read-out  must  be  noted  scale.  withdrawing  wished.  to  resistance  pressure  locating  Machine,  sample  the  back  back  effective  the  pressure,  withdrawn.  is increased  and  back  the  be  on  to  suddenly  This  that  pin  is illustrated  admitted  transferring  the  quite  stage,  slowly  locating  knowing  be  this  i s then  pressure load  at  to is  the  Assembled vertical  whatever then  value  allowed  to  Over  Sample.  locating  pin,  i s desired  the  for  consolidate for  153 APPENDIX CYCLIC  Haney natural limit of  5  Few  of or  26,  a  6.  tests  were  about  1.6  to  hours  at  5.27  per  with  cycles sq.  cm.  away, With  same  as  about  pressure only  Columbia  percent,  intention  clay.  with  and  i s 2.7  sq.  cm.  when  a  a  kg.  was  a  Its  plastic  sensitivity per  to  sq.  cm.  establish  kg.  to  the  about  which  strain  cyclic the  ±  stress  kg.  rose and  rise. test,  sq.  but  sq.  cm.  to  to  a  ±  0.6  on  strains  are  above  kg.  per  fell  (see  Fig.  45a).  40  clay  2/3  then  cm.  the  failure.  the  some  same  about  kg.  first,  the  and  to  0.47  threshold  but  at  sq.  rate  after  tests  shear  per  per  Dur-  sq.  cyclic rate  i t gradually  these be  then  switching off At  cm.  per  twenty  and  of  slightly  1.0  was  in.)  kg.  stress  increasing  hardening  ultimately  0.1  shear  rose  failure  below  shear  an  per  of  sq.  increased slightly  of  strength  consolidated for  l b . per  shear  pressure  appears  manner  cyclic  stress  of  shear  c o n s o l i d a t e d under  cyclic  There  than  then  pressure  pore  end  (75  increments  to  After  3.7  cm.  were  pore  points  A  44  CLAY  percent,  simple  sq.  continued  the  by  b)  42  the  strains  the  pressure  interesting  HANEY  that  i n c r e a s e d at  pore  a)  the  stress  shear  F i g . 45b).  at  per  With  the  (see  off  of  past  per  subjected  time  at  i s about  limit  as  kg.  kg.  cyclic  this  British  found  shear  deflections ing  was  sec.  while a  done,  Samples  the  local  maximum  shear  and  per  is a  content  1.7  minute.  pressure  ON  procedure.  It  loaded  TESTS  liquid  The  experimental  SHEAR  clay  moisture  F  cm. load,  was  the  levelled  minutes.  The  are:  cyclic  stress  ultimately which  reduced  failure  will  occur.  stress  which  leads  static  shear  s t r e n g t h , measured  already  described.  s t r e n g t h appears  to  to  failure  However, depend  on  as the  may  be  in  the  less  the "static"  rate  of  154  F i g . 45b. C y c l i c S i m p l e S h e a r on Haney C l a y . N o r m a l L o a d 5.28 k g / c m . S h e a r L o a d ± 1.0 k g / c m . 2  2  155 strain  at  failure  1  8  ,  this  i s , perhaps,  a  premature  conclusion. c)  The  mechanism  shear The and  are  agree tests  9  ,  trolled  Seed  or  of  Seed  and  Chan,  rise  of  1 9  2 0  clay  pore  lead  and  to  to  be  and  in  below the  pressure  one  above  who  who  cyclic  clay  shear  under  progressive based  on  from  would  cyclic remoulding.  very  e s p e c i a l l y as  concluded  i t s normally  after  of  concluded  Seed  same  for  are  tentative,  Chan  , that  strength  might  only  Thiers  tests  failure  conclusions  therefore  with 1  appears  of  they  stress  from  few  tests  do  not  controlled  strain  con-  not  reduce  the  measured  static  strength.  reference, cessation  did of  note, cyclic  however, stress,  a which  failure.  S n e a d , D.E., Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, pending. S e e d , H.B., and C h a n , C.K., " C l a y S t r e n g t h Under E a r t h quake L o a d i n g C o n d i t i o n s " , J o u r n a l , of the S o i l M e c h a n i c s & F o u n d a t i o n s D i v i s i o n , A.S.C.E., March, 1966. T h i e r s , G.R., a n d S e e d , H.B., "Cyclic Stress-Strain. C h a r a c t e r i s t i c s of C l a y " . J o u r n a l , of the S o i l Mechanics & F o u n d a t i o n s D i v i s i o n , A.S.C.E., March, 1968.  156 LIST  A  ,  1  A  Functions  2  B  Average  Matrix  C s w  C  EX,  E  EY,  E  &  x y  of e l a s t i c  and w a t e r .  constants. elastic  constants.  of s o i l  Compressibility  of water.  Young's  modulus  i n the x - d i r e c t i o n .  Young's  modulus  i n the y - d i r e c t i o n .  skeleton.  ratio.  Shear  factor. modulus  Shear  i n a plane  Height  of simple  Half  K  Compliance  Ko  Co-efficient K  ,K  3  2  3  k  i n a n x-y  modulus  HT Hj h  1  of s o i l  at r i g h t  angles  plane.  shear  sample.  o f HT.  Functions  of pore  pressure  of earth  measuring  pressure  of e l a s t i c  apparatus.  at rest.  constants.  Permeability. Length  of simple  shear  sample.  Poisson's y-axis.  ratio  i n a plane  Poisson's  ratio  i n a n x-y  n  Porosity.  Term  p s  Pressure.  