UBC Theses and Dissertations

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UBC Theses and Dissertations

Elastic-viscoplastic response of earth structures to earthquake motion Byrne, Peter Michael 1969

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E L A S T I C - V I S C O P L A S T I C EARTH  STRUCTURES  TO-  RESPONSE  EARTHQUAKE  OF MOTION  by PETER M. BYRNE M.A.Sc. University of B r i t i s h Columbia, 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Ph.D. in the Department of C i v i l Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969  I  In p r e s e n t i n g an the  this thesis  advanced degree at  in p a r t i a l  f u l f i l m e n t of  the U n i v e r s i t y of B r i t i s h  the  requirements f o r  Columbia,  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of f o r s c h o l a r l y purposes may  be  granted by  by  his representatives.  of  t h i s thes,is f o r f i n a n c i a l gain  written  the  Head of my  It is understood that shall  not  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  C/£A/<</A/£&/?SA/fr Columbia  this  thesis  Department  copying or  that  Study.  or  publication  be allowed without  permission.  Department of  I agree  my  i  A B S T R A C T  A t h e o r y for. p r e d i c t i n g the dynamic response of structures subject  earth  to earthquake;..forces as. p r e s e n t e d h e r e i n .  posed t h e o r y e s s e n t i a l l y i n t r o d u c e s  a s t r e s s l i m i t i n t o the  e l a s t i c s h e a r - s l i c e or shear-beam analogy and  The  pro-  visco-  thus, i n essence,  the  s t r u c t u r e i s modelled" by a m u l t i d e g r e e o f freedom system w h i c h responds i n an e l a s t i c - v i s c o p l a s t i c manner.  The  s t r e s s l i m i t may  w i t h b o t h the magnitude of d i s p l a c e m e n t and  be  varied  the number of s t r e s s  so t h a t , i n p r a c t i c e , a complex s t r e s s - ' S t r a i n x e l a t i o n s h i p can  cycles  be  considered i n the a n a l y s i s . The method a l l o w s  the dynamic d i s p l a c e m e n t s , v e l o c i t i e s ,  a c c e l e r a t i o n s w i t h i n the s t r u c t u r e t o be c a l c u l a t e d . a t any when the base i s s u b j e c t e d  t o a known a c c e l e r a t i o n .  and'  time, t ,  I f magnitude of  d i s p l a c e m e n t s i s c o n s i d e r e d the c r i t e r i o n f o r earthquake d e s i g n , the proposed method of a n a l y s i s g i v e s a d i r e c t measure of  then  these  displacements. The  a n a l y s i s i s a p p l i e d t o a number of e a r t h s t r u c t u r e s  a base a c c e l e r a t i o n c o r r e s p o n d i n g t o the 0-10 earthquake.  D i s p l a c e m e n t s , v e l o c i t i e s and  at d i s c r e e t time i n t e r v a l s .  Results  s e e s , of E l C e n t r p ,  accelerations  are  1940  calculated  are compared w i t h t h o s e o b t a i n e d  from a v i s c o e l a s t i c dynamic a n a l y s i s and causes l a r g e r d i s p l a c e m e n t s and  using  smaller  i t i s found t h a t p l a s t i c accelerations.  The  action  location  of p l a s t i c y i e l d i s dependent on the s t r e n g t h p r o p e r t i e s of the  material.  ii  For m a t e r i a l o f u n i f o r m s t r e n g t h , y i e l d w i l l o c c u r e s s e n t i a l l y a t t h e base o f t h e s t r u c t u r e , w h i l e f o r i d e a l f r i c t i o n a l m a t e r i a l , y i e l d  will  take p l a c e throughout the s t r u c t u r e but w i l l be g r e a t e s t a t the t o p . L a t e r a l f o r c e o r s e i s m i c , c o e f f i c i e n t s from v i s c o e l a s t i c and e l a s t i c - v i s c o p l a s t i c a r e compared.  R e s u l t s suggest t h a t t h e common  p r a c t i c e o f a l l o w i n g f o r p l a s t i c - a c t i o n by assuming some h i g h  viscous  damping f a c t o r such as 20% o f c r i t i c a l i n a v i s c o e l a s t i c a n a l y s i s i s u n l i k e l y to give correct r e s u l t s .  iii TABLE OF CONTENTS PAGE NO.  ABSTRACT  i  TABLE OF CONTENTS  i i i  LIST OF TABLES  v  LIST OF FIGURES  v i  LIST OF SYMBOLS  xiv  ACKNOWLEDGEMENT  xvii  CHAPTER 1  INTRODUCTION  1  CHAPTER 2  PURPOSE AND SCOPE  5  2.1 Purpose  5  2.2 Scope  6  CHAPTER 3 CHAPTER 4  REVIEW OF STABILITY ANALYSES OF EARTH STRUCTURES SUBJECTED TO EARTHQUAKE MOTION REVIEW OF VISCOELASTIC RESPONSE THEORY  9 25  4.1 I n t r o d u c t i o n  25  4.2 I n f i n i t e s m a l Shear S l i c e Theory.  26  4.3 Lumped Parameter System  38  4.4 Normal Mode Theory  41  4.5 N u m e r i c a l A n a l y s i s o f Coupled E q u a t i o n s o f M o t i o n  45  4.6 Comparison o f t h e Shear S l i c e and F i n i t e  Element  Methods o f A n a l y s i s  48  4.7 C o n c l u s i o n s CHAPTER 5  51  PROPOSED ELASTIC VISCOPLASTIC DYNAMIC RESPONSE THEORY  53  5.1 I n t r o d u c t i o n 5.2 E l a s t i c - V i s c o p l a s t i c  53 Response Theory  54  iv  PAGE NO. 5.3 E s t i m a t i o n o f S t a t i c Shear S t r e s s e s 5.4 S t r e s s - S t r a i n and Force-Displacement of S o i l Under Dynamic Loading  64 Characteristics 68  5.5 D i s c u s s i o n o f Damping  79  5.6 D e t e r m i n a t i o n o f the Damping M a t r i x from the Percentage o f C r i t i c a l Damping  83  CHAPTER 6  APPLICATION OF ELASTIC-VISCOPLASTIC THEORY TO THE DYNAMIC RESPONSE ANALYSES OF EARTH STRUCTURES  6.1 I n t r o d u c t i o n  87 87  6.2 Dynamic Response o f H o r i z o n t a l and S l o p i n g Layers o f Soil  87  6.3 Dynamic Response o f an E a r t h Dam  106  6.4 Average Dynamic S e i s m i c C o e f f i c i e n t s  128  CHAPTER 7  SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH  139  7.1 Summary  139  7.2 C o n c l u s i o n s  141  7.3 Suggestions f o r F u r t h e r Research  142  LITERATURE CITED DERIVATION OF THE EXPRESSION FOR AVERAGE DYNAMIC SEISMIC COEFFICIENTS OF A WEDGE OF AN EARTH STRUCTURE APPENDIX I I COMPUTER PROGRAMS AND TEST PROBLEMS  144  APPENDIX I  147 149  V  LIST OF TABLES  PAGE NO.  TABLE I  Physical properties  associated  110  w i t h the e a r t h dam model of ( F i g . 29b) TABLE I I  Physical properties  associated  w i t h t h e unsymmetrical of dam TABLE I I I  Seismic  120  division  ( F i g . 35a) coefficients  from a n a l y s i s  of 285 f t . h i g h e a r t h dam  subject  t o 0-10 s e e s , of the N-S component of E l Centro earthquake  135  vi LIST OF FIGURES Figure No.  la lb  Page No.  Block-Type Motion of a Dam to Earthquake  Subject 13  Surface movement In Granular Unsaturated Material  13  Model of Single-Degree of Freedom R i g i d - P l a s t i c System  13  2b  Assumed Pulse  13  3  Model Showing Unsymmetrical Y i e l d Acceleration  16  Model Considered by Goodman and Seed (1966)  16  5a  Section of Dam  27  5b  Forces on Infinitesmal S l i c e  27  6  V e l o c i t y Spectrum f o r E l Centro, C a l i f o r n i a , Earthquake, May 18, 1940, Component N-S  32  2a  4  7a  Earthquake Force on Potential C y l i n d r i c a l Sliding Surface  36  7b  Earthquake Force on Idealized Wedge  36  7c  Forces on Element  36  8a 8b  Section of Homogeneous Dam Dynamic Shear Stress D i s t r i b u t i o n on Plane 60 F t . from Base, F i n i t e Element Analysis  49  49  Comparison of Average Dynamic Shear Stresses on a Plane 60 F t . above the Base, 0-10 Sec. N-S. Component of E l Centro Earthquake  49  Idealized Shear Structure  55  8c  9a  Shear S p r i n g C h a r a c t e r i s t i c s of Idealized Structure Forces on T y p i c a l Mass of I d e a l i z e d Shear S t r u c t u r e Uniform S t a t i c Shear Conditions Non-Uniform Conditions  Straus  S t a t i c Shear S t r e s s  S t a t i c P l u s Dynamic Shear S t r e s s C o n d i t i o n f o r the Case of Non-Uniform S t a t i c Shear S t r e s s Proposed Model o f E a r t h  Dam  Force-Displacement C h a r a c t e r i s t i c of H o r i z o n t a l S p r i n g Homogeneous E a r t h  Dam  Non-Homogeneous E a r t h  Dam  Shear S t r e s s D i s t r i b u t i o n on H o r i z o n t a l Planes Unsymmetrical D i v i s i o n of E a r t h S t a t i c Forces  Dam  on Element of Dam  Stress-Strain Relationship, Constant Y i e l d S t r e s s Stress-Strain Relationship f o r Strain Softening Material Dynamic S t r e s s - S t r a i n R e l a t i o n s h i p from S t r a i n C o n t r o l l e d T r i a x i a l T e s t s on Loose S a t u r a t e d Sand (Seed and Lee, 1966) S t i f f n e s s of R e c t a n g u l a r Element S t i f f n e s s of T r a p e z o i d a l Force-Displacement  Element  Relationship  Force-Displacement R e l a t i o n s h i p i n D i m e n s i o n l e s s Form  VXll.  F i g u r e No. 17a  17b 17c  Page No. Force-Displacement. R e l a t i o n s h i p f o r S t r a i n S o f t e n i n g M a t e r i a l , NonD i m e n s i o n a l Form  79  F a i l u r e on. a S i n g l e P l a n e W i t h i n a Slice  79  F a i l u r e Uniform.Throughout  the  Slice  79  18a  S e c t i o n . a n d P r o p e r t i e s of C l a y L a y e r  88  18b  Model-and. Model P r o p e r t i e s  88  18c  Force — Displacement R e l a t i o n s h i p  88  18d 19a  19b  20a  20b  • -Yield. S t r e s s Versus. Depth Relationship S u r f a c e D i s p l a c e m e n t s of 300 F t . C l a y L a y e r . S u b j e c t e d to Base M o t i o n C o r r e s p o n d i n g to the 0-10 Seconds o f the N-S Compt. o f E l C e n t r o , 1940 Earthquake  88  90  S u r f a c e A c c e l e r a t i o n of .300 F t . C l a y Layer. Subj e c t e d . to Base M o t i o n C o r r e s p o n d i n g t o the 0-10 seconds o f t h e N-S Compt. of E l C e n t r o , 1940 Earthquake  90  Envelopes of Max. Shear S t r e s s f o r 300 F t . L a y e r S u b j e c t e d to E l Centro Earthquake  92  Shear S t r e n g t h Assumptions  f o r 300  Ft. Layer  '92  20c  Force - Displacement R e l a t i o n s h i p s  92  21a  E f f e c t of V a r i o u s S t r e n g t h Assumptions. on the S u r f a c e D i s p l a c e m e n t s of a. 300 F t . H o r i z o n t a l L a y e r o f S o i l when S u b j e c t e d t o the N-S Component o f E l C e n t r o , 1940 Earthquake  94  ix F i g u r e No. 21b  22  23  Page No. E f f e c t of V a r i o u s S t r e n g t h Assumptions on the S u r f a c e A c c e l e r a t i o n s o f a 300 F t . H o r i z o n t a l L a y e r . o f S o i l when S u b j e c t e d t o t h e N-S Component.of E l C e n t r o , 1940 Earthquake R e l a t i o n s h i p between D u c t i l i t y F a c t o r and Depth f o r 300 F t . L a y e r S u b j e c t e d to 0-10 Sees, o f t h e N-S Component of E l Centro Earthquake  97  R e s u l t s o f Power S p e c t r a l D e n s i t y Analyses  24a  S l o p i n g Layer of S o i l  24b  V a r i a t i o n o f Modulus E, w i t h i n Layer Shear Model and P r o p e r t i e s o f 50 F t . S o i l Layer  24c  94  99 100  100 100  24d  S t r e s s e s and S t r e n g t h  100  24e  S t a t i c and Combined S t a t i c P l u s Maximum Dynamic S t r e s s e s  100  E f f e c t o f V a r i o u s S t r e n g t h Assumpt i o n s on t h e S u r f a c e Displacement o f a 50 Ft„ I n c l i n e d L a y e r of S o i l when S u b j e c t e d t o t h e N-S Component o f E l Centro Earthquake  103  R e l a t i o n s h i p between F a c t o r of S a f e t y and D i s p l a c e m e n t f o r 50 F t . S l o p i n g L a y e r S u b j e c t e d to 0-10 Sees. of t h e N-S Component o f E l Centro Earthquake  104  R e l a t i o n s h i p Between D u c t i l i t y F a c t o r and Depth f o r 50 F t . S l o p i n g L a y e r S u b j e c t e d t o 0-10 Sees, o f t h e N-S Component o f E l Centro Earthquake  104  E f f e c t o f Damping on Time - D i s p l a c e ment R e l a t i o n s h i p s f o r a 50 F t . I n c l i n e d L a y e r S u b j e c t e d t o the N-S Component o f E l C e n t r o , 1940 Earthquake  105  25  26a  26b  27  E f f e c t on S u r f a c e D i s p l a c e m e n t s Caused by S t r e n g t h V a r y i n g w i t h Displacement 50 F t , I n c l i n e d L a y e r S u b j e c t e d to the N-S Component o f E l Centro Earthquake S e c t i o n of E a r t h Model o f E a r t h  Dam  Dam  Method.for O b t a i n i n g H o r i z o n t a l Spring S t i f f n e s s S t a t i c Shear S t r e s s on H o r i z o n t a l P l a n e 100 F t . Above the Base S t a t i c F o r c e on T y p i c a l Elements Y i e l d S t r e s s Versus Depth R e l a t i o n s h i p f o r 285 F t . H i g h E a r t h Dam Having a S t a t i c F a c t o r of S a f e t y = 1.5 Displacements R e s u l t i n g at Crest of 285 F t . H i g h Dam when the Base i s S u b j e c t e d t o the N-S Component o f E l Centro Earthquake, Symm e t r i c a l D i v i s i o n and Cohesive Material F o r c e i n Top H o r i z o n t a l S p r i n g as a F u n c t i o n of Time R e l a t i o n s h i p s Between D u c t i l i t y F a c t o r and Depth f o r 285 F t . H i g h E a r t h Dam S u b j e c t e d t o the 0-10 Sec, of the N-S Component o f El. Centro Earthquake D i s p l a c e m e n t s R e s u l t i n g a t the C r e s t of 285 F t . H i g h Dam when t h e Base i s S u b j e c t e d t o the,N-S Compt. of E l Centro Earthquake. M a t e r i a l Assumed to be F r i c t i o n a l F o r c e i n Top H o r i z o n t a l S p r i n g as a F u n c t i o n o f Time  F i g u r e No.  Page No  35a  Unsymmetrical S u b d i v i s i o n of Dam  11.9  35b  Y i e l d S t r e s s e s f o r Unsymmetrical D i v i s i o n o f Dam  H9  35c  S t a t i c Shear S t r e s s e s f o r Unsymm e t r i c a l D i v i s i o n o f Dam  36  Displacements R e s u l t i n g at Crest of 285 F t o High Dam when t h e Base S u b j e c t e d to the N-S Component o f E l Centro Earthquake. U n s y m m e t r i c a l D i v i s i o n and F r i c t i o n a l M a t e r i a l  122  Displacements R e s u l t i n g at Crest of 285 F t . High Dam when the Base S u b j e c t e d to the N-S Component o f E l Centro Earthquake. Unsymmetrical D i v i s i o n and F r i c t i o n a l M a t e r i a l  124  R e l a t i o n s h i p s Between D u c t i l i t y F a c t o r and Depth f o r 285 Ft.. H i g h E a r t h Dam S u b j e c t e d to 10 Seconds of t h e N-S Component of E l Centro Earthquake  125  H o r i z o n t a l D i s p l a c e m e n t s t o the L e f t and R i g h t o f the D i v i s i o n f o r a 285 F t . H i g h Dam S u b j e c t e d t o the 0-10 Sees, of the N-S Component of E l Centro Earthquake  127  Average Dynamic S e i s m i c C o e f f i c i e n t s f o r 285 F t . H i g h Dam S u b j e c t e d t o the 0-10 Seconds of E l C e n t r o , 1940 E a r t h q u a k e , V i s c o e l a s t i c Response  129  Average Dynamic S e i s m i c C o e f f i c i e n t s f o r 285 F t . H i g h Dam S u b j e c t e d to the N-S Component o f E l C e n t r o , 1940 Earthquake. Cohesive M a t e r i a l  131  Average Dynamic S e i s m i c C o e f f i c i e n t s f o r 285 F t . H i g h Dam S u b j e c t e d t o the N-S Component of E l C e n t r o , 1940 Earthquake. F r i c t i o n a l M a t e r i a l and R i g i d H o r i z o n t a l Springs  132  37  38  39  40  41  42  F i g u r e No.  43  Page No  Average Dynamic S e i s m i c C o e f f i c i e n t s f o r 285 F t . High E a r t h Dam S u b j e c t e d to t h e N-S Component o f E l Centro 1960 Earthquake. F r i c t i o n a l M a t e r i a l and F l e x i b l e H o r i z o n t a l Springs  134  V a r i a t i o n i n Maximum Average S e i s m i c C o e f f i c i e n t w i t h Depth f o r a 285 F t . High Dam S u b j e c t e d to E l Centro 1940 Earthquake  137  45  PROGRAM 1  150  46  PROGRAM 2  151  47  PROGRAM 3  152  48  PROGRAM 4  153  49a  E l a s t i c - P l a s t i c , System, A f t e r Biggs (1964)  156  44  49b  50a  50b  50c  Response o f E l a s t i c - P l a s t i c of F i g . 49a  System 156  S e c t i o n and P r o p e r t i e s of.300 F t . Clay Layer  157  Model and P r o p e r t y Values o f C l a y Layer  157  Displacement o f Top S u r f a c e Versus Time f o r Base M o t i o n Corresponding to t h e N-S Component o f E l Centro 1940 Earthquake  15'7  XXII  F i g u r e No. 51  Page No. Model S o l v e d by Program 3  159  LIST OF SYMBOLS  a constant  associated with i n i t i a l v e l o c i t y  a c c e l e r a t i o n impulse i n g r a v i t y u n i t s a constant  associated with i n i t i a l  displacement  s t r e n g t h parameter damping c o e f f i c i e n t per u n i t mass damping c o e f f i c i e n t damping m a t r i x depth of element p e r t i n e n t sample d i m e n s i o n Young's modulus force inertia  force  damping f o r c e yield  force  a c c e l e r a t i o n of g r a v i t y shear modulus horizontal spring  force  i n e r t i a f o r c e on element B e s s e l f u n c t i o n of the f i r s t  k i n d and o r d e r  zero  B e s s e l f u n c t i o n of the f i r s t  k i n d and o r d e r  one  shear s t i f f n e s s seismic c o e f f i c i e n t y i e l d acceleration i n gravity units  average s e i s m i c c o e f f i c i e n t stiffness matrix mass index mass m a t r i x index s t r e n g t h parameter normal f o r c e shear f o r c e spectral  velocity  time increment o f time displacement i n x d i r e c t i o n velocity i n x direction acceleration  i n x direction  displacement vector non-dimensional  displacement  nonr-dimensional v e l o c i t y non-dimensional  acceleration  non-dimensional y i e l d superposition  integral  shear wave v e l o c i t y w e i g h t o f wedge cartesian coordinate cartesian coordinate  displacement  XVI  a  slope angle  a  phase a n g l e a s s o c i a t e d w i t h s t r u c t u r a l damping  a  damping c o e f f i c i e n t  3  damping c o e f f i c i e n t  3 n  n r o o t o f J (.3 ) = 0 o n  8  partial derivative  {n}  displacement v e c t o r  Y  unit-weight of s o i l  Y  shear s t r a i n  ,  M  Y  shear s t r a i n to cause y i e l d y  X  % c r i t i c a l damping  u  Poissons r a t i o  u  ductility  {£}  normal c o o r d i n a t e d i s p l a c e m e n t v e c t o r  p  density  a  normal, s t r e s s  x  shear s t r e s s  <J>  f r i c t i o n angle  <j>  e  factor  equivalent f r i c t i o n angle  ;  A C K N O W L E D G E M E N T  T h e ' w r i t e r ' s - i m m e d i a t e . a d v i s o r was Dr. W. D. Liam F i n n whose guidance and'many.helpful s u g g e s t i o n s . i n t h e p r e p a r a t i o n o f t h i s work i s s i n c e r e l y a p p r e c i a t e d . .The-.writer .would .also ..like-to-acknowledge  h e l p f u l suggest-  i o n s and t o e x p r e s s ' a p p r e c i a t i o n t o Drs. R. G. Campanella and D.L.  Anderson.  The  r e s e a r c h .was  supported-by funds p r o v i d e d by t h e  N a t i o n a l Research C o u n c i l , o f . C a n a d a . f i n a n c i a l support.for the.writer . -  expressed and  These funds a l s o i n c l u d e d Grateful appreciationi s  f o r t h i s a s s i s t a n c e w i t h o u t w h i c h t h e graduate s t u d i e s  t h i s t h e s i s c o u l d .not have been  accomplished.  - 1 CHAPTER 1  INTRODUCTION  C o n s i d e r a b l e advances have been made i n u n d e r s t a n d i n g the response of s t e e l and c o n c r e t e s t r u c t u r e s t o earthquake f o r c e s i n the past t h i r t y years.  The performance  of b u i l d i n g s d u r i n g E l C e n t r o ,  Tehachapi and the A l a s k a n earthquakes i n d i c a t e that such s t r u c t u r e s can be designed t o s a t i s f a c t o r i l y r e s i s t major earthquakes Records of the performance f a r more meager.  (Barnes 1965).  of e a r t h s t r u c t u r e s d u r i n g earthquake a r e  Three cases a r e known where an e a r t h dam has f a i l e d  c o m p l e t e l y d u r i n g an earthquake.  In a l l r e c o r d e d cases of damaged dams,  c o n s t r u c t i o n was c a r r i e d out w i t h o u t t h e use of modern compaction equipment.  T h i s has l e d some d e s i g n e r s t o the c o n c l u s i o n t h a t p r e s e n t  earthquake d e s i g n procedures f o r e a r t h dams a r e adequate. Seed  However,  (1966) has s t a t e d t h a t no l a r g e dam has as y e t been s u b j e c t e d to a  major earthquake.  S i n c e e a r t h dams a r e b e i n g c o n s t r u c t e d t o i n c r e a s i n g  h e i g h t s , i t i s important t h a t the f a c t o r s g o v e r n i n g t h e earthquake s t a b i l i t y of these s t r u c t u r e s be understood.  I t i s to this  problem  that t h i s t h e s i s i s d i r e c t e d . An earthquake i s a ground v i b r a t i o n phenomenon.  S i n c e the  e a r t h ' s c r u s t i s comprised of m a t e r i a l which i s e l a s t i c i n i t s gross c h a r a c t e r i s t i c s and p o s s e s s e s mass, i t w i l l v i b r a t e when s u b j e c t e d t o shock l o a d i n g .  Thus, when sudden s l i p p a g e o c c u r s a t a f a u l t zone,  shock  waves a r e propagated i n a l l d i r e c t i o n s and when these waves pass any g i v e n p o i n t on t h e e a r t h ' s s u r f a c e , i t i s caused t o v i b r a t e .  - 2 Ground v i b r a t i o n s i n d u c e i n e r t i a f o r c e s w i t h i n a s t r u c t u r e . Thus, d u r i n g  an e a r t h q u a k e , a s t r u c t u r e i s s u b j e c t e d t o dynamic f o r c e s .  i n a d d i t i o n t o s t a t i c forces„ i n e r t i a forces  F o r many problems  the h o r i z o n t a l  induced a r e more c r i t i c a l than t h e v e r t i c a l f o r c e s and i n  most dynamic a n a l y s e s o f s t r u c t u r e o n l y the h o r i z o n t a l f o r c e s a r e considered„ I t i s common p r a c t i c e when d e s i g n i n g  earthquake r e s i s t a n t  s t r u c t u r e s t o a l l o w f o r t h e e f f e c t o f an earthquake by t h e i n c l u s i o n of a h o r i z o n t a l s t a t i c force„  T h i s f o r c e i s u s u a l l y e x p r e s s e d as t h e  product of the weight times a l a t e r a l f o r c e or seismic  coefficient„  The s t a b i l i t y of t h e s t r u c t u r e i s then a n a l y z e d as a s t a t i c problem„ I n e a r t h s t r u c t u r e s , t h e s t a b i l i t y i s u s u a l l y e x p r e s s e d i n terms o f a f a c t o r o f s a f e t y , where t h e term f a c t o r o f s a f e t y e x p r e s s e s t h e r a t i o of t h e s t r e n g t h  t o t h e s t r e s s on a p o t e n t i a l s l i d i n g  surface.  However, an earthquake a p p l i e s a t r a n s i e n t f o r c e system t o a structure rather  t h a n a s t a t i c f o r c e system,,  T h e r e f o r e , i f t h e con-  v e n t i o n a l method o f a n a l y s i s i s t o have any r a t i o n a l b a s i s , a h o r i z o n t a l f o r c e w h i c h i s i n some way e q u i v a l e n t must be chosen.  to the transient force  system  I f t h e m a t e r i a l o f t h e s t r u c t u r e i s b r i t t l e , then an  a b r u p t f a i l u r e w i l l occur whenever t h e combined s t a t i c and dynamic s t r e s s r e a c h e s t h e s t r e n g t h o f t h e material„  I n t h i s case a s t r u c t u r e  s h o u l d be d e s i g n e d t o r e s i s t a h o r i z o n t a l f o r c e c o r r e s p o n d i n g t o t h e maximum i n e r t i a f o r c e a c t i n g on t h e structure,,. I f , however  s  t h e m a t e r i a l possesses s o m e , p l a s t i c i t y ,  plastic  y i e l d r a t h e r t h a n an a b r u p t f a i l u r e w i l l o c c u r whenever t h e s t r e n g t h of  -  the m a t e r i a l i s r e a c h e d .  1  S i n c e t h e maximum i n e r t i a f o r c e w i l l a c t  for only a very short period of time  only small p l a s t i c  B  deformations  can occur i n t h i s p e r i o d and t h e s e may be q u i t e acceptable„ T h e r e f o r e , f o r s t r u c t u r e s comprised o f m a t e r i a l s w h i c h p l a s t i c i t y , i t would seem more r e a s o n a b l e t o judge t h e i r performance  earthquake  i n te-r-ms o f d i s p l a c e m e n t s prgdyggd r a t h e r t h a n i n terms o f  a f a c t o r o f s a f e t y based on s t r e s s e s The  possess  ?  i m p o r t a n t concept t h a t t h e earthquake r e s i s t a n c e o f an  e a r t h s t r u c t u r e s h o u l d be e v a l u a t e d i n terms o f d i s p l a c e m e n t s r a t h e r t h a n f a c t o r of s a f e t y was f i r s t proposed  by Newmark (1963)„  Subse-  q u e n t l y Newmark (1965) and Goodman and Seed (1966) p r e s e n t e d methods f o r o b t a i n i n g t h e d i s p l a c e m e n t s o f a wedge o r b l o c k of an embankment subject t o earthquake.  These methods m o d e l l e d t h e b e h a v i o u r o f t h e  s l i d i n g b l o c k by a s i n g l e - d e g r e e o f freedom r i g i d - p l a s t i c system a mass r e s t i n g on an i n c l i n e d p l a n e ) .  (i.e.,,  T h e i r a n a l y s i s assumed t h a t a l l  p l a s t i c y i e l d i n g o c c u r r e d a l o n g a s i n g l e s u r f a c e o r p l a n e and t h a t no. r e l a t i v e displacements occurred other than across t h i s  potential  f a i l u r e surface. F o r c e d v i b r a t i o n t e s t s on e a r t h dams (Seed 1966) i n d i c a t e t h a t r e l a t i v e m o t i o n throughout a s t r u c t u r e t a k e s place,,  For small ampli-  tude v i b r a t i o n s such as those o b t a i n e d from t e s t s , d i s p l a c e m e n t s a r e r e c o v e r a b l e and c a n be s i m u l a t e d by assuming t h e s t r u c t u r e t o be m o d e l l e d by a m u l t i d e g r e e o f freedom v i s c o e l a s t i c system,.  For larger  v i b r a t i o n s c o r r e s p o n d i n g t o t h o s e caused by a major e a r t h q u a k e ,  plastic  d i s p l a c e m e n t s i n a d d i t i o n t o e l a s t i c d i s p l a c e m e n t s c a n be expected (Ambraseys 1960).  An a n a l y s i s which would c o n s i d e r an e a r t h s t r u c t u r e  3  -  - 4 -  as a m u l t i d e g r e e o f freedom v i s c o e l a s t i c system, b u t w h i c h , i n a d d i t i o n , would a l l o w p l a s t i c y i e l d a t h i g h s t r e s s l e v e l s would f o r e seem v e r y d e s i r a b l e . of such an a n a l y s i s .  T h i s t h e s i s i s concerned w i t h the  there-  development  CHAPTER 2  - 5 -  PURPOSE AND SCOPE  2.1  PURPOSE The purpose o f t h i s t h e s i s was t o d e v e l o p and a p p l y a response  t h e o r y t o model t h e performance o f an e a r t h s t r u c t u r e s u b j e c t e d t o a strong motion earthquake.  The t h e o r y p r e s e n t e d h e r e i n c o n s i d e r s t h e  s t r u c t u r e as a m u l t i d e g r e e o f freedom system w h i c h responds i n an elastic-viscoplastic  manner.  response w i l l be v i s c o e l a s t i c ,  That i s , f o r s m a l l s c a l e v i b r a t i o n s t h e while f o r large scale vibrations,  yield-  i n g o r p l a s t i c d i s p l a c e m e n t s w i l l occur i n a d d i t i o n t o e l a s t i c displacements. • Most dynamic response a n a l y s e s performed o n . e a r t h s t r u c t u r e s t o d a t e (1967) have assumed t h e s t r u c t u r e t o respond i n a manner.  viscoelastic  The e f f e c t of p l a s t i c d e f o r m a t i o n has been a l l o w e d f o r t o  some e x t e n t i n t h i s t y p e o f a n a l y s i s by u s i n g h i g h v i s c o u s damping factors.  However, V e l e t s o s and Newmark (1960) have shown t h a t  plastic  a c t i o n may be c o n s i d e r a b l y more e f f e c t i v e t h a n v i s c o u s damping i n . r e d u c i n g t h e c a l c u l a t e d l a t e r a l earthquake f o r c e s , a c t i n g on a s t e e l or concrete structure.  They suggest t h a t when compared w i t h  viscoelastic  r e s p o n s e , e l a s t i c - p l a s t i c b e h a v i o u r may reduce t h e l a t e r a l f o r c e by a f a c t o r of f o u r or more.  I t c o u l d be e x p e c t e d t h a t p l a s t i c a c t i o n would  a l s o g r e a t l y m o d i f y t h e dynamic•response o f e a r t h s t r u c t u r e s , and hence a method o f a n a l y s i s w h i c h c o n s i d e r s p l a s t i c a c t i o n would be an advance i n t h e f i e l d o f earthquake response o f e a r t h s t r u c t u r e s .  - 6 2o2  SCOPE  The proposed method o f a n a l y s i s e s s e n t i a l l y i n t r o d u c e s p l a s t i c i t y i n t o t h e v i s c o e l a s t i c s h e a r - s l i c e method suggested by Mononoke, Takata and Matamura (1936).  I t s u f f e r s from t h e r e s t r i c t i o n t h a t d i s -  placements a r e assumed t o occur i n a d i r e c t i o n p a r a l l e l t o the base of t h e s t r u c t u r e and t o be caused s o l e l y by s h e a r i n g s t r a i n s , , I n t h e s h e a r - s l i c e method, a s t r u c t u r e i s c o n s i d e r e d t o comprise of a f i n i t e number o f s l i c e s p a r a l l e l t o t h e base.  Slices are ideal-  i z e d by p o i n t masses connected by i n t e r - s t o r e y s p r i n g s and d a s h p o t s  0  P l a s t i c i t y i s i n t r o d u c e d by c o n s i d e r i n g t h e s p r i n g s t o have a f o r c e l i m i t c o r r e s p o n d i n g t o t h e y i e l d s t r e s s of t h e material„ . Thus, r e l a t i v e displacements g r e a t e r than those r e q u i r e d to reach the y i e l d s t r e s s , cause p l a s t i c action.,  T h i s type o f model i s thought t o a p p l y  to h o r i z o n t a l and s l o p i n g l a y e r s o f s o i l where t h e shear s t r e s s on p l a n e s p a r a l l e l t o t h e base i s uniform,, In a s t r u c t u r e such as an e a r t h dam^ t h e s t a t i c shear s t r e s s on h o r i z o n t a l planes i s not uniform,  I n f a c t , i n t h e absence o f water  f o r c e s , t h e shear f o r c e on h o r i z o n t a l p l a n e s t o t h e l e f t and r i g h t of t h e c e n t r e - l i n e i s e q u a l and opposite„  T h e r e f o r e , d u r i n g earthquake  m o t i o n , t h e l e f t hand s i d e o f a dam may-move,into t h e p l a s t i c  range  w h i l e t h e r i g h t hand s i d e i s s t i l l i n t h e e l a s t i c range, and v i c e v e r s a . I t was thought t h a t t h e presence of t h e s e s t a t i c shear s t r e s s e s which a r e o f o p p o s i t e s i g n would tend t o cause a dam t o spread d u r i n g earthquake^ w i t h t h e p o s s i b l e f o r m a t i o n o f cracks„ t h i s type o f b e h a v i o u r t o o c c u r  s  laterally To a l l o w  each h o r i z o n t a l s l i c e was c o n s i d e r e d  -  t o comprise of two  p a r t s s e p a r a t e d by a h o r i z o n t a l s p r i n g ,  the h o r i z o n t a l s t i f f n e s s of the s l i c e . c o m p r e s s i o n i n the s p r i n g . s p r i n g f o r c e may  However  become t e n s i l e .  s p r i n g to represent  9  The  representing  s t a t i c f o r c e system causes  under dynamic c o n d i t i o n s ,  the  A t e n s i l e l i m i t can be imposed on  the t e n s i l e s t r e n g t h of the m a t e r i a l .  l i m i t has been reached a c r a c k i s assumed t o occur and l i m i t drops t o z e r o .  7 -  the  Once t h i s  the s p r i n g  tensile  In granular m a t e r i a l a crack i s n o t . p o s s i b l e  as  the m a t e r i a l w i l l s p r e a d , so t h a t the h o r i z o n t a l f o r c e cannot remain zero.  T h i s t y p e of b e h a v i o u r can be m o d e l l e d by assuming a more f l e x -  i b l e s p r i n g or by l i m i t i n g the minimum c o m p r e s s i v e f o r c e i n the The constant  s t r e s s - s t r a i n b e h a v i o u r of the m a t e r i a l i s not r e s t r i c t e d  values  s t r e n g t h and  spring.  of the e l a s t i c - v i s c o p l a s t i c p r o p e r t i e s .  the s t i f f n e s s of the m a t e r i a l may  Both t h e  be c o n s i d e r e d  to  to  yield  vary  w i t h b o t h magnitude of s t r a i n and w i t h the number of s t r a i n c y c l e s . I n t h i s way The  a complex s t r e s s - s t r a i n r e l a t i o n s h i p can,be  proposed e l a s t i c - v i s c o p l a s t i c . r e s p o n s e t h e o r y has  a p p l i e d h e r e i n b o t h t o h o r i z o n t a l and e a r t h damsi  considered.  s l o p i n g l a y e r s of s o i l and  A base a c c e l e r a t i o n c o r r e s p o n d i n g to the N-S  E l C e n t r o 1940  earthquake was  used f o r a l l a n a l y s e s .  The  = o t a n <(>) or a u n i f o r m s t r e n g t h m a t e r i a l (x  s o i l was  con-  frictional  = a  constant).  In a d d i t i o n , some examples were examined where the s t r e n g t h was s i d e r e d t o be a f u n c t i o n of d i s p l a c e m e n t .  to  component of  s i d e r e d t o have a shear s t r e n g t h c o r r e s p o n d i n g t o e i t h e r a m a t e r i a l (x  been  con-  Dynamic a c c e l e r a t i o n s  and  d i s p l a c e m e n t s were c a l c u l a t e d a t d i s c r e e t i n t e r v a l s of time w i t h  the  a i d of an e l e c t r o n i c d i g i t a l computer and  s i g n i f i c a n t r e s u l t s are  shown i n . g r a p h i c a l form. discussed  Average dynamic s e i s m i c  by Seed and M a r t i n  coefficients,  (1966), were a l s o c a l c u l a t e d and  as  are  compared w i t h those o b t a i n e d from a v i s c o e l a s t i c response analysis„ t h i s way,  a measure of the e f f e c t of p l a s t i c a c t i o n on the i n e r t i a  f o r c e s g e n e r a t e d w i t h i n a s t r u c t u r e can be  obtained.  - 9 CHAPTER 3 REVIEW OF STABILITY ANALYSES OF EARTH STRUCTURES SUBJECTED TO EARTHQUAKE MOTION The  s t a b i l i t y o f an e a r t h dam s u b j e c t  t o earthquake f o r c e s has  g e n e r a l l y been a s s e s s e d i n terms of t h e f a c t o r of s a f e t y o f a p o t e n t i a l s l i d i n g mass o f soilo defined  The term f a c t o r of s a f e t y i s u s u a l l y  as t h e r a t i o o f t h e shear s t r e n g t h t o t h e shear s t r e s s a t  p o i n t s on t h e p o t e n t i a l f a i l u r e surface,,  Methods o f s t a b i l i t y  analy-  s i s such as those proposed by B i s h o p (1955), T a y l o r and Lowe (1959), and M o r g e n s t e r n (1965) have t h e common s u p p o s i t i o n  that the f a c t o r  of s a f e t y a t every p o i n t on t h e p o t e n t i a l f a i l u r e s u r f a c e  i s t h e same.  A c a l c u l a t e d f a c t o r o f s a f e t y l e s s than u n i t y i m p l i e s t h a t t h e d r i v i n g f o r c e s exceed t h e r e s i s t i n g f o r c e s and hence t h e mass o f s o i l would accelerate.  I n a s t a t i c a n a l y s i s , where l o a d s a c t c o n t i n u o u s l y ,  would n o t be an a c c e p t a b l e s t a b i l i t y c o n d i t i o n .  this  However, i n a dynamic  a n a l y s i s where t h e i n e r t i a f o r c e s a c t f o r a f i n i t e t i m e , a f a c t o r of s a f e t y l e s s t h a n u n i t y may be q u i t e The  acceptable.  e f f e c t o f earthquake m o t i o n i s a l l o w e d  f o r i n a conven-  t i o n a l s t a b i l i t y a n a l y s i s by t h e i n t r o d u c t i o n o f a h o r i z o n t a l f o r c e system w h i c h i s u s u a l l y s p e c i f i e d i n terms of t h e product o f t h e w e i g h t , W, t i m e s a l a t e r a l f o r c e c o e f f i c i e n t or a s e i s m i c c o e f f i c i e n t , k.  The earthquake does n o t j i n f a c t , a p p l y a c o n s t a n t f o r c e t o t h e  mass, but r a t h e r a t r a n s i e n t f o r c e system w h i c h l a s t s f o r a f i n i t e length of time.  T h e r e f o r e , t h e magnitude of t h e s t a t i c f o r c e  implied  - 10 by the term s e i s m i c c o e f f i c i e n t must f i r s t be d e c i d e d upon b e f o r e i t can be r a t i o n a l l y  evaluated.  P o s s i b l e v a l u e s of the f o r c e i m p l i e d by the term s e i s m i c e f f i c i e n t were d i s c u s s e d  i n d e t a i l by Seed and M a r t i n  the s e i s m i c c o e f f i c i e n t may 1)  2)  (1966). B r i e f l y ,  designate:  The maximum i n e r t i a f o r c e t h a t the dam thereof  co-  i s subjected  to d u r i n g an  or p o r t i o n  earthquake.  A s t a t i c f o r c e which i s equivalent  i n e f f e c t to  the  t r a n s i e n t f o r c e system, i . e . , would cause the same displacements. 3)  An.empirical vative  f o r c e t h a t w i l l produce a more c o n s e r -  design.  I t would appear t h a t (1) i s u l t r a - c o n s e r v a t i v e when the m a t e r i a l of the s t r u t u r e p o s s e s s e s p l a s t i c i t y . only a very  S i n c e the maximum f o r c e w i l l a c t f o r  s h o r t p e r i o d of time, then i f i t does cause the  strength  t o be r e a c h e d , o n l y s m a l l p l a s t i c d e f o r m a t i o n s can occur and be q u i t e a c c e p t a b l e .  A s t a t i c f o r c e which i s equivalent  i s such a f o r c e t o be o b t a i n e d ?  of t h e earthquake t h a t i s of i n t e r e s t . no l o n g e r r e q u i r e d .  The  necessary f o r s t a t i c Seismic one  one  I t i s r e a l l y the  to  use.  effect  I f t h i s i s known then (2) i s  concept i n v o l v e d i n (3) would appear t o  r e a l i z e d by u s i n g a h i g h e r  may  i n e f f e c t to  the t r a n s i e n t f o r c e system would seem the most a p p r o p r i a t e However, how  these  f a c t o r of s a f e t y than n o r m a l l y  be  considered  conditions.  c o e f f i c i e n t s may  of t h e f o l l o w i n g means:  be o b t a i n e d  f o r an e a r t h s t r u c t u r e by  - 11 1)  Empirical,  2)  R i g i d body dynamic r e s p o n s e a n a l y s i s ,  3)  V i s c o e l a s t i c dynamic response a n a l y s i s .  These methods a r e a g a i n d i s c u s s e d Empirical values  i n d e t a i l by Seed and M a r t i n  of the s e i s m i c c o e f f i c i e n t r a n g i n g  a r e commonly used i n N o r t h A m e r i c a . between 0.12  and  0.25  from 0.05  (1966), to  Higher v a l u e s , g e n e r a l l y  are used i n Japan.  The  0.15  ranging  R u s s i a n Code r e f e r r e d t o  by Ambraseys (1960) i s the o n l y one which c u r r e n t l y c o n s i d e r s a t i o n of the s e i s m i c c o e f f i c i e n t w i t h depth w i t h i n a  a vari-  dam.  R i g i d body response i m p l i e s t h a t the s t r u c t u r e moves e x a c t l y as the ground, so t h a t a l l elements of the s t r u c t u r e a r e s u b j e c t to same a c c e l e r a t i o n as the ground.  the  Thus the maximum a c c e l e r a t i o n would  the same as the ground a c c e l e r a t i o n .  Seed and M a r t i n  (1966) c o n s i d -  ered t h a t low s t i f f embankments and dams i n narrow canyons might respond i n t h i s manner. Seismic  c o e f f i c i e n t s can be c a l c u l a t e d from v i s c o e l a s t i c r e -  sponse t h e o r y and  t h i s w i l l be d i s c u s s e d  i n Chapter 4.2.  