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Vibration analysis of dry sand models Aoki, Yoshinori 1969

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VIBRATION ANALYSIS OF DPY SAND MODELS  by Yoshinori  Aoki  B E « , Tokyo M e t r o p o l i t a n U n i v e r s i t y , Japan, .1961 t  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A«3c, in  the Department of  Civil  We accept  Engineering-  t h i s t h e s i s as conforming t o the. r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA March., .1969  ii  ABSTRACT  Usiwg the s h a k i n g t a b l e , two types o f t e s t s have been made w i t h 8 f e e t l o n g by 1-1/2 f e e t wide and from 1/2 f o o t to 1 f o o t h i g h d r y Wedson sand models'. One o f them was a response  study o f the h o r i z o n t a l model,  which the a f f e c t s o f boundary r e s t r a i n t s * soil  the frequency-response of  l a y e r . a n d the•dynanic p r o p e r t i e s o f d r y sand were s t u d i e d .  measured r e s u l t s w i t h r e g a r d t o the e f f e c t s of boundary and  from  t h e frequency response o f s o i l  the l i n e a r v i s c o - e l a s t i c  theory.  restraints  l a y e r agreed w i t h p r e d i c t i o n s by The measured shear wave moduli and  damping r a t i o a l s o agreed, w i t h those o b t a i n e d by p r e v i o u s The  o t h e r type o f t e s t s performed  using t i l t e d  The '  workers.  was a study o f s l o p e  models, i n which the a c c u m u l a t i v e d i s p l a c e m e n t  stability of slope  was induced, by a s i n u s o i d a l base motion,  the c r i t i c a l  slope angle  and  The measured  accumulative  the s t a b l e s l o p e angle were s t u d i e d .  displacement  agreed w i t h the t h e o r y suggested by Goodman and Seed (.1966),  I t has been found  t h a t t h e r e a r e two d i s t i n c t l y  a n g l e s o f s l o p e a s s o c i a t e d w i t h dynamic s t a b i l i t y are the c r i t i c a l  s i n u s o i d a l base motion  characteristic  f o r a slope.  s l o p e angle and the s t a b l e s l o p e a n g l e .  the s t a b l e s l o p e angle i s unique  slope.  different  These  Moreover,  f o r a m a t e r i a l and a f r e q u e n c y o f the  and independent  o f the i n i t i a l c o n d i t i o n o f the  In p r e s e n t i n g  this thesis  i n p a r t i a l fulfilment of the requirements f o r  an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , the  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  I further agree that permission  f o r extensive  I agree  that  and Study.  copying of this  thesis  f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s .  It i s understood that copying o r p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l written  permission.  Department o f  Civil  Engineering  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date  g a i n s h a l l n o t b e a l l o w e d w i t h o u t my  Columbia  TABLE. OF jCpNTBNTS t'a&e CHAPTER 1  INTRODUCTION  ,  1  CHAPTER 2  DESCRIPTION OF TEST EQUIPMENT AND.THE SOIL TESTED 3  CHAPTER' 3  DYNAMIC RESPONSE OF HORIZONTAL MODELS  , , , . ,  3.1  Introduction  3.2  Theoretical Consideration  3.3  Boundary E f f e c t s  3.4  Frequency-Response Curve , « « , . « < . « • . «  26  3.5  Measurement  31  3.6  Summary and C o n c l u s i o n  CHAPTER 4  « « .  11 11  . « « , « . « « « « .  . , , « , «  13  « < « .  o f Dynamic Modulus  « « , , « « , , ,  • « < < < <  . « < > « . «  24  45  SLOPE STABILITY DUPING VIBRATION  47  4.1  Introduction  47  4.2  C o n s i d e r a t i o n of Boundary E f f e c t s  4.3  Preliminary Tests  4.4  Measurement  of Accumulative Displacement . . . .  55  4.5  Measurement  of S t a b l e Slope Angle  65  . 4.6  Measurement  of C r i t i c a l  4.7  « « « « < « « « < « « « « « « « « , , « , , , ,  .,  Summary and C o n c l u s i o n CONCLUSION  48 .  • ,  Slope Angle  . « « . , .  51  70  . , , « , . . . . . . . .  74  . « . . . , , . . , . .  76  CHAPTER 5  SUMMARY AND  APPENDIX I  SOLUTION FOR VIBRATION OF FINITE MODEL . , , , ,  81  "  II  BOUNDARY EFFECTS  ON VIBRATION  . . . . . . . . .  88  "  III  BOUNDARY EFFECTS  ON PERMANENT DISPLACEMENT . . .  92  "  IV  THEORY OF ACCUMULATIVE DISPLACEMENT  "  V  SIMILITUDE  ,  . . . « . . . < . .  98 102  XV  LIST OF FIGURES  Figure  Page  2-1.  Photograph o f s h a k i n g t a b l e , , h o r i z o n t a l and c o n t r o l system , , . , ,  model . « « , , .  2-2  Photograph of s h a k i n g t a b l e w i t h t i l t e d model  2- 3  S k e t c h of c o n t r o l system f o r s h a k i n g t a b l e  3- 1  E f f e c t s o f v a r i a t i o n i n shear modulus  3-2  Dimension of model  3-3  Comparison o f f r e q u e n c y - r e s p o n s e curve  3-4  E x p l a n a t i o n o f h a l f power p o i n t  3-5  Explanation of l o g a r i t h m i c  3-6  Comparison of a c c e l e r a t i o n r a t i o  3-7  F r e q u e n c y - r e s p o n s e of h o r i z o n t a l model  3-S  •  Relation applied  between v o i d  . «  7  . , . .  8  . « . . « «  . « . « « . . «  20  . . . . . .  22  method , , « . . «  25  decrecent  r a t i o and  7  « . . « • • . . • . < . . . « < •  25 27 29  acceleration  during v i b r a t i o n « < « . . . . . . « « • «  3-9  An example o f f r e e v i b r a t i o n r e c o r d  3-10  An example of resonant f r e q u e n c y . . . • « « . . .  34  3-11  Fundamental frequency  . • . . . . . . . . . . . .  39  3-12  Shear wave v e l o c i t y  . . . . . . . . . . . . . . .  39  3-13  Shear wave v e l o c i t y of p r e v i o u s workers and t h i s research  3-14  g  . . • .  . , . < « • <  33  . . . . « • < « .  =i§= curve . . . .  34  41 44  3- 15  Logarithmic decrecent  4- 1  F r i c t i o n measurement t e s t  . . . . . . . . . . . .  50  4-2  An example o f the (2) type f a i l u r e p l a n e . . . . .  53  4-3  D e f o r m a t i o n of a v e r t i c a l marker a t the s i d e wall  . . . < « . . « < . . « « •  f o r t h e (3) type f a i l u r e  . . . . . . . . . .  44  54  Displacement  and shape of f a i l u r e p l a n e  Mechanism o f f a i l u r e p l a n e  , * • » * * ,  * * * , , , * * , , * *  C o n d i t i o n at f a5.1ure f o r s l i c i n g element Permanent  displacement  S t a b l e s l o p e angle Critical  i n one c y c l e  * * * * * * * * * *  R e l a t i o n between-, s l o p e angle and  • • « •  amplitude  , , , , * * * * , * * * * * * * * ,  C o o r d i n a t e o f model Effect  * * * * * * * *  , , , , * , * , , , . * , * * ,  s l o p e angle  of a c c e l e r a t i o n  . < . . ,  * * * * * * * * * * * * * * * *  of l e n g t h of model on i t s n a t u r a l  E f f e c t of width o f model  , ,  Force a c t i n g on a f a i l u r e p l a n e A s e c t i o n of the model  frequency * , » , * *  * * * * * * * * * *  , . * * , , , * * , , , * *  F o r c e s a c t i n g on the toe  * * * * * * * * * * * * *  Simplified  , * , * * * * * * , * , ,  f a i l u r e plane  F o r c e s a c t i n g on a s l i d i . n g element Acceleration  and v e l o c i t y  Acceleration  and v e l o c i t y when  , , * , * * , ,  .  * , , * , ?A  /  * * * *  LIST OF TABLES  jjj&SS.  Table 2- 1  Composition of Wedron sand i'4098  10  2~2  Angle of i n t e r n a l  • « « * * « « • « « « «  10  3- 1  R e l a t i o n between frequency and t i c c e l e r a t i o n r a t i o  28  3- 2  Fundamental frequency  4- 1  F r i c t i o n between sand and p l e x i g l a s s  4-2  List  4-3  L i s t o f angles o f i n t e r n a l  friction  .«  o . . « . . . < < « <  of d i s p l a c e m e n t measurement  . • « < « < «  tests  friction  <  37 50  . < « . < <  62  , . « < « « «  64  10-1 '  Similitude  f o r dynamic model t e s t  (1)  • • • • • •  105  10-2  Similitude  f o r dynamic model t e s t  (2)  . « » « . .  105  LIST OF SYMBOLS  A r e a of h y s t e r e s i s Acceleration  loop  a m p l i t u d e of base motion  Width o f model C o e f f i c i e n t of f r i c t i o n Damping  coefficient  Cohesion E q u i v a l e n t damping  coefficient  Depth o f f a i l u r e p l a n e Young's modulus Void  ratio  Friction  f o r c e a c t i n g on the f a i l u r e p l a n e  C o n f i n i n g f o r c e at toe F r i c t i o n f o r c e a c t i n g on the s i d e E x c i t a t i o n forces  wall  i n the d i r e c t i o n i n d i c a t e d by i n d e c e s  Frequency i n terms c f cps Resonant f r e q u e n c y Shear modulus Acceleration  of g r a v i t y  Thickness of l a y e r H e i g h t o f the model Bessel  function  Acceleration  ratio  to the a c c e l e r a t i o n o f g r a v i t y  Acceleration  of base motion i n terms o f (g)  (g)  A c c e l e r a t i o n at base, mid-height and top of the model r e s p e c t i v e l y i n terms of (g) Y i e l d a c c e l e r a t i o n i n terms o f (g)  k  Stress  D  r a t i o between v e r t i c a l and h o r i z o n t a l  in soil  JLJC  Wave l e n g t h  o f Love wave  £  Length o f the model o r sample  M  Acceleration  ^  Mass  n  V o i d r a t i o to t o t a l volume  "P  F o r c e a c t i n g on toe wedge  £  * tan ^ k  7?  Frequency r a t i o o f s e m i - f i n i t e l a y e r t o r e s t r i c t e d model  P  R a t i o o f wave v e l o c i t y r =  ratio  ^/uk  E q u i v a l e n t d i s t r i b u t e d f o r c e due t o t o e c o n f i n i n g  7  7T- 2  t, ~ti t  T  Times a t which f a i l u r e s t a r t s and s t o p s i  m  e  o f maximum v e l o c i t y  W  Weight o f s o i l mass  U^w_c*r  Displacements  U  Displacement i n one c y c l e  U$  Amplitude o f d i s p l a c e m e n t o f base motion  lf  c  '  Velocity  o f d i l a t a t i o n a l wave  Velocity  of l o n g i t u d i n a l wave o f r o d  ^  V e l o c i t y o f Love wave  k£  V e l o c i t y o f R a l e i g h wave  Us  V e l o c i t y o f Shear wave V e l o c i t y o f t o r s i o n a l wave  '  ^C^^  Rectangular  coordinate  C*  Angle o f s l o p e  &c  Critical  angle o f s l o p e  S t a b l e angle o f s l o p e  respectively  ix  Constant.  F  Roots  r r  .  of J-b(p„)=0  Gainnia f u n c t i o n U n i t weight o f s o i l  r  Shear  strain  Logarithmic  decrecent  0  Angle between t h e s u r f a c e o f s l o p e and f a i l u r e p l a n e  4>  Angle o f i n t e r n a l  friction-  A n g l e o f f r i c t i o n on the p l a n e  o f t o e wedge  Angle o f f r i c t i o n m o b i l i z e d a t i n i t i a t i o n Angle o f f r i c t i o n m o b i l i z e d d u r i n g Phase  X  of y i e l d  sliding  angle  Density  o f mass  Lame's  constant  -  ;  S c a l e f a c t o r of l e n g t h //  Poisson's  ratio  cr  Confining  pressure  Principal stresses Mean normal s t r e s s = cos cx •!- sin<?< tar. 0  CO  Damping  ratio  Angular  frequency  Undamped n a t u r a l  1  frequency  S c a l e f a c t o r -for an i t e m i n d i c a t e d by  indeces  a t toe  X  ACKNOWLEDGE_MS_MT  The i n v e s t i g a t i o n r e p o r t e d provided  h e r e i n has been s u p p o r t e d by funds  by the N a t i o n a l Research C o u n c i l o f Canada,  also included  f i n a n c i a l support f o r t h e w r i t e r .  i s e x p r e s s e d f o r t h i s a s s i s t a n c e w i t h o u t which and t h i s  These  funds  Grateful appreciation the graduate  studies  t h e s i s c o u l d n o t have been a c c o m p l i s h e d .  The w r i t e r wishes Dr. R.G. Campanella  to e x p r e s s h i s thanks t o D r . K,D, Liam  and P r o f e s s o r  Finn,  P.M. Byrne f o r t h e i r guidance and  c o n s t r u c t i v e c r i t i c i s m during the preparation  of this  thesis.  The t e c h n i c a l a s s i s t a n c e s u p p l i e d by the s t a f f o f t h e C i v i l Engineering  Department i s g r a t e f u l l y  acknowledged.  1.  2IM>ATIQN..ANAMSIS ..QF, PRY .SAND MODELS. by  Aoki  CHAPTER 1  Introduction In  r e c e n t y e a r s t h e r e have been g r e a t advances made i n the  i n v e s t i g a t i o n of the dynamic p r o p e r t i e s of s o i l s and s o i l  structures.  T h i s a r e a of s o i l mechanics, however, i s so nex* and c o m p l i c a t e d t h a t many a s p e c t s are s t i l l Richart  (1963) extended  left  to be s t u d i e d .  For example, H a r d i n  and  l a b o r a t o r y i n v e s t i g a t i o n s of wave p r o p a g a t i o n  phenomena i n s o i l but t h e i r samples were t e s t e d under a h y d r o s t a t i c c o n f i n i n g p r e s s u r e i n a t o r s i o n a l mode v i b r a t i o n to measure shear modulus.  I n o r d e r to e s t i m a t e the v e l o c i t y of wave p r o p a g a t i o n i n  the ground, o r i n some e a r t h s t r u c t u r e , i n v e s t i g a t i o n s which i n v o l v e t e s t c o n d i t i o n s under p l a n e s t r a i n and of  simple shear and  v a r i a t i o n of i n d i v i d u a l s t r e s s e s are needed.  used  to a n a l y t i c a l l y determine  advanced due  approach  s t r u c t u r e s i s quite,  elemeiit methods of  i s known c o n c e r n i n g dynamic p r o p e r t i e s of  purpose of t h i s r e s e a r c h i s to examine the s u i t a b i l i t y of s h a k i n g t a b l e from an e x p e r i m e n t a l p o i n t of view and  some b a s i c d a t a f o r f u t u r e developments u s i n g t h i s k i n d of in  the  insitu.  The the new  little  Although  of s o i l  to the c o n t r i b u t i o n of the f i n i t e  analysis, relatively soils,  the response  the e f f e c t s  s o i l mechanics.  b e h a v i o u r may  to o b t a i n apparatus  Fundamental but b r o a d l y d i f f e r e n t s t u d i e s of model  p r o v i d e v a l u a b l e i n f o r m a t i o n c o n c e r n i n g dynamic p r o p e r t i e s  2.  of s o i l ,  a t l e a s t f o r the model.  T h e r e f o r e , one  o f the  important  p o i n t s of the i n v e s t i g a t i o n r e p o r t e d h e r e i n i s to compare the behaviour Dry  o f the model w i t h sand was  those p r e d i c t e d by e x i s t i n g t h e o r i e s .  chosen as the model m a t e r i a l f o r t h i s i n i t i a l  because of i t s r e l a t i v e ease oJ: h a n d l i n g . purposes o f t h i s  observed  Moreover, one  of  study  the  r e s e a r c h i s to i n v e s t i g a t e the dynamic p r o p e r t i e s of  dry sand, which i s a fundamental m a t e r i a l , and  to p r o v i d e data f o r f u t u r  investigations involving i t . The  s i z e o f the model i s one  o f the more important  f a c t o r s of  model s t u d i e s which i s a s s o c i a t e d w i t h boundary e f f e c t . r e s e a r c h an 8' l o n g by to house the s o i l .  1,5'  wide and  2* h i g h r i g i d  In  c o n t a i n e r was  The boundary e f f e c t s of the r e s t r i c t i n g  of the model were examined both  experimentally  and  this built  dimensions  theoretically in  •order to o b t a i n i n f o r m a t i o n c o n c e r n i n g boundary e f f e c t s on model behaviour  for future investigations,  T e s t s performed i n t h i s r e s e a r c h are b a s i c a l l y as f o l l o w s ; (1) the dynamic response of h o r i z o n t a l models ( i n Chapter 3 ) , and the s t a b i l i t y  of sand model s l o p e s i n u s o i d a l v i b r a t i o n  Because t h i s r e s e a r c h i n v o l v e s q u i t e broad  and  different  throughout the a r e a o f dynamic problems o f s o i l s it  is difficult  to review  i n each c o r r e s p o n d i n g l i t e r a t u r e w i l l be  the l i t e r a t u r e i n one  chapter  reviewed.  (Chapter  and  (2),  4),  aspects  s o i l structures,  chapter.  Therefore,  c r r e l a t e d s e c t i o n the p e r t i n e n t  3. CHAPTER 2  DESCRIPTION OF TEST EQUIPMENT AND THE SOIL TESTED  2.1  D e s c r i p t i o n o f .test, eiuipmgnt and instrur.imtation., The  s h a k i n g t a b l e and connected i n s t r u m e n t a t i o n s  been r e p o r t e d  (1968).  by F i n n , Campanella and A o k i  have  already  These a r e as f o l l o w s .  The.photograph i n Fig,2-1 shows the s i n g l e - a x i s s h a k i n g together  w i t h a frame s u p p o r t i n g  a model ready f o r t e s t i n g .  shown i n the f i g u r e a r e the h y d r a u l i c l o a d i n g p i s t o n , c o n t r o l and  m o n i t o r i n g and r e c o r d i n g The  up  table  provide  ( 6 f t , wide by 9 f t , l o n g and 6-3/4  very  by 2 x 6 i n , h o l l o w r e c t a n g u l a r  stiffness.  stiff  i n order  to transmit  tubing  i n , deep ) i s made top and bottom  i n a g r i d pattern to  u n i f o r m base a c c e l e r a t i o n s and be as  t o make optimum use o f the l o a d i n g ram. The  t a b l e , which weighs about 1000 l b s , 0  V - s l o t t e d needle bearings only  console,  The e s s e n t i a l f e a t u r e o f the t a b l e i s t h a t i t be  l i g h t as p o s s i b l e i n o r d e r  allow  Also  equipment.  o f welded aluminum s e c t i o n s w i t h 3/8 i n , p l a t e s  separated  table  i s mounted on a s e t o f 4 h o r i z o n t a l ,  (manufactured by Schneeberger, S w i t z e r l a n d ) which  one degree o f motion.  The b e a r i n g s a r e mounted on a welded  s t e e l frame made up o f wide f l a n g e s e c t i o n s w h i c h , i n t u r n , i s b o l t e d to the f l o o r .  Great c a r e and p r e c i s i o n i s r e q u i r e d  i n the alignment  of the f o u r s e t s o f b e a r i n g s . The  h y d r a u l i c ram i s mounted i n a v e r y  stiff,  a l l welded frame  made up o f 3/4 i n , p l a t e s t e e l which, i n t u r n , i s r i g i d l y mounted t o the base frame. 2  The ram i s a double-ended h y d r a u l i c p i s t o n w i t h a  i n , s t r o k e , a dynamic c a p a c i t y o f  of about 17 i n , p e r s e c , min,  at f u l l  load,  2500  l b s , and a maximum v e l o c i t y  A r e l a t i v e l y small  pump (not shown i n F i g . l ) s u p p l i e s o i l a t  3000  3 g a l . per  p s i to the ram through  4.  