MUXX, y MUXY, y  XX xy  Change  SIG  X ,Y  Time.  u  Elastic Pore  V  i n V,  Abbreviation  t  number  x  displacement  angles  plane.  in a  due t o u n i t fora  at right  series.  pressure  change.  y  x-direction.  in  pressure.  Compliance  of a piezometer  cell.  Volume. V v x  3  to the  y-axls.  G xy  K  constants.  of a n i s o t r o p i c  Intake  G XX  of e l a s t i c  Compressibility  Void  F  SYMBOLS  compressibility  Function C  OF  Volume l  Elastic y,  z  of pore  pressure  displacement  Directions  measuring  device.  in y-direction.  of co-ordinate  axes  in  space.  to the  157 LIST  OF  SYMBOLS  Continued  Functions  of e l a s t i c  constants.  Functions  of e l a s t i c  constants.  Shear  strain.  Density. Linear Axial  strain. linear  Function  strain.  of e l a s t i c  constants.  Angle between normal to plane of a p p l i e d shear s t r e s s a n d r e s u l t a n t s t r e s s o n s a m a p l a n e , same q u a n t i t y measured i n terms o f e f f e c t i v e stress. A  coefficient  Poisson's Function  of stress  ratio.  Also  of e l a s t i c  Normal  stress.  Major,  intermediate  Effective Shear An  see u  xx  , u  xy  constants.  and m i n o r  principal  stresses.  stress.  stress.  Airey  Functions Angle  normal  uniformity  stress  f u n c t i o n . Angle  of e l a s t i c  of deformation  o f max.  stress  constants. of simple  shear  sample.  obliq  15 8 LIST  OF  REFERENCES  1.  C o r n f o r t h , D.H., "Some E x p e r i m e n t s o n t h e I n f l u e n c e o f S t r a i n C o n d i t i o n s on t h e S t r e n g t h o f Sand". Geotechn i q u e V o l . X I V , No. 2, p . 1 4 3 , J u n e , 1 9 6 4 .  2.  K j e l l m a n , W. , " R e p o r t o n a n A p p a r a t u s f o r C o n s u m m a t e I n v e s t i g a t i o n of the M e c h a n i c a l P r o p e r t i e s of S o i l s " . P r o c e e d i n g s I n t e r n a t i o n a l C o n f . on S o i l M e c h a n i c s , Cambridge, Mass. 1936.  3.  K o , H.-Y., a n d S c o t t , R . F . , "A New S o i l T e s t i n g A p p a r a tus". G e o t e c h n i q u e V o l . X V I I , No. 1, M a r c h 1 9 6 7 .  4.  H a r d i n , B.O., a n d R i c h a r t , J r . , F . E . , " E l a s t i c Wave V e l o c i t i e s i n G r a n u l a r S o i l s " , J o u r n a l of the 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 . No. SMI, F e b r u a r y 1 9 6 3 .  89,  5.  R o s c o e , K.H., "An A p p a r a t u s f o r t h e A p p l i c a t i o n o f S i m p l e Shear to S o i l Samples", P r o c e e d i n g s , 3 r d I n t e r n a t i o n a l C o n f e r e n c e o n S o i l M e c h a n i c s a n d F o u n d a t i o n s , V o l . 1, Z u r i c h 1953.  6.  R o s c o e , K.H., B a s s e t t , R.H., a n d C o l e , E . R . L . , " P r i n c i p a l Axes O b s e r v e d D u r i n g S i m p l e Shear o f a Sand". P r o c e e d i n g s C o n f e r e n c e on Shear S t r e n g t h P r o p e r t i e s o f N a t u r a l Soils and R o c k s . V o l . I , p. 231-237. O s l o 1967.  7.  Kjellman, Sweden".  8.  B j e r r u m , L . , a n d L a n d v a , A., " D i r e c t S i m p l e - S h e a r T e s t s on a N o r w e g i a n Q u i c k C l a y " . G e o t e c h n i q u e V o l . X V I , No. 1, p . 1, M a r c h 1 9 6 6 .  9.  P e a c o c k , W.H., a n d S e e d , H.B., " L i q u e f a c t i o n o f S a t u r a t e d Sand u n d e r C y c l i c L o a d i n g S i m p l e S h e a r C o n d i 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 4 , No. SM3, May 1 9 6 8 .  10.  Rowe, P.W., and B a r d e n , L . , " I m p o r t a n c e o f F r e e Ends i n Triaxial Testing". 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 0 , No. SMI, J a n u a r y 1964 .  11.  H v o r s l e v , M.J., "Time Water O b s e r v a t i o n s " . Experimental Station,  12.  Z i e n k i e w i c z , O.C., C h e u n g , Y.K., a n d S t a g g , K.G., " S t r e s s e s i n A n i s o t r o p i c Media w i t h P a r t i c u l a r R e f e r e n c e to problems o f R o c k M e c h a n i c s " . J o u r n a l o f S t r a i n A n a l y s i s V o l . 1, No. 2, 1 9 6 6 .  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V o l . 6, No. 1 1 , N o v e m b e r 1968.  18.  S n e a d , D.E., pending.  19.  S e e d , H.B., a n d C h a n , C.K., "Clay Strength quake L o a d i n g C o n d i t i o n s " . J o u r n a l of the and F o u n d a t i o n s D i v i s i o n , A . S . C . E . , M a r c h ,  20.  T h i e r s , G.R., and S e e d , H.B., "Cyclic Stress-Strain C h a r a c t e r i s t i c s of C l a y " . J o u r n a l of the S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , A . S . C . E . , M a r c h , 1968.  D.W.,  "Fundamentals "A T r e a t i s e 1892.  Ph.D.  Thesis,  of  on  Soil  the  Mechanics",  1948.  Mathematical Theory  University  of  British  of  Columbia,  Under EarthSoil Mechanics 1966.  


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