Ambraseys  (1960), c a l c u l a t e d s e i s m i c c o e f f i c i e n t s c o r r e s p o n d i n g t o the maximum i n e r t i a f o r c e a t the v a r i o u s l e v e l s w i t h i n a dam advocated t h a t t h e s e f o r c e s be i n t r o d u c e d  (Chapter 4.2).  He  i n t o the s t a b i l i t y a n a l y s i s  proposed by B i s h o p (1955) , a l l o w i n g a f a c t o r of s a f e t y a g a i n s t  earth-  quake f a i l u r e t o be deduced. Newmark (1963), Newmark (1965) and t h a t f a c t o r . o f s a f e t y i s a poor way  Seed (1966) have suggested  t o a s s e s s the s t a b i l i t y of  s t r u c t u r e s s u b j e c t to earthquake f o r c e s .  earth  R a t h e r , the performance  be  -  s h o u l d be judged quake.  i n terms o f t h e d i s p l a c e m e n t s  I f i t i s e s t i m a t e d t h a t d i s p l a c e m e n t s w i l l remain w i t h i n  t o l e r a b l e l i m i t s under t h e d e s i g n earthquake, s i d e r e d t h a t t h e earthquake The  then i t c o u l d be con-  s t a b i l i t y of t h e dam i s s a t i s f a c t o r y .  t y p e s o f m o t i o n o f e a r t h and r o c k - f i l l dams, when sub-  j e c t e d t o earthquake as  produced by the e a r t h -  f o r c e s , were summarized by Newmark (1965) t o be  follows: 1)  M o t i o n of a wedge or s l i c e o f t h e upstream o r downstream s l o p e , g e n e r a l l y out and d o w n h i l l as shown i n F i g . l a by " a " and "b"o  In granular unsaturated  s l o p e s , a s u r f a c e o r near s u r f a c e downslope movement o c c u r s as shown i n F i g . l b .  This, although not  s t r i c t l y a b l o c k - t y p e movement, may be c o n s i d e r e d as such from t h e p o i n t o f v i e w o f a n a l y s i s , 2)  M o t i o n o f t h e dam as a b l o c k w i t h r e l a t i v e m o t i o n occurring along a h o r i z o n t a l plane. Fig, at  l a by " c " .  T h i s p l a n e o f weakness need n o t be  t h e base o f t h e dam as shown, but may be w i t h i n  the f o u n d a t i o n s 3)  T h i s i s shown i n  o r  a t  some l e v e l w i t h i n t h e dam,  R e l a t i v e m o t i o n w i t h i n t h e dam o r f o u n d a t i o n o f such a n a t u r e as t o cause f i s s u r e s t o open.  These  f i s s u r e s a r e g e n e r a l l y v e r t i c a l and, i n a d d i t i o n , a r e u s u a l l y p a r a l l e l t o t h e a x i s o f the dam (Duke 1960). Newmark (1965) c o n s i d e r e d t h a t t h e c h a r a c t e r o f t h e m o t i o n would depend on t h e t y p e o f m a t e r i a l .  In general, for  non-cohesive  - 13 -  FIG.  2a  MODEL OF SINGLE— DEGREE OF FREEDOM SYSTEM  RIGID-PLASTIC  FIG. 2b  ASSUMED  PULSE  - 14 material and for cohesive material where a well defined plane of weakness can develop, the motion occurs essentially along arcs or planes.  Where the material is highly cohesive, the motion is more  elastic or nearly elastic i n character and a general deformation field w i l l develop. Newmark (1965) presented an analysis which allows displacements to be calculated where movements take place along arcs or planes. The acceleration, k^g, which w i l l just cause yielding to occur along the boundary of the surface is calculated. The net resistance to dynamic forces is therefore m k^g.  The sliding mass i s then consider-  ed to act as a rigid-plastic single-degree of freedom system, which may be modelled by a mass resting on a plane, as shown in Fig. 2a. The direction of the dynamic force mk(t)g, the resisting force, mk^g, and the relative motion of the block, u, are a l l assumed to be colinear.  The displacement of the mass relative to the base can be cal-  culated by assuming the base to be fixed and the mass subjected to force mk(t)g as shown. For an acceleration pulse of magnitude Ag which acts for a duration t  (Fig. 2b) the equation of motion is  o m li = mAg - mk^g  (3.1)  where U = the acceleration i n the u direction by integration u = Agt - k^gt (t > t ) Q  Thus from which,  Q  (3.2)  Agt - k gt = o ° o y m  (3.3)  At t = m k y  (3.4)  - 15 -  The maximum ground v e l o c i t y , v , w i l l o c c u r a t t = t V The s u p p o s i t i o n  Q  = Agt  and i s g i v e n by (3.5)  Q  of a s i n g l e pulse leads to a constant v e l o c i t y a t the  end' o f the p u l s e ( F i g . 2 b ) . The maximum d i s p l a c e m e n t , u^, i s g i v e n -  by  •™ " ' " m u = / (vdt - k gdt) m v o v k c  (3.6)  J  J  2  from w h i c h  u = T T T — (1 m zgk y  )  (3.7)  A  Thus t h e maximum d i s p l a c e m e n t i s a f u n c t i o n o f t h e square o f t h e maximum ground v e l o c i t y , v  2  , t h e y i e l d a c c e l e r a t i o n , k^, and t h e  maximum ground a c c e l e r a t i o n , Ag. Newmark s t a t e s t h a t Eq.„ 3.7 would g i v e a c o n s e r v a t i v e  result  s i n c e i n a c t u a l f a c t an earthquake c o n s i s t s o f many p u l s e s some o f w h i c h a r e p o s i t i v e and some n e g a t i v e .  The a r e a under t h e a c c e l e r -  a t i o n curve must, i n f a c t , be z e r o s i n c e the ground comes to r e s t a f t e r t h e earthquake m o t i o n c e a s e s .  However, t h e y i e l d  acceleration  i s n o t l i k e l y t o be t h e same f o r a r e l a t i v e d i s p l a c e m e n t , u, o f opposite sign.  The c o n d i t i o n o f unequal y i e l d a c c e l e r a t i o n can be  m o d e l l e d by a mass on an i n c l i n e d p l a n e ( F i g . 3 ) . The l i m i t i n g f r i c t i o n f o r c e p e r u n i t normal f o r c e i s d e s i g n a t e d by t a n <j).  The  c o n d i t i o n f o r l i m i t i n g s t a t i c e q u i l i b r i u m f o r p o t e n t i a l m o t i o n down, the p l a n e i s from w h i c h  me  i g ( k y ) ^ = mg cos a t a n $ (k ) , = s i n a y'd tana v  _ i)  - mg s i n a  (3.8) (3.9)  - 16 -  FIG.  4  MODEL  CONSIDERED  BY  GOODMAN  AND  SEED  (1966)  - 17 where ( k ) d = t h e y i e l d a c c e l e r a t i o n i n g r a v i t y u n i t s y f o r p o t e n t i a l m o t i o n down t h e p l a n e The y i e l d a c c e l e r a t i o n f o r p o t e n t i a l m o t i o n up t h e p l a n e i s g i v e n by (k ) = s i n a(r^ y u . tana  +- 1)  (3,10)  .  S i n c e t h e y i e l d a c c e l e r a t i o n f o r p o t e n t i a l m o t i o n up t h e p l a n e  will  be many times g r e a t e r t h a n t h a t f o r m o t i o n down t h e p l a n e f o r most problems, i t i s g e n e r a l l y  s u f f i c i e n t l y a c c u r a t e t o assume t h a t no  m o t i o n can o c c u r up...the p l a n e , i , e . , .an i n f i n i t e y i e l d  acceleration  f o r m o t i o n up t h e p l a n e . U s i n g t h e s i n g l e - d e g r e e o f freedom model Newmark c a l c u l a t e d t h e maximum d i s p l a c e m e n t s f o r f o u r earthquakes the y i e l d a c c e l e r a t i o n , k^.  assuming v a r i o u s v a l u e s o f  The earthquakes were f i r s t n o r m a l i z e d t o  a maximum a c c e l e r a t i o n o f 0.5g and a. maximum ground v e l o c i t y o f 30 in.^ec.  Two cases were c o n s i d e r e d :  Case 1, i n which  the y i e l d  a c c e l e r a t i o n was t h e same i n b o t h d i r e c t i o n s , termed s y m m e t r i c a l r e s i s t ance, and Case 2, i n which t h e y i e l d a c c e l e r a t i o n was i n f i n i t e i n one d i r e c t i o n , termed u n s y m m e t r i c a l r e s i s t a n c e .  Maximum d i s p l a c e m e n t s  were p l o t t e d v e r s u s t h e r a t i o k^/A, where A i s t h e maximum  ground  a c c e l e r a t i o n , and showed t h a t f o r s y m m e t r i c a l r e s i s t a n c e Eq. 3.7 y i e l d s a good a p p r o x i m a t i o n f o r d i s p l a c e m e n t s f o r v a l u e s o f k^/A g r e a t e r than 0.1.  Maximum d i s p l a c e m e n t s a r e c o n s i d e r a b l y g r e a t e r f o r t h e case o f  u n s y m m e t r i c a l r e s i s t a n c e and may be e s t i m a t e d from t h e d a t a p r e s e n t e d by Newmark. Goodman and Seed (1966) p r e s e n t e d a method f o r d e t e r m i n i n g t h e d i s p l a c e m e n t s o f a r i g i d - p l a s t i c s i n g l e - d e g r e e o f freedom system  - 18 -  w h e r e i n t h e s t r e n g t h was a f u n c t i o n o f t h e r e l a t i v e d i s p l a c e m e n t , u. The  model was s i m i l a r t o t h a t v i s u a l i z e d by Newmark e x c e p t t h a t t h e  e x c i t i n g a c c e l e r a t i o n was c o n s i d e r e d t o be h o r i z o n t a l w h i l e  displace-  ments o c c u r r e d a t an a n g l e a to t h e h o r i z o n t a l ( F i g . 4 ) . The s t r e n g t h or r e s i s t i n g f o r c e , F , was c o n s i d e r e d t o be d e r i v e d or equivalent  f r i c t i o n so t h a t F  where t a n <}>  from f r i c t i o n  w  a  However, T a y l o r  s  =  y  N t a n <j> eq  (3.11)  considered to vary w i t h  displacement.  (1948) has shown t h a t <j> i s a f u n c t i o n o f the normal  stress i n a d d i t i o n to the void, r a t i o .  T h e r e f o r e <t> s h o u l d be a eq  f u n c t i o n o f b o t h d i s p l a c e m e n t u and normal f o r c e N.  From F i g . 4, t h e  normal f o r c e , N, i s g i v e n by N = mg cosa-mg k ( t ) s i n a  (3.12)  Thus t h e normal f o r c e v a r i e s w i t h t h e i n e r t i a f o r c e mgk(t) and t h e r e f o r e so a l s o s h o u l d cb . eq .  T h i s problem would not a r i s e i f t h e i n e r t i a  f o r c e were taken to b e . p a r a l l e l to t h e d i r e c t i o n o f motion as suggested by Newmark (1965).  S i n c e t h e earthquake has v e r t i c a l as w e l l as  h o r i z o n t a l components t h e r e seems l i t t l e p o i n t i n assuming t h e e a r t h quake f o r c e t o be h o r i z o n t a l i f i t c o m p l i c a t e s t h e a n a l y s i s . However, t e s t s performed by Goodman and Seed were c a r r i e d out u s i n g a h o r i z o n t a l e x c i t a t i o n f o r c e and t h e r e f o r e a model such as shown i n F i g . 4 would be a p p l i c a b l e . model i s g i v e n by k g = t a n (<j> - a) g y eq  The y i e l d a c c e l e r a t i o n f o r such a  (3.13)  - 19 and  the e q u a t i o n - o f • m o t i o n can-be-.shown to ii = B (x) | k ( t ) . - k  be  (x)}  y  (3.14)  where B (x) = g ( s i n a tancfi  + cosa) eq  and  i s a f u n c t i o n o f d i s p l a c e m e n t , u, s i n c e <j>  Eq.  3.14  i s a f u n c t i o n of  can be i n t e g r a t e d to o b t a i n the d i s p l a c e m e n t a t any  u.  time, t ,  f o r the p a r t i c u l a r f u n c t i o n k ( t ) , known or assumed. Seed (1966) c o n s i d e r e d  t h a t the method of d i r e c t c a l c u l a t i o n  o f d i s p l a c e m e n t s would apply o n l y to u n s a t u r a t e d  granular slopes.  s u g g e s t e d t h a t i n s a t u r a t e d m a t e r i a l where pore p r e s s u r e  He  changes w i l l  o c c u r d u r i n g earthquake m o t i o n , i t would be p r e s e n t l y i m p o s s i b l e determine y i e l d a c c e l e r a t i o n s .  The y i e l d a c c e l e r a t i o n at any  to  instant  of time depends on the i n s t a n t a n e o u s s t r e n g t h , w h i c h i n t u r n depends pn the e f f e c t i v e s t r e s s e s and hence the t o t a l s t r e s s e s minus the pore pressure. has  I t i s the f a c t t h a t the s t r e n g t h cannot be p r e d i c t e d  that  l e d Seed to the c o n c l u s i o n t h a t the d i s p l a c e m e n t s cannot be  estimated  by the method o u t l i n e d by Newmark.  Seed s u g g e s t s , i n s t e a d , t h a t the dynamic s t r e s s e s w i t h i n e a r t h s t r u c t u r e can be e s t i m a t e d .  the  I f these s t r e s s e s are a p p l i e d to  samples w h i c h have been l o a d e d w i t h the p r e - e a r t h q u a k e s t a t i c s t r e s s e s , t h e n s t r a i n s can be measured. the s t r u c t u r e are o b t a i n e d ,  U n l e s s the s t r a i n s a t a l l p o i n t s w i t h i n  the d i s p l a c e m e n t s cannot be c a l c u l a t e d .  However, he s u g g e s t s t h a t an i n d i c a t i o n of the magnitude of the placements may  be o b t a i n e d  -  from a knowledge of the s t r a i n s on  a  dis-  - 20 potential slip  surface.  He proposes t h a t an e s t i m a t e  o f t h e dynamic s t r e s s e s can be  obtained! i n t h e f o l l o w i n g manners 1)  Determine.. •inertiarf^rces.-'.w&thiji- -the- dam ..due. t o a g i v e n  earthquake u s i n g v i s c o e l a s t i c r e s p o n s e 2)  theory.  I d e a l i z e t h e i n e r t i a f o r c e i n t o a number o f s i g n i f i c a n t c y c l e s represented  by a "maximum e q u i v a l e n t  seismic  c o e f f i c i e n t " (Chapter 4.2) a t a p a r t i c u l a r f r e q u e n c y . 3)  Determine t h e . s t a t i c s t r e s s e s on a p o t e n t i a l s l i p  sur-  face, u s i n g t h e s l i p s u r f a c e a n a l y s i s proposed by T a y l o r and Lowe (1959) and e f f e c t i v e s t r e n g t h p a r a m e t e r s . 4)  Determine the. maximum dynamic, s t r e s s e s on t h e p o t e n t i a l s l i p s u r f a c e by i n c o r p o r a t i n g a h o r i z o n t a l f o r c e system designated  by. t h e "maximum-equivalent s e i s m i c  i n t o the a n a l y s i s o f Step 3. assumption w i t h regard  coefficient"  T h i s w i l l r e q u i r e an  t o the strength of the m a t e r i a l .  S i n c e t h e p o r e , p r e s s u r e cannot p r e s e n t l y be p r e d i c t e d , the s t r e n g t h must•be determined from u n d r a i n e d t e s t s . Seed appears t o suggest t h a t t h e " s t a t i c " u n d r a i n e d s t r e n g t h o f the m a t e r i a l be used f o r t h i s purpose. The  philosophy  p r e s e n t e d by Seed i s that.dynamic s t r e s s e s can  be c a l c u l a t e d and t h e n a p p l i e d t o a p p r o p r i a t e t o be d e t e r m i n e d .  samples a l l o w i n g s t r a i n s  However, i n p r a c t i c e he s u g g e s t s t h a t i t i s more  c o n v e n i e n t t o f i r s t d e t e r m i n e t h e dynamic s t r e n g t h o f t h e m a t e r i a l a t p o i n t s a l o n g a p o t e n t i a l s l i p s u r f a c e j and t o then p e r f o r m a s t a b i l i t y a n a l y s i s i n c o r p o r a t i n g an earthquake f o r c e r e p r e s e n t e d  by a  - 21 -  "maximum e q u i v a l e n t  seismic c o e f f i c i e n t " .  This allows a f a c t o r of  s a f e t y a g a i n s t earthquake i n s t a b i l i t y t o be determined,, s t r e n g t h used i n t h e a n a l y s i s may a b r u p t f a i l u r e or i t may  The dynamic  be t h a t w h i c h c o r r e s p o n d s t o an  c o r r e s p o n d t o some l i m i t i n g s t r a i n c o n d i t i o n . \  I n t h e l a t t e r case t h e f a c t o r o f s a f e t y as c a l c u l a t e d by Seed c o u l d t h e n be i n t e r p r e t e d as a f a c t o r o f - s a f e t y a g a i n s t the o c c u r r e n c e of undesirable s t r a i n s . The w r i t e r f e e l s t h a t s i n c e the proposed method of a n a l y s i s a c t u a l l y i n v o l v e s the pre-determination  of s t r e n g t h , t h e n , i n f a c t ,  y i e l d a c c e l e r a t i o n s c o u l d be d e t e r m i n e d .  Therefore,  displacements  c o u l d be d e t e r m i n e d by the p r i n c i p l e s o u t l i n e d by Newmark.  The  s t r e n g t h c o u l d be e x p e c t e d t o v a r y w i t h t h e number of s t r e s s c y c l e s and w i t h s t r a i n and hence the y i e l d a c c e l e r a t i o n would v a r y , but so l o n g as i t i s a known f u n c t i o n , t h i s v a r i a t i o n would p r e s e n t l i t t l e problem i n a numerical  analysis.  The v i s c o e l a s t i c r e s p o n s e of e a r t h s t r u c t u r e s to earthquake f o r c e s has, u n t i l 1966, g e n e r a l l y been determined u s i n g the i n f i n i t e s m a l s h e a r - s l i c e method w h i c h i s d i s c u s s e d  i n d e t a i l i n Chapter 4.  More  r e c e n t l y , however, t h e f i n i t e element method of a n a l y s i s , a l s o i n Chapter 4, has been used f o r t h i s purpose.  discussed  Clough and Chopra  used t h i s method t o d e t e r m i n e d y n a m i c • s t r e s s e s i n e a r t h dams.  (1966)  Finn  (1966a) has proposed t h a t the dynamic s t r e s s e s c a l c u l a t e d from the f i n i t e element a n a l y s i s s h o u l d be a p p l i e d d i r e c t l y t o samples of e a r t h s t r u c t u r e w h i c h have been p r e l o a d e d w i t h the p r e - e a r t h q u a k e s t r e s s e s , and dynamic s t r a i n s measured. t h e s e dynamic s t r a i n s to t h e i r a s s o c i a t e d  static  He suggests t h a t by r e l a t i n g f i n i t e e l e m e n t s , t h e changes  - 22 i n the embankment c o n f i g u r a t i o n d u r i n g earthquake may  be  obtained.  The method of a n a l y s i s proposed by F i n n (1966a) d i f f e r s  sig-  n i f i c a n t l y from t h a t proposed by Seed (1966) i n t h a t F i n n s u g g e s t s t h a t t h e dynamic  s t r e s s e s be c a l c u l a t e d d i r e c t l y from a,dynamic a n a l y s i s .  Seed, on t h e o t h e r hand, proposed t h a t o n l y the i n e r t i a  forces  should be c a l c u l a t e d from the dynamic a n a l y s i s and dynamic then o b t a i n e d  stresses  using a s l i p c i r c l e a n a l y s i s which a l l o w s n o n - e l a s t i c  t r a n s f e r of l o a d .  T h u s , . a l t h o u g h the method proposed by Seed assumes  v i s c o e l a s t i c m a t e r i a l when c a l c u l a t i n g dynamic f o r c e s , i t a l l o w s e l a s t i c t r a n s f e r of l o a d when c a l c u l a t i n g the dynamic  non-  stresses.  I f the dynamic s t r e s s e s can be. a c c u r a t e l y d e t e r m i n e d , t h e n the methods o u t l i n e d by Seed (1966) and F i n n (1966a) a r e always However, t h e dynamic  appropriate.  s t r e s s e s depend g r e a t l y on the amount o f damping  or energy d i s s i p a t e d w i t h i n t h e v i b r a t i n g system. depends on the amount of n o n - e l a s t i c a c t i o n .  I f s t r a i n s tend t o be  l a r g e , then p l a s t i c a c t i o n w i l l be h i g h , c a u s i n g w h i c h i n t u r n t e n d s t o reduce s t r e s s e s .  This i n turn  l a r g e damping  Thus s t r e s s and s t r a i n a r e  i n t i m a t e l y r e l a t e d and t o attempt t o d e t e r m i n e s t r e s s e s and  forces  w i t h o u t c o n s i d e r i n g s t r a i n s and t o then use t h e s e s t r e s s e s t o o b t a i n s t r a i n s seems u n d e s i r a b l e . Finn  (1966b) proposed a method f o r o b t a i n i n g the r e s p o n s e , o f a  continuum t o a u n i f o r m a c c e l e r a t i o n .  The method a l l o w s p l a s t i c but  not e l a s t i c d e f o r m a t i o n s t o o c c u r t h r o u g h o u t . t h e r e g i o n .  I t appears  t h a t i t would be p o s s i b l e t o adapt t h i s method t o a l l o w the v i s c o p l a s t i c r e s p o n s e of an e a r t h s t r u c t u r e t o be o b t a i n e d .  However, t o the  -  23  w r i t e r ' s knowledge, t h i s has not been done. An e a r t h s t r u c t u r e responds e s s e n t i a l l y i n a v i s c o e l a s t i c manner t o s m a l l s c a l e v i b r a t i o n s .  