an e l e c t r o n i c c o n t r o l l e d servo-valve« force  The ram, which p r o v i d e s the  t o move the t a b l e , i s connected t o the t a b l e  through a r i g i d  A s k e t c h o f the c o n t r o l system f o r the s h a k i n g t a b l e i n Fig.2-'3  The system i n c l u d i n g e l e c t r o n i c c o n t r o l s ,  S  link,  i s shown  hydraulic  pump and ram were o b t a i n e d from MTS Systems C o r p o r a t i o n , M i n n e a p o l i s , Minn,  F i g , 2 shows t h a t  t h e c o n t r o l l e r w i l l a c c e p t a v o l t a g e v s , time  s i g n a l from e i t h e r a f u n c t i o n tape.  g e n e r a t o r , curve f o l l o w e r  For shaking table operation t h i s input  d i s p l a c e m e n t - t i m e command. i s housed w i t h i n  o r magnetic  voltage representes a  An LVDT ( d i s p l a c e m e n t t r a n s d u c e r ) ,  the h y d r a u l i c  which  ram a l s o t r a n s m i t s an e l e c t r o n i c  signal  i n t o the c o n t r o l l e r which r e p r e s e n t s t h e a c t u a l movement o f the ram and  therefore  the t a b l e  The command s i g n a l , V^, i s compared t o the  c  feedback s i g n a l from t h e d i s p l a c e m e n t t r a n s d u c e r , difference,  - V^, i s t r a n s m i t t e d  valve located allowing  i n the h y d r a u l i c  t o the e l e c t r o n i c c o n t r o l l e d  servo-  The e l e c t r o n i c v a l v e responds by  the ram to move i n such a manner as to make the feedback  v o l t a g e from t h e d i s p l a c e m e n t voltage,  ram.  and the v o l t a g e  X^,  This  t r a n s d u c e r , X^, e q u a l t o the. command  primary r o u t e o f c o n t r o l i s i n d i c a t e d  as a S e r v o -  Loop i n F i g , 2 - 3 . The  system i s a l s o  as one o f a c c e l e r a t i o n  c a p a b l e o f t r e a t i n g the i n p u t vs,  time by i n p u t t i n g a s i g n a l t o the c o n t r o l l e r  which i s the double i n t e g r a l o f the a c c e l e r a t i o n - t i m e integration  i s done e l e c t r i c a l l y ) .  displacement w i t h i n actual  table  command s i g n a l  command,  Thus, the system always  controls  the s e r v o - l o o p f o r s h a k i n g t a b l e o p e r a t i o n .  accelerations  (The  The  a r e measured w i t h a K i s t l e r Model 305A/515  S e r v o - A c c e l e r o m e t e r which i s a l s o used i n a secondary feedback loopo The by  command s i g n a l , t a b l e  d i s p l a c e m e n t and a c c e l e r a t i o n  any r e l a t i v e l y h i g h i n p u t  such as an o s c i l l o s c o p e .  impedance v o l t a g e r e c o r d i n g  can be measured instrument  The  response of the s h a k i n g t a b l e i s a f u n c t i o n of the  a m p l i t u d e , the  desired  response of the s e x v o - v a l v e , the pumping c a p a c i t y  the h y d r a u l i c pump, the volume compliance of the h y d r a u l i c ram, s t i f f n e s s of the ram Tests  cps  about 15  ( c y c l e s per  i t was  above 20  p o s s i b l e to use  as 60  s h a k i n g system has  cps  and  cps the  still  l o n g and was  reported  operating  along  This  container  occurs.  as  by curve.  rigid  container  i s 2 f t . high by The  8 ft, base  i n , plywood w i t h a l a y e r of sand bonded to the upper frictional  resistance for  A K i s t l e r Model 505A/515 a c c e l e r o r c e t e r was  d e s i g n e d so i t c o u l d be tilt  However,  the r o l l o f f  (see F i g . l ) ,  the plywood base to measure base a c c e l e r a t i o n s .  a given  to about 2700 l b s .  herein a r e l a t i v e l y  plywood s u r f a c e w i t h epoxy to p r o v i d e  at  lbs,  s h a k i n g t a b l e at s i n u s o i d a l f r e q u e n c i e s  f t , wide w i t h p l e x i g l a s s s i d e s  made of 3/4  sand models,  a r e s o n a n t frequency of  a characteristic rolloff  to house s o i l models, 1-1/2  model.  o b t a i n smooth a c c e l e r a t i o n s i n e curves  For the model s t u d i e s built  t a b l e and  the  second) f o r a moving mass of about 1000  i n c r e a s i n g the i n p u t amplitude and  was  the mass of the  cps when the moving mass i s i n c r e a s e d  At f r e q u e n c i e s  high  t a b l e and  show t h a t the p r e s e n t  about 17 and  and  of  tilted  angle up  under dynamic c o n d i t i o n s ,  to 40° (See  for static  The  the  mounted beneath  container  t e s t i n g , as w e l l as  at i n t e r v a l s of 5 degrees and  was fixed tested  Fig,2-2),  Instrumentation In order  to measure the response of the s o i l models d u r i n g  a c c e l e r a t i o n s i t was mass. be v e r y  The  n e c e s s a r y to measure a c c e l e r a t i o n s w i t h i n  base the  soil  e s s e n t i a l requirements of such an a c c e l e r o m e t e r are t h a t i t  small  s e n s i t i v e and  and  have the same u n i t weight as the s o i l .  accurate  The  highly  a c c e l e r o m e t e r s used to measure base motions were  too  l a r g e and too heavy, as a r e the p r e s e n t l y a v a i l a b l e s t r a i n gage  a c c e l e r o m e t e r s which a r e n o r m a l l y used f o r s e i s m i c s t u d i e s .  Piezoelectric  a c c e l e r o m e t e r s appear to be s m a l l enough to be embedded i n the  soil.  U n f o r t u n a t e l y , t h e s m a l l e r the ac.celerom.eter the g r e a t e r i s i t s f u l l s c a l e a c c e l e r a t i o n and the p o o r e r i s i t s low f r e q u e n c y response thereby, reducing i t s s e n s i t i v i t y example,  i n the 0 to l g and DC  to 10 cps range.  For  a t y p i c a l m i n i a t u r e p i e z o e l e c t r i c a c c e l e r o m e t e r might have a  frequency response from 5 t o 10,000 cps and a range o f lOOOg's, F o r t u n a t e l y , some p i e z o e l e c t r i c a c c e l e r o m e t e r s are e x t r e m e l y s e n s i t i v e and have a r e s o l u t i o n o f ,01 g's o r b e t t e r .  Thus, i t was  decided to  o b t a i n m i n i a t u r e a c c e l e r o m e t e r s from s e v e r a l m a n u f a c t u r e r s and their suitability to  f o r embedment i n s o i l  i n seismic testing.  determine  In o r d e r  check the c a l i b r a t i o n and response of a c c e l e r o m e t e r s the K i s t l e r  505A/515 s e r v o - a c c e l e r o m e t e r was type used on t a b l e ) . a c c u r a c y of about ,005g. full  used as the p r i m a r y r e f e r e n c e  (same  T h i s e x t r e m e l y s e n s i t i v e i n s t r u m e n t has an  ,01% i n c l u d i n g l i n e a r i t y  I t a l s o has a f l a t  s c a l e range o f +  and h y s t e r e s i s which  f r e q u e n c y response o f DC to 500  overall represents  cps and a  50g,  A K i s t l e r model 504A charge A m p l i f i e r was  used w i t h a l l o f t h e  following p i e z o e l e c t r i c accelerometers, CEC Model 4-274-001, weight 6 gm, KISTLER Model  u n i t weight about 450 p c f ,  803A , weight 20 gm,  METRIX Model 502 WILCOXON Model  127,  , weight 6 gm, weight 1-3/4  u n i t weight about 430 p c f , u n i t weight about 3.30 p c f , gm,  Of these 4 t e s t e d , the M a t r i x , which was sensitivity,  u n i t weight about 185 p c f .  judged b e s t , had  gave the most n o i s e - f r e e s i g n a l s a t low  the h i g h e s t  accelerations  and f r e q u e n c i e s above about 1 c p s , and had a u n i t weight v e r y c l o s e to  that of s o i l .  The W i l c o x o n was  a l s o q u i t e good and had the  7.  Fig.2-2.  PHOTOGRAPH OF SHAKING TABLE WITH TILTED MODEL,  •' 3o00p.s.i  PlSPLACEM&ST  TBED&QCK  ~\  TABLE „ ACCELERATION  Z  TZBOBflCk  _  11  \ACCBL  o  scope  Fla 2-3 SKETCH OF CONTROL SYSTEM] FOR SHAKIAjtr TABLE  advantage of b e i n g s e v e r a l times s m a l l e r i t was  not  f o r low  f r e q u e n c y - low  reported  2-2,  as s e n s i t i v e .  herein  only  Description The  The  CEC  and  than the M e t r i x ,  K i s t l e r were judged l e a s t s u i t a b l e  acceleration testing.  the M e t r i x and  of the s o i l  although  For  a l l of  the  test^  Wilcoxon a c c e l e r o m e t e r s were used,  tested,  models were comprised of Wedron sand, which i s c l e a n ,  rounded, u n i f o r m , medium s i l i c a sand and sand 20-30,  The  well  almost i d e n t i c a l to Ottawa  c o m p o s i t i o n of the' sand i s shown i n T a b l e  2-1,  F i v e samples were t e s t e d by vacuum t r i a x i a l r e s t i n o r d e r determine the  angle o f i n t e r n a l , f r i c t i o n o f the sand.  are shown i n T a b l e 2-2, peak s t r e n g t h of the  and  the  The  to  results  In t h i s t e s t d i s t i n c t d i f f e r e n c e s between  r e s i d u a l strength  were not  observed up  to  the  6-8%  axial strain.  D u r i n g the experiment o f the dynamic response o f h o r i z o n t a l models the  r e l a t i o n s h i p between the  maximum amplitude of the obtained  (see  average v o i d r a t i o of models and  acceleration applied  Fig,3-8 i n Chapter 3 ) .  r a t i o s of models t e s t e d were 0,47 models r e s p e c t i v e l y .  and  during  v i b r a t i o n has  From these r e s u l t s , the 0,65  the been  void  f o r dense models and  loose  7a Me %  Ret  C0F7m5/T/D,V  2-f  on U.S.  Of'  UfeDROA/  2.2  A/o. 20 30  38-6  4-0  S7.6  so "  Tah/e  2-2  /.4 O. Z  70  AMfrlZ Wold ratio .SOB . S/o  OF Con-f.  /AJTFRA/AL  2.5 P**  1  S-O /O.O s.o  FK/CT/Otf 3x/ai  pressure 'r  •far ^snaK  47-5 °  a L  38-7°  2.S /*« . 773  #4090  S/)MO  ^  36. <?° 35. 6°  6 3  % %  4- % B  %  11.  • CHAPTER 3  DYNAMIC RESPONSE OF HORIZONTAL MODELS  3-1„  I n t r o d u c t i pn In g e n e r a l ,  soils  a r e n o t l i n e a r l y e l a s t i c m a t e r i a l , but i t i s  -4 known t h a t under s m a l l amplitude o f s t r a i n , say l e s s than 10 K e l v i i v - V o i g t model which c o n s i s t s o f a s i m p l e l i n e a r e l a s t i c and  pure v i s c o u s  damping assembled p a r a l l e l  t o each o t h e r  spring  may  s a t i s f a c t o r i l y e x p l a i n the dynamic b e h a v i o u r o f some s o i l s . Kelvin-Voigt  , the  F o r the  model, three parameters a r e enough to d e s c r i b e the  dynamic, b e h a v i o u r o f s o i l s , which a r e , f o r i n s t a n c e , the sh velocity, V the d i l a t a t i o n a l wave v e l o c i t y , V and the e q u i v a l e n t s c 9  9  damping r a t i o  £, o r e i t h e r the shear modulus, G, o r Young's modulus E ,  P o i s o n ' s r a t i o , u,and £ , Methods o f measurement o f dynamic p r o p e r t i e s o f s o i l s which have been r e p o r t e d (1)  can be c l a s s i f i e d  as f o l l o w s :  F i e l d o r i n - s i t u measurement - Resonant method. - D i r e c t measurement o f wave v e l o c i t y ,  (2)  L a b o r a t o r y t e s t - Resonant method, - Free v i b r a t i o n method, - D i r e c t measurement o f wave v e l o c i t y , - Amplitude r a t i o and phase s h i f t  measurement.  Because o f the s i z e r e s t r i c t i o n o f the t e s t model and t h e type o f instrumentation  used, o n l y  the l a b o r a t o r y  r e s o n a n t and f r e e . v i b r a t i o n  methods were used f o r the i n v e s t i g a t i o n r e p o r t e d are  herein.  These methods  a s s o c i a t e d w i t h the frequency e q u a t i o n which i n v o l v e s o n l y the  wave v e l o c i t y o r dynamic modulus as an unknown f a c t o r i f t h e damping  12. r a t i o i s s m a l l enough to be i g n o r e d . measured f o r s o i l models or s o i l  I f the n a t u r a l f r e q u e n c i e 3 are  structures.whose  frequency e q u a t i o n i s  known, the wave v e l o c i t y o r the dynamic modulus can be c a l c u l a t e d the frequency e q u a t i o n by s u b s t i t u t i n g the measured f r e q u e n c y . resonant method* of  resonant f r e q u e n c i e s a r e found by changing  e x c i t i n g motion,  The f r e e v i b r a t i o n method i s based  t h a t models o f s t r u c t u r e s v i b r a t e i n t h e i r fundamental f r e e v i b r a t i o n , ( ( t h e r e f o r e an e x c i t a t i o n impulse o r an disturbance i s applied  In the  the frequency  on the f a c t mode d u r i n g initial  to the model, and the r e s u l t i n g f r e q u e n c y of  f r e e v i b r a t i o n i s measured. have used  from  H a r d i n and R i c h a r t (1963), f o r example,  the l a b o r a t o r y r e s o n a n t method and Z e e v a e r t  l a b o r a t o r y f r e e v i b r a t i o n method.  (1967) used the  F u r t h e r d e t a i l s w i t h r e g a r d to the  frequency e q u a t i o n o f the model t e s t e d h e r e i n a r e d i s c u s s e d i n the next section. Other methods o f measurement d i s c u s s e d elsewhere, M a r t i n and Seed methods,.  A t MIT  measurement  f o r example, Jones  (1958), Barnhard  (1959) and  (1966) have d i s c u s s e d f i e l d o r i n - s i t u measurement ( T a y l o r and Whitman  (1954)), the method o f d i r e c t  o f wave v e l o c i t y has been developed  and S t a l l e t a l  and Seli'g and Vey  (1965) have a l s o used t h i s method,  d i s c u s s e d the amplitude  Chae  r a t i o and phase s h i f t measurement  T h i s c h a p t e r concerns of  o f dynamic p r o p e r t i e s o f s o i l s are  (1968) has method.  i t s e l f w i t h the study of the dynamic  h o r i z o n t a l models, whose purpose  i s . t o observe  (1965)  response  the b e h v a i o u r o f s o i l  models and compare i t w i t h e x i s t i n g t h e o r i e s and t o i n v e s t i g a t e the dynamic modulus o f s o i l s  through  the model study and compare i t w i t h  those o b t a i n e d by o t h e r i n v e s t i g a t o r s , carried out:  The f o l l o w i n g experiments  were  (1)  study o f boundary base motion  (2)  e f f e c t s on the response o f models to s i n u s o i d a l  ( S e c t i o n 3-3),  study of the f r e q u e n c y - r e s p o n s e curve o f t h e h o r i z o n t a l model t o s i n u s o i d a l base motion  (3)  soil  ( S e c t i o n 3-4).  measurement o f dynamic modulus o f d r y sand by the f r e e v i b r a t i o n method and by the resonant method w i t h h o r i z o n t a l s o i l  models  ( S e c t i o n 3-5),  3-2,  Theoretical  consideration,,  B e f o r e the e x p e r i m e n t a l r e s u l t s a r e d i s c u s s e d , r e l a t e d  theories  of t h e dynamic response o f h o r i z o n t a l s o i l models on which  the a n a l y s i s  of t h i s r e s e a r c h i s based w i l l  A solution  be b r i e f l y p r e s e n t e d h e r e ,  of the response o f a h o r i z o n t a l model to a s i n u s o i d a l base motion, i n which boundary  e f f e c t s due t o the l i m i t e d l e n g t h  were taken i n t o a c c o u n t , w i l l  a l s o be p r e s e n t e d .  There a r e f o u r fundamental the  ground.  These  and w i d t h o f the model  types o f waves which propagate  through  a r e as f o l l o w s : Shear wave  Body waves  — D i l a t a t i o n a l wave R a l e i g h wave  S u r f a c e wavesLove wave  The s h e a r wave i n v o l v e s particles  no v o l u m e t r i c change and the movement o f s o i l  i s p e r p e n d i c u l a r t o the d i r e c t i o n o f t h e p r o p a g a t i o n , and  the v e l o c i t y i s g i v e n by  3-1  14.  where p stands  f o r the mass d e n s i t y o f s o i l s .  wave the movement o f s o i l p a r t i c l e s  I n the d i l a t a t i o n a l  i s back and f o r t h , i n the d i r e c t i o n  of p r o p a g a t i o n , t h e r e f o r e compressions and e x t e n s i o n s a r e i n v o l v e d . Its velocity  i s g i v e n by  2J2 =f(?\ * £-)/[>  where  ?\ =  3-2  /-r^X' -2/J)  •  p a r t i c l e movements o f e l l i p t i c a l  The R a l e i g h wave c o n s i s t s o f  shape which i s on the p l a n e  p e r p e n d i c u l a r t o the s u r f a c e and p a r a l l e l The v e l o c i t y  to the d i r e c t i o n o f p r o p a g a t i o n .  i s g i v e n by  IT* = r V7  3-3  where r i s ,a c o n s t a n t which depends cn P o i s o n ' s u » 0.2  r » 0,911  . • y =» 0.3  r = 0.928  y = 0.4  r » 0,942  y « 0.5  r = 0.955  The Love wave i s sometimes  referred  to as h o r i z o n t a l  The movement o f s o i l p a r t i c l e s i s h o r i z o n t a l of  the p r o p a g a t i o n .  T h i s wave e x i s t s  of  the s u r f a c e l a y e r  i s less  velocity  r a t i o , u , f o r example  SR,  shear wave  and normal t o the d i r e c t i o n  o n l y i f the shear wave v e l o c i t y  than t h a t o f the lower  layer.  The  of the Love wave depends upon i t s wave l e n g t h and the l o n g e r  the wave l e n g t h the g r e a t e r the v e l o c i t y .  The maximum v e l o c i t y  wave i s e q u a l to the shear wave v e l o c i t y o f the lower l a y e r minimum i s e q u a l to the shear wave v e l o c i t y  o f the s u r f a c e  o f the Love  and t h e layer.  I f the h o r i z o n t a l s o i l model i s assumed to be.an i n f i n i t e i t s response to a . h o r i z o n t a l  base motion can be  layer,  a n a l y z e d as a pure  shear-wave p r o p a g a t i o n problem i n a s e m i - i n f i n i t e . m e d i a .  The  equation  of motion f o r v i b r a t i o n of a s e m i - i n f i n i t e h o r i z o n t a l l a y e r w i t h u n i f o r m shear modulus, G,  s u b j e c t e d a t i t s base t c a h o r i z o n t a l  motion, ii , i s g  r  dt  ^dtf*  r u  3  where c i s a damping c o e f f i c i e n t . s i n u s o i d a l base motion, i . e . u  g  •= a  3-4  The  S  s o l u t i o n of t h i s e q u a t i o n f o r  s i n ut i s  3-5  where YnC^i-COS^O-Ji-  H i s the  thickness  an  ?4  -  3-6  of l a y e r ,  V  £  ~  /^ft-i  f  **"'Tl3tf  3-9  In f a c t , models t e s t e d here were n e i t h e r nor w i t h u n i f o r m shear modulus, facts  that  the l e n g t h  and  > hese two  semi-infinite layers,  conditions;,  t h a t i s , the  w i d t h of the model were s i g n i f i c a n t l y s m a l l  compared w i t h i t s depth i n c r e a s e s w i t h depth, time.  and t h a t the shear modulus of the  are q u i t e d i f f i c u l t  to account  T h e r e f o r e , the one which l e a s t a f f e c t s  modo.l  f o r at the rtame  the a n a l y s i s w i l l be  ignored. I f the model i s assumed to be a s e m i - i n f i n i t e l a y e r and i t s sheax, modulus i n c r e a s e s w i t h depth by  3-10  then the e q u a t i o n o f motion i s ( I d r i s s and Seed, 1966)  3-11  If  p^.  , the s t e a d y - s t a t e s o l u t i o n f o r s i n u s o i d a l base motion o f  e q u a t i o n 3-11 i s  ^ ^ y ; = J> YnCy)- , »->- ,  3-12  2  and 3-13  where  ./3„ = r o o t s o f J^  f i r s t k i n d of o r d e r «= -£> and p  =0  »  J-l  a  B  e  s  s  e  l  f u n c t i o n of the  = gamma f u n c t i o n .  Also,  /-^  3  -  1 5  17 /  3-17  c m )  s and 0 a r e c o n s t a n t s r e l a t e d to  by  /Q0-0-r25 = O  In  o r d e r to examine t h e e f f e c t of t h e v a r i a t i o n o f the shear modulus  of  models, the fundamental  f r e q u e n c y and mode shape o f a 1 f t . deep  l a y e r were c a l c u l a t e d by u s i n g s o l u t i o n s f o r both a s e m i - i n f i n i t e w i t h u n i f o r m shear modulus, e q u a t i o n s infinite  layer  (3-8) and ( 3 - 5 ) , and f o r a semi-  l a y e r whose shear modulus i n c r e a s e s w i t h depth by e q u a t i o n s  (3-10), (3-15) and (3-12).  