However, f o r l a r g e s c a l e v i b r a t i o n s ,  c o r r e s p o n d i n g to s t r o n g m o t i o n e a r t h q u a k e s , c o n s i d e r a b l e  plastic  can be expected (Ambraseys I 9 6 0 ) ,  action  The  amount of p l a s t i c  o c c u r r i n g i n a g i v e n s t r u c t u r e f o r a g i v e n earthquake w i l l be a  action  function  of the dynamic s t r e s s ~ s t r a i n c h a r a c t e r i s t i c s o f the m a t e r i a l , Veletsos  and Newmark (1960) have shown t h a t p l a s t i c a c t i o n may  m o d i f y the dynamic r e s p o n s e of a s t r u c t u r e and  hence the w r i t e r con-  s i d e r s i t e s s e n t i a l t h a t p l a s t i c a c t i o n be c o n s i d e r e d dynamic a n a l y s i s of an e a r t h  greatly  i n any r a t i o n a l  structure,  P e n z i e n (1960) p r e s e n t e d a method f o r the dynamic a n a l y s i s of a s t r u c t u r e which allowed  p l a s t i c d i s p l a c e m e n t s to o c c u r i n a d d i t i o n  to e l a s t i c displacements,  P e n z i e n m o d e l l e d the s t r u c t u r e by a number  of p o i n t masses connected by i n t e r - s t o r e y shear s p r i n g s and The  dashpots.  i n t e r - s t o r e y shear s p r i n g s were assumed t o have a s t i f f n e s s  c o r r e s p o n d i n g to the shear s t i f f n e s s of the i n t e r - m a s s m a t e r i a l . such the model i s s i m i l a r t o t h a t o f the "shear-beam" or s l i c e " method which i s d i s c u s s e d sidered  i n Chapter 4,  greater  "shear-  However, P e n z i e n c o n -  the shear s p r i n g s t o have a f o r c e l i m i t c o r r e s p o n d i n g t o  y i e l d s t r e s s of the i n t e r - m a s s m a t e r i a l . t h a n those r e q u i r e d  As  the  Inter-mass d e f o r m a t i o n s  t o cause y i e l d i n g would cause no i n c r e a s e  the shear s p r i n g f o r c e , . The method a l l o w s the dynamic response i n terms of d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n of m a s s . p o i n t s t o c a l c u l a t e d a t any acceleration.  times, t , f o r a, s t r u c t u r e s u b j e c t e d  be  t o a known base  in  -  - 24  P e n z i e n (1960) was  concerned w i t h the response o f s t e e l and  c o n c r e t e s t r u c t u r e s r a t h e r than s t r u c t u r e s comprised o f a continuum m a t e r i a l such as a . s o i l .  However, t h e model proposed by P e n z i e n can  be' c o n s i d e r e d . t o . a p p l y to any s t r u c t u r e p r o v i d e d the l i m i t a t i o n s on d i s p l a c e m e n t s and s t r a i n s a r e c o n s i d e r e d to be t o l e r a b l e f o r the structure in.question. S i n c e the d i s p l a c e m e n t f i e l d i s l e s s r e s t r i c t e d i n the f i n i t e element method of a n a l y s i s i n comparison to the " s h e a r - s l i c e " method, i t would be d e s i r a b l e t o i n t r o d u c e p l a s t i c i t y i n t o method.  this  However, i n t r o d u c i n g p l a s t i c i t y i n t o a f i n i t e element  dyn-  amic a n a l y s i s i s a complex problem and to" date (1967) no such a n a l y s i s has been p r e s e n t e d to the w r i t e r ' s knowledge.  T h i s t h e s i s i s con-  cerned w i t h i n t r o d u c i n g p l a s t i c i t y i n t o the " s h e a r - s l i c e " method and i s e s s e n t i a l l y an e x t e n s i o n o f the method of a n a l y s i s p r e s e n t e d by P e n z i e n (1960). To i n v e s t i g a t e the e f f e c t of p l a s t i c i t y on r e s p o n s e , v i s c o e l a s t i c and e l a s t i c - v i s c o p l a s t i c responses w i l l be compared.  Conse-  q u e n t l y , i t i s e s s e n t i a l t h a t v i s c o e l a s t i c response t h e o r y be  fully  u n d e r s t o o d and t h i s w i l l f i r s t be r e v i e w e d i n Chapter 4.  -  - 25 CHAPTER 4 REVIEW OF VISCOELASTIC RESPONSE THEORY 4.1  INTRODUCTION An e a r t h s t r u c t u r e i s a t h r e e d i m e n s i o n a l continuum  composed  of n o n - i s o t r o p i c , non-homogeneous and n o n - e l a s t i c m a t e r i a l .  The  complete a n a l y s i s o f the dynamic response of such a s t r u c t u r e to earthquake e x c i t a t i o n appears to be beyond the p r e s e n t (1967) a b i l i t i e s of the b e s t d i g i t a l computers.  cap-  I n t h i s c h a p t e r the d i s -  c u s s i o n w i l l be l i m i t e d to the assumption o f v i s c o e l a s t i c m a t e r i a l . Each p o i n t i n an e a r t h s t r u c t u r e s u b j e c t e d to earthquake f o r c e s has t h r e e degrees o f freedom.  S i n c e a s t r u c t u r e i s comprised  of an i n f i n i t e number o f p o i n t s , i t has an i n f i n i t e number of degrees of  freedom.  The dynamic response o f such a s t r u c t u r e , even f o r  e l a s t i c m a t e r i a l , i s a complex problem a n d . r e q u i r e s f u r t h e r assumptions . The f i r s t m a t h e m a t i c a l t r e a t m e n t o f the dynamic response of an e a r t h dam reduced the problem to a o n e - d i m e n s i o n a l form. s t r u c t u r e was  The  c o n s i d e r e d t o be of t r i a n g u l a r c r o s s - s e c t i o n and o f  i n f i n i t e length.  The d i s p l a c e m e n t s were assumed to be h o r i z o n t a l  o n l y , and due t o d i s t o r t i o n a l s t r a i n s .  These r e s t r i c t i o n s l e a d to a  s t r u c t u r e w h i c h can be m o d e l l e d by the " i n f i n i t e s m a l shear s l i c e t h e o r y " and t h i s t h e o r y w i l l be d i s c u s s e d i n S e c t i o n  4.2.  I f an e a r t h s t r u c t u r e can be c o n s i d e r e d to comprise of a number of f i n i t e s i z e d elements r a t h e r than a continuum, then the  - 26 response i n w h i c h b o t h l o n g i t u d i n a l and s h e a r i n g s t r a i n s a r e a l l o w e d can be determined.  I n a dynamic a n a l y s i s , the m a t h e m a t i c a l model of  such a s t r u c t u r e reduces to a f i n i t e  number o f p o i n t masses i n t e r -  connected by s p r i n g s , i . e . , a lumped parameter system. response o f such a system w i l l be c o n s i d e r e d i n S e c t i o n  The dynamic 4.3.  The s o l u t i o n o f dynamic problems r e q u i r e s the i n t e g r a t i o n of the  e q u a t i o n s of m o t i o n o f the system.  coupled.  These e q u a t i o n s a r e g e n e r a l l y  They can be s o l v e d by "normal mode t h e o r y " w h i c h i n v o l v e s  f i r s t u n c o u p l i n g t h e e q u a t i o n s and t h i s w i l l be c o n s i d e r e d i n S e c t i o n 4.4.  They may  a l s o be s o l v e d by d i r e c t i n t e g r a t i o n o f the  c o u p l e d e q u a t i o n s and t h i s w i l l be c o n s i d e r e d i n S e c t i o n  4.2  4.5.  INFINITESMAL SHEAR SLICE THEORY The f i r s t dynamic response a n a l y s i s of an e a r t h dam was  peri-  formed by Mononobe, T a k a t a and Matumura (1936) and i n v o l v e d the following  assumptions:  1)  The dam  c o n s i s t s of an i n f i n i t e l y  long uniform  t r i a n g u l a r s e c t i o n r e s t i n g on r o c k . 2)  Homogeneous l i n e a r - e l a s t i c m a t e r i a l .  3)  Only shear s t r a i n s o c c u r .  4)  The shear s t r e s s i s u n i f o r m on h o r i z o n t a l p l a n e s .  5)  The i n f l u e n c e of s t o r e d water i s n e g l i g i b l e .  A s e c t i o n o f such a dam  i s shown i n F i g . 5a.  The f o r c e s a c t i n g  on a t y p i c a l i n f i n i t e s m a l element due t o earthquake m o t i o n a r e shown i n F i g . 5b. acceleration.  The i n e r t i a f o r c e i s the mass m u l t i p l i e d by the a b s o l u t e For dynamic e q u i l i b r i u m i t i s r e a d i l y determined t h a t  -  27 -  ,  Fj  =  Inertia  Q  -  Shear  C  =  Coefficient  />U  ^  +  5 y  c  Force = Force  Damping  :  When  °  Force u  -  |  bt From h  FIG.  5b  FORCES  ON  I N F I N I T E S M AL  i  +  Equilibrium  °  6t  SLICE  by  ° y  2  paydy ~~^f~  = 0t y T  Of  6 Un  x y  = a y  G-2  Damping  =  Per'llnit  Mass  cpa  - 28 a u 2  a  7T7~ 3t^  where  , 3u G ! 3 u + c — = — { r—7 9t p 3y^  H  1  u  1  z  y  T3u— ) 1  d y  ,  (4.1).  1  =  d i s p l a c e m e n t i n x d i r e c t i o n r e l a t i v e to the ground  =  displacement i n x d i r e c t i o n  t  =  time  p  =  mass d e n s i t y o f the m a t e r i a l  G  =  shear modulus o f t h e m a t e r i a l  c  =  damping f o r c e p e r u n i t mass f o r u n i t v e l o c i t y = u and Eq. 4.1  I n t h e case o f f r e e v i b r a t i o n s , u  becomes  a 2  2  3u  3*  .3u  + c  St  -  G  p  ,3 u , H u d y  -  .  7  (  //  ^  o\  7a7  }  The s o l u t i o n of t h i s p a r t i a l d i f f e r e n t i a l e q u a t i o n i s o b t a i n e d  by  s e p a r a t i o n of v a r i a b l e s and f o r the boundary c o n d i t i o n s u = 0 when 3u y = h.and — = 0 when y = 0, i s a y  \ -A OJ t u ( y , t ) = \e {A S i n u> / l - A t + B C O S O J / l - A t } j (g ^ ) / n n n n n n o n h n  n  z  z  l  J  (4.3)  n=l where  J 8 CJ  and A  n  A and B  o n n n n  =  B e s s e l f u n c t i o n o f t h e f i r s t k i n d and o r d e r  =  the n r o o t s o f the e q u a t i o n J  =  3 7-^- /G, , the n n a t u r a l frequencies h /p  =  c/2co , the f r a c t i o n o f c r i t i c a l damping n'  (6 ) = 0 o n o f the system  a r e c o n s t a n t s which a r e determined from the i n i t i a l v e l o c i t y  and d i s p l a c e m e n t o f the system  - 29 The  term.J  o  (B if-) g i v e s nh  t h e mode shapes o f t h e system,  These a r e a c h a r a c t e r i s t i c o f a g i v e n s t r u c t u r e and a r e independent of t h e f o r c i n g f u n c t i o n . properties which  They p o s s e s s t h e f o l l o w i n g o r t h o g o n a l  a r e u s e f u l when c o n s i d e r i n g  1  forced v i b r a t i o n s ,  h /pay o  J (S £) 0  n  J ( B ^ ) dy = 0 0  m  = 1/2 {pah  where  = Bessel  2  J! (6 )} 2  n  ^  n  m = n  f u n c t i o n o f t h e f i r s t k i n d and f i r s t  (  ^  4  )  order.  When earthquake m o t i o n i s c o n s i d e r e d and t h e ground moves, Eq.  4.1 can be w r i t t e n a  3u  ^  2  3u  G, r 3 u ^ 1 2  3u  T  where u a  =  u + u g  and  =  t h e ground d i s p l a c e m e n t  u g  Making use o f t h e o r t h o g o n a l p r o p e r t i e s shown t h a t f o r i n i t i a l c o n d i t i o n s Eq.  2 e  (4.5)  o f t h e mode shapes i t can be  equal zero, the general s o l u t i o n of  4.5 i s  r  -X w (t-x)  u (T) e ~  n  n  Sin u v/TT (t-T)dx n n 2  (4.6) where = x  a dummy i n t e g r a t i o n v a r i a b l e  I n the case o f z e r o damping i t can be shown a f t e r s e p a r a t i o n o f variables  that  - 308 u 2  where u an-  =  an _  2  the absolute displacement  i n the n  r  th  mode th  and  u^  =  t h e d i s p l a c e m e n t w i t h r e s p e c t t o t h e base i n t h e n mode  E q u a t i o n 4.7 may a l s o be c o n s i d e r e d t o be a r e a s o n a b l e app r o x i m a t i o n f o r s m a l l v a l u e s o f damping. -Thus from Eq. 4.6 assuming co / l - A = oo , t h e a b s o l u t e a c c e l e r a t i o n i n any mode i s g i v e n by n n n' z  3  Jo(S*) "an^^  '  " ^nT^F)  -Vn V0  j  *  (  t  ~  T  6  3  )  S i n ^ ( t - x ) dx  ° and U ( y , t ) = 7. = ii ( y , t ) a .. a n n=l n=c  3  L  w  I t i s more c o n v e n i e n t  n  2 J o ( e  where  4> (y) = n  n  (  n  1  (h q^i ^'* }  t o w r i t e Eq. 4.8 i n t h e f o l l o w i n g form  u ( y , t ) = u) * ( y ) V ( t ) a n  (4.8)  n  n  (4.10)  ^ )  « n  }  p  t —A co ( t - x ) and V ( t ) = / i i (x) e S i n co ( t - x ) d x n g n o n  n  J  5  The  term ^ ( y ) e x p r e s s e s  t h e modal p a r t i c i p a t i o n and t h e d i s t r i b u t i o n  o f d i s p l a c e m e n t , v e l o c i t y and a b s o l u t e a c c e l e r a t i o n w i t h i n t h e dam. The  i n t e g r a l V ( t ) i s c a l l e d t h e Duhammel o r c o n v o l u t i o n i n t e g r a l .  I t s v a l u e , w h i c h has t h e u n i t s o f v e l o c i t y i s dependent on f o u r  - 31 -  p a r a m e t e r s , namely: 1)  t h e ground m o t i o n c h a r a c t e r i s t i c , u  2)  t h e n a t u r a l f r e q u e n c y , UJ  3)  the f r a c t i o n of c r i t i c a l  4)  the time, t  damping,  I n dynamic a n a l y s e s o f s t e e l and c o n c r e t e s t r u c t u r e s i t has been common p r a c t i c e t o d e s i g n f o r l a t e r a l f o r c e s deduced from i n e r t i a f o r c e s based on maximum a b s o l u t e a c c e l e r a t i o n s . ing  Design i n v o l v -  maximums has l e a d t o t h e concept o f s p e c t r a l a n a l y s i s . The maximum a b s o l u t e a c c e l e r a t i o n i n any mode w i l l o c c u r when  V (t) n  i s a maximum.  F o r any g i v e n ground m o t i o n  v n  ( t ) w i l l have a  maximum w h i c h i s r e f e r r e d t o as t h e s p e c t r a l v e l o c i t y , S  .  A  r e l a t i o n s h i p between t h e s p e c t r a l v e l o c i t y and t h e n a t u r a l f r e q u e n c y for  v a r i o u s p e r c e n t a g e s o f c r i t i c a l damping can be determined f o r any  one earthquake.  Such a r e l a t i o n s h i p f o r t h e N-S component o f E l  Centro earthquake i s shown i n F i g . 6.  T h i s a l l o w s t h e maximum a b s o l u t e  a c c e l e r a t i o n s i n each mode t o be determined from  u  an  (y) max  = 'o> 4> (y) S n n vn  J  J  (4.11)  From t h e d i s t r i b u t i o n o f a c c e l e r a t i o n s determined from Eq. 4.11 t h e maximum i n e r t i a f o r c e i n any mode can be determined f o r any element of w e i g h t , AW, by t h e f o l l o w i n g e x p r e s s i o n (Al  n  )max = AW ii (y) — an max g  where g  =  the a c c e l e r a t i o n of g r a v i t y  (4.12)  - 32 -  FIG.  6  VELOCITY MAY  16,  SPECTRUM 1940.  FOR  EL  COMPONENT  CENTRO, N-S.  CALIF.  E A R T H Q U A K E ,  - 33 o r , i f t h e i n e r t i a f o r c e i s e x p r e s s e d i n terms o f a s e i s m i c c o e f f i c i e n t , k, then  (Al where  ) n max  = AW k (y) n  '  J  (4.13)  k (y) = — u (y) nV g an max J  However, i n g e n e r a l , maximum responses i n d i f f e r e n t modes w i l l o c c u r a t d i f f e r e n t . t i m e s , so t h a t d i r e c t s u p e r p o s i t i o n o f maximum v a l u e s i n each mode may be i n v a l i d .  Ambraseys (1960) s u g g e s t e d t h a t t h e  l a t e r a l f o r c e s a c t i n g on a dam c o u l d be e x p r e s s e d by a s t a t i c f o r c e w i t h magnitude  determined by a s e i s m i c c o e f f i c i e n t e v a l u a t e d by one  o f t h e f o l l o w i n g methods: 1.  The s e i s m i c c o e f f i c i e n t a t any depth c o u l d be t a k e n as t h e square r o o t o f t h e sum o f t h e squares o f t h e s e i s m i c c o e f f i c i e n t s f o r peak response i n each o f t h e f i r s t f o u r modes, i . e .  k(y) =  n=4  I  n=l  2.  2 1/2 {k ( y ) l  (4.14)  The s e i s m i c c o e f f i c i e n t a t any depth c o u l d be t a k e n as t h e maximum v a l u e a t t h a t depth f o r any one o f t h e modal d i s t r i b u t i o n s , i . e . ,  k  ^  =  max  S p e c t r a l a n a l y s i s has been concerned w i t h p r e d i c t i n g t h e  ( 4  -  1 5 )  34  -  -  maximum i n e r t i a f o r c e s t h a t the s t r u c t u r e i s l i k e l y to be s u b j e c t e d t o under a g i v e n earthquake.  Ambraseys has suggested  that  these  i n e r t i a f o r c e s can be used i n d e s i g n by i n c o r p o r a t i n g them i n the s t a b i l i t y a n a l y s i s proposed by Bishop  (1955), thus a l l o w i n g a f a c t o r  o f s a f e t y a g a i n s t f a i l u r e due to earthquake (Chapter  calculated  3).  I n Chapter way  t o be  3 i t was  argued t h a t f a c t o r o f s a f e t y i s a poor  t o a s s e s s the s t a b i l i t y o f an e a r t h s t r u c t u r e .  I t was  proposed,  i n s t e a d , t h a t the performance s h o u l d be e v a l u a t e d i n terms of d i s placements caused by the earthquake. an i n d i c a t i o n o f the d i s p l a c e m e n t  Seed (1966) has suggested  o f a p o t e n t i a l s l i d i n g mass can  o b t a i n e d i f the dynamic s t r e s s h i s t o r y on the p o t e n t i a l s u r f a c e i s known (Chapter 3 ) .  that :  sliding  For the purposes o f a n a l y s i s he  posed t h a t the e f f e c t of the earthquake  be  pro-  on a p o t e n t i a l s l i d i n g mass  can be r e p r e s e n t e d by a number o f s i g n i f i c a n t a c c e l e r a t i o n p u l s e s of c o n s t a n t magnitude a t some f r e q u e n c y .  An e s t i m a t e o f t h e s e p u l s e s  can be o b t a i n e d by d e t e r m i n i n g "average dynamic s e i s m i c c o e f f i c i e n t s " from the e l a s t i c shear s l i c e t h e o r y (Seed and M a r t i n 1966).  The  method o f o b t a i n i n g t h e s e average dynamic c o e f f i c i e n t s i s d i s c u s s e d below. The a b s o l u t e a c c e l e r a t i o n w i t h i n a dam  i s g i v e n by Eq.  4.9,  namely n=°° U (y,t) = I n=l a  where  u  U  (y a n  '  t}  ( y , t ) = ui <>j (y) V ( t ) ann n n  (4.16)  J  Here maximums i n each mode are not b e i n g summed as was  done f o r the  - 35 case of spectral.analysis but rather the accelerations in each mode at a given.time, t, are added.  Using these accelerations the variation  in the lateral force with time can be estimated. If a potential failure surface such as that shown in Fig. 7a is considered, then the lateral inertia force, F (t), acting on the body due to earthquake motion i s i=n F(t) = I m.ii .(t) . -, i ai  .  1=1  (4.17)  :  The lateral force, F(t), could be evaluated i n this manner, but i f the sliding surface i s idealized into a wedge that extends to the crest of the dam, then the calculation i s greatly simplified.  Such a  wedge i s shown (dashed lines) i n Fig. 7b and the forces acting on a horizontal slice of mass, nu, as shown,in Fig. 7c. The equation of dynamic equilibrium for the element  is  m.ii. (t) = (Q. - Q ) - C.u (t) i ai i l - i il where  (4.18)  Q. i x  =  the shear force on the base of the i*"* slice  Q_^_^  =  the shear force on the top of the i  =  the damping coefficient = c nu  =  damping coefficient per unit mass  c  1  slice  The lateral force, F(t), i s given by  F(t) = i=l  ^.(t) = \ \ - Q i=l  - I ^ C.u.  (4.19)  1=1  i=n (Q.- Q. ,) = (Q - Q ) and i f the wedge extends to the top of now Ly x i - l n o i=n  - 36 -  Acceleration Distribution at Time , t  i=n F(t)=  2  i=l  mjU-itt)  Fig, 7a  Fig. 7b  Earthquake Force on Potential  Earthquake  Force  Cylindrical  on Idealized Wedge  'I-I  mj u (t) Qi  Fig.7c  7Forces  Cj  on Element  Uj(t)  Sliding Surface  -37  the dam,  then 0 = 0 o  and  -  the l a t e r a l f o r c e i s  i=n F(t) = Q where Q  n  -I  n  . ,  L  1=1  cm.u. l l  (4.20)  i s the shear f o r c e on the base of the wedge w h i c h i t i s b  more c o n v e n i e n t to r e f e r t o as F f o r c e , F^,  a p p l i e d to the wedge.  F ( t ) = (F  determining  1=1  i s the damping  . - . 1 1  So t h a t Eq.  4.20  becomes  + F.) d  s Seed and M a r t i n  S  i=n and - £ cm.u.  now  (4.21)  (1966) d i d not  the f o r c e F ( t ) .  consider  The  the damping f o r c e , F^, when  error involved i n neglecting  the  damping term i s not of the same o r d e r as t h a t i n v o l v e d i n o b t a i n i n g the a b s o l u t e  acceleration, U  , from the r e l a t i v e d i s p l a c e m e n t , u , an n assumed t h a t the r e l a t i o n s h i p was r  by Eq.  4.7. ii  I t was  This equation = - oj u n n 2  an  mentioned p r e v i o u s l y t h a t t h i s e x p r e s s i o n  the case of zero damping.  i s only correct for  When damping i s p r e s e n t ,  Newmark (1965) terms the p s e u d o - a c c e l e r a t i o n .  The  i t y i e l d s what actual  expression  f o r a c c e l e r a t i o n which i n c l u d e s damping i s ii  an  = - o/u - c u n n n  (4.22)  which i s seen to i n v o l v e a damping term. The  d e r i v a t i o n of the e x p r e s s i o n  c o e f f i c i e n t based on the e x p r e s s i o n  f o r the dynamic s e i s m i c  f o r a c c e l e r a t i o n g i v e n by Eq.  r a t h e r than on the p s e u d o - a c c e l e r a t i o n  i s p r e s e n t e d i n Appendix I .  4.22  - 38 I t i s shown t h a t the average dynamic s e i s m i c c o e f f i c i e n t f o r a wedge e x t e n d i n g from the apex o f the dam  t o a d e p t h , y, i s g i v e n by  n=°°  }  (4.23)  The second term i n t h e b r a c k e t r e p r e s e n t s the e f f e c t of the damping force.  I t w i l l i n f l u e n c e the v a l u e o f the dynamic s e i s m i c co-  e f f i c i e n t at a l l times. V a l u e s o f the average s e i s m i c c o e f f i c i e n t f o r a g i v e n wedge i n a dam  s u b j e c t e d to a p a r t i c u l a r earthquake  u s i n g Eq. 4.23. i n t h i s equation.  I t may  c o u l d be e v a l u a t e d  be seen t h a t the base w i d t h i s not  contained  Thus a l l . w e d g e s w i t h t h e i r bases a t a g i v e n l e v e l  w i l l have the same v a l u e s of average s e i s m i c c o e f f i c i e n t .  Seed and  M a r t i n (1966), n e g l e c t i n g the damping term i n v o l v e d i n . E q . 4.23 u s i n g the N-S  component o f E l Centro earthquake,  s e i s m i c c o e f f i c i e n t v e r s u s time r e l a t i o n s h i p s v a r i o u s depths f o r a.number of d i f f e r e n t dams. can be used i n d e s i g n i n the manner suggested cussed i n Chapter 4.3  and  o b t a i n e d average  f o r wedges.extending to These  relationships  by Seed (1966) and  dis-  3.  LUMPED PARAMETER SYSTEM The e q u a t i o n s o f m o t i o n o f an i d e a l i z e d l i n e a r e l a s t i c  s t r u c t u r e i n m a t r i x form a r e TMJ  (u} + [C] {u} + [K] {u} =  jf(t)}  (4.24)  - 39  where fMj  =  the d i a g o n a l mass m a t r i x  [C]  =  the damping m a t r i x  [K]  =  the s t i f f n e s s m a t r i x  {u}  =  the d i s p l a c e m e n t of the system r e l a t i v e t o the ground  {f(t)} I n the i d e a l i z e d a)  =  the f o r c e a c t i n g on the system  s t r u c t u r e the f o l l o w i n g assumptions a r e made  The s t r u c t u r e can be t r e a t e d as a lumped parameter system, i . e . , - i t can be r e p r e s e n t e d by p o i n t masses connected by s p r i n g s .  b)  The i n t e r - m a s s s p r i n g s behave i n a l i n e a r  elastic  fashion. c)  Damping i s o f the v i s c o u s t y p e .  The shear s l i c e approach d i s c u s s e d i n S e c t i o n 4.2 c o u l d a l s o be a n a l y z e d as a lumped parameter system i f the s l i c e s a r e c o n s i d e r e d t o be o f f i n i t e t h i c k n e s s r a t h e r than o f i n f i n i t e s m a l ness. slice.  thick-  The mass of each s l i c e c o u l d be lumped a t the c e n t r o i d of the These lumped p o i n t masses c o u l d then be c o n s i d e r e d t o be  connected by s p r i n g s .  The s t i f f n e s s o f t h e s e s p r i n g s c o u l d be  o b t a i n e d from the s h e a r s t i f f n e s s of the elements of s o i l between p o i n t masses.  T h i s t y p e o f lumped parameter system w i l l be used when  considering e l a s t i c - v i s c o p l a s t i c  response i n Chapter 5.  The s h e a r s l i c e method a l l o w s o n l y o n e - d i m e n s i o n a l motion to d i s t o r t i o n a l s t r a i n s .  due  The f i n i t e element method o f a n a l y s i s , which  a l s o reduces t o a lumped parameter system, a l l o w s t w o - d i m e n s i o n a l  -  - 40 motion w i t h l o n g i t u d i n a l i n a d d i t i o n to d i s t o r t i o n a l s t r a i n s . method o f a n a l y s i s has been d e s c r i b e d by Clough (1965) and  This  was  f i r s t used t o a n a l y z e the dynamic response of e a r t h dams by Clough and Chopra  (1966) and F i n n (1966a) .  The b a s i c concept o f the f i n i t e  element p r o c e d u r e i s the  i d e a l i z a t i o n of a continuum by an assemblage of d i s c r e e t . e l e m e n t s interconnected at nodal p o i n t s . dimensional s t r e s s f i e l d ,  F o r the a n a l y s i s of a two-  r e c t a n g u l a r o r t r i a n g u l a r p l a t e elements  have g e n e r a l l y been used.  To m a i n t a i n c o m p a t i b i l i t y between edges of  a d j a c e n t e l e m e n t s , Clough (1966) assumed t h a t t h e d i s p l a c e m e n t s l i n e a r l y w i t h i n an element.  vary  On the b a s i s o f t h i s assumption i t i s  p o s s i b l e t o c a l c u l a t e t h e s t i f f n e s s p r o p e r t i e s of the element, i . e . , the nodal force-displacement r e l a t i o n s h i p .  F i n a l l y , the s t i f f n e s s  m a t r i x , [K] , o f t h e complete s t r u c t u r a l assemblage i s o b t a i n e d by s u p e r p o s i n g t h e a p p r o p r i a t e s t i f f n e s s c o e f f i c i e n t s of the i n d i v i d u a l elements c o n n e c t i n g t o each n o d a l p o i n t . The mass m a t r i x , [ M ] , i s o b t a i n e d by assuming t h a t the mass of each element i s c o n c e n t r a t e d a t t h e n o d a l p o i n t s . elements Clough and Chopra  For t r i a n g u l a r  (1966) and F i n n (1966) lumped o n e - t h i r d o f  the mass o f each element a t i t s n o d a l p o i n t s . The damping m a t r i x , [ C ] , depends on the v i s c o u s damping p r o p e r t i e s o f the m a t e r i a l .  I n most dynamic problems a n . e s t i m a t e  o f the  p e r c e n t a g e o f c r i t i c a l damping thought t o be p r e s e n t i n the s t r u c t u r e can be made.  I n the normal mode t h e o r y p r e s e n t e d i n S e c t i o n 4.4 i t  w i l l be shown t h a t w i t h c e r t a i n a s s u m p t i o n s , the damping m a t r i x can be  - 41 e x p r e s s e d i n terms o f t h e p e r c e n t a g e o f c r i t i c a l damping p r e s e n t i n each mode. The system o f e q u a t i o n s . r e p r e s e n t e d by Eq. 4.24 may be s o l v e d by e i t h e r "normal mode t h e o r y " o r by " n u m e r i c a l a n a l y s i s o f c o u p l e d e q u a t i o n s " and t h e s e w i l l b e ' d i s c u s s e d i n S e c t i o n s 4,4 and  4.5. 4.4  NORMAL MODE THEORY E q u a t i o n 4.24 r e p r e s e n t s a system o f c o u p l e d l i n e a r  order equations.  second  F o r f r e e v i b r a t i o n s w i t h zero damping, t h i s  e q u a t i o n reduces t o  fMj  {ii} + [K] {u} = 0  (4:25)  or -1  {ii} + [M] [K] {u} = 0  (4.26)  For a , s i n g l e - d e g r e e o f freedom system, t h i s f u r t h e r reduces t o  ii + - u =. 0 m for which the s o l u t i o n i s u = A Sinwt + B c o s u t  (4.27)  where to = /k/m and A and B a r e c o n s t a n t s t o be determined  from i n i t i a l  conditions.  The p r i n c i p l e o f t h e normal mode t h e o r y i s t h a t by a s u i t a b l e f o r m a t i o n o f c o - o r d i n a t e s , Eq. 4.25 may be t r a n s f o r m e d i n t o n  trans-  - 42 -  u n c o u p l e d e q u a t i o n s o f the form o f Eq. 4.26,  the s o l u t i o n f o r w h i c h i s  known. U n c o u p l i n g o f . t h e e q u a t i o n s . o f m o t i o n o f the system c o r r e s p o n d s -1 to n o r m a l i z i n g o r d i a g o n a l i z i n g the m a t r i x [MJ [K] o f Eq. 4.26. problem o f d i a g o n a l i z i n g a m a t r i x i s a complex one. symmetric m a t r i x can always be d i a g o n a l i z e d .  The  However, a r e a l  A l t h o u g h the m a t r i x [K]  -1  i s symmetric the m a t r i x [M] [K] i s not i n g e n e r a l symmetric. the system o f e q u a t i o n s r e p r e s e n t e d by Eq. 4.25  However,  can be uncoupled by  the f o l l o w i n g p r o c e d u r e : -1/2 L e t {u} = [Ml {rj (4.28) I f Eq. 4.28 i s s u b s t i t u t e d i n Eq. 4.25 w h i c h i s then p r e m u l t i p l i e d by -1/2 [Ml , then -1/2 [M.] m  -1/2 -1/2 -1/2 [M] (n> + [Ml [ K l [M] {n>  = 0  (4.29)  w h i c h reduces to [I]{fi} + [K] {r,} = 0 where  _ -1/2 -1/2 _ [K] = M [K] M = [K] because  (4.30) T  [Kl i s symmetric.  S i n c e [K] = [ K ] t h e m a t r i x [K] i s t h e r e f o r e symmetric and can always T  be d i a g o n a l i z e d by a normal o r t h o g o n a l t r a n s f o r m a t i o n m a t r i x . Suppose [<))] i s such a m a t r i x , then l e t {n} = [<fr] {£}  (4.31)  T S u b s t i t u t i n g Eq. 31 i n Eq. 30 and p r e - m u l t i p l y i n g by [<j>] , Eq.  4.30  becomes [<j>] M {'0  + [<t»][Kl M U }  T  Since  (4.32)  [<f>] i s the m a t r i x t h a t d i a g o n a l i z e d [K] and remembering  T [<f>]  = 0  T  =  [<£]  where  1  s i n c e [<(>] i s normal o r t h o g o n a l , Eq. 4.32 reduces to ih  tU  that  + [w ]  U>  2  = 0  (4.33)  |> J = [<f>] [K] [<|>] 2  T  The s o l u t i o n w i l l be o b t a i n e d i n terms o f the normal c o o r d i n a t e s , {£}. However, i t i s the s p a t i a l c o - o r d i n a t e s , {u}, t h a t a r e r e q u i r e d .  The  r e l a t i o n s h i p between {u} and {£} i s -1/2 {u} = [M] [«)>] {£} = W  {E}  The n column v e c t o r s of the t r a n s f o r m a t i o n m a t r i x  (4.34) [ip] o f Eq. 4.34  give  the n c h a r a c t e r i s t i c mode shapes o f t h e system, w h i l e the n v a l u e s of co i n Eq. 4.33 r e p r e s e n t the n n a t u r a l f r e q u e n c i e s of the system. I f damping i s i n t r o d u c e d , t h e n Eq. 25 under the same t r a n s f o r m a t i o n becomes:  [ I ] (O  + [<t>]  T  -1/2 -1/2 . [M] [C][M] [<(,] (?) + [u 2 ]{U  I f the damping m a t r i x  = 0  (4.35)  [C] i s a l i n e a r c o m b i n a t i o n of the mass  and the s t i f f n e s s m a t r i x , t h e n as shown below, the damping m a t r i x be n o r m a l i z e d by the same t r a n s f o r m a t i o n t h a t n o r m a l i z e s matrix.  may  the s t i f f n e s s  T h i s i s a s u f f i c i e n t c o n d i t i o n but not a n e c e s s a r y  one.  Suppose  [C] = a[M] + 3[K]  (4.36)  Then, from Eq. 4.35 -1/2 -1/2 -1/2 [M] [C] [M] = [M]  {o[M]  -1/2 + B[K] }[M]  -1/2 -1/2 -1/2 -1/2 -1/2 -1/2 [M] [C] [M] = a[M] [M] [M] + g[M] [K] [M] -1/2 -1/2 ^ [M] [C] [M] = a [ I ] + B[K]  and  m m  T  ~  ~  1 / Z  [C]  T ,  1 / 2  [M]  [ + ]=.•[<!)]  {a[I] +  - ,  3[K]}'  [<j>]  -1/2 -1/2 [*] [M] [C] [M] [*] = a [ I ] + 0[to ] T  2  T h e r e f o r e Eq. 4.35 reduces t o [I]  {h + { a [ I ] +  B[co ]} {?} + 2  [a) ](U 2  = 0  Eq. 4.37 r e p r e s e n t s n uncoupled e q u a t i o n s , t h e i  (4.37) e q u a t i o n o f which  is E. + (a + Bco. ) E. + oof E = 0  (4.38)  2  1  1  1  1  o r i n s t a n d a r d form  E. +  l  where  2A.0J.E. + to  l i i  2  l  E. = 0  A^ i s t h e f r a c t i o n o f c r i t i c a l damping i n t h e i * " *  g i v e n by  (4.39)  l  1  mode and i s  Again the solution of Eq. 4.39 i s well known. The forced v i b r a t i o n system represented by Eq. 4.24  can be  uncoupled i n exactly the same manner and becomes [I] {'£} +  [2Xur {'0 ±  + [o»»] {?}  - [<|)] [MT T  1/2  {f(t)>  = {g(t)}  (4.41)  which can be solved by the Duhamel or superposition i n t e g r a l y i e l d i n g i  E  =  ' -X / g (T) e 0 1  X  (Aj/l-X^  where E^ i s the i ' *  1  U) X  (t-T)  Sin co A - X *  dt  (4.42)  modal displacement or normal co-ordinate.  Having obtained the E's the actual displacements {u} are determined from Eq. 3.5  4.34.  NUMERICAL ANALYSIS OF COUPLED EQUATIONS OF MOTION Numerical analysis of the coupled equations of motion have  been investigated by many researchers including Newmark (1959), Melin (1958) and Penzien (1960).  But f o r the general case, a method pro-  posed by Wilson and Clough (1962) appears to have advantage over other methods i n that i t i s a matrix approach applicable to both e l a s t i c and i n e l a s t i c systems.  The method proposed by Clough i s a step-by-  step matrix analysis of Eq. 4.24, wherein with known values of d i s placement, v e l o c i t y and acceleration at time, t - At, these same quantities can be determined at time t .  It i s assumed that the  acceleration i n the time i n t e r v a l , At, varies i n either a l i n e a r or parabolic manner.  If a l i n e a r v a r i a t i o n of acceleration i s assumed,  - 46 then t h e e x p r e s s i o n s f o r v e l o c i t y and d i s p l a c e m e n t a t time t a r e  H:  {  u  }  {  t  =  { u }  t-At  +  A t  {  a  t - A t  }  " t-At }  +  +  A  !  ¥  {  U  { U )  }  (4.43)  t  t-At  +  ¥ <  U  }  t  ^  from t h e dynamic e q u i l i b r i u m Eq. 4.22 -1 {U>  { f ( t ) } - [C]{u}  = tMj  t  t  (4.45)  - [K]{u}  s u b s t i t u t e f o r {u} and ( u > from Eqs. 4.43 and 4.44 fc  {ii}  t  = [F]  t  { f ( t ) } - [C] {a}  - [K] {b}  (4.46)  where -1 [F] = {[M]  {a  >  + f-  " »>t-At {  (4.47)  [K]}  [C]  (4.48)  f t-At  +  {U}  and {b} = { u } ^  +  At { u } _ t  A t  + ^-  {U} _ t  A t  (4.49)  S u b s t i t u t i n g Eqs. 4.48 and 4.49 i n Eqs* 4.43 and 4.44 t h e s e e q u a t i o n s reduce t o  {u}  {u>  t  t  =  {a} +Y"  =  {b} +  ^  t  ^ { t G o t  (4.50)  (4.51)  - 47 The solution proceeds as follows:  The i n i t i a l v e l o c i t i e s and d i s -  placements are given as the i n i t i a l conditions of the problem (these w i l l be zero f o r earthquake problems). obtained from Eq. 4.46 with t =» 0.  The i n i t i a l accelerations are  The step-by-step response i s  given by repeated applications of Equations 4.46,  4.50  and  4.51.  In c l a s s i c a l mode theory i t i s usual to assume that the damping  matrix i s a l i n e a r combination of the mass and s t i f f n e s s matrix,  that i s [C] o [ M ] 0  +  3 [K]  and t h i s can be substituted i n Eqs. 4.46  and  4.47.  For earthquake problems i t can be shown that the dynamic force f(t)  applied to each mass i s equal to minus the product of the mass  times the ground acceleration i n the d i r e c t i o n being considered.  Thus,  the force matrix becomes {f(t)> - - {m^ where  (t)}  (4.52)  u (t) • the ground acceleration i n the d i r e c t i o n being considered. 8  In general, an earthquake can be considered to comprise of three component accelerations, two horizontal and one v e r t i c a l .  Present  analyses are confined to plane s t r a i n problems, thus only two components can be considered.  I f two components are considered, Eq. 4.8 can be  written 0  l 0  m  m.  °2 (4.53)  -u (t) > gv <  {f(t)> - - u ( t ) g h  m m  - 48 where u , ( t ) ii  =  (t) = gv^  t h e h o r i z o n t a l ground t h e v e r t i c a l ground  acceleration  acceleration  Thus b o t h h o r i z o n t a l and v e r t i c a l components o f t h e earthquake can be c o n s i d e r e d i n t h i s method. The s t e p - b y - s t e p p r o c e d u r e can be a p p l i e d t o an "n" degree of freedom system, t h e r e f o r e i t can a l s o be a p p l i e d t o a s i n g l e degree o f freedom system, i . e . , i t c o u l d be used t o s o l v e t h e unc o u p l e d e q u a t i o n s o f Eq. 4.41 and t h i s was done by F i n n (1966a). I n t h e normal mode.theory matrix i s constant.  i t i s assumed t h a t t h e s t i f f n e s s  I f t h e m a t e r i a l i s n o n - l i n e a r , then t h e s t i f f n e s s  m a t r i x i s 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 s , and t h e r e f o r e . n o t c o n s t a n t . Thus, f o r n o n - l i n e a r m a t e r i a l t h e e q u a t i o n s o f m o t i o n cannot be u n i q u e l y uncoupled and t h e normal mode t h e o r y cannot be used.  The  method proposed by W i l s o n and Clough (1962), however, which does n o t i n v o l v e u n c o u p l i n g o f e q u a t i o n s c o u l d be used t o o b t a i n t h e response of n o n - l i n e a r m a t e r i a l . 4.6  COMPARISON OF THE SHEAR-SLICE AND FINITE ELEMENT METHODS OF ANALYSIS The v i s c o e l a s t i c response o f a 300 f t . h i g h t r i a n g u l a r dam  when s u b j e c t e d t o a base m o t i o n c o r r e s p o n d i n g t o t h e f i r s t t e n seconds of t h e N-S component o f t h e E l C e n t r o , 1940 earthquake was determined from b o t h a s h e a r s l i c e and a f i n i t e element a n a l y s i s . shown  i n F i g . 8a.  a modulus,  The dam i s  I t was assumed t o be homogeneous and i s o t r o p i c w i t h  E = 81,300 p . s . i . , a P o i s s o n ' s r a t i o u = 0.45 and a u n i t  co o:  40  40  30  30  20  CO  60'  =)IU=)ll(=)lll=)IIHIK=)lll=)IK=)lH= u, FIG.  8a  SECTION  OF  co  FIG.  DAM  T  =  3.1s  20 Sec; 10  400  300  200  100  0  DISTANCE  HOMOGENEOUS  = 5 ° /'o  10  «  JHHIK'J }  A  8b  DYNAMIC  100  FROM  200 IN  SHEAR 60  FINITE  ELEMENT  FT. F R O M  400  FT.  STRESS  PLANE  300  DISTRIBUTION  ON  BASE,  ANALYSIS.  o m  «  1.  > o  EL  2. X  <  CENTRO = 5 %  of  Scaled  O  w a> c o c o m v> a>  w  CO w a •  .c to o  6 o c  COMPARISON 0 - 1 0  SEC.  OF A V E R A G E N-S  DYNAMIC  COMPONENT  OF  SHEAR EL  STRESSES  CENTRO  ON A  PLANE  EARTHQUAKE..  60  FT.  ABOVE  THE  by  Critical.  BASE,  0.889.  -  weight y  =  130 p . c . f .  For the s h e a r s l i c e a n a l y s i s the dam  s i d e r e d t o comprise of 14 h o r i z o n t a l s l i c e s .  was  50  con-  T h i s method assumes the  dynamic shear s t r e s s t o be u n i f o r m on h o r i z o n t a l p l a n e s .  The  dynamic  s h e a r s t r e s s on a h o r i z o n t a l p l a n e at a h e i g h t of 60 f t . above the base of the dam  was  c a l c u l a t e d and  i s shown,as a s o l i d l i n e i n F i g . 8c.  I n the f i n i t e element a n a l y s i s the dam p r i s e d o f t r i a n g u l a r elements.  The  was  assumed to be com-  dynamic shear s t r e s s e s on  t a l p l a n e a t a h e i g h t of 60 f t . above the base of the dam a t e d as a f u n c t i o n of time. p l a n e at t i m e , t = 3.6  The  were c a l c u l -  shear s t r e s s d i s t r i b u t i o n on  sec. i s shown i n F i g . 8b.  I t may  this  be seen t h a t  dynamic shear s t r e s s e s are h i g h e s t a t the c e n t r e o f the dam a t the f a c e .  a,horizon-  Average dynamic shear s t r e s s e s were o b t a i n e d  and by  lowest inte-  g r a t i n g the s t r e s s o v e r t h e a r e a and t h e s e average s t r e s s e s are shown i n F i g . 8c as a dashed l i n e . dynamic shear s t r e s s e s o b t a i n e d s i m i l a r t o those o b t a i n e d I n S e c t i o n 4.2 i t was  I t may  also  be seen t h a t the average  by the f i n i t e element method a r e  quite  f o r the shear s l i c e method.  d e a l i n g w i t h the shear s l i c e method of a n a l y s i s  shown t h a t average dynamic s e i s m i c c o e f f i c i e n t s are  directly  r e l a t e d to the dynamic s h e a r s t r e s s e s on the base of a p o t e n t i a l s l i d i n g wedge.  S i n c e the dynamic shear s t r e s s e s on p a r a l l e l p l a n e s  are assumed c o n s t a n t  i n the shear s l i c e method, the l e n g t h of the base  of the wedge does not a f f e c t the c a l c u l a t e d average dynamic s e i s m i c coefficients.  