R e s u l t s a r e 'shown i n Fig«3-1«  The assumed  shear, moduli which were o b t a i n e d by e x t r a p o l a t i n g from H a r d i n and R i c h a r t ' s e q u a t i o n (see Eq.3-27) a r e shown i n F i g , 3 - 1 , the s o i l ,  /" = 113 p s f damping r a t i o ,  ^  U n i t weight o f  0.02 and ft f 1/2 were assumed.  From F i g , 3 - 1 ( a ) , i t can be seen t h a t the fundamental  frequency of the  l a y e r w i t h u n i f o r m shear modulus G - 2,20xl0~* p s f , i . e , 1520 p s i , corresponds to  t h a t of t h e l a y e r whose -shear modulus v a r i e s w i t h depth by \ •• r  G = (2.76x10^),y^/2,psf. if  Moreover, i t i s found from F i g , 3 - l ( b )  a u n i f o r m shear modulus i s chosen  that,  i n such a way t h a t the chosen  u n i f o r m shear modulus g i v e s the same fundamental  f r e q u e n c y as t h a t  c a l c u l a t e d by t h e e q u a t i o n which takes the v a r i a t i o n o f shear modulus i n t o a c c o u n t , mode shapes  c a l c u l a t e d by both e q u a t i o n s agree w i t h each o t h e r .  T h e r e f o r e , e f f e c t s o f v a r i a t i o n of t h e shear modulus w i t h depth a r e n o t very important i f G average On  as a u n i f o r m  f ( y ^ / 2 ) and can be overcome by t a k i n g a s u i t a b l e shear-modulus,  the o t h e r hand, e f f e c t s on the v i b r a t i o n o f models due t o the  restricted as f o l l o w s .  l e n g t h and w i d t h o f the model compared w i t h i t s depth a r e These e f f e c t s w i l l be c a l l e d  "boundary e f f e c t s on the  IS.  ._<> FOR. SHEAR  H0DULU5  •By  I  62 -  ©  FOfr  UMFORM  2,7Sx-/C ' !  SHER  MODULUS  (3-8)  e%-  6/  &.'l/£A/  ef. (3-/SJ  ?.8x/cf  CbJ PTPOE  SHBAf? Br  •75  •©  MO DULL/5  C£o  FOR  e?.  V/J/FO/ZM SH£Z  ^=2.2x/o (pJF) t  <j-/t/£,</  (3-/2)  rroDuu/s  ef(3-£)  {"= //3 PCF % .= o.oZ 7 P£GU&VCy oT EXC/TAT/O// oJ=SOQoJ J  :  /.o  1.0  2.0  3.0  4.D  P/5PLACEMEA/T PAT/0 UMTH RE5PECT TO BASE  p.  3  -f  EFFECTS  OF. MRIAT/OA/  //V WEAK  M0PUW5  vibration  hereafter.  T a k i n g boundary e f f e c t s on  the v i b r a t i o n  account; the more g e n e r a l e q u a t i o n of motion of a h o r i z o n t a l s u b j e c t at i t s base to a h o r i z o n t a l i n Appendix I ,  and  sinusoidal  The. s t e a d y - s t a t e p a r t  l/cz,  5/» J?^sm%^  64-  a? si»(^£  of the  motion has  solution  into  layer  been s o l v e d  is  sJr>-£^.  3.19  + &?rr)  3-20  3-21  where p . q . r , » 1,3,5 wave v e l o c i t y Y  « b,  and  , ^ p ^ r  ^J-2LLiJ3L  h are  t  ~ phase a n g l e , k£  • }f  a  s  are  as  sides,  b o u n d a r i e s are  (2)  the  (3)  d i r e c t i o n of e x c i t a t i o n ( i n the  and  the  relative  s o i l movements  horizontal,  Shear modulus i s u n i f o r m throughout the  model.  The  observation i n  assumption (1)  looks v a l i d  amplitude o f s t r a i n i s s m a l l .  from the  as shown on  Fig,3~6, provided  Namely, r a t i o s of  the  assumption (2)  i t has  been shown by  the  of p l a n e s t r a i n c o n d i t i o n  tests the  acceleration  s o i l s a d j a c e n t to b o u n d a r i e s w i t h r e s p e c t to the base are  element a n a l y s i s  *  zero,  d i r e c t i o n of  d i r e c t i o n of y i n Fig,3-2) and  d e s c r i b e d i n the next s e c t i o n  For  a u c  follows:  movements between s o i l p a r t i c l e s and  is uniaxial  t  dimensions of the model as shown i n F i g , 3 - 2 ,  At each boundary, bottom, both ends and  The  j  shear wave v e l o c i t y  Assumptions made i n t h i s s o l u t i o n (1)  » dilatational  of  almost U n i t y ,  t e c h n i q u e o f the  finite  that when the model i s  F/j  3-2  DIMEA/S/OA/S  OF  M O D E L  21.  e x c i t e d by  a u n i a x i a l h o r i z o n t a l base motion the predominant motion  of s o i l p a r t i c l e s i s h o r i z o n t a l and . and n e g l i g i b l e .  Assumption (3) has  namely i f a s u i t a b l e average  the v e r t i c a l movement i s q u i t e s m a l l a l r e a d y been shown to be  shear modulus i s chosen,  the model w i t h a u n i f o r m shear modulus w i l l approximation.  It is interesting  to note  give a  the e q u a t i o n f o r  reasonable  that equation  (3-18) i s no  l o n g e r f o r a s i m p l e shear v i b r a t i o n b u t c o n s i s t s of shear i n two all  p l a n e s , which are h o r i z o n t a l and v e r t i c a l ,  i n the d i r e c t i o n of the  vibrations  and a d i l a t a t i o n a l wave,  excitation.  In o r d e r to examine the boundary e f f e c t s on response  the v i b r a t i o n ,  frequency  curves i n terms of a c c e l e r a t i o n r a t i o a t the c e n t e r of the  s u r f a c e w i t h r e s p e c t t o the base were c a l c u l a t e d from both of s e m i - i n f i n i t e  l a y e r w i t h u n i f o r m shear modulus, e q u a t i o n  the e q u a t i o n of the f i n i t e model, e q u a t i o n (1) and  satisfied,  the curve  (3-18), and  the  equation  ( 3 - 5 ) , and  t h e s e , the  (2) r e s p e c t i v e l y , are shown on F i g , 3 - 3 ,  curve  In t h i s  f i g u r e , the n e g a t i v e r a t i o means t h a t when the base moves towards a p o s i t i v e d i r e c t i o n , the s u r f a c e moves towards an o p p o s i t e or n e g a t i v e direction.  T h i s happens, f o r example, a t mode shapes of odd numbers f o r  a s i m p l e h o r i z o n t a l shear v i b r a t i o n ,  Two  F i g . 3 - 3 show q u i t e d i f f e r e n t shapes,  These d i f f e r e n c e s a r e as f o l l o w s ;  (i)  (1) and  (2) i n  fundamental f r e q u e n c i e s are d i f f e r e n t ; 63 cps f o r a s e m i - I n f i n i t e  l a y e r , curve  ( 1 ) , w h i l e 108  a g i v e n dimension,  cps was  curve. ( 2 ) ,  cause a h i g h e r s t i f f n e s s and (ii)  c u r v e s , curve  the maximum response  o b t a i n e d f o r the f i n i t e  T h i s i s because the boundary  restraints  t h e r e f o r e a h i g h e r fundamental  frequency,  a c c e l e r a t i o n r a t i o s are d i f f e r e n t , more  than 20 f o r the m a g n i f i c a t i o n f a c t o r f o r curve 15 f o r curve movement.  (2),  layer of  (1) w h i l e l e s s  T h i s i s because the b o u n d a r i e s  restrict  the  than soil  22.  ^  <5>  V,  ^  ^  (iii)  the shape of the curves i t d i f f e r e n t , w i t h a more c o m p l i c a t e d  shape f o r curve (2) than t h a t f o r curve ( 1 ) , p r e v i o u s l y , because  T h i s i s , as  mentioned  e q u a t i o n (3-18) i n v o l v e s n o t o n l y the shear  v i b r a t i o n i n the h o r i z o n t a l p l a n e but a l s o the shear v i b r a t i o n i n the v e r t i c a l p l a n e and the d i l a t a t i o r . a l v i b r a t i o n i n the d i r e c t i o n o f the e x c i t a t i o n motion.  For example, peak " a " on curve ( 2 ) , which i s  g i v e n by e q u a t i o n (3-18), corresponds to the peak " a " ' on curve  (1)  which i s g i v e n by e q u a t i o n ( 3 - 5 ) , and "b" on curve. (2) corresponds to "b"  1  on c u r v e ( 1 ) .  Chese are the fundamental  frequency and the second  r e s o n a n t f r e q u e n c y , r e s p e c t i v e l y a s s o c i a t e d w i t h the h o r i z o n t a l vibration.  On curve (2) between " a " and "b" t h e r e are two  shear  distinct  peaks which cannot be found on curve ( 1 ) ; these are r e s o n a n t f r e q u e n c i e s a s s o c i a t e d w i t h the 3rd and 5th d i l a t a t i o n a l v i b r a t i o n i n l o n g i t u d i n a l direction.  I n c i d e n t l y , mode shapes  o f even number do n o t  to the v i b r a t i o n o f the model, because hence f o r c e s i n the model due  those mode shapes  contribute are  symmetric,  t o these modes o f v i b r a t i o n c a n c e l out  each o t h e r and have no e f f e c t on e x t e r n a l f o r c e s .  As d i s c u s s e d p r e v i o u s l y  the boundary e f f e c t s on the v i b r a t i o n are much more s i g n i f i c a n t the e f f e c t of the v a r i a t i o n of shear modulus w i t h depth.  than  T h e r e f o r e , the  r e s t r a i n i n g boundary e f f e c t s w i l l be taken i n t o account and e q u a t i o n (3-18) w i l l be used i n the a n a l y s i s h e r e a f t e r .  Boundary e f f e c t s  are  e v a l u a t e d and d i s c u s s e d in.more d e t a i l i n Appendix I I , Another  point  to be d i s c u s s e d here i s the d e t e r m i n a t i o n o f damping  r a t i o from e x p e r i m e n t a l r e s u l t s . determined  from experiment  The e q u i v a l e n t damping r a t i o can be  by v a r i o u s methods (see H a l l and R i c h a r t  1963),  Of t h e s e methods o n l y those a p p l i c a b l e t o t h i s r e s e a r c h w i l l be d i s c u s s e d . If  the f r e q u e n c y - r e s p o n s e  e q u i v a l e n t damping r a t i o ,  curve i s o b t a i n e d f o r f o r c e d v i b r a t i o n ^  , i s g i v e n by  the  where times  ^ f  of f r e q u e n c y  the maximum a m n l i t u d e , J  resonant this  i s the w i d t h  frequency,  i n their f i e l d  of. •—=  U - ,as shown i n F i n , 3 - 4 , and max' T  M a r t i n and test  a t the amplitude  1  » 0,707  f i s the o  Seed (1966), f o r example, have a p p l i e d  data.  In c o n n e c t i o n w i t h the f r e e v i b r a t i o n of the model, the e q u i v a l e n t damping c h a r a c t e r i s t i c i n terms of l o g a r i t h m i c d e c r e c e n t ,  £  , is  g i v e n by  ^ - i r ^ i t ^  and  the damping r a t i o ,  3-23  ^  , i s g i v e n by  3-24  J -  where u's  are amplitudes  on F i g , 3 - 5 ,  o f motion a t n t h and  U s i n g t h i s method, H a l l and  the damping c h a r a c t e r i s t i c s o f  3-3,  Boundary e f f e c t s on  (n-l-m) t h c y c l e as shown  R i c h a r t (1963) have  sand,  the v i b r a t i o n o f the model.  In o r d e r t o examine the v a l i d i t y  o f the s o l u t i o n  (3-18) o f  e q u a t i o n of motion o f the model which takes boundary e f f e c t s account, The was  s e v e r a l p r e l i m i n a r y t e s t s were e x p e r i m e n t a l procedure  v i b r a t e d by  i s as f o l l o w s .  i n t o a h o r i z o n t a l model,  About 700  While  a base motion of c o n s t a n t a c c e l e r a t i o n and  acceleration distribution  throughout  the  into  performed.  p l a c e d i n the c o n t a i n e r to a h e i g h t o f 6 i n c h e s .  compacted by v i b r a t i o n  determined  l b s of  sand  Then-the sand the model frequency,  the s u r f a c e of the model  measured by moving an a c c e l e r o m e t e r from p o i n t to p o i n t .  Then  was  was the  was acceleration  £}j  3-<r  FXPL4AI/)TlOAl  Of  LOGARITHMIC  VBCRSCEA/T.  26.  r a t i o s a t the s u r f a c e w i t h r e s p e c t to the base were c a l c u l a t e d . amplitude  r a t i o s are shown i n Fig,3-6 f o r a f r e q u e n c y o f 50 c p s .  shown are t h e o r e t i c a l v a l u e s , which agree values.  The  E = 1300  p s i , u - 0,3  fairly  3-4,  Also  c l o s e l y w i t h measured  t h e o r e t i c a l v a l u e s were c a l c u l a t e d by e q u a t i o n (3-18) w i t h and  5=3%,  Also acceleration r a t i o s  to both s i d e s and end w a l l s were almost u n i t y , which was the  Measured  adjacent  assumed i n  solution,  Frequency-response  curve of a h o r i z o n t a l model.  In o r d e r to o b t a i n the f r e q u e n c y - r e s p o n s e  curve o f a h o r i z o n t a l  model, a c o u p l e o f t e s t s were done w i t h 1 f o o t h i g h models. The  e x p e r i m e n t a l procedure  i s as f o l l o w s .  Two  a c c e l e r o m e t e r s were  p l a c e d i n the model which had been compacted by v i b r a t i o n , one model 502) 127)  a t the c e n t e r of the s u r f a c e and  the second  at* the c e n t e r o f the m i d - h e i g h t , namely 1/2'  (metrix  (Wilcoxon model  from t h e s u r f a c e .  A f t e r the a c c e l e r o m e t e r s were p l a c e d , a v i b r a t i o n w i t h s m a l l a c c e l e r a t i o n was  a p p l i e d so t h a t the sand and embedded a c c e l e r o m e t e r s reached  c o n f i g u r a t i o n r e l a t i v e to each o t h e r , • Lhen the model was  a stable  vibrated  by  a s e r i e s o f s i n u s o i d a l base motions of v a r i o u s f r e q u e n c i e s from 7,5 to 54 c p s . surface, k a t each  cps  A c c e l e r a t i o n s a t t h r e e p o i n t s o f the model, i«e, at the t  , a t the m i d - h e i g h t , k « aud  * nr  a t the base, k, , were measured " b*  f r e q u e n c y o f base motions (see T a b l e 3-1),  The  acceleration  r a t i o a t the s u r f a c e w i t h r e s p e c t to the b a s e , a c c e l e r a t i o n a t the mid h e i g h t w i t h r e s p e c t to the base  , and M=  were c a l c u l a t e d and p l o t t e d a g a i n s t frequency i n F i g . 3 - 7 , m a g n i f i c a t i o n s o f the r a t i o f o r both M^ of a p p r o x i m a t e l y  48 c y c l e s per  »  2  Pronounced  and M^ were n o t e d a t a frequency  second.  From these r e s u l t s dynamic moduli can be c a l c u l a t e d .  the  and  the e q u i v a l e n t damping  S u b s t i t u t i n g the l e n g t h of the model  £ <* 8',  ratio and  the  HE/G-h'T OP DSO'SB l\Z£OR0/J  •- /.ao  . /,  20-  — y£ PT  FPEQUEA/CY  = 52?cpsC^JNUSDID/U-')  •  /.<$0A  60 A-7Z  <P)  TrtEOP£7?C4/~  RAT/o.  AT  P>ISTR/3UrtOA/ SUREQCE  WITH  OE  RESPECT  ACCELEfcqTf TO  T^SB  OA/  28,  Tah/e  3-/  FREQU&VCYAMD  'fiEL/)770AI'3E7WE&A7  ACC£L£RAT/OA/_  FAT/D  & (3)  fc„ (3) .36  . 4o  .39 . 412  /3.5  . 7D  . 72  .67  /7. 9  .83  • 76  .73  .  . 7B .34  .74 .32  .76  . 45  274  .70 .72 .32  .34  . 942  /.OO  333  - 74  ./B  ./~7  .79  .945  /.o5  37. O 38.5  . /O  ./Z  ./3  ./5  . /5  3?.Z 40.0  - 73 ./z  ./3  ./3  .78 ./6 . 73  .  • o?  .o9  . 70  a/.o  .07  .08  .085  . 70  42.0  . /Z . //  .'25  .75  ./2  . , 6 *  y. 04 /.o?  43S  . /O . // .72  ./25  . 76  /.oB  44.5  .21  .25  ./35 .30  45. 5  .24-  .34  . 4-3  . 6o*~  /.26  /.  .62*  A 37  2.06  .32  ,44  A&5  7. OO  .74 . 38  .  2222  . 46  /.55 A 27  •26  .30  -fee  r.9)  PS)  7,5  23.  Z  42.5  416.5  O0  •  .21  b  . 30  47S  . 76  .22  40.7 zo.o 52.5 54*. O  .  .09 .30.  a/7 (zveraja  08  .2Z ./?5 ./6  .30  •3?  .ar  .  20  .36 .38  . 9J5  .925  .  A/OTG-  M2  Mr ?3  .905  . 93 9^5  /.  08  /.  5  00  •.'£>*  .975  A. 5 /. OO  /.  O0  /.  06  /. / / /.3f  20  f.25  /. 45 /.28  Sine C6£) /n/'st>ehai.'£. 1  A 6 77  7. 53  7.#2  .a  bad l/a/u€  /.o7  / . OO  2.  20  . 86  . 535  A poor-  bad fa/>/e /fc/ejt^-frA W-S-4*  29.  F,g  3~7  •7=RE0UJ£A7CY PZ5pDf/SB OT  HDR/ZOMTAL SO/^L MODEL.  30, w i d t h b = 1-1/2' i n t o e q u a t i o n  (3-21), and s e t t i n g p.q, and r •? 1,  Let us assume y = 0,3., i . e .  Then  '3-25a  . HE  =  S u b s t i t u t i n g f = 48 c p f and h => 1 f t , ° o '  and s u b s t i t u t i n g  f J^d 3=  J  f.  "  113^°^ and g => 32.2 f e e t / s e c .  The shear wave v e l o c i t y o r modulus i s r e l a t i v e l y other data  low compared w i t h  (see Fig,3-14) and w i t h p r e v i o u s i n v e s t i g a t o r ' s  results.  F u r t h e r d e t a i l s w i l l be d i s c u s s e d i n the next s e c t i o n t o g e t h e r w i t h  other  results. If  the l o g a r i t h m i c d e c r e c e n t e q u a t i o n  (3-22) i n the p r e v i o u s  section  i s a p p l i e d to the r e s u l t s i n F i g , 3 - 7 , the e q u i v a l e n t damping r a t i o i s e v a l u a t e d as f o l l o w s , s i n c e M = 2,26 •' • max  then 0,707 M  = max  1.60,  31. A  f « 5.7 and £  o  ~- 48 p c f ,  The damping r a t i o , 6%, (see F i g , 3 - 1 5 ) c  i s s l i g h t l y h i g h compared w i t h o t h e r d a t a  The o r d e r o f the damping r a t i o o b t a i n e d h e r e ,  i s q u i t e reasonable.  At the r e s o n a n t f r e q u e n c y  however,  the maximum r e l a t i v e  d i s p l a c e m e n t between the s u r f a c e and the base i s g i v e n by  S i n c e the h e i g h t h = 1 f o o t , the average s t r a i n i s  p  (0.004%)  = 4x'O*  The magnitude of s t r a i n o f 4x10  i s , i n g e n e r a l , s m a l l enough to  c o n s i d e r the model as a l i n e a r v i s c o - e l a s t i c m a t e r i a l , which i s one o f the b a s i c assumptions of the a n a l y s i s . The shape o f the f r e q u e n c y - r e s p o n s e to curve  curves i n F i g , 3 - 7 are s i m i l a r  (2) i n F i g . 3 - 3 which i s c a l c u l a t e d by e q u a t i o n  (3-18) ( f o r a  finite  l a y e r ) r a t h e r than the c u r v e  (!) by e q u a t i o n  finite  layer).  the r e s o n a n t f r e q u e n c y  Namely,  o c c u r s and the amplitude is  substituted  observed  3-5,  right after  r a t i o goes below u n i t y ,  i n t o equation  (3-5) ( f o r semia marked  rolloff  t h e r e f o r e i f G = 285 p s i  (3-18) the t h e o r e t i c a l  curve agrees  with  curve i n F i g , 3 - 7 ,  Measurement  of dynamic modulus.  U s i n g h o r i z o n t a l models a s e r i e s o f t e s t s were c a r r i e d out i n o r d e r to measure  the dynamic p r o p e r t i e s o f the model.  model were  changed i n the range o f 0,47  9" h i g h models were t e s t e d . v i b r a t i o n method  Two  and the second  The v o i d r a t i o s of the  to .68, and 6" h i g h models and  types of t e s t s ,  the f i r s t  the f r e e  the r e s o n a n t method, were done  sequentially  32,  w i t h an i d e n t i c a l model. The  p r o c e d u r e f o r the p r e p a r a t i o n  sand was  weighed when i t was  o f models 'was  poured i n t o the  weight of the model would be known.  as  container  Then the  sand  was  follows. so  that, the  placed  v i b r a t i o n of amplitudes o f a c c e l e r a t i o n  in  range of 0,3  and  the  o  a frequency o f 15  of the l a y e r was  read by  of  at e i g h t p o i n t s  the  container  the s c a l e s  the  t o t a l w e i g h t , the h e i g h t ,  the  average v o i d  void  r a t i o , was  r a t i o was  an average was  a r e a , and  calculated.  The  model i n o r d e r t h a t  applied  obtained.  b e i n g dug  the model would be  uniform,  I t i s p l o t t e d on  Fig,3-3,  (1967) have r e p o r t e d  of v i b r a t i o n and  that  the v o i d  This  of  around over the  r a t i o of the  v i b r a t i o n has  r e l a t i o n s h i p has  15  soils  entire  model  been  been o b t a i n e d  cps, however G r e e n f i e l d  the predominant f a c t o r o f the amplitude o f  and  void  acceleration frequency  vibration.  