However, the f i n i t e element method of a n a l y s i s i n d i c a t e s  t h a t the dynamic s h e a r s t r e s s e s do v a r y a l o n g  the base o f a wedge and  s i n c e the dynamic s t r e s s e s are l o w e s t near the f a c e of the s t r u c t u r e ,  - 51 i t c o u l d be e x p e c t e d t h a t average dynamic s e i s m i c  c o e f f i c i e n t s would  be l o w e s t f o r wedges c l o s e t o t h e f a c e o f t h e s t r u c t u r e . c a l c u l a t e d average dynamic s e i s m i c a n a l y s i s and method. seismic  4.7  Chopra (1967)  c o e f f i c i e n t s from t h e f i n i t e element  compared them w i t h v a l u e s o b t a i n e d f o r t h e s h e a r - s l i c e  He c o n c l u d e d t h a t t h e s h e a r - s l i c e method may o v e r - e s t i m a t e c o e f f i c i e n t s by from 10 t o 20 p e r c e n t .  CONCLUSIONS The  slice  p r i n c i p a l l i m i t i n g conditions  on t h e i n f i n i t e s m a l s h e a r -  t h e o r y a r e t h a t i t assumes d e f o r m a t i o n s a r e h o r i z o n t a l and due  s o l e l y t o s h e a r s t r a i n s and t h a t the m a t e r i a l i s homogeneous. p o s s i b l e to consider (Rashid,. 1961). are r e l a x e d . and  some s p e c i a l cases o f non-homogeneous m a t e r i a l  I n t h e f i n i t e element a n a l y s i s t h e above r e s t r i c t i o n s  Both v e r t i c a l . a n d h o r i z o n t a l deformations are allowed  the m a t e r i a l  analyses using  It is  can be non-homogeneous and a n i s t r o p i c .  t h e f i n i t e element approach may be s o l v e d  Dynamic e i t h e r by the  normal mode t h e o r y o r by t h e s o l u t i o n o f t h e c o u p l e d e q u a t i o n s o f motion.  T h i s l a t t e r approach l e a d s t o t h e p o s s i b i l i t y o f c o n s i d e r i n g  non-linear  m a t e r i a l and t h i s w i l l be d i s c u s s e d  i n Chapter 5.  A comparison o f t h e v i s c o e l a s t i c response o f a 300 f t . dam from f i n i t e element and s h e a r - s l i c e a n a l y s i s i n d i c a t e s t h a t f o r h o r i z o n t a l ground a c c e l e r a t i o n t h e s h e a r - s l i c e method g i v e s a good a p p r o x i m a t i o n o f t h e average dynamic shear s t r e s s on h o r i z o n t a l p l a n e s . The  f i n i t e element a n a l y s i s i n d i c a t e s t h a t t h e a c t u a l dynamic shear  s t r e s s w i l l n o t be u n i f o r m on h o r i z o n t a l p l a n e s .  However, i f t h e  -52 s t r e n g t h o f t h e m a t e r i a l i s u n i f o r m on h o r i z o n t a l p l a n e s then i t c o u l d be e x p e c t e d t h a t p l a s t i c a c t i o n would cause the combined s t a t i c and dynamic shear s t r e s s e s t o be u n i f o r m on h o r i z o n t a l p l a n e s d u r i n g earthquake l o a d i n g c o n d i t i o n s .  I t i s t h e r e f o r e suggested t h a t  a c t i o n c o u l d have a marked e f f e c t on dynamic stres_s.es.  peak  plastic  -  - 53 CHAPTER 5 PROPOSED E L A S T I C - V I S C O P L A S T I C DYNAMIC RESPONSE THEORY  INTRODUCTION  5.1  The r e s p o n s e non-elastic system of  [C]  {u} +  used i n t h i s  The s t i f f n e s s  matrix,  the s t r u c t u r e  comprised of m a t e r i a l  relations  coupled equations  The t e r m s  stated  a structure  stress-strain  tM] {ii} +  of  of  requires  represented  [K] {u}  the s o l u t i o n of  b y E q . 4.24,  the  only a constant  f i n i t e element  cations  to  analysis  However,  non-elastic  the a n a l y s i s  response  and t o d a t e  incorporating non-elastic  i n Chapter  provided a l l range.  method o f  the b e s t a p p r o x i m a t i o n to the dynamic b e h a v i o u r of continuum.  namely  (5.1)  remain w i t h i n the l i n e a r e l a s t i c  i n Chapter 4 that  the  = {f(t)}  e q u a t i o n have been d e s c r i b e d [K], i s  possessing  adds  4.3.  elements It  was  analysis  gives  a linear  elastic  considerable  compli-  (1967) no f i n i t e e l e m e n t stress-strain  dynamic  r e l a t i o n s has  been  presented. N o n - e l a s t i c b e h a v i o u r may h a v e c o n s i d e r a b l e ated dynamic s t r e s s e s  and s e i s m i c  (1960) h a v e p o i n t e d o u t seconds  and s u b j e c t e d  efficients cent  of  are of  that  for  seismic coefficient  Yet of  Veletsos  on c a l c u l a n d Newmark  a s t r u c t u r e w i t h a p e r i o d of  to E l Centro e a r t h q u a k e ,  the order of  critical.  coefficients.  effect  0.6  even  such s t r u c t u r e s  maximum s e i s m i c  f o r d a m p i n g a s h i g h as are  0.1 a n d s t r u c t u r e s  0.5 co10 p e r  commonly d e s i g n e d w i t h a so d e s i g n e d have  performed  - 54 s u c c e s s f u l l y under earthquakes of about the same o r d e r o f magnitude as E l Centro.  V e l e t s o s and Newmark suggest t h a t t h i s phenomenon can  be  e x p l a i n e d by p l a s t i c a c t i o n . R e s u l t s of the study p r e s e n t e d  by V e l e t s o s and Newmark (1960)  i n d i c a t e t h a t the i n t r o d u c t i o n o f p l a s t i c a c t i o n may  reduce the  maximum s e i s m i c c o e f f i c i e n t by a f a c t o r of 4 o r more.  The work o f  Chopra (1967) s u g g e s t s t h a t f o r v i s c o e l a s t i c r e s p o n s e , the s l i c e method may 20 per c e n t . important  over-estimate  shear-  the s e i s m i c c o e f f i c i e n t by from 10 to  I t t h e r e f o r e seems to the w r i t e r t h a t i t i s f a r more  t o i n c l u d e the e f f e c t of p l a s t i c a c t i o n than to be  concerned  by the r e s t r i c t i o n s imposed by the s h e a r - s l i c e method. A method o f i n c o r p o r a t i n g p l a s t i c b e h a v i o u r s l i c e a n a l y s i s was presented 5.2  developed by P e n z i e n  (1960).  into a  shear-  T h i s method i s  and adapted t o e a r t h s t r u c t u r e s i n t h i s  chapter.  ELASTIC-VISCOPLASTIC RESPONSE THEORY A continuum m a t e r i a l such as t h a t o f an e a r t h s t r u c t u r e may  c o n s i d e r e d as an assemblege of f i n i t e s l i c e s . represented  These s l i c e s may  be  be  by p o i n t masses i n t e r c o n n e c t e d by shear s p r i n g s w i t h  s t i f f n e s s corresponding a d d i t i o n , dashpots may  to t h a t o f the i n t e r - m a s s m a t e r i a l .  In  be i n c l u d e d to a l l o w f o r energy d i s s i p a t e d i n  v i s c o u s damping. A model of a s t r u c t u r e i n w h i c h o n l y s h e a r i n g d e f o r m a t i o n o c c u r i s shown i n F i g . 9a.  The  f o r c e , Q,  f u n c t i o n o f the r e l a t i v e d i s p l a c e m e n t  can  i n the s h e a r s p r i n g s i s a  between a d j a c e n t masses.  In  - 55 m  n  L _3-,  1 WWvVwVV-i 3  U  fnT  *  *  *-  3  —r-|_  '—\\VVVvVvVvVV—j  u  n  ,  JU  m,  2  1  '—=>-,  2  L—^^wv^^vvw--f u  —  3  C  '.  C z  — Tn =  1  v  Moving Base Reference  h C  ^ A A A W v V v N  FIG 9 0  Fig. 9b  IDEALIZED  SHEAR  SHEAR  SPRING  j  C  |  STRUCTURE  CHARACTERISTICS  Dimensionless Form  OF  IDEALIZED  STRUCTURE  i+i'l"i+r"i) —» > < mj (ii, •»• tig)  c  Citiij-Uj.,)  FIG.  9C  FORCES  ON TYPICAL  MASS OF  IDEALIZED  SHEAR  STRUCTURE  - 56 the l i n e a r range o f the m a t e r i a l , the s p r i n g f o r c e i s d i r e c t l y t i o n a l t o the d i s p l a c e m e n t .  propor-  However, once the y i e l d s t r e n g t h o f the  m a t e r i a l has been r e a c h e d , f u r t h e r d e f o r m a t i o n causes no i n c r e a s e i n the s p r i n g force  ( F i g . 9b).  The f o r c e i n the dashpots  is directly  p r o p o r t i o n a l t o the r e l a t i v e v e l o c i t y between a d j a c e n t masses. The f o r c e s a c t i n g on a t y p i c a l mass, nu, a r e shown i n F i g . 9c. The e q u a t i o n of m o t i o n o f such a mass i s m.(ii. + i i ) + C.(u. - u. .,) - C...(u..- - u.) + Q.i i g i i l - l l+l l+l l l where  Q... = 0 l+l  m. l  =  . th the l mass  u. l  =  the d i s p l a c e m e n t o f m. r e l a t i v e to the base l  u. l  =  the v e l o c i t y  •  r  o f m. r e l a t i v e t o the base, i  J  ii. = l  the a c c e l e r a t i o n o f m.  ii  =  the ground  =  the damping f o r c e between m.  g  C. 1  I  r e l a t i v e to the base  acceleration and m.  1  c o  =  (5.2)  _ when 1-1  the f o r c e i n the s p r i n g  The s p r i n g f o r c e , Q, i s determined r e l a t i o n s h i p of F i g . 9b.  from the f o r c e - d i s p l a c e m e n t  I n d e a l i n g w i t h Eq. 5.2  i t i s more c o n v e n i e n t  t o use a n o n - d i m e n s i o n a l r e l a t i v e d i s p l a c e m e n t , U, which i s g i v e n by U. = (u. - u. .) / (u. - u. ) l l l - l l l-l y where  ,(5.3)  (u. - u. ,) i s the r e l a t i v e d i s p l a c e m e n t i n the i l l-l y .  causes y i e l d i n g  ( F i g . 9b).  f o l l o w s from-Eq. 5.3  that  Since (u^ - j_-j_) u  i s  a  til  spring  constant, i t -  that  (  U  i " i-1> " U  <V-  i  U  (  i - i-l>y  U  U  ( 5  V l > " V i- i-l>y U  U  '  4 )  ( 5  -  5 )  and ( U  i " i-1> = U  i 1 " i-l>y  U  ( U  U  ( 5  '  6 )  The d i s p l a c e m e n t o f nu r e l a t i v e t o i t s moving base, u^, i s the sum o f the r e l a t i v e shear d i s p l a c e m e n t s below n u , i . e . ,  u. = 1  r=i  L  I  (u - u J . r r-1 r=l  r=i = y U . r r=l L  (u - u ) r r-1 y  (5.7)  and i n t h e same way t h e a c c e l e r a t i o n o f nu r e l a t i v e t o t h e base i s g i v e n by tt. = i  r=i„ U « r r=l  J  (u - u .) r r-1 y  (5.8)  J  S u b s t i t u t i n g Eqs. 5.5, 5.6 and 5.8 i n t o Eq. 5.2 t h e f o l l o w i n g  result  i s obtained r=i m. J U (u - u ) + Q. - Q. i . r r r-1 y x x+l r=l L  n  H  x  jn  J  ~  x+1  + C.U.(u. - u. ,) x l x x-l'y U.^Oi.,.. - u.) = x+1 i + l i y  An e x p r e s s i o n f o r t h e n o n - d i m e n s i o n a l a c c e l e r a t i o n f o r e , removing  rn.ii i g  i s required,  u\ from t h e summation, Eq. 5.9 y i e l d s  (5.9) there-  U. = 1  -,— m.(u. 1  -7- {(Q. - Q.,.) - u. -) ^1 ^1+1/ l-l y  ~ C..,  l+l  + m.tt. + C.U.(u. - u. .) 1 g 1 1 1 i-1'y  1  1  U.-TCU..,- U.)  l+l  l+l  1  y  - m.  1  6  LI  3  r=i-l (u n  r  - u  .)  r-l y  U }  r  J  (5.10)  E q u a t i o n 5.10 must be s o l v e d by n u m e r i c a l methods. I f the q u a n t i t i e s U, U and U a r e known f o r a l l n masses a t time t , t h e n these same q u a n t i t i e s can be determined a t time t + At by the " m i d - a c c e l e r a t i o n method" ( P e n z i e n 1960). follows:  The method proceeds  as  L e t IL , U, and U, be the known v a l u e s o f the n o n - d i m e n s i o n a l b' b b  displacement, v e l o c i t y i n t e r v a l , At.  and a c c e l e r a t i o n a t the b e g i n n i n g of a time  The m i d - i n t e r v a l v a l u e s of U and U (U, U) a r e  determined from the f o l l o w i n g e q u a t i o n s : U = U  b  u  + U, At/2 b  (5.11)  U = IL + U. At/2 + U, t /8 b b b  (5.12)  Knowing U and U, the shear r e s i s t a n c e at the m i d - i n t e r v a l , Q, i s o b t a i n e d from the known e l a s t i c - p l a s t i c f o r c e - d i s p l a c e m e n t r e l a t i o n s h i p ( F i g . 9b).  The n o n - d i m e n s i o n a l m i d - a c c e l e r a t i o n , U, can now  o b t a i n e d from Eq. 5.10.  be  Due t o the presence of the summation term on  the r i g h t . h a n d s i d e o f Eq. 5.10, n o n - d i m e n s i o n a l a c c e l e r a t i o n v a l u e s must be o b t a i n e d i n sequence, i . e . ,  must f i r s t be d e t e r m i n e d , t h e n  etc. Having o b t a i n e d a l l n v a l u e s o f , U , t h e n o n - d i m e n s i o n a l d i s placements and v e l o c i t i e s a t t h e end of the i n t e r v a l , U and U , a r e ^ e e o b t a i n e d , f r o m the f o l l o w i n g e q u a t i o n s :  - 59 U  and  U  e  = U, + UAt b  (5.13)  e  = t l + U, At + U A t / 2 b b  (5.14)  2  A g a i n , knowing  and U^,  the shear f o r c e , Q, can be o b t a i n e d from the  f o r c e - d i s p l a c e m e n t r e l a t i o n s h i p as b e f o r e .  The n o n - d i m e n s i o n a l  accel-  e r a t i o n a t t h e end o f the p e r i o d , U , i s o b t a i n e d by a p p l y i n g Eq..5.10 as p r e v i o u s l y . J  The known end v a l u e s U , U and U can now be c o n s i d e e e  e r e d as i n i t i a l v a l u e s f o r the next time increment  and the  process  repeated. The a c t u a l d i s p l a c e m e n t s o b t a i n e d from the n o n - d i m e n s i o n a l a p p l y i n g Eqs. 5.7  and 5.8  a l l o w s the displacements  and a c c e l e r a t i o n s a t any t i m e , t , are displacements  respectively.  and a c c e l e r a t i o n s by  Thus t h e above a n a l y s i s  and a c c e l e r a t i o n s a t any  t i m e , t , t o be  cal-  c u l a t e d w i t h i n the model s t r u c t u r e when t h e base i s s u b j e c t e d t o a known a c c e l e r a t i o n . In  s t a r t i n g the above procedure  i n i t i a l displacements  and v e l o c i t i e s .  i t i s necessary The  to know the  i n i t i a l displacements  be o b t a i n e d from a s t a t i c a n a l y s i s , w h i l e the i n i t i a l v e l o c i t i e s be z e r o , i . e . , s t r u c t u r e i n i t i a l l y a t - r e s t . may If  be o b t a i n e d from Eq. 5.10  as  may will  The i n i t i a l a c c e l e r a t i o n s  follows:  1 = 1, then U i = - u / (u i - u ) g o y  (5.15)  i 4 1, then U  (5.16)  ±  = 0  The model shown,in F i g . 9a would a p p l y to e a r t h s t r u c t u r e s such  - 60 as h o r i z o n t a l and  s l o p i n g l a y e r s of s o i l u n d e r l a i n by bedrock ( F i g .  where s t a t i c shear s t r e s s o n . p a r a l l e l planes' w i l l be u n i f o r m . for a.structure  such as an e a r t h dam  u n i f o r m on h o r i z o n t a l p l a n e s .  10a)  However,  the s t a t i c s t r e s s e s w i l l not  be  I n f a c t , i n the absence of boundary water  f o r c e s the shear f o r c e s t o the l e f t . a n d r i g h t of the c e n t r e l i n e  will  be e q u a l i n magnitude but o p p o s i t e  and  10c.  i n s i g n as shown i n F i g s . 10b  Under earthquake c o n d i t i o n s , and  assuming r i g i d h o r i z o n t a l  s l i c e s , i t i s p o s s i b l e f o r the shear f o r c e on the r i g h t hand s i d e t o move i n t o the p l a s t i c range w h i l e the shear f o r c e on the l e f t hand s i d e remains i n the e l a s t i c range.  This condition i s depicted  10c where the s t a t i c e q u i l i b r i u m c o n d i t i o n r e p r e s e n t e d by the Q  i n Fig. points  and Q ^nove to the dynamic e q u i l i b r i u m c o n d i t i o n r e p r e s e n t e d by L  and  Q  R  1  ij •  . Review of earthquake damage to embankments by Ambraseys (1960)  and  Duke (1960) i n d i c a t e s t h a t a dominant f e a t u r e i s the f o r m a t i o n  c r a c k s p a r a l l e l to the a x i s of a . s t r u c t u r e . m a t t e r of some i n c h e s wide and may  be  e x t e n d i n g to a depth of 70  To a l l o w f o r the p o s s i b i l i t y of c r a c k  h o r i z o n t a l s l i c e t o be comprised of two  f o r c e i n the s p r i n g w i l l be c o m p r e s s i v e due  tensile.  A tensile limit, H , f  each  p a r t s s e p a r a t e d by a h o r i -  Such a model i s shown i n F i g . 11a.  However, under dynamic c o n d i t i o n s  feet  formation  the model shown i n F i g . 9a can be f u r t h e r r e f i n e d by c o n s i d e r i n g  zontal spring.  a  e x t e n d t o a depth of many f e e t .  Ambraseys (1960) mentions t h a t c r a c k s have been r e c o r d e d .  These c r a c k s may  of  The  horizontal  to the s t a t i c  stresses.  the f o r c e i n the s p r i n g may  i s p l a c e d on the s p r i n g  become  corresponding  - 61 -  HORIZONTAL  FIG.  Shear  IOQ  SOIL  LAYER  UNIFORM  SLOPING  STATIC  SHEAR  STRESS  SOIL  LAYER  CONDITIONS  Stress  SECTION  FIG.  10b  FIG.  IOC  OF  EARTH  NON-UNIFORM  STATIC FOR  PLUS  THE  DAM  STATIC  DYNAMIC  CASE  OF  SHEAR  SHEAR  NON-  STRESS  CONDITIONS  STRESS  CONDITION  UNIFORM  STATIC  SHEAR  STRESS  FIG.  Ilo  PROPOSED  Differentiol  MODEL  OF  Displacement  EARTH  DAM  ( UR-UL)  e  c.e  —  Hp  ( Tension Failure )  O <A N C  a>  e x FIG.  lib  FORCE OF  C,  +  ,(U  I  +  L  -Ui)  DISPLACEMENT  HORIZONTAL  C|+, (Qi-H-Clt)  QJ-M  m, ( U i + uf)  H,-  RIGHT  L E F T  lie  m,(Ui +  c , ( u , - a,.,)  C l ( u , - u,.,)  FIG.  CHARACTERISTIC  SPRING  FORCES THE  ON  TYPICAL  STRUCTURE  ELEMENT  OF  Qi ft  ufl)  - 63 -  to the t e n s i l e s t r e n g t h  o f the s l i c e  been reached, i t i s assumed t h a t  ( F i g . l i b ) . Once t h i s l i m i t  a crack occurs.  immediately drops to zero and no f u r t h e r t e n s i o n  The t e n s i l e  has  force  can be taken i n t h i s  spring. The f o r c e s a c t i n g on t y p i c a l l e f t shown i n F i g . 11c.  The c o n d i t i o n  m.(U, + ii ) + Q. - Q... i i g x a+1  and r i g h t hand masses are  f o r e q u i l i b r i u m o f the l e f t mass i s  + C.(u.- u. _) - C . ( u . . . - u . ) x x l - l x+1 x+1 x  x  + H. = 0 x (5.17)  and  f o r the r i g h t mass i s  m.(U. x x  + tt ) + Q. - Q.,g x ^i+1 x  + C.(u. - u. .,) - C...(u...- u.) - H. = 0 l x x-1 x+1 x+1 x x (5.18) th  Where IL = the f o r c e i n the i  h o r i z o n t a l s p r i n g and the remaining  terms have the same meaning as d e s c r i b e d same n o t a t i o n  used to a v o i d  the  case of the s i m p l e r  and  accelerations  the  l e f t mass then becomes  U  i  - -  and r i g h t masses.  T h i s , however, does not  these w i l l have the same n u m e r i c a l v a l u e s .  n o t a t i o n was  The  i s used f o r mass, d i s p l a c e m e n t , e t c . f o r the dynamic  e q u i l i b r i u m o f both the l e f t imply that  f o r the model o f F i g . 9.  using  doubly s u b s c r i p t e d  The same  variables.  model, non-dimensional d i s p l a c e m e n t s ,  are introduced.  As i n velocities  The dynamic e q u i l i b r i u m e q u a t i o n f o r  ; r {m.U +(Q.-Q _ )+C.U. (u -u, . ).-C. ,,._,_, (u. ,.-u. ) m.(u.- u. ..) x g ii+1 i i i i - l ' y i+1 x + l x+1 i y l I x-1 y ° u  1  x  H  JJ  n  v  v  J  r-i-1 + H. - m. J (u - u ) V \ x l , r r-1 y r / r=l L  n  (5.19)  - 64 and f o r t h e r i g h t mass  U.=  x  — r - {m.u + ( Q . - Q . ) + C U . ( u . - u. .) -C. U. m.(u.-u. x g ^x x + l x i x x-1 y x+1 x+1 x x l - ly e> • J 1  x  (u.  - u. )  x+1  x y  r-i-.l - H.- m. J (u - u .) U } x i i' r r-ly r r=l '  J  (5.20)  J  J  The o n l y d i f f e r e n c e between the above e q u a t i o n s and Eq. 10 i s the h o r i z o n t a l f o r c e term, R\. placement evaluation  between l e f t  T h i s f o r c e w i l l depend on t h e r e l a t i v e d i s -  and r i g h t masses and hence w i l l  o f t h e term (u  - u ).  require  the  I n o t h e r r e s p e c t s the i n t e g r a t i o n o f  Eqs. 5.19 and 5.20 proceeds i n t h e same manner as t h a t used f o r Eq. 10. The n u m e r i c a l i n t e g r a t i o n p r o c e d u r e s o u t l i n e d above can be c a r r i e d o u t u s i n g t h e e l e c t r o n i c computer and t h i s has been done by the w r i t e r f o r b o t h t h e s i m p l e and the more complex models o f F i g s , 9a and 1.1a.  However, b e f o r e such an a n a l y s i s  can be u n d e r t a k e n , i t i s  n e c e s s a r y t o know t h e f o l l o w i n g : 1.  