model 502)  was  v i b r a t i o n was reach a s t a b l e The  embedded at the made so  that  the  configuration  p r e p a r e d , an a c c e l e r o m e t e r  c e n t e r of the model s u r f a c e , sand and  (Metric then a  embedded a c c e l e r o m e t e r would  r e l a t i v e to each o t h e r .  f r e e v i b r a t i o n method has  been made by  the  A square wave h a v i n g a frequency of p r e c i s e l y 2,5 an  Knowing  r a t i o i s almost independent o f the  A f t e r a h o r i z o n t a l model was  and  wall  s p e c i f i c g r a v i t y of  amplitude of -acceleration, d u r i n g  r a t i o of the v i b r a t i o n a l sand mass i s the  of the  taken.  average v o i d  f o r the v i b r a t i o n of a frequency of o n l y Misiaszek  plexiglass  height  next model, h a v i n g a d i f f e r e n t  sand l a y e r was  A unique r e l a t i o n s h i p between the the  the  that,  the  p r e p a r e d by v i b r a t i o n o f a d i f f e r e n t amplitude  a c c e l e r a t i o n , w h i l e the  and  After  a t t a c h e d to the  and  the  cps.  total  in a  h o r i z o n t a l l a y e r by - l 0g  The  amplitude of d i s p l a c e m e n t of 0,02"  was  following  c y c l e s per  applied,  and  procedure, second  both  the  33.  F>?3-B  RELAT/OA/ BE7WEBA/ MD RA770 AA/P ACCELEMT/OA/ APPL/ED PUPJMfr V/3RS7'/OA/  35.  e x c i t a t i o n square wave and  the response  a c c e l e r a t i o n a t the s u r f a c e of  the model were r e c o r d e d by an o s c i l l o g r a p h . o s c i l l o g r a p h r e c o r d s i s shown i n F i g , 3 - 9 .  An example o f these From the o s c i l l o g r a p h r e c o r d s ,  the frequency o f the f r e e v i b r a t i o n of the model was  determined,  using  the p e r i o d of the e x c i t a t i o n square wave as a time r e f e r e n c e . From these r e c o r d s , the l o g a r i t h m i c d a c r e c e n t was by  t a k i n g the decrement of the amplitude w i t h i n s e v e r a l c y c l e s  using equation Another  t e s t by  i s as f o l l o w s .  The  s h a k i n g t a b l e system  the resonant method..was c a r r i e d out i n t u r n u s i n g  d i s p l a c e m e n t was  chosen  p o s s i b l e t o o p e r a t e by  the d i s p l a c e m e n t  quite d i f f i c u l t  command s i g n a l ,  procedure  to  a c c e l e r o m e t e r s a t the base and  connected  In f i g u r e 3-10  a r e s o n a n t frequency i s shown,  the  to both  the s u r f a c e o f the model was  When the resonant frequency was  While  g r a d u a l l y i n c r e a s e d from a  low enough v a l u e , the o s c i l l o s c o p e which was  recorded.  test  the a c c e l e r a t i o n command s i g n a l , w h i l e i t  frequency of the s i n u s o i d a l base motion was  the frequency was  The  as a command s i g n a l o f the  i n t h i s t e s t because i t was  o p e r a t e a t h i g h frequency by  observed.  and  3-23.  the'..same model as used f o r the f r e e v i b r a t i o n method.  was  also obtained  carefully  r e c o g n i z e d on the o s c i l l o s c o p e , an example o f the r e c o r d s a t  W i t h i n a v e r y narrow band  resonant  f r e q u e n c i e s were r e c o g n i z e d a t which the a c c e l e r a t i o n r a t i o at the s u r f a c e o f the model w i t h r e s p e c t to the base of the c o n t a i n e r reached as h i g h as  12,  Fundamental f r e q u e n c i e s measured by both and  the resonant method are shown i n T a b l e 3-2  v o i d r a t i o of models.  F i g u r e 3-11  appears  the f r e e v i b r a t i o n method and F i g u r e 3-11,  with  t o be v e r y s c a t t e r e d ,  e x p e c i a l l y a t low v o i d r a t i o , however, c o n s i d e r i n g the f a c t  that  the  s c a l e o f the frequency c o o r d i n a t e i s l a r g e and t h a t the range o f s c a t t e r i s about  15%,  i t i s not too bad.  V o i d r a t i o s shown i n T a b l e 3-2  and  on  36.  the v e r t i c a l the model.  c o o r d i n a t e i n Fig,3-11 are i n i t i a l Because the maximum amplitude  s i n u s o i d a l base motion was  at most 0,20,  average v o i d r a t i o s  of a c c e l e r a t i o n of the a p p l i e d as r e c o g n i z e d  from  the v o i d rc.tio of the model would h a r d l y change d u r i n g the Therefore  these v o i d r a t i o s can be  F i g u r e 3-11  considered  as those  140  test.  d u r i n g the  c y c l e s per second f o r the dense 6" h i g h model,  c y c l e s per second f o r the dense 9" h i g h model and f o r l o o s e models, and model were s l i g h t l y  tests.  90 c y c l e s per  t h a t the fundamental f r e q u e n c i e s  lower than  those  130 second  f o r the 9"  high  f o r the 6" h i g h model a t the same  ratio. Using Equation  3-26,  p l o t t e d i n F i g u r e 3-12 that  Fig,3-8,  shows t h a t the fundamental f r e q u e i i c i e s of models were  approximately  void  of  shear wave v e l o c i t i e s were c a l c u l a t e d and  w i t h v o i d r a t i o of models.  T h i s F i g u r e shows  the average shear wave v e l o c i t y of the model i n c r e a s e s as the v o i d  r a t i o decreases  and  the h e i g h t of the model i n c r e a s e s , and  v a l u e s were 1 6 0 ~ 2 3 0 and  that  those  2 0 0 ~ 2 9 0 f e e t / s e c . f o r 6" h i g h models and  9"  h i g h models r e s p e c t i v e l y . These r e s u l t s as d i s c u s s e d above can be facts; g i v e n by  the fundamental f r e q u e n c y ,  f , for a horizontal s o i l Q  (see Eq,3-8) i , e , the fundamental f r e q u e n c y  p r o p o r t i o n a l to the h e i g h t of the model. Richart  i n t e r p r e t e d from the f o l l o w i n g layer is i s inversely  A l s o , a c c o r d i n g to H a r d i n  (1963), the shear wave v e l o c i t y of s o i l s ,  L7  s  }  and  decreases  3 w i t h v o i d r a t i o and  i s p r o p o r t i o n a l to  p r e s s u r e , <r , as shown in-Eq,3-27,  /10  power of the c o n f i n i n g  Therefore  the r e s u l t s shown i n  Fig,3-11 shows t h a t the e f f e c t o f the h e i g h t of the model on fundamental frequency  was  g r e a t e r than  which i s a s s o c i a t e d w i t h v o i d r a t i o and  the  t h a t of the shear wave v e l o c i t y confining pressure,  Because  the c o n f i n i n g pressure, i s l a r g e r i n the h i g h e r model, r e s u l t s shown i n  TM a)  3-2  FWSOA/lBA/mL  FR£QL/BA/£Y  /?&4J!>s)a/7i'- Method-far 6"rf/jh/tfodeT?  (FEBT)  . 60s  7 i0 (CPS) /2?^0  .53&  /.7o5 2/6—B  • 56>  A35-6  . 521  A 63  7-2-2  .55  /34~5  .578  7-3-/  .525  /4a-f  7-3-2  . -57  /.42~"5  e. 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A39 • 55  /32  . 5/7  A &7  220  4.09  53.1  72-3D-2  .5/  /30  .7o5  A 64  273  4-05  52. 5  77-3/-/  • 49  /37  . 498  A 625  2/3  4.00  53.2  P-3/-2  .4g  .495  A 62  224  3.W  567  Z>-3/-3  .65  /3B /o4-.5  .552  /. 74  4.22  4.3. f  4'5  52.5  Z>-?/-4 .6o5  / J>7. 5 . 536,  A 7#$  /&2 2/8  J--2-'  .76  /2B  . 521  A.68  2/5  4./0  57.0  7-2-2  . 55  /3?  . 57Z  /.67  233  4.69  53.5  J--3-/  .525  /37  . $/o  A657  2/7  4.05  58. O  7-3-2  .7!  /45  . 5o5  /.64  235  4-oS  77.8  /4/.7  . 49S  A 62  237  3.99  53.5  7-3-?  . <Z77  C)  P&j-o/iait /rje/fiod -forrf"d/jh  MO.  e  2  17*  Vh  CCPS)  IS A 49-3^ $35 38.4  .1/9  2.02  24o ^ 2£r8  4.8/  . 80  2./2  /9S  S.ol  . 175  2.09  S2S  .776  2.o?  2&Z  5.O0  $-25  .525  s40  . 704  2.o5  287  4.39  S&.T  7-/1-3  .50  . 736  2.o4  275  4,87  7-//-4  .485  s35 /4/~243  .729  2.o3  $6.5 590^ <T9.8  7-3^4  .470  7-9-1  . 630  7-9-2  . 53  7-/1-/  .53  J - / / - ?  92 S 2 0  5. DO  4.34  4-9sro.2  <d) Free u/brn-f/o/) weft/ad -Toy <?"fi/ybmpc/el 7-3-4  .490  /28  .7; 9  7-9- f  . 630  S8.7  .Boo  7-tt-f  .380  yo7  • 775  /30  J--/1-Z .525 7-//-3 • 50 7- / / - 4 .485  /a3  258  4.81  /SB  5.07  53.5 37.2  2.0?  224  5.00  44.8  .744  2.o5  267  4.89  54.5  .736  2.o4  286  4?87  S£.6  .778  2.03  2 90  4.84  60.O  2.o2  /&££ WBt?GT/0/L> METHOD, £ " AfOPEL " R&SOMA/T METHOD „  9o  O  9 "MOOB-L. ,  e  rtOOZL O  *? " MODJrL  P O  CO Q  /30  /OO  7^  o  •f. (CP.?) 3-/f  7=C/A/OAM&VfAL  3-/z S H Z A R  mva  FREtQUBA/CY  \/&iocf7Y  /40  40.  Fig*3-12 are r e a s o n a b l e f o r the range of low v o i d r a t i o s .  It i s  i n t e r e s t i n g to note t h a t f o r h i g h v o i d r a t i o s , the e f f e c t o f h e i g h t of the model on the shear wave v e l o c i t y i s not d i s t i n c t , and both  the  6"  h i g h mode; and 9" h i g h model g i v e the same shear wave v e l o c i t y f o r a given  ratio, A s u b s t a n t i a l number of l a b o r a t o r y t e s t s have been c a r r i e d  to measure the dynamic moduli reviewed  of s o i l s ,  these p r e v i o u s works f u l l y .  H a r d i n and R i c h a r t (1963) have  From these works the  shear  wave v e l o c i t y of v a r i o u s dry sands i s p l o t t e d i n F i g u r e 3-13, r e s u l t s of t h i s s e r i e s of t e s t s .  (r~  f o r s m a l l amplitude  -S6  e  )  3  -  2  i s the c o n f i n i n g p r e s s u r e l e s s than 2000 p s f .  R i c h a r t used  7  Harding  e q u a t i o n 3-27  was  In F i g u r e  extrapolated to very small c o n f i n i n g pressures,  a l t h o u g h the range of c o n f i n i n g p r e s s u r e s of t h e i r experiments 500~2000 psf f o r t h i s equation, out a s e r i e s of v e r y s i m i l a r t e s t s  2,0'  and 2,7'  to those r e p o r t e d h e r e .  except  The  About  for loose state  17' and  f o r dense s t a t e , were t e s t e d by a s h a k i n g t a b l e  u s i n g s i n u s o i d a l base motion, i n Fig,3-13,  was  A r a i and Umehara (1966) have c a r r i e d  l o n g by 5' wide models, whose h e i g h t s were 1,25' 1,3',  and  c y l i n d r i c a l specimens t e s t e d by the resonant method i n  t o r s i o n a l v i b r a t i o n under h y d r o s t a t i c c o n f i n i n g p r e s s u r e s . 3-13  of  be g i v e n by  = (//9  where  w i t h the'  A c c o r d i n g to H a r d i n and R i c h a r t (1963)  the shear wave v e l o c i t y of Ottawa dry sand s t r a i n may  out  A r a i and Umehara's r e s u l t s are a l s o p l o t t e d  agreement of r e s u l t s w i t h p r e v i o u s works i s v e r y good  t h a t the v a l u e f o r e » ,68  as r e c o g n i z e d i n F i g , 3 - 1 3 ,  41.  HARD/A/ AVO R/&/ART 0$£3), DTTAU/A  o-o-o- JIDA • Q  SAA/D , DRY.  £rR/$D£-JD QUART? SAAJD,  6'= CTA/^A/OU/A/;  uT=/4.g% ,<r.TO  Vf. oT  _ k//t30/J AW M/JULER (/?<£2), MED.F/ME, CtEA-A/ SAA/O €=.7/ uT^ /3.2jg fj tfYVR ARAI A/v>£> UrtEHfr&A OV£6) 7?VE ErRA//j£D PAY, (T7Cue- 7V u/T. OE SPBCIM&A/ y  of Trf/S RBSEARCH OB7A/A/EP 7/J 7Hf5 5ECT/OA/  @  RESULTS  ©  RESULTS OT TH/S RESEARCH  OT3TflUVE&  FROM FZE&U&A/CY- f?ESPO/JSe  CURVE  '7v'f 3-/3 SHEAR WAVE t/E-LOUTY oT- FXE-WOUS U/ORKER5 AA/D  T/I'/S  RESEARCH  42.  In  F i g , 3 ~ 1 2 , wave v e l o c i t i e s  calculated  a r e a l s o p l o t t e d f o r the model  from e q u a t i o n 3-27 f o r 0~- « 27.5 and 48 p s f , which p r e s s u r e s a t the mid-height  corresponds  to  the overburden  o f 6" and 9", r e s p e c t i v e l y .  It  i s r e c o g n i z e d from the f i g u r e t h a t shear wave v e l o c i t i e s  of  v o i d r a t i o s l e s s than 0.6 agree r e a s o n a b l y w i t h the e q u a t i o n 3-27,  i n the range  but f o r v o i d ra.-.ios g r e a t e r than 0,6 the wave v e l o c i t i e s o b t a i n e d h e r e are  much l e s s than those c a l c u l a t e d by the e q u a t i o n 3-27,  r e a s o n f o r t h i s d i s c r e p a n c y i s presumably  the f a c t  One p o s s i b l e  t h a t the (T i n the  e q u a t i o n 3-27 i s a h y d r o s t a t i c c o n f i n i n g p r e s s u r e i n the range o f 500 p s f ~ 2000 p s f , w h i l e i n t h i s t e s t burden  the c o n f i n i n g p r e s s u r e i s due to the o v e r -  p r e s s u r e o f the sand i t s e l f  As mentioned  and i s v e r y  small.  i n the p r e v i o u s s e c t i o n , i t i s found on F i g u r e 3-13  t h a t the shear wave v e l o c i t y  (108 f e e t / s e c , ) o b t a i n e d from t h e f r e q u e n c y  response curve ( F i g u r e 3-7) was much lower than the o t h e r s f e e t / s e c . ) Fig,3-12. of  strain.  amplitude  One p o s s i b l e reason i s presumably  I t i s known t h a t t h e shear wave v e l o c i t y increases,  D r n e v i c h , H a l l and R i c h a r t  (230^-280  t h e magnitude  d e c r e a s e s as the  (1967) have r e p o r t e d  some t e s t r e s u l t s r e g a r d i n g the e f f e c t s o f amplitude o f v i b r a t i o n on the shear modulus of sand,  Zeevaert  (1967) has t e s t e d some Ottawa sand  specimens by the f r e e v i b r a t i o n method, i n which he used r a t h e r h i g h amplitude o f s t r a i n because slightly  o f the n a t u r e o f h i s a p p a r a t u s , and o b t a i n e d  low shear wave v e l o c i t i e s .  In these t e s t s , c y l i n d r i c a l  specimens  were t e s t e d I n t o r s i o n a l mode and the s t r a i n was e x p r e s s e d i n terms o f radian, therefore i t i s d i f f i c u l t It  t o compare  directly.  i s i n t e r e s t i n g t o n o t e t h a t A r a l and Umehara (1966) have o b t a i n e d  a shear wave v e l o c i t y c f about  570 f e e t / s e c . In dense s t a t e and 400  f e e t / s e c , in. l o o s e s t a t e i n the s m a l l s t r a i n range  (5x10  , but i n  both cases i t decreased c o n t i n u o u s l y as t h e shear s t r a i n i n c r e a s e d and at  a s t r a i n of lx.10-3 i t reached as low as 100 f e e t / s e c .  The amplitude  43.  of t - t r a i n a t the r e s o n a n t f r e q u e n c y i n the f r e q u e n c y - r e s p o n s e was 4>0x10 and  , w h i l e i t was around  those o f t h e  range o f around  Another  i n the f r e e v i b r a t i o n method,  r e s o n a n t method v a r i e d i n each  1.3x10  t e s t b u t were i n the  T h e r e f o r e , the s h e a r wave v e l o c i t y o b t a i n e d  from the f r e q u e n c y - r e s p o n s e o t h e r methods  4,0x10  curve  curve can be s m a l l e r than t h a t o b t a i n e d by  4  r e a s o n may be t h e shape o f t h e base m o t i o n .  i n T a b l e 3-1, the shape o f the base motions  As i n d i c a t e d  which were supposed  t o be  s i n u s o i d a l was n o t q u i t e s o , e s p e c i a l l y i n the range o f h i g h f r e q u e n c i e s , T h e r e f o r e , i f t h e response spectrum actually  applied  base motion,  o f the base, motion which was  to the model was d i f f e r e n t  then i t i s p o s s i b l e t h a t  curve w i l l be d i f f e r e n t  from t h a t o f a s i n u s o i d a l  the measured f r e q u e n c y  response  from t h a t p r e d i c t e d by a t h e o r y , i n which a  s i n u s o i d a l motion was assumed. In F i g u r e 3-14 d i m e n s i o n l e s s q u a n t i t i e s are p l o t t e d  against void  r a t i o , e.  —  I t cannot be s a i d  =  I—3L.  definitely  because  the d a t a a r e n o t good o r e x t e n s i v e enough, but i t l o o k s as though / <5~  1rf) is  constant f o r a given void r a t i o  i n the range o f low v o i d  f a r as the d a t a o b t a i n e d here i s concerned.  I t follows  r a t i o as  t h a t an average  shear modulus o f d r y sand l a y e r i s p r o p o r t i o n a l t o i t s t h i c k n e s s .  This  means the d i s t r i b u t i o n o f t h e shear modulus o f a d r y sand l a y e r i s a l s o l i n e a r w i t h r e s p e c t to the depth. A c c o r d i n g t o H a r d i n and R i c h a r t shear wave v e l o c i t y o f dry sand and s t r a i n and  (1963) predominant  are void  f a c t o r s o f the  r a t i o , e, c o n f i n i n g p r e s s u r e , cr ,  amplitude, f , and o t h e r f a c t o r s such as g r a i n s i z e and shape  f r e q u e n c y have minor e f f e c t s , and the dynamic s h e a r modulus i s  1 1 ~ 1 2 % h i g h e r than, s t a t i c , .  ©  O  n o a  O e  FX££ V/&mr/PA>/iE7rtODj <L '' A70DBL  O  <f" /10OSL-  *  o O  ES3  ©a  O  ea O  O  55  5£  30  -A e- J 2 L  &j-3-/4  A/&&TOF weogoA/  o  MOP&L ^ 6 waves  §.3  s  I-'  Cl  •5 yo/D  Fty  -6  -7  RAT/Oe  3-/S- LO$-A/?/77irifC  DEOZECBAfT  6o  As i s shown on F i g u r e 3-15, in  t h i s s e r i e s of t e s t s was  the l o g a r i t h m i c d e c r e c e n t  i n the range o f 0 , 1 ~ 0 , 2 ,  e f f e c t of v o i d r a t i o of the model was  observed,  No  observed significant  Thesewalues  give  an e q u i v a l e n t damping r a t i o i n the range of 0,02-^-0.03, i , e , 2-^3%, Although  the frequency  and  c o n f i n i n g p r e s s u r e are q u i t e d i f f e r e n t ,  the  o r d e r of the l o g a r i t h m i c d e c r e c e n t agrees w i t h H a r d i n and R i c h a r t ' s results  (1963),  damping r a t i o  A r a i and Umehara (1966) have, o b t a i n e d a v a l u e of  l e s s than 0,1  f o r the s m a l l s t r a i n and  0.4  the  at s t r a i n of  _3 1x10  .  According to Hardin  as the amplitude 3-6.  (1966) , the l o g a r i t h m i c d e c r e c e n t i n c r e a s e s  of s t r a i n i n c r e a s e s and  decrease  Summary and c o n c l u s i o n . U s i n g the s h a k i n g t a b l e and  of  the c o n f i n i n g p r e s s u r e  a finite  a s o l u t i o n of the e q u a t i o n o f motion  s o i l l a y e r s u b j e c t a t i t s base t o a h o r i z o n t a l motion, a  s e r i e s of t e s t s x^ere c a r r i e d out i n o r d e r t o observe the h o r i z o n t a l dry sand model d u r i n g v i b r a t i o n and dynamic p r o p e r t i e s .  The  the b e h a v i o u r  of  to measure the  t e s t i n g t e c h n i q u e used h e r e i n has  the f o l l o w i n g  advantages: (1)  The  s t r e s s e s i n the model are overburden  p r e s s u r e and k  condition o  xtfhich r e p r e s e n t the s t r e s s c o n d i t i o n s of s o i l (2)  in situ,  The predominant v i b r a t i o n of the model i s a h o r i z o n t a l  v i b r a t i o n x^hich i s an important  component of the earthquake  shear motions of  the ground, and which governs the v i b r a t i o n mode of dams, (3)  Because the model i s c o n s i d e r a b l e l a r g e , the e f f e c t s o f d e t e c t o r s  are s m a l l , C o n c l u s i o n s o b t a i n e d are as f o l l o w s : (1)  The b e h a v i o u r  p r e d i c t e d by  o f the model durin.g v i b r a t i o n agreed w i t h t h a t  the e l a s t i c  t h e o r y , and  the s o l u t i o n f o r the e q u a t i o n of  46.  