The s t a t i c s h e a r s t r e s s e s  e x i s t i n g i n the s t r u c t u r e  prior  to t h e e a r t h q u a k e . 2.  The s t r e s s - s t r a i n c h a r a c t e r i s t i c s o f t h e m a t e r i a l dynamic  under  conditions.  3. ' V a l u e s f o r t h e damping f a c t o r s , The above i t e m s w i l l be b r i e f l y d i s c u s s e d i n t h e s e c t i o n s  5.3  which  follow.  ESTIMATION OF STATIC SHEAR STRESSES  For h o r i z o n t a l and s l o p i n g  l a y e r s o f s o i l such as those shown  i n F i g . 10a i t . s e e m s r e a s o n a b l e t o assume t h a t the s t a t i c shear s t r e s s e s on p l a n e s p a r a l l e l t o the base are u n i f o r m .  Hence, the  static  shear s t r e s s e s on t h e s e p l a n e s can be determined from the p r i n c i p l e s of statics.  However, f o r an e a r t h dam  ( F i g . 10b),  the s t a t i c shear s t r e s s  d i s t r i b u t i o n on h o r i z o n t a l p l a n e s i s not u n i f o r m and hence the shear s t r e s s e s cannot be determined from the p r i n c i p l e s of alone.  The  considered.  static  statics  s t r e s s - s t r a i n c h a r a c t e r i s t i c s of the m a t e r i a l must a l s o The  be  s t a t i c shear s t r e s s on h o r i z o n t a l p l a n e s f o r t h i s  type of s t r u c t u r e are b e s t determined from a . f i n i t e element a n a l y s i s ing  the s t a t i c s t r e s s - s t r a i n c h a r a c t e r i s t i c s of the m a t e r i a l .  of such a n a l y s e s  are g i v e n . b y F i n n  (1966a) and  us-  Examples  Clough and Woodward,  (1967). For s t r u c t u r e s a n a l y z e d  i n t h i s t h e s i s where the s t a t i c , s h e a r  s t r e s s e s c o u l d not be determined by the p r i n c i p l e s of s t a t i c s , f i n i t e element method of a n a l y s i s was a n a l y s i s t h a t the s t r u c t u r e was isotropic material. e l a s t i c constants.  used.  I t was  assumed f o r the  comprised of l a y e r s of homogeneous and  Thus each l a y e r would have o n l y two These c o n s t a n t s  independent  were e x p r e s s e d i n terms of  s t r e s s e s r a t h e r than e f f e c t i v e s t r e s s e s .  i n e f f e c t i v e r a t h e r than t o t a l s t r e s s e s .  total  I t i s r e a l i z e d that s o i l , i n  g e n e r a l , i s n e i t h e r homogeneous nor i s o t r o p i c and  responds to changes  However, the  s i s of the s t a t i c s t r e s s e s would be a.major u n d e r t a k i n g scope of t h i s t h e s i s .  the  rigorous,analyand  Hence s i m p l e assumptions w i t h r e g a r d  outside to  the  the  b e h a v i o u r of the s o i l under s t a t i c s t r e s s e s were made. The  e l a s t i c constants  Poisson's r a t i o  y.  used were Young's modulus, E,  and  The modulus, E, depends b a s i c a l l y upon the type  -  of  s o i l and the mean normal e f f e c t i v e s t r e s s .  Values o f the modulus,  E, used i n t h i s t h e s i s were based on d a t a p r e s e n t e d by W i l s o n Dietrich 0.5  (1960).  Poisson's r a t i o  f o r most s o i l s .  and  u, g e n e r a l l y ranges between 0.3  and  I n t h i s t h e s i s , v a l u e s o f u = 0.5 were used f o r  s a t u r a t e d c l a y s o i l s , w h i l e v a l u e s o f u r a n g i n g between 0.4 were used f o r sands.  66  and  0.5  The shear modulus, G, i s more a p p r o p r i a t e f o r the  s h e a r - s l i c e method and i s r e l a t e d t o E and u by the f o l l o w i n g f o r m u l a :  G  = za^rT  <5-21>  T y p i c a l shear s t r e s s d i s t r i b u t i o n s on h o r i z o n t a l p l a n e s determined by use o f the f i n i t e element method f o r the 285 f t . h i g h s y m m e t r i c a l dam  o f F i g . 12a are shown i n F i g . 12c.  The dam was  first  assumed to be homogeneous w i t h a modulus, E = 68,000 p . s . i . and a P o i s s o n ' s r a t i o , u = 0.4. assumption  The shear s t r e s s d i s t r i b u t i o n s f o r t h i s  a r e shown as s o l i d l i n e s i n F i g . 12c.  In actual fact i t  c o u l d be expected t h a t t h e modulus would i n c r e a s e w i t h c o n f i n i n g p r e s s u r e , i . e . , w i t h depth below the c r e s t .  In f i g .  12b t h i s dam  c o n s i d e r e d t o comprise of s i x l a y e r s w i t h a modulus v a r i a t i o n 38,000 p . s . i . t o 90,000 p . s . i . from c r e s t to base and r a t i o c o n s t a n t and e q u a l t o 0.4.  was  from  Poisson's  The shear s t r e s s e s f o r t h i s assump-  t i o n a r e shown as dashed l i n e s i n F i g . 12c and a r e seen t o be v e r y s i m i l a r to those o b t a i n e d f o r the homogeneous case.  Therefore v a r i a t i o n  i n modulus makes l i t t l e d i f f e r e n c e t o the shear s t r e s s on h o r i z o n t a l planes. in  However, the s h e a r s t r e s s e s a r e g r e a t l y a f f e c t e d by changes  the v a l u e of P o i s s o n ' s r a t i o , and t h e r e f o r e i n p r a c t i c e c a r e must  -  Shear  S t r e s s in  PS I  a ro cr  o z X O  2  m z m o O  G)  ro a  X  z o m z o c O  <7) rn w  c  X  JO  o > 2  2  285  - 68 -  be t a k e n to a s s u r e t h a t s u i t a b l e v a l u e s o f P o i s s o n ' s r a t i o a r e used. For t h e dynamic a n a l y s i s t h e dam i s c o n s i d e r e d t o comprise o f a number o f h o r i z o n t a l s l i c e s each o f w h i c h c o n s i s t s o f two s e p a r a t e masses.  Thus t h e dam c o u l d be c o n s i d e r e d as d i v i d e d a l o n g a l i n e  such as A - B i n F i g . 13a.  The s t a t i c shear f o r c e on h o r i z o n t a l  p l a n e s t o t h e l e f t and r i g h t o f t h e l i n e A - B i s r e q u i r e d .  The  shear f o r c e on a h o r i z o n t a l p l a n e to t h e l e f t o f t h i s l i n e , Q , may be o b t a i n e d by i n t e g r a t i n g t h e shear s t r e s s over t h e a r e a . force to the r i g h t , Q , i s equal to Q K  from s t a t i c s .  The shear  The h o r i z o n t a l  L  s p r i n g f o r c e , H^, i s e q u a l to t h e d i f f e r e n c e i n shear f o r c e s on t h e top and bottom o f a s l i c e as shown i n F i g . 13b. the r e q u i r e d 5.4  acting  I n t h i s manner  s t a t i c stresses are obtained.  STRESS-STRAIN AND FORCE-DISPLACEMENT UNDER DYNAMIC LOADING  CHARACTERISTICS OF SOIL-  The s t r e s s - s t r a i n c h a r a c t e r i s t i c s o f s o i l under s i m u l a t e d \  earthquake c o n d i t i o n s  has been t h e s u b j e c t  o f r e c e n t s t u d y (Seed and  Chan 1966, Seed and Lee 1966, and Seed and L e e , 1967).  I t appears  t h a t s t r e s s - s t r a i n r e l a t i o n s depend on: 1.  The n a t u r e o f t h e s o i l .  2.  The s t a t i c s t r e s s  3.  The d r a i n a g e  4.  The c h a r a c t e r i s t i c , n a t u r e o f t h e t r a n s i e n t  condition.  condition. forces  a c t i n g on elements o f s o i l . I f an e s t i m a t e o f t h e s t r e s s - s t r a i n c h a r a c t e r i s t i c s o f . e l e m e n t s of s o i l a t v a r i o u s  p o i n t s w i t h i n an e a r t h s t r u c t u r e i s d e s i r e d ,  then  FIG.  I3Q  UNSYMMETRICAL  (Qi+  DIVISION  PL  (Qi)R  RIGHT  LEFT  13b  DAM  (Qi+I)R  (Qi)L  FIG.  OF E A R T H  STATIC  FORCES  ON  ELEMENT  OF DAM  conditions  1 and 2 r e q u i r e t h a t samples of the m a t e r i a l be brought t o  e q u i l i b r i u m under the s t a t i c s t r e s s e s e x i s t i n g p r i o r to the earthquake. These samples, under the a p p r o p r i a t e  d r a i n a g e c o n d i t i o n , are then sub-  j e c t e d to the t r a n s i e n t f o r c e system a t the p o i n t i n q u e s t i o n ( c o n d i t i o n s 3 and 4) and t h e s t r e s s - s t r a i n r e l a t i o n s h i p s determined. However, i n an e a r t h s t r u c t u r e , the t r a n s i e n t f o r c e s a t each p o i n t w i l l depend on: 1.  The earthquake a c c e l e r a t i o n s a t the base of the s t r u c t u r e .  2.  The geometry of the s t r u c t u r e .  3.  The mass d e n s i t y of the m a t e r i a l .  4.  The s t r e s s - s t r a i n c h a r a c t e r i s t i c s of the m a t e r i a l .  Thus the t r a n s i e n t f o r c e s depend on the s t r e s s - s t r a i n c h a r a c t e r i s t i c s of the m a t e r i a l and s i n c e i t has a l r e a d y been s t a t e d t h a t the s t r e s s s t r a i n c h a r a c t e r i s t i c s depend on the t r a n s i e n t f o r c e system, a r a t i o n a l dynamic a n a l y s i s would r e q u i r e an i t e r a t i v e s o l u t i o n , i . e . s u c c e s s i v e adjustments t o the s t r e s s - s t r a i n r e l a t i o n s u n t i l t h e observed  (from  t e s t s ) and c a l c u l a t e d s t r e s s - s t r a i n r e l a t i o n s were i n agreement. For problems a n a l y z e d out.  i n t h i s t h e s i s , no i t e r a t i o n was c a r r i e d  An i t e r a t i v e s o l u t i o n would cause no f u r t h e r c o m p l i c a t i o n  a n a l y s i s as i t would merely i n v o l v e s u c c e s s i v e strain relations.  r e - r u n s w i t h new s t r e s s -  Assumed s t r e s s - s t r a i n r e l a t i o n s were l i m i t e d t o  types t h a t c o u l d be i n c o r p o r a t e d  i n the n u m e r i c a l procedure o u t l i n e d  i n S e c t i o n 5.2 and these w i l l now be d i s c u s s e d The  t o the  i n some d e t a i l .  s t r e s s - s t r a i n r e l a t i o n s f o r an e l a s t i c - p l a s t i c m a t e r i a l are  shown i n F i g . l 4 a by the s o l i d l i n e .  I f v i s c o u s damping i s a l s o  Fig-14b  Stress - Strain  Relationship  for  Strain Softening Material.  - 72  i n c l u d e d , t h e n the r e l a t i o n s w i l l be the form shwon by the l i n e o f F i g . 14a.  dashed  Such r e l a t i o n s might w e l l a p p l y to some compacted  c o h e s i v e s o i l s and t o l o o s e and medium sands of low s a t u r a t i o n . S t r e s s - s t r a i n r e l a t i o n s f o r e l a s t i c - p l a s t i c m a t e r i a l f o r which the y i e l d s t r e s s decreases w i t h the a b s o l u t e v a l u e o f the maximum s t r a i n a r e shown i n F i g . 14b.  The s t r e s s - s t r a i n r e l a t i o n s i n the  p l a s t i c r e g i o n a r e c o n t r o l l e d by the parameters y i , Y2»  a n c  * n.  By a  s u i t a b l e c h o i c e o f t h e s e parameters the b e h a v i o u r o f a wide range o f s o i l types may  be m o d e l l e d .  V i s c o u s damping can a l s o be i n c l u d e d as  before. S t r e s s - s t r a i n r e l a t i o n s o f the form shown i n F i g . 14a and c o u l d be used to model the b e h a v i o u r o f most s o i l s . s a t u r a t e d sands p r e s e n t s p e c i a l  14b  However, some  problems.  C y c l i c l o a d i n g t e s t s on s a t u r a t e d u n d r a i n e d specimens were f i r s t r e p o r t e d on by Seed and Lee  (1966).  on a u n i f o r m f i n e sand under s t a n d a r d t r i a x i a l  of sand  T e s t s were performed conditions ( o  2  =03).  More r e c e n t l y r e s u l t s o f t e s t s on a medium t o f i n e u n i f o r m sand u s i n g a s i m p l e shear d e v i c e have been r e p o r t e d on by Peacock and Seed (1968). The l a t t e r t y p e o f t e s t i s thought to cause s t r a i n s which more c l o s e l y model those o f an element i n a l a y e r o f s o i l s u b j e c t e d to earthquake motion.  R e s u l t s from b o t h types o f t e s t i n d i c a t e t h a t the s t r e n g t h of  a g i v e n sand sample i s a f u n c t i o n o f the number o f c y c l e s o f shear stress applied.  I t appears t h a t t h e c y c l i c shear s t r e s s e s i n d u c e a  tendency f o r volume decreases which i s r e f l e c t e d i n a r i s e i n pore water p r e s s u r e and a consequent f o r the l o s s i n s t r e n g t h .  reduction i n e f f e c t i v e stress accounting  -  - 73  Most dynamic t e s t s have been o f t h e c y c l i c l o a d type w h e r e i n the l o a d i s c y c l e d between two l i m i t s .  T h i s method does not u s u a l l y  a l l o w an a c c u r a t e e s t i m a t e o f the s t r e s s - s t r a i n r e l a t i o n s to be o b t a i n e d because, a l t h o u g h t h e maximum and minimum v a l u e o f t h e s t r e s s can be a c c u r a t e l y d e t e r m i n e d , t h e v a l u e o f t h e s t r e s s d u r i n g the l o a d change i s n o t u s u a l l y w e l l d e f i n e d .  Cyclic strain controlled tests  w h e r e i n t h e s t r a i n i s c y c l e d between two l i m i t s a r e c o n s i d e r e d t o y i e l d a more r e l i a b l e e s t i m a t e o f s t r e s s - s t r a i n r e l a t i o n s .  Such  r e l a t i o n s f o r a l o o s e u n i f o r m s a t u r a t e d f i n e sand s u b j e c t e d t o s t a n d a r d s t r e s s t r i a x i a l c o n d i t i o n s a r e shown i n F i g . 15. t h a t , although  I t may be seen  the r e l a t i o n s a r e complex, b o t h the Young's modulus  and t h e shear s t r e n g t h d e c r e a s e w i t h the number o f c y c l e s o f s t r a i n . C y c l i c s t r a i n c o n t r o l l e d s i m p l e shear t e s t s on s a t u r a t e d sands have not been r e p o r t e d on i n the l i t e r a t u r e t o date (1968) but  examination  o f s i m p l e shear c y c l i c s t r e s s c o n t r o l l e d d a t a suggest t h a t b o t h the shear modulus and t h e s t r e n g t h decreases w i t h the magnitude and number o f c y c l e s o f s t r a i n , and such d e c r e a s e s c o u l d be i n c o r p o r a t e d i n the analysis outlined i n Section  5=2.  The a n a l y s i s o u t l i n e d i n S e c t i o n 5.2 r e q u i r e s f o r c e displacement structure.  r e l a t i o n s t o be f i r s t determined f o r t h e elements of a These r e l a t i o n s can be determined from the known o r assumed  stress-strain relations.  Consider  an e l a s t i c - v i s c o p l a s t i c m a t e r i a l  w i t h s t r e s s s t r a i n r e l a t i o n s as shown i n F i g . 14a.  I n the e l a s t i c  range t h e shear s t r e s s , r i s g i v e n by r = G(y-Y ) + Cy  (5.22)  -  - 74 -  FIG.  15  DYNAMIC  STRESS  CONTROLLED SAND  ( S E E D  - STRAIN  TRI A X I A L AND  LOE,  RELATIONSHIP  TESTS 1966.)  ON  LOOSE  FROM  STRAIN  SATURATED  - 75 and i n t h e p l a s t i c range  x = x where  G  + Cy  (5.23)  =  t h e s h e a r modulus  =  t h e i n t e r c e p t on t h e s t r a i n  y  =  t h e shear s t r a i n  Y  =  the r a t e of shear s t r a i n  =  t h e y i e l d s t r e s s o r s t r e n g t h when y  y  o  axis  =  0.  The s h e a r s t i f f n e s s , k, i s the f o r c e r e q u i r e d ment when t h e s t r a i n r a t e i s z e r o .  f o runit  F o r the r e c t a n g u l a r  displace-  element shown,  i n F i g . 16a t h e s h e a r s t i f f n e s s i s g i v e n by k = Gb/d where  (5.24)  b =  the w i d t h of the s l i c e  d =  t h e depth o f t h e s l i c e  and f o r t h e t r a p e z o i d a l element ( F i g . 16b) t h e shear s t i f f n e s s i s  k = —-5 ±d i n y /y b  where  ay^ = top w i d t h o f s l i c e  and  ay^ = bottom w i d t h o f s l i c e The y i e l d f o r c e , Q , f o r a r e c t a n g u l a r Q  (5.25)  t  element i s g i v e n by  = bx y  (5.26) y  and f o r t h e t r a p e z o i d a l element i t i s assumed t o be g i v e n by  Fig. 16c  Force-'Displacement Relationship  Fig. 16 d  Force-Displacement Relationship in Dimensionless Form.  - 77  Q  = 1/2  y  a(y  b  + y ) i fc  -  (5.27)  y  The s h e a r s t i f f n e s s and y i e l d f o r c e v a l u e s d e t e r m i n e the f o r c e d i s p l a c e m e n t c h a r a c t e r i s t i c s of an element as shown i n F i g . 16c. the proposed a n a l y s i s i t i s u s e f u l to use a n o n - d i m e n s i o n a l  For  strain  parameter, U, g i v e n by U = (u - u ) t  where  and  u^_  b  / (u  t  - u ) b  (5.28)  y  =  the d i s p l a c e m e n t a t the top o f the  =  the d i s p l a c e m e n t a t the bottom of the  (u^ - u ^ )  y  =  element element  the r e l a t i v e d i s p l a c e m e n t t h a t causes y i e l d to o c c u r .  The f o r c e - d i s p l a c e m e n t r e l a t i o n i n n o n - d i m e n s i o n a l form i s shown i n Fig.  16d. I f the y i e l d s t r e s s o f a m a t e r i a l v a r i e s with;.the magnitude o f  the s t r a i n as was  shown i n F i g . 14b, where y i and Y2 a r e the s t r a i n s  between w h i c h the y i e l d s t r e s s v a r i e s l i n e a r l y , then i t c o u l d be exp e c t e d t h a t the f o r c e - d i s p l a c e m e n t r e l a t i o n s i n n o n - d i m e n s i o n a l  form  f o r an element of t h i s m a t e r i a l would be as shown i n F i g . ' 1 7 a , where Ui  and U  2  a r e t h e n o n - d i m e n s i o n a l s t r a i n s c o r r e s p o n d i n g to Y i a n d Y2-  However, t h e v a l u e s of  and U  2  depend on whether y i e l d i n g t a k e s  p l a c e on a s i n g l e p l a n e o r o v e r a broad y i e l d zone as shown i n F i g s . 17b and c.  I t may  be seen from F i g . 17b t h a t where y i e l d t a k e s p l a c e  on a s i n g l e p l a n e the magnitude of the p l a s t i c d i s p l a c e m e n t , 6 ^ , f o r a g i v e n shear f o r c e , Q, i s independent o f the depth of the  element,  whereas i n F i g . 17c, where p l a s t i c d i s p l a c e m e n t i s o c c u r r i n g  throughout  - 78 -  Fig. 17a  Force - Displacement Relationship for Strain Softening Material, Non-Dimensional Form.  Q  Fig. |7c  Failure  Uniform  Throughout  the  Slice.  - 79 the element, the magnitude o f the p l a s t i c d i s p l a c e m e n t i s d i r e c t l y proportional  t o the depth of the element.  to a specimen o f s o i l under t e s t . shown on F i g . 14a were c a l c u l a t e d  These same arguments  I f the s t r e s s - s t r a i n  apply  relations  by d i v i d i n g a d i s p l a c e m e n t by a  c h a r a c t e r i s t i c l e n g t h (average s t r a i n ) , then f o r the case o f y i e l d on a s i n g l e plane  Ui = 1 + f-  ( —  - 1)  (5.29)  y  U  2  = 1 +  ( —  - 1)  (5.30)  y  where  d  =  t h e , d e p t h o f the element  d' = and YI»  Y2»  the sample d i m e n s i o n from w h i c h the s t r a i n was anc  ^ Yy  calculated  the shear s t r a i n parameters shown i n F i g . 14a.  a r e  I f , on the o t h e r hand, t h e m a t e r i a l  i s y i e l d i n g u n i f o r m l y throughout  b o t h the element and the t e s t specimen, then  and  Ui =  Y1 /Yy  (5.31)  U  Y2/Y  (5.32)  2  =  y  The f o r c e - d i s p l a c e m e n t r e l a t i o n s . c a n  t h e r e f o r e be o b t a i n e d from the  s t r e s s - s t r a i n r e l a t i o n s , p r o v i d e d y i e l d i n g o c c u r s e i t h e r on a s i n g l e p l a n e o r u n i f o r m l y throughout t h e r e g i o n .  5.5  DISCUSSION OF DAMPING Damping r e p r e s e n t s a form o f energy d i s s i p a t i o n and i s u s u a l l y  d e s c r i b e d as one of the  following:  - 80 -  1.  V i s c o u s damping  2.  S t r u c t u r a l damping  3.  