motion of the model which takes boundary e f f e c t s i n t o account c o n f i r m e d t o be v a l i d .  But the f a c t t h a t the sand has  a non  was linear  modulus, namely t h a t the shear modulus depends on the a m p l i t u d e o f  strain,  has been o b s e r v e d , (2)  For the v o i d r a t i o l e s s than ,6, the o b t a i n e d shear wave v e l o c i t i e s  agreed with t h o s e e x t r a p o l a t e d from r e s u l t s by -Hardin and R i c h a r t t a k i n g overburden g r e a t e r than  p r e s s u r e as c o n f i n i n g p r e s s u r e , but f o r the void, r a t i o  ,6 the r e s u l t s were q u i t e d i f f e r e n t .  s h e a r wave v e l o c i t y of the a c t u a l ground  T h e r e f o r e , when the  o r some s o i l s t r u c t u r e s  e s t i m a t e d , the v a l u e o f the k w i l l be v e r y * o (3)  (1963),  are  important, '  I f the v o i d r a t i o of a dry sand l a y e r i s a c o n s t a n t and i s g i v e n ,  then the s h e a r modulus o f the l a y e r i s l i n e a r l y p r o p o r t i o n a l t o i t s depth.  T h i s c o n c l u s i o n i s n o t . d e f i n i t e because  of the s c a t t e r e d  data  and i n s u b s t a n t i a l number o f d a t a w i t h r e s p e c t to h e i g h t o f models,, (4)  Damping r a t i o o f dry sand under low c o n f i n i n g p r e s s u r e and  f r e q u e n c y i n the range o f 1 0 0 ~ 1 4 0 cps was  around  2~-3%,  '  47.  CHAPTER.4  STABILITY OF MODEL SAND SLOPES DURING VIBRATION  4T1.  Introduction The  stability  d u r i n g an earthquake of an e a r t h o r r o c k  fill  dam,  embankment, c u t t i n g of the e x i s t i n g ground o r even e x i s t i n g n a t u r a l ground s l o p e , i s one Slope  stability  depending on with  of the most important  the m a t e r i a l s c o m p r i s i n g  comprised of dry  does not  affect  mechanics.  the s l o p e , which i s a s s o c i a t e d The  becomes an important  f i r s t mechanism d e a l s  pressure  iThe second d e a l s w i t h  slopes  f i n e sand i n which l i q u e f a c t i o n phenomenon  factor.  of f a i l u r e  and  completely  d i f f e r e n t from one  The  last  i s cohesive  slopes.  The  the s t r e n g t h c h a r a c t e r i s t i c s of these m a t e r i a l s  mechanism are  another.  c o n v e t i t i o n a l method f o r the p r e d i c t i o n of the s t a b i l i t y  slopes  comprised of dry m a t e r i a l i s fundamentally  static  s l o p e s t a b i l i t y , except t h a t h o r i z o n t a l i n e r t i a  i n t o account. difficult  f o r c e s are  procedures: of the d e s i g n  a c c e l e r a t i o n of the  (2)  determination  of the d e s i g n  s t r e n g t h of the m a t e r i a l d u r i n g  earthquake  s l i c e method by Mononobe and T a k a t a (1936) (reviewed  Martin  of s o i l  taken  In p r a c t i c e , however, t h i s method i n v o l v e s the f o l l o w i n g •  determination  and  of  the same as t h a t of^ the  (1)  The  with  c o h e s i o n l e s s m a t e r i a l i n which the pore  the s t a b i l i t y o f s l o p e s ,  comprised of s a t u r a t e d  The  of s o i l  d u r i n g an earthquake i n v o l v e s q u i t e d i f f e r e n t mechanisms  the r o l e of pore water p r e s s u r e ,  slopes  fields  (1966)) and  by  the earthquake Seed  the f i n i t e element method o f the response a n a l y s i s  s t r u c t u r e s (Clough  Khanua (1966), I d r i s s and  and Seed  Chopra (1966) , F i n n (1967) and  Idrisc  (1966b) , F i n n  and  (1968)), have made  48,  g r e a t c o n t r i b u t i o n s t o the d e t e r m i n a t i o n Unfortunately,  there has n o t been much p r o g r e s s  of s l o p e s t a b i l i t y concerning  of design a c c e l e r a t i o n .  d u r i n g earthquakes because r e l a t i v e l y l i t t l e  the dynamic c h a r a c t e r i s t i c s o f s o i l s .  (1968) has developed  Although  i s known  Seed  t e s t i n g methods and r e p o r t e d some r e s u l t s ,  extensive i n v e s t i g a t i o n s are required i n t h i s The  i n the whole problem  purpose o f t h i s c h a p t e r  (1966) other  area.  i s to observe t h e mechanism o f f a i l u r e  of d r y sand s l o p e s and t o i n v e s t i g a t e t h e dynamic s t r e n g t h  characteristics  of d r y sand and t o compare the r e s u l t s w i t h e x i s t i n g t h e o r i e s . A f t e r the e f f e c t s o f the s i d e s and ends o f the model cn the s l o p e s t a b i l i t y were examined  ( s e c t i o n 4-2) a s e r i e s o f p r e l i m i n a r y t e s t s  w e r e made i n o r d e r t o get an i d e a o f the b e h a v i o u r 4-3).  Then the measurements o f t h e a c c u m u l a t i v e  the base motion were made ( s e c t i o n 4-4) , tests  t h a t t h e r e were two d i s t i n c t  dry sand f o r a g i v e n amplitude of t e s t s  concerned w i t h  each c o r r e s p o n d i n g  4-2,  o f t h e model ( s e c t i o n  displacement  I t was found  due t o  i n the p r e l i m i n a r y  c h a r a c t e r i s t i c s l o p e angles  of a c c e l e r a t i o n .  o f the  T h e r e f o r e , two s e r i e s  these slope.:angles, which w i l l be d e f i n e d i n  s e c t i o n , was made ( s e c t i o n 4-5, 4 - 6 ) ,  C o n s i d e r a t i o n o f t h e boundary  effects.  Because the c o n t a i n e r has a r e s t r i c t e d w i d t h and l e n g t h , e f f e c t s s h o u l d be examined i n o r d e r p l a n e s t r a i n and i n f i n i t e s l o p e . Chapter 3, i s concerned w i t h  these  to determine i f t h e problem i s one o f  The c o n s i d e r a t i o n in s e c t i o n 3-3,  the boundary e f f e c t s on e l a s t i c  vibration,  which was a s s o c i a t e d w i t h "forces due t o e l a s t i c s t r a i n caused by the boundary r e s t r i c t i o n s , but i n t h i s s e c t i o n e l a s t i c d e f o r m a t i o n  w i l l be  n e g l e c t e d and o n l y the f r i c t i o n f o r c e between the sand and s i d e w a l l s and  the c o n f i n i n g f o r c e a t the toe. o f t h e model w i l l  be c o n s i d e r e d ,  49.  Therefore,  i n t h i s s e c t i o n , l a r g e i r r e v e r s i b l e d e f o r m a t i o n s are  i n t o account w h i l e i n Chapter 3 the • and  b o u r d a r i e s were assumed to be A few  follows;  r e l a t i v e ' d e f o r m a t i o n between sand zero  0  t e s t s were made i n o r d e r to measure the  the p l e x i g l a s s s i d e p l a t e s  and  the sand*  The  a p l e x i g l a s s p l a t e , l ' square x 1/2"  vertically  i n a h a l f - f o o t high  friction  t h i c k , was  sand l a y e r , under the  sand l a y e r was  measured by weights through a p u l l y and  Results  friction  earth with  to p u l l the p l a t e out  shown i n T a b l e 4-1,  c y c l e s per  second ( s i n u s o i d a l ) ,  c o e f f i c i e n t and  pressure,  the  a piece  the of wire  plate  ^f^  0  *  s  was  and  P °duct  a  a °f  r  r a t i o ' o f l a t e r a l p r e s s u r e to overburden  assuming a l i n e a r l a t e r a l e a r t h p r e s s u r e d i s t r i b u t i o n  depth,  between the p l e x i g l a s s w a l l s evaluated. than 1%  of  o f the  I t shows t h a t the the  total friction  the e f f e c t w i l l be  only  a few  container  e f f e c t of the  per  and  the  the  the  evaluated  r e s t r i c t e d length i n Appendix I I I ,  l e s s than a few  per  f a i l u r e p l a n e i f 1/4"  been  even f o r 1-1/4" deep f a i l u r e cent.  Thus, these e r r o r s can  of the model. I t shows t h a t  cent of the  sand has  total  force  be  f o r c e a c t i n g on  t h i c k f a i l u r e p l a n e i s assumed, and  ignored.  force  a l s o been  toe c o n f i n i n g f o r c e  friction  1/4"  plane  confining  T h i s e f f e c t has the  less  f a i l u r e p l a n e when  Another component of the boundary- e f f e c t i s a t o e to the  friction  s i d e f r i c t i o n w i l l be  f o r c e a c t i n g on  deep f a i l u r e p l a n e i s assumed, and  be  of  In dynamic t e s t s , the  U s i n g the v a l u e s of f r i c t i o n shown i n T a b l e 4-1  due  shown  v i b r a t i o n o f an amplitude of a c c e l e r a t i o n o f 0,4g  frequency o f 10 the  embedded  top of the p l a t e , as shown i n F i g * 4 - 1 .  are  p u l l e d during  as  conditions  Then the  to the  force required  between  t e s t p r o c e d u r e was  i n T a b l e 4-1*  attached  taken  i t will  will the be  50. K  7u6fe 4-1  FfZ/CT/OA/fiETweEA/ RWCE  SJa.&'c , -£oos--c sa/>4  2.$/^  Py/wm/c  6.07^  c/eos-e Sa/7#7  7*f'  FAEXIG-JLASS  SA/SD A/SP  PLEXl£rt-AS5 Q fa  mueiReo  3.3/  p./z  /bs  6.42  '*  s  A 37 ~ 0 . 4 O (2.23 ~-<?, 24  . u/e/^f-tT  v ,  \  fry <z-/  —a /4  f7?/C7?&A7 r~/£ASui?EMEA7T TBST  51.  s e v e r a l per cent f o r 1-1/4" t h i c k f a i l u r e In  plane,  c o n c l u s i o n , the boundary e f f e c t s due  t o r e s t r i c t e d width  l e n g t h of the model are so s m a l l i n terms of percentage t h a t they can be i g n o r e d and s t r a i n and  4-3,  infinite  and  o f the f o r c e s  the a n a l y s i s can be r e p r e s e n t e d as a p l ^ n e  s l o p e problem,  Preliminary tests. In  o r d e r to o b t a i n a q u a l i t a t i v e i d e a about the b e h a v i o u r  of  the  dry sand model s l o p e d u r i n g induced v i b r a t i o n , a s e r i e s of p r e l i m i n a r y t e s t s have been c a r r i e d o u t . follows;  about 700  l b s , of sand was  compacted h e i g h t of 6 i n c h e s . w i t h an amplitude p e r second  The b a s i c procedure  The  sand was  compacted by  and  vibration  a frequency  of 10 c y c l e s  V a r i o u s k i n d s of markers, f o r  example t r a n s v e r s e l i n e s on the s u r f a c e , v e r t i c a l l i n e s on the s i d e w a l l o r . t h i n columns i n s i d e the model, were used measure d i s p l a c e m e n t  o r t o observe  was  made of the same sand  was  used.  The  sand  itself  the f a i l u r e p l a n e .  i n o r d e r to Dyed sand, which  a p p r o p r i a t e markers were p l a c e d a f t e r the model  the sand  was  compacted a g a i n by v i b r a t i o n  of a c c e l e r a t i o n of around and  plexiglass  as the: model by means of d y i n g w i t h b l u e i n k ,  compacted, then the whole mode was an amplitude  s u r r o u n d i n g the marker, which was  tilted  p a r t o f the model.  to a proposed  angle  A f t e r the model was  (see F i g u r e 2-2),  h o r i z o n t a l motions were a p p l i e d by and  the f r e q u e n c y .  The  having  ,7-^--.8g, i n o r d e r t h a t the marker disturbed  when the markers were p l a c e d , would have the same c o n s i s t e n c y as remaining  as  p l a c e d i n the c o n t a i n e r to a  of a c c e l e r a t i o n of 1.0'g  i n the h o r i z o n t a l model,  of the t e s t s was  changing  the  p r e p a r e d , i t was  Then v a r i o u s the amplitude  sinusoidal of  acceleration  s l o p e a n g l e s of the model were a l s o changed by  52.  tilting. failure  The b e h a v i o u r of the model d u r i n g v i b r a t i o n , the mode.; o f -  and the f a i l u r e p l a n e developed were observed«  v i b r a t i o n was  stopped and the model was  Then, a f t ^ r  taken down to the h o r i z o n t a l ,  the a c c u m u l a t i v e d i s p l a c e m e n t s were measured. The modes o f the s l o p e f a i l u r e in  this  of dry sand which have been  s e r i e s o f t e s t s are as f o l l o w s ;  f a i l u r e ; s t a r t i n g with a l o c a l i z e d  observed'  (1) p r o g r e s s i v e type o f  motion of p a r t i c l e s on the s u r f a c e , •<  the f a i l u r e propagates  up the s l o p e and outward throughout  the e n t i r e  s u r f a c e of the s l o p e .  T h i s type of f a i l u r e has been observed d u r i n g  t e s t s on a s t e e p , dense s l o p e s u b j e c t e d t o s m a l l amplitudes o f a c c e l e r a t i o n , (2) f a i l u r e w i t h d i s t i n c t  shear p l a n e s ; a l o n g a f a i l u r e p l a n e o r  s e v e r a l f a i l u r e p l a n e s , a mass of m a t e r i a l moves down as a u n i t .  This  type o f f a i l u r e has been observed d u r i n g the t e s t s i n which a dense s l o p e i s s u b j e c t e d to a c c e l e r a t i o n s o f l a r g e a m p l i t u d e s , c h a r a c t e r i s t i c of t h i s f a i l u r e plane. i s shown, at  type o f f a i l u r e  In F i g u r e 4-2  A  distinct  i s to have a s i n g l e o r compound  an example o f the type (2) k i n d o f f a i l u r e  (3) f a i l u r e xcithout shear p l a n e s ; the d i s p l a c e m e n t i s l a r g e s t  the s u r f a c e and d e c r e a s e s g r a d u a l l y w i t h d e p t h , t h e r e f o r e t h e r e i s  no d i s t i n c t  f a i l u r e p l a n e (see F i g u r e 4-3),  T h i s type of f a i l u r e has  been observed d u r i n g t e s t s on l o o s e sand s l o p e s , Bustamate (1965) has • r e p o r t e d types  (1) and (2) modes o f f a i l u r e b u t n o t mode ( 3 ) ,  Another i n t e r e s t i n g p o i n t the p r e l i m i n a r y of  to be d i s c u s s e d r e g a r d i n g r e s u l t s o f  t e s t s i s the f o l l o w i n g ' t e s t .  I n o r d e r to get the shape  the f a i l u r e p l a n e i n s i d e the model and examine the e f f e c t s o f f i n i t e  b o u n d a r i e s , a t e s t was made by u s i n g 32 v e r t i c a l markers not o n l y a t the s i d e w a l l s but a l s o i n s i d e the model.  Measuring  the d i s p l a c e m e n t of  the marker columns i n s i d e the model by g r a d u a l l y d i g g i n g the sand around  8  4  3;  vb  CN  I Uj Si  N I  F/j  4-3  VBFDRtfffnOA/  OF  l/ERT/CAL.  .P/A.QKER AT THE SfPE K//1LL f&g 7H£. F3J TYPY miVPE  55.'  the marker, d i s p l a c e m e n t s i i i s i d e the model were measured and shape o f the f a i l u r e plane i n s i d e the model was shown i i .  F i g u r e 4-4,  obtained.  T e s t c o n d i t i o n s are as f o l l o w s ;  <X =» 1 5 ° , the a p p l i e d amplitude of a c c e l e r a t i o n , k  the  R e s u l t s are  the s l o p e a n g l e  = 0,73g the f r e q u e n c y S  of a p p l i e d motion, cycles.  f - 10 cps and the number o f c y c l e s a p p l i e d , N ~ 26  In. t h i s t e s t two  a depth o f about  1/4"  and the deeper  1-1/4", were o b s e r v e d . i n F i g u r e 4-4,  f a i l u r e p l a n e s , the upper  f a i l u r e plane at  f a i l u r e p l a n e a t a depth o f  These are denoted by " 1 s t " and "2nd"  Shaded areas on F i g u r e 4-4  a r e c o n s i d e r e d as  zones, i n which the marker column p o s i t i o n c o u l d not be  respectively failure  confirmed.  From t h i s f i g u r e i t i s r e c o g n i z e d t h a t the e f f e c t o f s i d e affects deeper  the shape o f the deeper ("2nd" i n f i g u r e )  about  friction  f a i l u r e p l a n e , n>amely the depth o f the  f a i l u r e p l a n e a d j a c e n t to the s i d e w a l l i s  s h a l l o w e r than a t the c e n t e r (see F i g u r e 4 - 4 ( b ) ) , the. magnitude o f d i s p l a c e m e n t , however, i s almost c o n s t a n t throughout F i g u r e 4 - 4 ( c ) , s e c t i o n d^-d^, or e ^ - e ^ ) ,  I t was  f a i l u r e plane  5 i n c h e s and f o r deeper  i n f i n i t e slope i s v a l i d  the  the same, f o r  ("1st" on the f i g u r e ) i t i s about  f a i l u r e plane  From these f a c t s i t can be s a i d t h a t  (see  a l s o found t h a t  magnitude o f d i s p l a c e m e n t at a l l s e c t i o n s i s almost example f o r the upper  a section  ("2nd" on the f i g u r e ) about the assumption  3-1/4".  of plane s t r a i n  f o r the model t e s t e d as had been  and  theoretically  concluded i n the p r e v i o u s s e c t i o n . 4-4.  m  Measurement o f a c c u m u l a t i v e disjda_cejnent, A v e r y i m p o r t a n t concept c o n c e r n i n g the s t a b i l i t y  comprised and  of s l o p e s  o f dry sand d u r i n g v i b r a t i o n has been s u g g e s t e d by Newmark (1963)  (1965) and a s e r i e s o f r e l a t e d experiments  Seed and Goodman (1964) and  (1966),  have been r e p o r t e d by  As d e s c r i b e d p r e v i o u s l y , p a s t p r a c t i c e  a) /.OA/Sr/TUDWAL. SHAPE- Op PA/llSSE  PAA/E  56. S/.OPE Amt-E, <*^/£-° fiQCEl E/SAT/OA/j • 7Z$3  6  tt£Q(/E/JCy,  -f^/pcpS  CYCIES Xl///, //=2&cycles  t>) Tm.VSyEtfSS D15TPJ6Ur/OA/ OfPJ5PLACEH&A/T  b>  C) Tf?A/JSVe#SE SHAPE SerflM  A,  C,  &E F4ILL/PE b,~bz  3-' <urt jerif 5-ecfy'on di ~d?  7  4-4  V/5PLA)CEM£A/r  /WD  sHAV op  miURB  e,  PLAve  plA//Z  57.  i n the d e s i g n of s l o p e s a g a i n s t earthquake the s i m p l e computation  i n the s t c t i c s t a b i l i t y a n a l y s i s . f o r c e s may be s u f f i c i e n t l y  of  D u r i n g an e a r t h q u a k e ,  the i n e r t i a ,  l a r g e t o drop the f a c t o r o f s a f e t y f o r The  w i l l be a r r e s t e d when the magnitude o f a c c e l e r a t i o n  decreases o r i s reversed. brief  ground motions, i s i n c l u d e d  p e r i o d s c f time and permanent d i s p l a c e m e n t s may o c c u r .  movements, however,  involved  o f a f a c t o r of s a f e t y a g a i n s t s l i d i n g when an  i n e r t i a - f o r c e , generated by the earthquake  brief  f o r c e s has u s u a l l y  The o v e r a l l e f f e c t o f a s e r i e s o f l a r g e but  i n e r t i a f o r c e s may w e l l be an a c c u m u l a t i v e permanent d i s p l a c e m e n t  a s e c t i o n o f t h e embankment, b u t once the ground motions g e n e r a t i n g  the i n e r t i a f o r c e s have ceased, no f u r t h e r d e f o r m a t i o n w i l l o c c u r u n l e s s t h e r e has been a marked l o s s i n s t r e n g t h by the d e f o r m a t i o n s . significant of  Thus,  d i s p l a c e m e n t may have o c c u r r e d but the f a c t o r o f s a f e t y  the s e c t i o n a f t e r earthquake may be a p p r o x i m a t e l y the same as i t  was b e f o r e the earthquake.  T h e r e f o r e , the e f f e c t s o f earthquakes on  s l o p e s t a b i l i t y s h o u l d be a s s e s s e d i n terms of the d e f o r m a t i o n they produce r a t h e r than the minimum f a c t o r o f s a f e t y d e v e l o p e d , concept o f y i e l d  a c c e l e r a t i o n , k^, i s i n t r o d u c e d , the d e f o r m a t i o n o f  s l o p e s d u r i n g earthquake The  concept o f y i e l d  acceleration deformation.  can be e v a l u a t e d as i s shown i n F i g u r e 4-5, a c c e l e r a t i o n i s d e f i n e d t o be the maximum  which the s l o p e can r e s i s t w i t h o u t any permanent I n o t h e r words, an a c c e l e r a t i o n g r e a t e r than the y i e l d  a c c e l e r a t i o n induces permanent d e f o r m a t i o n i n the s l o p e . as f o l l o w s ;  I f the  I t i s derived  c o n s i d e r t h e . s i m p l i f i e d model as shown i n F i g u r e 4-6,  the a c c e l e r a t i o n i s a p p l i e d i n the d i r e c t i o n o f a n g l e G above the h o r i z o n t a l , the f o r c e s a r e r e l a t e d as f o l l o w s ;  by W J/nC^-ci)  =  w 5in(qo°-t-&-f-cX- 4)  _  4  1  If  ACCB1.-ERATIDA/  J-  4-S  MECHANISM  Op P/SPLACEflEA/T  < <5ooDH/W AMD SEED^ iq6£,)  COA/0T/O,V AT  FA/LUXE  FOR  -si/P/A/^  ELEMENT  59.  i.e.  ^  Sin (J,-*)  •_ 4  2  where 0 i s a n g l e of f r i c t i o n , W i s weight of the mass r e s t i n g on s l o p e , and 6  0 - (X  CX  i s angle of the s l o p e ,  and y i e l d s s i n ( 0 - a  h o r i z o n t a l anJ  ),  at r i g h t angle to  has  i t s maximum v a l u e when  I f 9 » 0 (when s e i s m i c  stability.  ^ 3  i s u s u a l l y a p p l i e d i n the a n a l y s i s of dynamic  I t should  be noted t h a t the r e s u l t which F i n n  developed f o r y i e l d a c c e l e r a t i o n , c o n s i d e r i n g the e q u i l i b r i u m i n the m a t e r i a l c o m p r i s i n g 4-3  can be  applied f o r cohesionless  g e n e r a l i t y , and  i f the m a t e r i a l has  a c c e l e r a t i o n i s given  by  (1966a) has  viscc-elastic  the s l o p e , shows t h a t  equation  m a t e r i a l v/ithcut l o s i n g any some cohesion  component, the  yield  P i s u n i t weight o f s o i l , y^ i s the  depth at which down s l o p e v e l o c i t y reaches a maximum and In appendix IV,  slope  by  where c i s magnitude of c o h e s i o n ,  induced  force i s  gravity)  = tan (4  This equation  the  7^^7cos&-ts/>ic(Tan<fi,  the permanent, d i s p l a c e m e n t o f the s l o p e i n one  a s i n u s o i d a l d i s p l a c e m e n t was  t h e o r e t i c a l l y estimated,  the angle of i n t e r n a l f r i c t i o n of the sand was  assumed to be  f a i l u r e because i t has  the sand t e s t e d h e r e i n has  d i s t i n c t d i f f e r e n c e between the peak  strength  and  the r e s i d u a l s t r e n g t h .  c y c l e , u, i s g i v e n  by  The  where  fully  developed w i t h no p r o g r e s s i v e no  cycle  been found  that  permanent d i s p l a c e m e n t i n  one  60.  where f - frequency i n terms of c y c l e s per second, tan 0),  k  ^~=^(cos&  +  sincr  i s the amplitude o f a c c e l e r a t i o n o f a h o r i z o n t a l base  —1 k  and 1 ~ 7T - 2 s i n  _v_ k  motion  " A p r o d u c t o f the square o f the f r e q u e n c y  «  and the permanent d i s p l a c e m e n t i n one c y c l e , f ^ u , i s independent  of  the f r e q u e n c y . I n o r d e r to measure the a c c u m u l a t i v e d i s p l a c e m e n t o f the model and  to compare r e s u l t s w i t h e q u a t i o n 4 - 5 , a s e r i e s o f t e s t s was  The procedures  made.  f o r the p r e p a r a t i o n o f the model were e x a c t l y the same  as those c a r r i e d out f o r the p r e l i m i n a r y t e s t s d e s c r i b e d i n the S e c t i o n 4 - 3 , except t h a t f i v e dot type markers were used.  1/4"  F i v e s m a l l , about  d i a m e t e r , dot type markers were p l a c e d at 1 f t , i n t e r v a l s a l o n g the c e n t e r l i n e on the s u r f a c e o f the model.  A f t e r the model was  to a s l o p e a n g l e o f 15 degrees, a s i n u s o i d a l v i b r a t i o n was which had a frequency of 10 c y c l e s per second of a c c e l e r a t i o n which was  a p p l i e d d u r i n g v i b r a t i o n was  counted.  d i s p l a c e m e n t of the markers was a c c u m u l a t i v e d i s p l a c e m e n t was  applied  and a p a r t i c u l a r  changed f o r each t e s t .  A f t e r the v i b r a t i o n was  obtained.  stopped,  taken.  d i v i d e d by the number o f c y c l e s c y c l e was  amplitude  The number o f c y c l e s  measured and an average was  and the permanent displacement i n one  tilted  This  applied  Results i n  terms of the permanent d i s p l a c e m e n t in"one c y c l e , u , and the p r o d u c t s 2— of the square o f the frequency and the d i s p l a c e m e n t i n one  cycle, f u ,  are p l o t t e d a g a i n s t the amplitude of a c c e l e r a t i o n i n F i g u r e 4 - 7 , measured r e s u l t s are l i s t e d  i n Table 4-2,  The  t h e o r e t i c a l values  c a l c u l a t e d by E q u a t i o n 4-5 are a l s o shown on F i g u r e 4 - 7 . the v e r t i c a l  The  The v a l u e on  c o o r d i n a t e i n F i g , 4 - 7 , i . e . k = ,51g and x - 0 , was  taken  N  TLiWe  4-2  l/ST OF  D/SpiAc&MEA/T  . /7E6SUJ?EME*/T TEST  7EST A/0.  TOTAL A7o. OECYCIES 7/OV20A/T/4Z77SPL4CEM&/T APPLIED PJSP16CB4EA/J W 6A/B CYCLE 8 A S E ACC. . $34- &J  2. 3 O'nches). 07-37(7»c7j s e  »  2.  . <790  30  4.2(7  ./4/5  -  3  .  652  3f  6./P  • 7 99  "  4  . 70S  24  6-40  . 22 D  «  5  .-7S8  32  S.35  • 26 f  "  6  .8S4  26  7.90  . 3 0 4  '<  7  .  20  7-'f  . 3ZS  930 AA/4L£  OT- SLOPE J  c2=/jr  63.  from the r e s u l t o f the measurement o f the c r i t i c a l (see F i g , 4 - 8 i n the next s e c t i o n ) . the y i e l d 0,  angle o f s l o p e  From t h i s f i g u r e i t i s found  a c c e l e r a t i o n , k^, o r the m o b i l i z e d a n g l e o f i n t e r n a l  depends on t h e amplitude o f a c c e l e r a t i o n a p p l i e d .  that  friction,  For small  a c c e l e r a t i o n s 0 was as h i g h as 42 degrees; then 0 d e c r e a s e d w i t h i n c r e a s i n g amplitude o f a c c e l e r a t i o n .  F o r amplitudes o f a c c e l e r a t i o n  g r e a t e r than 0,6, 0 reached a r e s i d u a l v a l u e o f 27 d e g r e e s . I t i s i n t e r e s t i n g t o note t h a t , f o r amplitudes o f a c c e l e r a t i o n less  than 0,6, t h e r e s i d u a l angle o f i n t e r n a l f r i c t i o n was n o t r e a c h e d ,  even though  t o t a l displacemeixt was as h i g h as 2,3 i n c h e s which i s l a r g e  enough to r e a c h the r e s i d u a l angle o f i n t e r n a l shear t e s t .  f r i c t i o n i n the o r d i n a r y  From t h i s e v i d e n c e the f a i l u r e mechanism on the f a i l u r e  p l a n e i s presumably  as f o l l o w s ;  base motion  the y i e l d  reaches  until  t h e amplitude o f a c c e l e r a t i o n o f  a c c e l e r a t i o n , k^, o f the model, no  permanent d i s p l a c e m e n t w i l l o c c u r , b b u t once the amplitude o f a p p l i e d a c c e l e r a t i o n exceeds  k^ f o r some p e r i o d o f t i m e , t h e m o b i l i z e d a n g l e  of i n t e r n a l f r i c t i o n  d e c r e a s e s and reaches i t s r e s i d u a l v a l u e .  After  the motion has ceased o r r e v e r s e d i t s d i r e c t i o n , however, the v o i d r a t i o o f the sand on the f a i l u r e p l a n e becomes denser than t h a t d u r i n g f a i l u r e and the angle o f i n t e r n a l f r i c t i o n increas-es a g a i n . p r o c e d u r e i s r e p e a t e d f o r each  c y c l e o f base motion.  This  T h e r e f o r e , the  observed a n g l e s o f i n t e r n a l f r i c t i o n a r e average v a l u e s i n one c y c l e , which i s presumably  a f u n c t i o n o f t h e d i s p l a c e m e n t i n one c y c l e  than t o t a l d i s p l a c e m e n t ,  rather  .For example, f o r t h e a m p l i t u d e o f a c c e l e r a t i o n  0.53g (see T a b l e 4-2) the angle o f i n t e r n a l f r i c t i o n d i d n o t reach t h e r e s i d u a l v a l u e because  the d i s p l a c e m e n t i n one c y c l e , 0,064 i n c h e s , was  n o t l a r g e enough even though  the t o t a l d i s p l a c e m e n t was 23 i n c h e s .  In  the p r e l i m i n a r y t e s t  the deeper f a i l u r e p l a n e , about 1-1/4  was  observed b e s i d e the t h i n  The  same a n a l y s i s as d i s c u s s e d p r e v i o u s l y was  f a i l u r e plane.  f a i l u r e p l a n e , which i s shown i n Fig.ire a p p l i e d to t h i s  In the i n s t a n c e when more than two  o c c u r i n the model, e q u a t i o n 4-5, to  inches,  failure  4-4,  deep  planes  however, cannot be a p p l i e d except  the s h a l l o w e s t f a i l u r e p l a n e , because the f r i c t i o n f o r c e t r a n s m i t t e d  from  the upper f a i l u r e p l a n e a f f e c t s  f a i l u r e plane, and or  the d i s p l a c e m e n t  i h e r e f o r e , assuming the sand has  the model comprised  of s l i c e s  of sand, and  displacement  f o l l o w i n g Penzien  w i t h the use o f the computer.  f r i c t i o n d u r i n g f a i l u r e was  p l a n e where any was  with results  R e s u l t s a r e as f o l l o w s ;  32 degrees.  calculated  The  angle o f i n t e r n a l  the a p p l i e d amplitude  at j u s t  below the a c t u a l  permanent d i s p l a c e m e n t has not taken p l a c e .  49 degrees.  These angles of i n t e r n a l  of the vacuum t r i a x i a l  TABLE 4 - 3 .  friction  i n c h e s deep, the m o b i l i z e d a n g l e  which must be m o b i l i z e d i n o r d e r to r e s i s t a c c e l e r a t i o n , 0,725g, was  internal  (1960)  were c a l c u l a t e d from i t s measured  a l o n g the deeper f a i l u r e , i , e , 1-1/4 of  lower  r i g i d plastic properties  F i n n and Byrne (196S) , the m o b i l i z e d a n g l e s o f  i n the model shown i n F i g u r e 4-5  o f the  friction, of  failure The  result  f r i c t i o n are l i s t e d i n Table  4-3  test.  LIST OF ANGLES OF  INTERNAL FRICTION  COSJD/T/OA/S  Cafai/tzted -fto/n displacement nt s/z/if/ice. -fa/fore, plane27°ffn~o —  —  '/  at  •  deep 7-al/ure p/a/ie 07j=.l /  32°  * at" lasf J?e/t>ur -Me deeper -/nu'/ut-e t>fa/ie 4-75° Vacuum'-/rt'axiafT W - tr% = e^-^o <r, =/o. e = -53 3S.7" p s L  p s i  m  - '?.5*  03  = 5.  p  s  i  si  e = . 64 e  -.77  35.6°  According  t o F i n n (1966) , i n dense f r i c t i o n a l m a t e r i a l the a n g l e  of f r i c t i o n , 0 , m o b i l i z e d d u r i n g s l i d i n g down the s l o p e w i l l be l e s s s than the angle o f f r i c t i o n , 0 „ m o b i l i z e d at; i n i t i a t i o n ° ' m B  o f y i e l d atJ  the l a t t e r i n c l u d e s the e f f e c t o f d i l a t a t i o n and the m a t e r i a l moving down a s l o p e i s l i k e l y volume.  to occur a t the c r i t i c a l v o i d r a t i o o r c o n s t a n t  From these p o s t u l a t i o n s the r e l a t i o n s h i p between angles o f  i n t e r n a l f r i c t i o n i n T a b l e 4-3 can be e x p l a i n e d . i.e.  27° and 3 2 ° , c o r r e s p o n d  to 0- ,  The f i r s t  two v a l u e s ,  Moreover, 27° may be r e p r e s e n t a t i v e  of t h e c r i t i c a l v o i d r a t i o under almost  zero c o n f i n i n g p r e s s u r e a t 'the  s u r f a c e o f the model and 32° may be r e p r e s e n t a t i v e o f the c r i t i c a l v o i d r a t i o under about 0,1 p s i , and c o n f i n i n g p r e s s u r e s o f the t r i a x i a l t e s t specimen were much g r e a t e r than model.  those on t h e f a i l u r e p l a n e o f the  T h e r e f o r e , t h e angles o f i n t e r n a l f r i c t i o n c a l c u l a t e d  displacement  o f t h e model s h o u l d be l e s s  *  o  the t r i a x i a l  test,  49  than t h e v a l u e o b t a i n e d  corresponds  to 0^ which agrees w i t h  r a t i o o f t h e model, 0,47, was lower  t h a t the v o i d  than t h a t o f t h e t r i a x i a l  specimen, ,50 and the c o n f i n i n g p r e s s u r e o f t h e model, . —  4-5.  the maximum  t e s t o f dense  specimens and low c o n f i n i n g p r e s s u r e , c o n s i d e r i n g the f a c t  than  from  .  angle o f i n t e r n a l f r i c t i o n o b t a i n e d by the t r i a x i a l  was l e s s  from  test  0.1 p s i ,  t h a t o f the t r i a x i a l specimen 2,5 p s i ,  Measurement o f stable__slope_._an_gle_.. As mentioned p r e v i o u s l y , i t has been found  t h a t t h e r e a r e two d i s t i n c t  c h a r a c t e r i s t i c s l o p e a n g l e s o f d r y sand f o r a g i v e n amplitude o f a c c e l e r a t i o n o f base motion. angle  a C  These a r e denoted by " t h e c r i t i c a l  " and the s t a b l e s l o p e a n g l e , •  ".  The c r i t i c a l s l o p e  angle  s  i s d e f i n e d as the maximum s l o p e angle which can r e s i s t a g i v e n of a c c e l e r a t i o n o f base motion w i t h o u t  slope  amplitude  any permanent d e f o r m a t i o n .  In  66.  other words the slope-whose slope angle i s greater than the c r i t i c a l slope angle undergoes permanent deformation f o r a given base acceleration. The y i e l c acceleration, k^, the angle of i n t e r n a l f r i c t i o n mobilized at i n i t i a t i o n of y i e l d of the slope, 0 , and the c r i t i c a l slope angle are, i n general, describing the same phenomenon, and these are looking at the phenomenon from d i f f e r e n t aspects.  A l l of them can be connected i n  one equation, which i s  T^y = -tan (<j>„ - CYc) The stable slope angle, however, i s a d i f f e r e n t concept from those described above.  I t i s defined as follows; when a dry sand slope i s  subjected to a base motion whose amplitude of acceleration i s greater than the y i e l d acceleration of the slope, some permanent displacement takes place and the angle of slope decreases gradually u n t i l i t reaches a p a r t i c u l a r angle of slope, a f t e r which permanent displacements are no longer induced unless the amplitude of acceleration i s increased. This angle of slope at which the slope f i n a l l y comes to rest i s referred to as the stable slope angle. A couple of s t a t i c t i l t i n g tests were made.  The ssad was  compacted  by v i b r a t i o n i n a horizontal layer then i t was t i l t e d gradually u n t i l f a i l u r e took.place.  The mode of f a i l u r e i n this test was the progressive  type of f a i l u r e which has been described i n section 4-3 as the f i r s t type of mode of f a i l u r e .  The angle of the slope at which the f a i l u r e  started and the angle of slope at which the slope s e t t l e d were measured. At  37 degrees the f a i l u r e started and kept going u n t i l the slope came to  rest at an angle of 30 degrees,  37 degrees of slope angle can be  considered to be a good example of the c r i t i c a l angle of slope under zero amplitude of acceleration, and 30 degrees of slope angle as an example of i t s stable slope angle, also f o r zero acceleration amplitude.  67.  In order to measure the stable slope angle two d i f f e r e n t kinds of  test were carried out.  follows;  The test procedure of one of the testt i s as  a slope of loose sand was b u i l t into the horizontal container  as indicated i n the sketch i n Fig,4-8, which was  Its i n i t i a l angle was  30 deg.-ees  the repose angle of the sand, i . e . the s t a t i c stable angle  of  slope,  nhen the model slope was subjected to sinusoidal v i b r a t i o n  of  an amplitude of acceleration of 0,05g and a frequency cf 10 cycles  per second.  The v i b r a t i o n was continuously applied u n t i l the slope  reached the stable angle of the slope. of  After the permanent movement  sand p a r t i c l e s ceased.the angle of slope was measured.  Then the  amplitude of acceleration of the v i b r a t i o n was increased to O.lg, the model slope was again vibrated u n t i l i t reached the. stable angle, keeping the frequency constant, and the stable slope angle was measured. procedure was  This  continued up to ,6g amplitude of acceleration and a dozen  of the stable slope angles were obtained, as shown i n Figure 4-8, This t e s t , however, involves the following problem; of  tests was  t h i s series  carried out sequentially with only one i n i t i a l model,  therefore the conditions of each model (or void ratio) were controlled by the previous t e s t .  The results might be connected with each other  as shown by the s o l i d dots and curve i n Figure 4-8, Another type of test was made to examine i f the stable slope angles obtained were unique and independent of the s t a r t i n g conditions, iThe test procedure i s as follows;  about 700 pounds of sand was placed i n  the container to a compacted height of 6 inches.  I t was  compacted by  v i b r a t i o n of an amplitude of acceleration of l,0g and a frequency of 10 cycles per second.  Then the model was  t i l t e d to a given angle of slope.  2 models were started from the slope angle of 30 degrees and 3 models were from 15 degrees, as shown i n Figure 4-8.  For each model a sinusoidal  VJBBZotJ SfiUP U'lTH  STGPTE-P UPfSPITlOfiJ  \  \  1  _l .2  •  I  I  I .3  .4  /)MPL/7VP£ OT ACCBUE&iT/Ofi/, f? (JJ  smeu?  S L O P E  MfrLe  I AO  69.  v i b r a t i o n o f t h e i n d i c a t e d amplitude c f a c c e l e r a t i o n shown In-F. .g,4-3 !  was a p p l i e d , a t a c o n s t a n t f r e q u e n c y o f 10 c y c l e s p e r second, v i b r a t i o n was c o n t i n u e d u n t i l  the s l o p e  then the a n g l e o f s l o p e was measured The  first  slope, only  s e t t l e d t o i t s s t a b l e angle,,  ( i n d i c a t e d as o p e n - c i r c l e  t h i n g t o be found from F i g u r e  the s t a b l e s l o p e  o f the a p p l i e d  and the i n i t i a l  i n Fig«4-8)«  4-8 i s the i n d i c a t i o n t h a t  angle may be independent o f i n i t i a l  such as the d e n s i t y  Th,*.  slope  conditions  of the  a n g l e , and a f u n c t i o n  amplitude o f a c c e l e r a t i o n and perhaps the m a t e r i a l  c o m p r i s i n g the s l o p e .  T h i s i s presumably because, d u r i n g  i n d u c e d by v i b r a t i o n , the i n i t i a l , c o n d i t i o n s d e s t r o y e d and new c o n d i t i o n s  the f a i l u r e  o f the s l o p e w i l l be c o m p l e t e l y  such as the v o i d r a t i o , t h e s l o p e  angle  and  t h e arrangement o f sand p a r t i c l e s , a r e formed, which a r e j u s t  for  the amplitude o f a p p l i e d  of t h e p o i n t s  acceleration,  Lherefore,  a l o n g t h e curve o f the s t a b l e s l o p e  stable  the v o i d r a t i o s  angle'in Figure  4-8  are a l l d i f f e r e n t . In t h i s s e r i e s o f t e s t s the f r e q u e n c y o f the v i b r a t i o n was kept constant, therefore has  been l e f t  t h e e f f e c t o f f r e q u e n c y on the s t a b l e s l o p e  f o r future  investigations.  In some t e s t s the s t a b l e s l o p e surface  angle was measured a l o n g the sand  2 i n c h e s each s i d e w a l l as w e l l as a l o n g the c e n t e r  the s l o p e .  The angles p l o t t e d on the. F i g u r e  center  l i n e of the s l o p e .  friction  angle a l o n g the. l i n e s c l o s e to the .  s i d e w a l l s were somewhat s t e e p e r than a l o n g the c e n t e r  the  l i n e of  A c c o r d i n g t o these r e s u l t s the e f f e c t o f the s i d e  was seen, namely, the s t a b l e s l o p e  slope.  angle  4-8  l i n e of the '  a r e a l l those measured a l o n g  70.  4-5.  Measurement of the c r i t i c a l  slope angle.  In order to measure the c r i t i c a l  angle of slope, which was defined  i n the previous section, a series of tests was made. i s as follows;  The testing procedure  the model, which was compacted by v i b r a t i o n at an  amplitude of acceleration of l.Og and a frequency  of 10 cycles per  second i n a 6 inch high h o r i z o n t a l l a y e r , was t i l t e d  to a given angle.  Then, the amplitude of s i n u s o i d a l acceleration was gradually increased at a constavit while thecsurface of the slope was c a r e f u l l y observed. When a permanent displacement  of sand p a r t i c l e s on the slope was f i r s t  recognized, the amplitude of acceleration of the applied s i n u s o i d a l v i b r a t i o n was recorded. These amplitudes of acceleration are plotted against the angle of slope i n Figure 4-9,  As i s shown i n Figure 4-9 during this series  of t e s t s , two d i f f e r e n t kinds of behaviour of the slope have been observed.  The f i r s t i s the progressive type of f a i l u r e which has been  described i n section 4-3,  This type of f a i l u r e was observed f o r slope  angles greater than 20 degrees.  The second i s as follows;  the sand  p a r t i c l e s on the slope surface underwent an i r r e v e r s i b l e movement, but the movement stopped a f t e r they moved down a l i t t l e , and any change i n the slope angle was hardly recognizeable, behaviour,  In the second type of  the slope presumably reached a d i f f e r e n t void r a t i o , or a  d i f f e r e n t arrangement of sand p a r t i c l e s on the slope surface which was made more stable by the sand p a r t i c l e s moving down a l i t t l e . The permanent deformation  of the slope i n this series of tests  was s l i g h t l y d i f f e r e n t from the usual f a i l u r e , because the f a i l u r e took place within the thickness of a few sand p a r t i c l e s r i g h t on the surface of the slope. sought.  Therefore, an appropriate method of analysis must be  Because i t was, however, very d i f f i c u l t  to describe the motion  71.  —O  TEST RBSUL 7"5, J=/OcpS THEORETICAL  Za/7(43  l/ALUE  ')  NOI/&V&UT  OE SiRFME PARTICLE: PPOtrfBSSED CAUSED COMPLETE PA/LURE AA/D A MARKED FL4TT&VTA/£T OE THE.SLOPE  AS?  MOl/BMSVrOE SURFACE PARTICLES 370PP&P AFTER  OA/L-Y  3L/&ttT  MCmEME*/T /ODHJ/J SLOPE  DBA/SB H/BDRO//  COMPACTED Ar  &YV/3RAVOAJ  -ft"  2  S/}AM)  ' HoertOAWAL.  p=/OCp5  .<£  .6  - AMPUTUPB OP  f/j  4-9  CPJT/CAL  •&  ACCBLERATWA/j £ (?)  SLOPE  AA/EFLT:  AO  72,  of  the p a r t i c l e s i n such a c o n d i t i o n , e q u a t i o n 4-3,  was.applied. It  In F i g u r e 4-9  i s found t h a t the observed curve In F i g u r e 4-9  plotted.  agrees f a i r l y welil  f o r 0 » 43° except i n the  g r e a t e r than 0 = 37° which  from the s t a t i c t i l t i n g The  ky = tan(0-A'),  range  I t i s i n t e r e s t i n g t o note t h a t 0 « 43° o b t a i n e d  small slope angles.  from dynamic t e s t s was  0  t h i s r e l a t i o n f o r 0 = 43° i s a l s o  w i t h the r e l a t i o n between k and & of  i,e  has been o b t a i n e d  tests.  curve of the c r i t i c a l  a n g l e o f s l o p e i n F i g u r e 4-9  depends  on the v o i d r a t i o of the s l o p e and. the arrangement o f the sand  particles  on the s u r f a c e o f the s l o p e , w h i l e the curve o f the s t a b l e s l o p e a n g l e i n F i g u r e 4-8 In  i s independent  o f t h e s e c o n d i t i o n s of the i n i t i a l s l o p e .  o t h e r words, i f o t h e r models w i t h h i g h e r i n i t i a l v o i d r a t i o  t e s t e d , the curve o f the c r i t i c a l shown i n F i g . 4 - 9 ,  s l o p e angle w i l l  Though the c u r v e i n F i g u r e 4-8  are  come below the curve i s also a  critical'  s l o p e a n g l e s i n c e , i f the amplitude o f a c c e l e r a t i o n i s i n c r e a s e d from the p o i n t of the curve permanent d i s p l a c e m e n t w i l l  o c c u r , each p o i n t  on the curve o f the s t a b l e s l o p e a n g l e i n F i g u r e 4-8 conditions.  I t i s i n t e r e s t i n g t o superimpose  curve i n F i g u r e 4-8  angle which  c u r v e s , i , e , the  But when  these.two slope  comes below the curve o f the s t a b l e s l o p e angle has  t h i s p a r t o f the c r i t i c a l  no  In o t h e r words, f o r  s l o p e a n g l e curve i f an angle of s l o p e  exceeds  s l o p e a n g l e no s i g n i f i c a n t permanent d i s p l a c e m e n t takes  p l a c e u n l e s s t h a t angle exceeds In  different  the p a r t o f the curve of the c r i t i c a l  s i g n i f i c a n c e from the p r a c t i c a l p o i n t of view.  the c r i t i c a l  these two  and the curve i n F i g u r e 4-9,  curves are superimposed,  has  F i g u r e 4-10  these two  the s t a b l e s l o p e a n g l e .  curves are superimposed,  d i v i d e d i n t o t h r e e zones, A, B and C,  • They are-  I f the r e l a t i o n between an  amplitude o f a c c e l e r a t i o n and a s l o p e a n g l e i s l o c a t e d i n the zone A,  73.  o I  :—-«— a <?  1  1  —t ^-5"  AtfPl/T(/D£ &P ACCEiapATfOA/y A 7V^? ^-/O tfBLAT/OA/ BS-7WBBU AMPJ-(TL/D£'ACCELERA770A/  ^  /}//&££: P^AAJ  74,  the s l o p e i s q u i t e s a f e .  I f i t i s l o c a t e d i n the zone C, permanent  d e f o r m a t i o n s w i l l n e c e s s a r i l y take p l a c e and t h e s l o p e w i l l settle  on a p o i n t on the curve  o f the s t a b l e slope angle.  finally I f the  r e l a t i o n between an amplitude o f a c c e l e r a t i o n and a s l o p e angle i s l o c a t e d i n the zone B, the s l o p e i s s a f e u n l e s s curve  o f the c r i t i c a l  the curve and  o f the c r i t i c a l  o f s l o p e , b u t even i f o n l y one p u l s e  angle o f s l o p e t h e f a i l u r e  i t w i l l n o t stop u n t i l  curve if  angle  the angle  o f the s t a b l e s l o p e a n g l e ,  the angle o f s l o p e i s s t e e p e r  loose c o n d i t i o n , the f a i l u r e motion has ceased, u n t i l  any p u l s e h i t s the hits  i s triggered,  o f s l o p e reaches a p o i n t on t h e  Bustamate (1965)  has r e p o r t e d  that  than t h e angle o f repose o f s o i l i n  w i l l n o t s t o p , even, when the earthquake  the angle o f s l o p e reaches t h e angle o f  repose i n loose c o n d i t i o n .  This c o n d i t i o n i s d e s c r i b i n g the hatched  zone i n F i g u r e 4-10, since, t h e a n g l e  o f repose o f s o i l  i n loose  c o n d i t i o n s c o r r e s p o n d s t o the i n t e r s e c t i o n o f the curve o f t h e s t a b l e s l o p e angle  4-6,  a t the v e r t i c a l  c o o r d i n a t e , namely s t a b l e s l o p e  angle.  Summary and c o n c l u s i o n ,  . From the s e r i e s o f t h e dynamic s l o p e s t a b i l i t y  t e s t s with  a dry  sand model, the f o l l o w i n g d i s c o v e r i e s have been made: (1)  The boundary e f f e c t s o f the model on the s l o p e s t a b i l i t y  not v e r y (2)  significant.  The angle  of i n t e r n a l  slope i s l i k e l y displacement  f r i c t i o n mobilized during s l i d i n g  i n one c y c l e r a t h e r than t o t a l  displacement,  For a large  i n one c y c l e , say 0.15 i n c h e s , the r e s i d u a l angle  of i n t e r n a l f r i c t i o n m o b i l i z e d d u r i n g s l i d i n g plane  down a  to be a f u n c t i o n o f the magnitude o f the permanent  enough displacement  failure  were  f o r the angle  was 27° a t 1/4 i n c h e s  o f s l o p e 15 d e g r e e s .  deep  75,  (3)  I n dense f r a c t i o n a l m a t e r i a l s the a n g l e o f f r i c t i o n m o b i l i z e d  d u r i n g s l i d i n g down the s l o p e was much l e s s than the angle o f f r i c t i o n m o b i l i z e d at i n i t i a t i o n o f y i e l d .  These were 32 degrees  r e s p e c t i v e l y a t 1-1/4  i n c h e s deep f a i l u r e p l a n e .  (4)  angle o f s l o p e , the r e l a t i o n o f k  F o r the c r i t i c a l  is likely of  to Le a p p l i c a b l e .  dynamic s l o p e s t a b i l i t y  degrees  = tan(0-<tf)  The v a l u e o f 0 c a l c u l a t e d from t h e r e s u l t s t e s t s was  o b t a i n e d by the s t a t i c t i l t i n g respectively,  and 49  test.  g r e a t e r than t h e a n g l e o f repose These were 43 degrees  and 37  degrees  76.  CHAPTER 5  SUMMARY AND  CONCLUSIONS  In osder t o i n v e s t i g a t e the fundamental  dynamic p r o p e r t i e s o f  dry sand through model s t u d i e s and. t o compare, t h e b e h a v i o u r o f models d u r i n g v i b r a t i o n w i t h e x i s t i n g t h e o r i e s , t h e f o l l o w i n g two types of experimental' s t u d i e s were performed; h o r i z o n t a l models and vibration.  (1) the dynamic response o f  (2) the s t a b i l i t y o f model sand s l o p e s d u r i n g  A l l t e s t s were c a r r i e d out on t h e s h a k i n g t a b l e u s i n g d r y  Wedron sand models housed  i n an 8 f o o t l o n g by 1-1/2 f o o t wide  rigid  container. C o n c l u s i o n s o b t a i n e d a r e as f o l l o w s : •5-1.. Dynamic response of h o r i z o n t a l models, (i)  The t e s t i n g t e c h n i q u e used h e r e i n has s e v e r a l  simplifying  advantages, namely the s t r e s s e s i n the model were overburden p r e s s u r e with k  o  c o n d i t i o n and t h e predominant .  motion o f the model was i n  h o r i z o n t a l shear v i b r a t i o n , ( i i ) The b e h a v i o u r o f the model d u r i n g v i b r a t i o n  agreed w i t h t h a t  p r e d i c t e d by e l a s t i c t h e o r y , and t h e s o l u t i o n f o r the e q u a t i o n o f motion of the model which  takes boundary  e f f e c t s i n t o account was found to  be a p p l i c a b l e t o the observed b e h a v i o u r , ( i i i ) ' For a v o i d r a t i o  l e s s than 0,6, the o b t a i n e d s h e a r wave v e l o c i t i e s  agreed w i t h those e x t r a p o l a t e d • from, r e s u l t s by H a r d i n and R i c h a r t Experimental r e s u l t s  (1963),  tend t o suggest t h a t f o r a d r y sand l a y e r a t a  given constant void r a t i o  the shear modulus of. t h e l a y e r may be  l i n e a r l y p r o p o r t i o n a l to i t s depth.  77.  (iv)  Damping r a t i o o f d r y sand  frequency  o f v i b r a t i o n i n the range o f 100  be about 2 5-2. (I)  under low c o n f i n i n g p r e s s u r e and a 140 cps was measured t o  3%.  Stability  o f model sand s l o p e duving  vibration.  The a n g l e o f i n t e r n a l f r i c t i o n m o b i l i z e d a l o n g s l i p p a g e plaices a t  relatively  s h a l l o w depths below s u r f a c e o f s l o p e s d u r i n g v i b r a t i o n  may be a f u n c t i o n o f the magnitude o f permanent d i s p l a c e m e n t c y c l e r a t h e r than t o t a l d i s p l a c e m e n t .  F o r a l a r g e enough  i n one  displacement  i n one c y c l e , say 0.15 i n c h e s , t h e r e s i d u a l a n g l e o f i ? i t e r n a l m o b i l i z e d d u r i n g s l i d i n g was found failure  friction  to be about 27° a t 1/4 i n c h e s deep  plane.  ( i i ) The a n g l e o f f r i c t i o n m o b i l i z e d d u r i n g s l i d i n g down the s l o p e was much l e s s than t h e angle o f f r i c t i o n m o b i l i z e d a t i n i t i a t i o n of y i e l d . These angles were 32° and 4 9 ° r e s p e c t i v e l y , a l o n g 1-1/4 i n c h e s deep failure (iii) is  planes.  For the c r i t i c a l  likely  angle o f s l o p e , the r e l a t i o n o f k  to be a p p l i c a b l e .  repose o b t a i n e d by s t a t i c  (iv)  I t was found  tan(0-  )  iThe v a l u e o f 0 c a l c u l a t e d from the  r e s u l t s o f dynamic s l o p e s t a b i l i t y of  =>  t e s t s was g r e a t e r than  tilting  the angle  t e s t , o r 43° and 37° r e s p e c t i v e l y ,  t h a t i t i s v e r y dangerous to d e s i g n a s l o p e w i t h an  a n g l e o f s l o p e g r e a t e r than the repose the s l o p e i n i t s l o o s e c o n d i t i o n .  angle o f t h e m a t e r i a l c o m p r i s i n g  78.  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Seed, I I , B , and M a r t i n , G.R., (1966) "The S e i s m i c C o e f f i c i e n t i n E a r t h Dam D e s i g n " , J o u r n a l o f S o i l Mechanics and Foundations Da*.vision, ASCE, No.SM3, May 1966, pp.25, S e l i g , E.T. and Vey, E.E, (1965) "Shock Induced S t r e s s Wave P r o p a g a t i o n On Sand", J o u r n a l of S o i l Mechanics and Foundations D i v i s i o n , ASCE, No.SM3, May 1965. S t a l l , R.D, e t a l (1965) "Shock Wave i n G r a n u l a r S o i l " , J o u r n a l o f S o i l Mechanics and Foundations D i v i s i o n , ASCE, No,SM , J u l y 1965, T a y l o r , D,W, and Whitman, R,V. (1954) "The B e h a v i o u r o f S o i l s under Dynamic L o a d i n g " , 3rd F i n a l Report on L a b o r a t o r y S t u d i e s , Report to O f f i c e o f t h e Chf, o f E n g r s , , Dept, o f C i v i l and S a n i t a r y Engrg,, S o i l Mechanics Lab., AFSWP-118, 1954. W i l s o n , S.D, and M i l l e r , R.P. (1962) " D i s c u s s i o n o f F o u n d a t i o n V i b r a t i o n " , by F.E, R i c h a r t , J r . , T r a n s a c t i o n , ASCE, Vol.127, P a r t I , 1962, Z e e v a e r t , L, (1967) " F r e e V i b r a t i o n T o r s i o n T e s t s t o Determine t h e Shear Modulus o f E l a s t i c i t y o f S o i l " , P r o c , o f 3 r d Panamerican Conf, on S o i l Mech. and F o u n d a t i o n Engrg,, V e n e z u e l a 1967, V o l I , p p . I l l ,  81,  Appendix- I - S o l u t i o n  for 3-dimensional vibration. m  The g o v e r n i n g e q u a t i o n s o f the v i b r a t i o n o f a continuum w i t h i n t e r n a l damping a r e as  follows:  y\ = E^/^uX/~2//).  where  jU - P o i s s o n ' s r a t i o ,  s/zCf+f)j  ,  E = Young's modulus,  e ~ volumetric s t r a i n ,  yz  ^ . -/- &*~ -r  u, v and w - d i s p l a c e m e n t i n the d i r e c t i o n of x, y and z as shown on F i g u r e  => d e n s i t y  e x c i t a t i o n body f o r c e s If  ,  respectively,  F x ( t ) , F y ( t ) and F z ( t ) a r e  ;  i n . the d i r e c t i o n of x, y and z  respectively.  t h e e x c i t a t i o n i s o n l y i n x - d i r e c t i o n , F y ( t ) and v can be  i g n o r e d w i t h o u t l o s i n g any g e n e r a l i t y  and, moreover, i t has been  coil f i r m e d by the f i n i t e element method t h a t w can be i g n o r e d i f the excitation force 2 ^ are  be ^t2(r)/f>  i s applied t  /^"/j°  a n c  only i n x d i r e c t i o n . ^  r  e  s  P  e  c  t  l  v  e  L e t IS j U C  and  s  l y > then the e q u a t i o n s  6-1  reduced to  *'yp  -  ^  ?  - &'  ^  6  '  z  The s o l u t i o n o f the homogeneous e q u a t i o n o f (6-2) can be o b t a i n e d by and  the s e p a r a t i o n  of v a r i a b l e s .  substituting into  Dividing  L e t u ( x , y , z , t ) = X(x) , Y ( y ) , Z ( z ) , T ( t )  (6-2) y i e l d s  through by' X.Y.Z.T and a r r a n g i n g  z(ur)  E;j 6~(  COORD/A/ATE OF  MODEL  83,  Then  j£<™.JJLCLTL_tf„M*jfe--^  (6-4)  anc, V~7dX  _ _j  jj; cf^Y  2  w h e r e i i and j a r e c o n s t a n t s . i n t o the f o l l o w i n g  +  four  <*>'-{')  Z  Then the e q u a t i o n  (6-2)  can be d i v i d e d  eauations  (6-5)  .0  and  The s o l u t i o n s  f o r t h e s e e q u a t i o n s a r e as  For the f i r s t  equation  and f o r f o u r t h  V(xJ =  follows*  equation  C sJn-L X-h C COS-4? XL  L e t us apply the boundary c o n d i t i o n w i t h r e g a r d t o the x d i r e c t i o n , that i s  then  C'=0  and --£r  / =  .-.  j = - ^ ^  hence  And -for the third egctaf/o/o Y(yj = Vsin HEI^  + p'^^EZTy  L e t us apply the boundary  c o n d i t i o n with  regard t o the y d i r e c t i o n ,  that i s  then  D'~o  and  J/j5=Jlb LT  3  = ?7?  .'.-fi^/:  *  •  *.  =  *  hence  and f o r the second  equation  is* L e t us apply the boundary  i/  s  c o n d i t i o n s w i t h r e g a r d t o the & d i r e c t i o n ,  that i s £/^ — D  O  at the base  E ' - O  then  f.^ ) ~ 7Z.--0 a t the s u r f a c e  and  u  then  Us  -2  77  hence  All  c o n s t a n t s i n the e x p r e s s i o n o f X ( x ) , Y ( y ) , and Z ( z ) can be  r e p r e s e n t e d by A and B, then the complete s o l u t i o n o f homogeneous equation i s  •e  where  "(Asintof^pi  13  costd//-ytJ  ( 6-6a )  Let  us  take  a  g  s i n wt  as e x c i t a t i o n ,  \  and e x p r e s s  this  in:terms  of t r i p l e F o u r i e S e r i e s , i . e .  f  g r  r  Z  i  2  t>  J  •hhtn  or  f  In  ^ ' ^ % r - : <  o r d e r to f i n d  the p a r t i c u l a r  f t  into  the o r i g i n a l  • - U/ ZZ  S  /  '  >:i,r.  ....  integral,  l e t us  <6-7)  substitute  r  d i f f e r e n t i a l equation  7J^^^'  y ^  jrTp (<W  + H yCos n  bit)  86.  Making t h i s r e l a t i o n i d e n t i c a l hy e q u a t i n g t h e c o e f f i c i e n t o f  J'  E  xi,n  '"jJi  5/  t  E  s u b s t i t u t i n g u> pgr  from e q u a t i o n *  The c o e f f i c i e n t  coscot  of  c  z e r  o  f o r each p, q and r , and  (6- 6 b ) ' * t  s h o u l d be zero  or  (6-8)  and r e g a r d i n g the c o e f f i c i e n t o f  (6-9)  or 2 ju>to . nt  H  p%h  T (to  r  ) ^  f  „ -f  From (6-8)  (6-9)  Substituting into  7=w (j then  jU>(Ajf, ) th  -f- (kJ x  U) fr) r  P  Z  y  =o  Now  the complete s o l u t i o n of the e q u a t i o n o f f o r c e d  vibration  of the model w i t h u n i f o r m shear modulus and i n t e r n a l damping i s ready, namely  (6-10)  The c o n s t a n t s A  pqr  and 33 are t o be determined pqr  Only steady s t a t e v i b r a t i o n i s of i n t e r e s t h e r e .  by i n i t i a l  condition,  Then the e x p r e s s i o n  of steady s t a t e v i b r a t i o n i s etc*.  - Z 212 tfo-Zf-x sm^u p f r  *  sfo-tg-s  b J  J-*>  '^^  3  p%r-7j  or Pit-  t?  J  --v 2 ;  (6-11)  where  <Pp<ir =  Lav  f  (6-12)  (6-13)  Appendix I I .  EfJ[g.c_tg...flj: Boundary R e s t r a i n t s on V i b r a t i o n .  E f f e c t s o f t h e r e s t r i c t e d l e n g t h o f the model compared w i t h i t s depth and e f f e c t s o f t h e r e s t r i c t e d width w i l l be e v a l u a t e d and discussed here,  a l s o the p r e f e r a b l e width o f the model f o r the  experiment c o n c e r n i n g  the v i b r a t i o n of the h o r i z o n t a l s o i l l a y e r i s  suggested. Resonant f r e q u e n c i e s  o f the model can be e v a l u a t e d  by e q u a t i o n  (6-13) i n Appendix I , i . e .  R g i v e s the r a t i o o f t h e f r e q u e n c i e s between a model h a v i n g l e n g t h and one w i t h "against ^  f  a semi-infinite layer.  but  In F i g u r e 7-1 R i s p l o t t e d  which shows t h a t i f t h e l e n g t h and depth o f a model a r e  8' and 1/2* r e s p e c t i v e l y , i , e , frequency  restricted  =  , the e r r o r i n the fundamental  w i l l be l e s s than 3% i f s e m i - i n f i n i t e l a y e r a n a l y s i s i s used,  i f t h e depth o f the mode i s i n c r e a s e d  the e r r o r w i l l  to 1', i . e . % = £ }  i n c r e a s e t o more than 10% and become q u i t e  A l s o , t h e e r r o r i n c r e a s e s as P o i s s o n ' s  t  then  significant.  r a t i o and mode number i n c r e a s e .  E f f e c t s o f t h e r e s t r i c t e d width on the fundamental frequency and the response amplitude o f the model were c a l c u l a t e d and .shown i n F i g u r e 7-2,  I n t h i s c a l c u l a t i o n , t h e l e n g t h and depth o f the model  were f i x e d as 8' and 1/2' r e s p e c t i v e l y and dynamic p r o p e r t i e s o f t h e soil  as s t a t e d on the. F i g u r e were assumed,  F i g u r e 7-2a shows the  d i s t r i b u t i o n o f t h e a c c e l e r a t i o n r a t i o on the t r a n s v e r s e c e n t e r at  the s u r f a c e w i t h  respect  to t h e base f o r s e v e r a l v a l u e s  line  o f b/h.  90.  c?J D&TP/BVT/OAA  e>TAccBA.E-pAr/CH/ /?A)T/0  8.00  -2^ 3 =. S'f-eef  J = 0.03  \p- S0CP5 /•3  -21-  A2  AO /•/ ACCEIERA T/OAV RA77P AT^Lf?FAC&  WITH RESPECT TV  b ) rVA/DAMEMTAL  PREQuEA/cr  Si §2  / 3  °  1 I  /<?£>  1  -  FiX /AJF/MFE MOTH AMP S^eL^  FtP fEFf/ JA/T/AJI  TEIAYER"^  //O  /OP  /£>  7yf  7-2 £FF£^T  cOF k/WFH OF  MODEL  //  I t i s found on the F i g u r e i.e.  *Vh = 3  that  the 1-1/2' wide by 1/2' h i g h  model,  There a r e no r e g i o n s where the a c c e l e r a t i o n  t  d i s t r i b u t i o n i s u n i f o r m , b u t i f ^/h i s i n c r e a s e d  up t o about 7, tl.e  a c c e l e r a t i o n i n the middle t h i r d o f the width w i l l be s t e a d y and not a f f e c t e d by s i d e r e s t r a i n t . response i n c r e a s e s has  This  f i g u r e shows t h a t the maximum  as ^/h i n c r e a s e s .  This  i s because every  curve  been c a l c u l a t e d f o r 50 c y c l e s / s e c , as a f r e q u e n c y o f e x c i t a t i o n ,  however t h e n a t u r a l frequency of the model d e c r e a s e s as the w i d t h o f the model i n c r e a s e s , as shown i n F i g u r e  7-2b, the r a t i o o f %j„ , t  namely the r a t i o o f frequency t o the n a t u r a l f r e q u e n c y o f t h e model, i s not constant is  f o r each curve,,  Therefore,  t h e shape o f the curve  importaiit r a t h e r than the maximum a m p l i t u d e .  p r e v i o u s l y , ^/h r a t i o o f a model g r e a t e r  As mentioned  than 7 i s p r e f e r a b l e t o  c a r r y out t e s t s i n v o l v i n g t h e response o f s o i l l a y e r s .  This value of  /h reduces a l s o the d e v i a t i o n o f the fundamental n a t u r a l  frequency  from t h a t o f i n f i n i t e l a y e r , as i s seen i n F i g u r e 7-2b, I t i s i n t e r e s t i n g t o note t h a t a c c o r d i n g wlfea the l e n g t h  t o Hatanaka  o f a dam i s about f o u r times the h e i g h t ,  (1955), the i n f l u e n c e  of end r e s t r a i n t has n e g l i g i b l e e f f e c t on the n a t u r a l frequency o f v i b r a t i o n and the magnitude o f response i n the c e n t r a l r e g i o n , hence the use o f an a n a l y s i s based on the assumption o f i n f i n i t e can be c o n s i d e r e d  sufficiently  accurate  fora l l practical  length purposes.  92  Appendix III...  Boundary E H e c t s on Permanent D i s p l a c e m e n t .  There a r e two d i f f e r e n t types of boundary e f f e c t s on the r e s u l t s of s t a b i l i t y and  t e s t s of model sand s l o p e ,  the f i r s t  i s the side  friction  the second i s the end c o n f i n i n g .  Side E f f e c t s . From the t e s t o f the measurement o f f r i c t i o n s between t h e p l e x i g l a s s p l a t e and the sand, the p r o d u c t s of the f r i c t i o n c o e f f i c i e n t , and  the c o e f f i c i e n t of h o r i z o n t a l e a r t h p r e s s u r e ,  C k , are as f o l l o w s f o' r  '  r  (see s e c t i o n 4-2), S t a t i c , dense '  -C k » .372 ~ ,396 - ,384 f o r  dynamic, dense • *  CJc =» ,231 ~- ,244 f o  C o n s i d e r a s e c t i o n o f model as i s shown i n Fig,8-1, the  In f a c t ,  f a i l u r e p l a n e was n o t s t r a i g h t but a curve as shown i n F i g u r e  (see F i g u r e give  =*• ,238  4-4),  The assumption of h o r i z o n t a l s t r a i g h t l i n e  a s l i g h t l y l a r g e r r e s i s t a n c e , but would n o t make a l a r g e  3-1  will difference  Then, a s t r a i g h t l i n e i s assumed here as the f a i l u r e p l a n e .  Then the  f o r c e s a c t i n g on the f a i l u r e p l a n e are shown i n F i g u r e  On t h e  o t h e r hand, the f r i c t i o n  F«  =i  rtQ*.  f o r c e on the w a l l s , Fw, i s  *z = rt*c,A  (8-D  0  S u b s t i t u t i n g the v a l u e s f o r E = 1,5', f the  8-2,  ra  113 p s f and C^k  ,238,  f r i c t i o n f o r c e a c t i n g on the f a i l u r e p l a n e , F^, i s  p^ '= /.5*//3 Acos&J-ao& = /70  ficos<?<teo<fi  (8-2)  SURFACE  • ACTUAL  FlAA/5  ASSUMEP FAILURE PLO/JE  ~/f 8 - /  A SECTION OF THE-MODEL.  I t P~/J S~Z  POPXES ACTfA/fT  OA/ PA71 L/J?E PIP/7B  and (8-3)  7v= //3 x 0.2386* «= z&f*  1  Then the p e r c e n t a g e  o f Fw w i t h  respect to F  (8-4) /TO  co5tX-fo/l<p'  For p a r t i c u l a r v a l u e s o f cx and 0 , say of ~ 15  and 0 = 30  (8-5)  /?o* .466X.S-77  S u b s t i t u t i n g h ~ 1-1/4 i n c h e s and 1/4 i n c h e s , the e f f e c t s i n terms of the p e r c e n t a g e  o f f o r c e s a r e 3% and 0,6%,  These a r e s m a l l enough  to be n e g l e c t e d .  End  confining. If  the f o r c e a c t i n g on the t o e o f the f a i l u r e p l a n e i s assumed  as i s shown i n F i g u r e 8-3, then the f o l l o w i n g r e l a t i o n i s o b t a i n e d from the f o r c e p o l y g o n .  Vf -f-  fiw  (8-6)  Co S(0r <£-<£>') where F  i s the c o n f i n i n g f o r c e a t t h e tow, 9 i s an angle between the  s u r f a c e o f t h e s l o p e and the f a i l u r e p l a n e , 0 i s the angle o f i n t e r n a l friction,  0'  i s the angle- o f f r i c t i o n on the t o e wedge as i s shown,  i n " F i g u r e 8-3, f-  t a n ^ k,!-7  and k i s the c o e f f i c i e n t  i s the t o t a l - w e i g h t o f t h e toe wedge,  o f earthquake.  Then  (8-7)  95.  Hence the component of the F  parallel  £  to the s u r f a c e o f the s l o p e , P,  which i s of i n t e r e s t , i s g i v e n by  p=([V+  &W)  —  ;  ;—  < <> 8  B  But  •w=  ^rhhcota  fin/ £££  =  opt a  cote  =  then nence  The  '  p. /JTT  P expressed  (8-9)  '  sc*+*+*j  t0  by  the e q u a t i o n 8-9  minimum v a l u e of 9,  has  a minimum v a l u e a t the  i n o t h e r words f a i l u r e p l a n e w i l l be a shape  of wedge p r o v i d e d t h a t o t h e r c o n d i t i o n s are homogeneous. the r e s u l t s o b t a i n e d were not s o . e f f e c t s , such 9 was  as s i d e f r i c t i o n .  around 5 ° , and  —0 P = 6,95  x 1,32  T h i s i s presumably clue to o t h e r For deeper, 1-1/4", f a i l u r e  substituting  h - 1-1/4", 0 » 35°, then  plane,  (X =» 1 5 ° , k •- 0,725, f - 113  cf = 36° and  assuming 0  1  » 0,  pcf,  i t follows  19x8?  — - — g ™  means the wedge s l i d e s by  <  0 ,  itself.  The  P turns out n e g a t i v e ,  and i t l e a d s the l e s s F ,  The  E  i n F i g u r e 3-8(c)  a c c e l e r a t i o n e q u a l to y i e l d a c c e l e r a t i o n  l e a d s the maximum r e s i s t a n c e which has  to be  t h i s case ky = t a n (0-#) Hence  .364,  = /TT7TTT  this  T h i s v a l u e i s , however, the minimum  r e s i s t a n c e because the l a r g e r the k the l a r g e r the  /TTT  However,  = tan 20° =  =. /.o£  taken  i n t o account.  In  7if  .B-'S FORCES ACT/A/tr OA/ THE TOE:  97.  The length of this wedge was 1.5' as shown i n Figure 8-4, then assuming the toe confining force uniformly d i s t r i b u t e s along entire f a i l u r e 2 plane, i . e . 6-1/2  feet (see Figure 8-4)', then 2.0/6,5' =» 0.31  lbs./ft, ,  While the f r i c t i o n force acting on the f a i l u r e plane i s  ^ =  cos* i a n  O. /of * O- 9U * O. 7 =  9  2  ( _j0) 8  The error w i l l be  E r r o r  =  .  T/TCx/ao =  - ^  0.3/  •  ^  4-%>  (8-11)  If the same magnitude of .9 i s assumed f o r the shallow f a i l u r e plane (h = 1/4"), the error w i l l be 1/5 of 4%, namely less than 1%,  because  the error i s proportional to the depth of the f a i l u r e plane (see Equation 8-11,  8-9 and 8-10).  Appendix IV. The  Thjgojry. of Accumulative ^Displacement,  permanent displacement of the slope per one cycle induced  by s i n u s o i d a l v i b r a t i o n can be calculated as follows;  simplifying  the problem as shown i n Figure 9-1, an aquation of dynamic equilibrium i s (see Goodman a7;id Seed (1966))  d l < L .  where  f&tj-  ^  fy")  ^(ui - J ( cosa! y stoc* -fa/7 <fi)  (9-1)  (9-2)  = -fan (<t>-ct )  ( 9  "  3 )  k^ i s referred to as the y i e l d acceleration, g i s the acceleration of gravity, o( i s the angle of slope and 0 i s the angle of f r i c t i o n . As shown i n Figure 9-2, the displacement of the slope per one cycle i s given by i n t e g r a t i n g (9-1) twice from the s t a r t i n g point of displacement, t ^ , to the termination  of displacement, t^.  And assumin  k(t) - k ^ s i n LO t and equating ky and k(t) ,  where ko i s the amplitude of acceleration of an e x c i t a t i o n base, motion., and tm i s the time when the v e l o c i t y i s maximum. i s independent of  (t)  If the jp u) (  with i n i t i a l condition  • ••  (t) - 0 at t « t ^ , then  (9-5)  t / W = f[-&(u,seot,-cosu>t)i--fijH -t)] /  Then integrating again u  "  J  =  /  =  umeft  cos«Jt,-cos<Jif&/-tj]  //- - A . CO  = -^L^-t/; . z  th en  £, - ^  dt .  trS/nu) I,)  100.  But t^. i s given by t/(t) - 0, i , e , from the equation 9-5 Oct) = '^{-^-Cecseot, - Ccscji) + tos coi, - u>su)'t-+  /mi,-iot)  Ct,-t)]=0 -o  3  f}€nc€  where of  ^ ( ~ ^ ~ ) i s a modification factor depending upon the value  JL  '  If ky/ko «  7/-|^J =  1, t  2  = 2t - t , . then m l  2Ct ~t,)--^Cr>m  (  7  J  2s;ro-'^L)  (  (9-8a) 9  -  8  b  )  This condition, i s shown i n Figure 9-3, Since to  or  ^  co&co~t/  t  equation (9-6) i s reduced  101.  Ffc? 9-2 /)CC&L&RAT/OA/ AMD VBLO/L/Ty  102.  Appendix V.  Similitude.  Clough (1958) has shown the s i m i l a r i t y relationship of the dynamic model test of s o i l structure, which.is as follows;  f i v e types  of forces are of importance i n v. the analysis, these are: 1,  Dead weight of material,  2,  I n e r t i a forces due to earthquake acceleration,  3,  Forces associated with e l a s t i c deformations i n the structure.  4,  External force,  5,  Forces connected with the ultimate strength of the structure.  "  .  the scale factor for each force i s :  dead weight force  ^  ^  i n e r t i a force  =  q  e l a s t i c force  M  m  cohesion force  Gjp  JOk? = fip  ^  4^  (10-2)  . ^ &p <q-p  (10-3)  (r^ )"  ^  1  =  where s u f f i x m and p represent respectively,  *"> =  =  €p  ^  (10-4)  the items cf model and prototype  ; scale factors for each item i s indicated by i t s  surface, /*; unit weight, V; volume, L; s p e c i f i c length, A «• m/Lp; L  l i n e a r ' s c a l e r a t i o , M; mass, k; acceleration, £ ; s t r a i n , G; shear modulus, Aj area, C; cohesion. In order to gei s i m i l a r i t y , a l l scale factors of the forces should equal. On T i  At f i r s t , equating (10-1) and _  _  „  far*  .  _ /_  (10-2) i t follows  ST**  _ ~> / t p  tfien  « fK  -  (10  5)  )  And  T h i s i s a v e r y important dynamic model t e s t .  r e l a t i o n s h i p i n c o n n e c t i o n w i t h the  Then, e q u a t i n g  (10-1) and (10-3) and l e t - f ^ -  because geometric  s i m i l i t u d e s h o u l d hi maintained  F i n a l l y , equating  (10-1) and (10-4), i t f o l l o w s ,  \3 _  frn  ——  Cm  *~p  as f o l l o w s ;  ~ 2  A ~ -JF-A. (10-7)  -p  v  >p  These r e l a t i o n s a r e shown i n T a b l e 10-1, The angle o f i n t e r n a l f r i c t i o n has n o t been d i s c u s s e d , but i t o b v i o u s l y s h o u l d remain c o n s t a n t , s i n c e i t i s o f d i m e n s i o n l e s s  value  Clough d i d n o t d i s c u s s f u r t h e r , b u t these s i m i l i t u d e s are f o r r a t h e r pseudo s t a t i c a n a l y s i s .  A c c o r d i n g t o the dimension a n a l y s i s ,  the e q u a t i o n o f v i b r a t i o n o f the continuous media i s g i v e n by  where  cOn I undamped n a t u r a l f r e q u e n c i e s o f the s t r u c t u r e ^  ; damping  ratio  I n o r d e r t o s a t i s f y - t h e s i m i l a r i t y r e l a t i o n s h i p between  model  and p r o t o t y p e , a l l terms i n e q u a t i o n (10-8) s h o u l d be constant r e g a r d l e s s o f the model o r p r o t o t y p e , g i v e s the f i r s t  i t e m o f T a b l e 10-1,  The l e f t The f i r s t  hand s i d e of (10-8) terra o f the r i g h t  hand side of the equation (10--8) gives  Ctin)? ^ (UJ)P  U)„ - Xt»)  since  .(V*)p\\  (10-9)  r  f-yr- - I(n)  *  w  n  e  r  ^  e  I  s  v e l o c i t y , i f only the shear v i b r a t i o n i s considered  shear wave and TfaJ i s  a constant depending on the mode number of v i b r a t i o n *  The equation  (10-9) means that, i f the same material as the prototype i s used for the model, the scale factor of time i s not /X  , but j u s t  X  t  and f o r a r b i t r a r y material the scale factor of time i s given by equation (10-9),  For a s p e c i a l case, i n which the s i m i l i t u d e becomes  most simple, i f the model material i s chosen i n such a way that the scale factor of modulus of r i g i d i t y i s X  a  s  shown i n Table 10-1, i n  other words the scale factor of shear wave v e l o c i t y i s ] scale factor of time i s reduced to /X provided  /  , the  » which agrees, with Table 10-1,  that the /!/ are the same.  S i m i l a r l y , the t h i r d and fourth term of the r i g h t hand side gives  =( * )  •M—) ^ivjx  '  I u>7x p  m  J  then  and  (10-10)  '(KJ?X)p  ftp  f  _ •  (V,)p  S\  '  (10-11)  In order to obtain the r e l a t i o n of equation 10-11, since the i n t e r n a l damping c o e f f i c i e n t ,  C  f= — —  , has to hold the following  relation,  •tSLzL - _A? Cp  =  (&  hr» rp (^i)p ^ fp  fi"  (U>)m  (Us)p*  I  (10—12)  - ?.t c/  105.  TaBk /O-/ Stf/L/TcVDE FOR PYA'AM/C MOPBL SCALE  TE5T (/J  FACTOR  X ?• T/sr> e •3-  dcc-ef-emt'/hn  4. MoJuCxs ofhW'f / 1  /  X  f  S~. fif/nfcrnaP-f/-;ct/o>r  /  6. (//ictm/'/oec/sAearstrc/yTi  7\  7ai>/e ZO-2  SM/L/TUP£  ITEMS  R)R DYA/AMIC/VODE-L  SCAIB  FACTOR  i 2 . Damp/'/ij -facfor 3. T/'/ne <?  /A/  (U~s) «A) /n  (LS)& /  p  -7 ^ A  7t^ST(2)  Therefore i f  - 1 and ^^/(l/^  fy*  &  o  r  ^%- ^'/\  the  t  p  scale factor of the damping c o e f f i c i e n t of the material i s  —  - = -=L=  In table 10-2  (10-13)  the scale factors for a r b i t r a r y model material  are  shown. Usually, the i d e a l material which s a t i s f i e s a l l s i m i l a r i t y relations i s very d i f f i c u l t to f i n d , and then some of them are ignored.  Villich condition should be taken depends upon the  of greatest  importance to the t e s t .  If any  resonant e f f e c t i s not  of i n t e r e s t and only the f a i l u r e or the strength of i n t e r e s t , then Table 10-1 priority.  condition  of the structure  should be considered as the  Is  first  For the response analysis, however, the table 10-2  should  be considered to be Important, As Seed and Clough (1963) have suggested, i n the discussed above, the e f f e c t of pore water pressure was  similitudes not  taken  into account. Roscoe (1968) has pressure.  The  discussed the similitude regarding pore water  gravity force of the s o l i d substance i s given by  ft (/-/>) 1/  (10-14)  where n i s the void r a t i o to t o t a l volume.  Letting  be  the  scale factor for the unit weight of the s o l i d material, i . e .  (tt)m =  (S*Jp *  equation (10-14)  anc  ^lf~ be the stress scale factor, then from  ^--^ns-~^r  -  (10  Likewise s c a l i n g the u p l i f e of the s o l i d phase,  (f (l-n)V  15)  where  i s unit weight of the pore l i q u i d .  / _  Where  ^  "  (10-16)  m  i s the scale factor f o r the unit weight of the pore  l i q u i d , and s c a l i n g the s e l f weight of the l i q u i d phase, / f ^ l A requires that  7 ^ <M>-^f-  (10-17)  From equation (10-15) and (10-16), i t i s evident that  7, =  and from equations  (10-18)  (10-16) and (10-17)  (10-19)  Now the bulk unit weight, f  , of a saturated media i s given by  (10-20)  103,  and i f ^  f  m  , the scale factor for the bulk unit weight i s defined by  =7 C  (10-21)  6  th en, provided n  = n_^ and y  = rj^, i t follows that  ( 1 0  -  2 2 )  If equations (10-19). and (10-22) are s a t i s f i e d , the requirements (10-15), (10-16) and (10-17) a l l reduce to the equation  These three equations, (10-18), (.10-22) and (10-23) specify the conditions that the material must s a t i s f y f o r s i m i l a r i t y to p r e v a i l .  

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