Coulomb o r f r i c t i o n a l damping  V i s c o u s damping i m p l i e s t h a t t h e damping f o r c e , F^ i s p r o p o r t i o n a l to the v e l o c i t y but opposite i n s i g n , i . e . F, = Cu d where  (5.33)  u  =  t h e v e l o c i t y o f t h e mass  C  =  t h e damping c o e f f i c i e n t  F o r a s i n u s o i d a l d i s p l a c e m e n t , u = A s i n cot, the f o r c e - d i s p l a c e m e n t curve d e s c r i b e s a h y s t e r e s i s l o o p .  The energy d i s s i p a t e d per c y c l e ,  i s the a r e a e n c l o s e d by the l o o p and i s g i v e n by  2  2TT/O)  W  d  =  /  . C u u d t = CA  o  (5.34)  COTT  ,  T h e r e f o r e , f o r v i s c o u s damping, the energy d i s s i p a t e d p e r c y c l e depends on.the a n g u l a r f r e q u e n c y , co, o f t h e d i s p l a c e m e n t  function.  S t r u c t u r a l damping i m p l i e s t h a t t h e energy d i s s i p a t e d p e r c y c l e i s independent o f the f r e q u e n c y o f v i b r a t i o n .  I f i t i s assumed t h a t  the f o r c e and t h e d i s p l a c e m e n t a r e out o f phase by an a n g l e a, i . e . the f o r c e , F, i s g i v e n by  F  =, k A S i n (cot-a)  t h e n i t can be shown t h a t t h e energy d i s s i p a t e d p e r c y c l e , W^,  (5.35) i s given  by  W, d  =  irA k S i n a  (5.36)  - 81 -  E q u a t i o n 5.36 i s independent o f t h e f r e q u e n c y , to, and hence the energy d i s s i p a t e d p e r c y c l e i s independent o f the f r e q u e n c y o f t h e d i s placement f u n c t i o n . Coulomb o r f r i c t i o n a l damping i m p l i e s t h a t t h e damping f o r c e i s p r o p o r t i o n a l t o t h e normal f o r c e and a c t s i n a d i r e c t i o n o p p o s i t e t o that of the v e l o c i t y .  I f t h e normal f o r c e i s c o n s t a n t ,  then f o r a,  s i n u s o i d a l d i s p l a c e m e n t f u n c t i o n , u = A S i n cot, t h e energy d i s s i p a t e d per c y c l e i s g i v e n by  W  d  J  =, 4uNA = 40 A  (5.37)  >  Where N i s t h e normal f o r c e and uN = of t h e m a t e r i a l . of t h e f r e q u e n c y .  i s the y i e l d  strength  A g a i n t h e energy d i s s i p a t e d p e r c y c l e i s independent T h i s t y p e o f energy d i s s i p a t i o n c o r r e s p o n d s t o t h a t  o c c u r r i n g d u r i n g p l a s t i c y i e l d and i s i n c l u d e d i n t h e e l a s t i c - v i s c o p l a s t i c a n a l y s i s o f Chapter 5.2. V i s c o u s damping i s t h e s i m p l e s t m a t i c a l analyses.  type t o i n c o r p o r a t e  i n mathe-  I t has been suggested t h a t t h e dynamic response o f  a s t r u c t u r e depends l a r g e l y on t h e amount o f energy d i s s i p a t e d than on t h e type o f damping mechanism p r e s e n t . concept o f e q u i v a l e n t per c y c l e , W^,  viscous  i s obtained  =  - r ^ -  T h i s has l e d t o t h e  damping, w h e r e i n t h e energy d i s s i p a t e d  from t h e h y s t e r e s i s l o o p and t h e e q u i v a l e n t  viscous.damping f a c t o r , C , obtained eq C  rather  from Eq. 5.34, namely  (5.38)  A tOTT The amount o f damping p r e s e n t i n a system i s . u s u a l l y e x p r e s s e d  - 82 i n terms o f t h e f r a c t i o n o f c r i t i c a l damping, A, r a t h e r than i n terms of  t h e damping c o e f f i c i e n t , C.  C r i t i c a l damping corresponds t o t h e  damping c o e f f i c i e n t , C^, which w i l l j u s t p r e v e n t o s c i l l a t i o n s o f a f r e e v i b r a t i o n system.  The damping f a c t o r , A, i s the r a t i o n o f t h e  a c t u a l damping c o e f f i c i e n t , C, t o the c r i t i c a l damping c o e f f i c i e n t ,  C , c  i.e. , A =  C  /C  (5.39) c  Very l i t t l e d a t a i s a v a i l a b l e r e l a t i n g t o measured v a l u e s o f . damping i n s o i l s . for  T e s t r e s u l t s p r e s e n t e d by H a l l and R i c h a r t  s m a l l a m p l i t u d e v i b r a t i o n . t e s t s on g r a n u l a r s o i l s i n d i c a t e damping  c o r r e s p o n d i n g t o A = 3%.  Damping v a l u e s were found t o i n c r e a s e w i t h  the a m p l i t u d e o f v i b r a t i o n . ing  (1963)  No attempt was made t o determine i f damp-  was f r e q u e n c y dependent, b u t t h e a u t h o r s s t a t e t h a t such t e s t s on  Amherst sandstone showed t h a t A was independent o f f r e q u e n c y i n t h e dry  c o n d i t i o n but was frequency dependent f o r t h i s same m a t e r i a l i n t h e  moist s t a t e . M a r t i n and Seed (1966) s t a t e t h a t f o r c e d v i b r a t i o n t e s t s on e a r t h dams i n d i c a t e t h a t e q u i v a l e n t v i s c o u s f a c t o r s o f 5 t o 10% a r e o p e r a t i v e d u r i n g low a m p l i t u d e e l a s t i c v i b r a t i o n s .  They s t a t e , however,  t h a t dynamic l a b o r a t o r y t e s t s suggest t h a t f o r t h e magnitude o f s t r a i n s l i k e l y t o o c c u r d u r i n g s e v e r e earthquake m o t i o n , v a l u e s o f about 20% would be more a p p r o p r i a t e . E v i d e n c e w i t h r e g a r d t o t h e type o f damping, v i s c o u s o r s t r u c t u r a l , p r e s e n t i n a dynamic system i s r a t h e r c o n f l i c t i n g .  I t has g e n e r a l l y  been c o n s i d e r e d t h a t f o r s t e e l and c o n c r e t e s t r u c t u r e s , damping i s o f  the s t r u c t u r a l type, i . e . , X i s independent performed of  of f r e q u e n c y .  Tests  by N i e l s o n (1964) i n d i c a t e v a l u e s of the c r i t i c a l damping, X  1 and 2% r e s p e c t i v e l y f o r s t e e l and c o n c r e t e s t r u c t u r e s .  The p e r -  centage of c r i t i c a l damping d i d i n c r e a s e w i t h the f o r c e l e v e l but appeared  to,be independent  of f r e q u e n c y , s u g g e s t i n g s t r u c t u r a l damping  r a t h e r than v i s c o u s damping.  However, more r e c e n t l y N i e l s o n (1966)  d e s c r i b e s r e s u l t s of t e s t s t o determine  the damping m a t r i x wherein t h e  damping does appear t o be of the r e l a t i v e v i s c o u s type. t h a t p o s s i b l y base movement o r poor  He suggests  t e s t i n g technique may have been  responsible f o r h i s previous r e s u l t s . It  seems, t h e r e f o r e , t h a t i n s t e e l and c o n c r e t e s t r u c t u r e s i t i s  not known whether the damping i s of the v i s c o u s or s t r u c t u r a l type, i.e.  frequency dependent o r n o t .  No e v i d e n c e i s p r e s e n t l y a v a i l a b l e t o  the w r i t e r w i t h r e g a r d t o the type of damping p r e s e n t i n e a r t h structures.  It will  t h e r e f o r e be assumed f o r example problems i n t h i s  t h e s i s t h a t damping i s of the v i s c o u s type.  5.6  DETERMINATION OF THE DAMPING MATRIX FROM THE PERCENTAGE OF CRITICAL DAMPING If  the c o u p l e d e q u a t i o n s of motion  s o l v e d , then the damping m a t r i x  (Chapter 5.1) a r e t o be  [C] must be known.  The damping m a t r i x  can be o b t a i n e d from dynamic t e s t s on the s t r u c t u r e ( N i e l s o n , 1964) but if  an e s t i m a t e of the percentage  of c r i t i c a l damping can be made, then  the damping m a t r i x can be determined In  d i r e c t l y without  tests.  Chapter 4.4 i t was s t a t e d t h a t t h e damping m a t r i x  always be n o r m a l i z e d i f i t i s assumed t h a t  [C] can  - 84 [C] = a fM>] + 8  [K]  (5.40)  where a and 8 a r e c o n s t a n t s  and  fMj  =  t h e d i a g o n a l mass m a t r i x  [K]  =  the s t i f f n e s s m a t r i x  The percentage (Eq.  o f c r i t i c a l damping, A^, i n any mode was shown t o be  4.40)  B  03.  1  The form,of  the damping m a t r i x [C] determines  p r e s e n t i n a system.  the type o f damping  I f the damping m a t r i x i s a d i a g o n a l m a t r i x , t h a t th  is  [C] = <x[M], then t h e damping f o r c e on t h e i  the v e l o c i t y o f the i  ^ mass.  mass depends o n l y on  T h i s t y p e . o f damping i s r e f e r r e d t o as  a b s o l u t e damping and can be r e p r e s e n t e d by dashpots c o n n e c t i n g the masses t o t h e base o f the s t r u c t u r e . that w i t h 8 = 0 ,  t h e percentage  p o r t i o n a l t o t h e f r e q u e n c y , w.  From Eq. 5.41 i t may be s e e n ,  o f c r i t i c a l damping i s i n v e r s e l y p r o I n t h e h i g h e r modes the frequency  be h i g h e r and hence t h e percentage  will  o f c r i t i c a l damping w i l l be l o w e r .  I f t h e damping m a t r i x , [ C ] , i s e q u a l t o B [ K ] , then the damping mechanism may be r e p r e s e n t e d by dashpots which i n t e r - c o n n e c t t h e same. masses as those i n t e r - c o n n e c t e d by the s p r i n g s o f the s t i f f n e s s m a t r i x . S i n c e t h e s t r e s s i n a continuum depends on r e l a t i v e  displacements  r a t h e r than d i s p l a c e m e n t s w i t h r e s p e c t to t h e b a s e , t h e system w i l l  be  m o d e l l e d by i n t e r - m a s s s p r i n g s r a t h e r than by s p r i n g s c o n n e c t i n g the th masses t o the base. The damping f o r c e on.the i mass w i l l t h e r e f o r e  - 85 depend on t h e r e l a t i v e v e l o c i t y a d j a c e n t masses.  o f the i * " * mass w i t h r e s p e c t t o 1  The damping mechanism may t h e r e f o r e be r e p r e s e n t e d  by i n t e r - m a s s d a s h p o t s , and t h i s type o f damping i s r e f e r r e d t o as r e l a t i v e - damping. percentage  From Eq. 5.41 i t may be.seen t h a t w i t h a = 0, the  o f c r i t i c a l damping i s p r o p o r t i o n a l to the frequency OJ.  T h e r e f o r e , f o r t h i s type o f damping, t h e percentage  of c r i t i c a l  damping w i l l be h i g h e r i n the h i g h e r modes. ..  E q u a t i o n 5.41 i n v o l v e s o n l y two unknowns a and B and hence  o n l y two independent c o n d i t i o n s can be a p p l i e d .  I f t h e v a l u e s of OK  and X. i n t h e f i r s t two modes a r e used » then i t i s e a s i l y shown t h a t 2toiO)2  a = —2 a>i  2" ( l  w  2 ^l)  (5.42)  2 ( " l *1 "  w  2 * )  (5.43)  w  -  co  2  and 2 B = —2  2  0 ) 1 - 0 ) 2  It i s interesting  t o note t h a t f o r t h e assumption t h a t the damping  m a t r i x i s a l i n e a r c o m b i n a t i o n o f the mass and s t i f f n e s s m a t r i c e s , i t i s o n l y p o s s i b l e t o have t h e percentage  o f c r i t i c a l damping the same  i n any two modes. I n t h e normal mode t h e o r y i t i s o f t e n assumed t h a t t h e p e r c e n t age o f c r i t i c a l damping i s the same i n every mode. can be a c h i e v e d by a s u i t a b l e for this particular  T h i s assumption  c h o i c e o f t h e damping m a t r i x  which,  c a s e , would n o t be a l i n e a r c o m b i n a t i o n o f t h e mass  and s t i f f n e s s m a t r i c e s b u t c o u l d n e v e r t h e l e s s be d i a g o n a l i z e d . I n t h e a n a l y s e s w h i c h f o l l o w i t w i l l be assumed t h a t the  - 86 -  damping i s r e l a t i v e , i . e . , the damping m a t r i x , [C] = 3 [ K ] , where 3 =  .The i n t e r - s t o r e y damping c o e f f i c i e n t s C. a r e t h e r e f o r e g i v e n 1  0)1  by  2A  X  C. =  k.  1  Ai  B]  (5.44)  1  =  f r a c t i o n o f c r i t i c a l damping i n t h e f i r s t mode  coi  =  the l o w e s t n a t u r a l f r e q u e n c y  k_^  =  the  inter-storey spring  stiffness.  Thus t h e damping f a c t o r s can be o b t a i n e d f o r t h e above assumption i f b o t h the p e r c e n t a g e o f c r i t i c a l damping and t h e fundamental of  the system a r e known.  frequency  - 87 CHAPTER 6 APPLICATION OF ELASTIC-VISCOPLASTIC THEORY TO THE DYNAMIC RESPONSE ANALYSIS OF EARTH STRUCTURES  6.1  INTRODUCTION The  response t h e o r y developed i n Chapter 5 a l l o w s  d i s p l a c e m e n t s , v e l o c i t i e s and a c c e l e r a t i o n s type s t r u c t u r e s s u b j e c t  t o be c a l c u l a t e d f o r shear  t o earthquake m o t i o n , where t h e s t r e s s - s t r a i n  r e l a t i o n s h i p s o f t h e m a t e r i a l may be o f a complex n a t u r e . g r a t i o n o f the equations of motion involved c a r r i e d out numerically The  t h e dynamic  The i n t e -  i n t h e a n a l y s i s was  w i t h t h e a i d o f an e l e c t r o n i c d i g i t a l computer.  computer programs, and some p r e l i m i n a r y  t e s t problems performed  to i n s u r e t h a t the programs were w o r k i n g p r o p e r l y ,  are included i n  Appendix I I Results  of the a n a l y s i s are presented i n three s e c t i o n s .  f i r s t s e c t i o n considers of s o i l .  t h e response o f h o r i z o n t a l and s l o p i n g  The  layers  The second s e c t i o n i s concerned w i t h dynamic d i s p l a c e m e n t s o f  e a r t h dams.  The t h i r d s e c t i o n s t u d i e s t h e e f f e c t o f p l a s t i c a c t i o n on  dynamic s e i s m i c c o e f f i c i e n t s . 6.2  DYNAMIC RESPONSE OF HORIZONTAL AND SLOPING LAYERS OF SOIL The  v i s c o e l a s t i c and e l a s t i c - v i s c o p l a s t i c , r e s p o n s e s o f a deep  l a y e r o f c l a y a r e f i r s t compared.  A 300 f o o t deep h o r i z o n t a l l a y e r o f  c l a y o v e r l y i n g b e d r o c k i s shown i n F i g . 18a. The c l a y i s assumed t o have a u n i t w e i g h t , y = 100 p . c . f . ,  a P o i s s o n ' s r a t i o , y = 0.5 and a  88 -  Modulus  F I G . 18a  f  SECTION  AND  PROPERTIES  niio  k.o 1—VvVV  1  2  1—vA'vv—^ m,  1—<vV\Jv*—j  FIG.  Mass  Number  1—wJv?—^ m  OF C L A Y  18b  kip  m.  »  e c  E  In K . S . I.  LAYER  Spring Stiff  ? | t . K In  k  Critical D a m p i n g Cc kip. s e a / f t .  'P/ft.  1  9.32  13(100  7, 907  2  9.32  6,060  3,523  c,  3  M  5,400  3, 210  4  M  4,800  2,832  c  4, 460  2,631  6  4,000  2,360  3,200  1,888  2,400  1,416  2  c,  MODEL  U  5  7  -  8  M  -  9  -  10  AND MODEL  1,600  • 66 472  eoo  PROPERTIES  _  100  200  o  300  0 Yield  F I G . 18c  FORCE — DEFORMATION RELATIONSHIP  FIG.  I8d  1  3  2  Stress  4  In K. S. F.  YIELD VERSUS  STRESS DEPTH  RELATIONSHIP  - 89 Young's modulus v a r i a t i o n , E, as shown i n F i g . 18a. A shear model o f the l a y e r and t h e v a l u e s o f t h e masses, s p r i n g s t i f f n e s s e s and c r i t i c a l damping f a c t o r s a r e shown i n . F i g . 18b.  Spring  damping f a c t o r s a r e o b t a i n e d as e x p l a i n e d  s t i f f n e s s and c r i t i c a l  i n Chapters 5.4 and 5.6. F o r  the e l a s t i c - v i s c o p l a s t i c , a n a l y s i s t h e y i e l d s t r e s s was assumed to be independent o f d i s p l a c e m e n t and t o v a r y l i n e a r l y w i t h depth as shown i n F i g s . 18c and d.  The term y i e l d s t r e s s i s used t o d e s i g n a t e  v e l o c i t y independent component o f t h e shear s t r e n g t h . s t r e s s due t o v i s c o u s  forces  was  the'''  A d d i t i o n a l shear  a l l o w e d f o r by t h e i n c l u s i o n o f damp-  i n g f a c t o r s c o r r e s p o n d i n g t o 5% o f c r i t i c a l i n the e l a s t i c range. . The  responses i n d u c e d a t t h e s u r f a c e  base a c c e l e r a t i o n s  o f t h e c l a y l a y e r due t o  c o r r e s p o n d i n g t o those o f t h e f i r s t 10 seconds o f .  the N-S component o f E l Centro earthquake a r e shown i n F i g s . 19a and b. Fig.  The d i s p l a c e m e n t s o f . t h e s u r f a c e  r e l a t i v e t o t h e base a r e shown i n  19a. I t i s seen t h a t d i s p l a c e m e n t s a r e g r e a t e r when p l a s t i c  deformation occurs.  The maximum d i s p l a c e m e n t i s about 0.6 f t . f o r v i s c o -  e l a s t i c response w i t h A = 5%.  Introduction  of the y i e l d s t r e s s of F i g .  18d  increases  t h e maximum d i s p l a c e m e n t t o 0.9 f t .  and  A = 20%, t h e maximum d i s p l a c e m e n t i s 0.4 f t .  of a y i e l d s t r e s s causes i n c r e a s e d viscous  F o r e l a s t i c response Thus t h e i n t r o d u t i o n  d i s p l a c e m e n t s , whereas  increased  damping causes reduced d i s p l a c e m e n t s . The  absolute accelerations  shown i n F i g . 19b.  of the top of the clay l a y e r are  I t may be seen t h a t the maximum a c c e l e r a t i o n f o r  2 A = 5 % i s 11.5 f t / s e c ., w h i c h i s s l i g h t l y h i g h e r than t h e peak a c c e l e r 2 a t i o n f o r E l Centro (10.1 f t . / s e c . ) . The i n t r o d u c t i o n o f t h e y i e l d /  FIG.  19a  SURFACE TO T H E  DISPLACEMENT 0 - 1 0  SECONDS  OF 3 0 0 OF T H E  FT. CLAY N-S  LAYER  COMPT.  OF  SUBJECTED EL  TO  CENTRO,  BASE  MOTION  1940  CORRESPONDING  EARTHQUAKE.  I FIG.  19b  SURFACE TO  THE  ACCELERATION 0 - 1 0  SECONDS  OF  300  OF T H E  FT CLAY N-S  LAYER  COMPT.  OF  SUBJECTED EL  CENTRO,  TO  BASE  1940  MOTION CORRESPONDING  EARTHQUAKE.  ^ C  - 91s t r e s s e s shown i n F i g . 18d reduces the maximum a c c e l e r a t i o n t o 5.0 2 ft./sec.  .  F o r e l a s t i c response and A = 20%, t h e maximum a c c e l e r a t i o n i s 2  about 5.5 f t , / s e c . .  Thus, f o r t h e p a r t i c u l a r y i e l d s t r e s s e s  chosen,  i n c r e a s i n g t h e damping f a c t o r from 5 t o 20% i n a v i s c o e l a s t i c a n a l y s i s produces a b s o l u t e obtained  accelerations at the surface  comparable t o t h o s e  from an e l a s t i c - v i s c o p l a s t i c a n a l y s i s .  Had lower y i e l d  s t r e s s e s been used, t h e n t h e a c c e l e r a t i o n s would be l o w e r t h a n those shown. absolute  In a v i s c o e l a s t i c analysis increased accelerations.  damping causes d e c r e a s e d  However, i t w i l l s u b s e q u e n t l y be shown t h a t i n  an e l a s t i c - v i s c o p l a s t i c a n a l y s i s t h e i n t r o d u c t i o n o f v i s c o u s a c t u a l l y causes i n c r e a s e d  absolute  a c c e l e r a t i o n s , provided  damping  that  plastic  a c t i o n does o c c u r . The maximum v a l u e s o f s h e a r s t r e s s a t t h e v a r i o u s  levels within  the l a y e r a r e compared i n F i g . 20a. I f p l a s t i c a c t i o n i s t o o c c u r then the v i s c o e l a s t i c s t r e s s e s o b t a i n e d exceed t h e y i e l d s t r e s s e s .  f o r t h e same damping c o e f f i c i e n t s must  The chosen y i e l d s t r e s s v a r i a t i o n w i t h  i s shown i n F i g . 20a and ranges from one q u a r t e r stresses obtained  to three quarters  from t h e v i s c o e l a s t i c a n a l y s i s (A = 5%).  of the  The maximum  dynamic s t r e s s e s f o r the e l a s t i c - v i s c o p l a s t i c system a r e h i g h e r y i e l d s t r e s s e s because t h e dynamic shear s t r e s s i n c l u d e s  depth  than t h e  the viscous  r e s i s t a n c e o f t h e dashpot i n a d d i t i o n t o t h e y i e l d s t r e s s .  This i s  p a r t i c u l a r l y n o t i c e a b l e near t h e t o p where t h e y i e l d s t r e s s i s s m a l l . S i n c e , f o r earthquake m o t i o n . t h e r e a r e , i n f a c t , no e x t e r n a l forces the absolute i s g i v e n by  a c c e l e r a t i o n o f t h e t o p mass o f . t h e s t r u c t u r e model  - 92 -  Yield  Stress  Max.  U.  Stress  Elastic —  Viscoplastic  o ID  <  Max.  100  A = 5  u. %  o < u.  Stress  -Viscoelastic, \  = 5 %  II.  3  Max. S t r e s s Viscoelastic,\ =  CO _J UJ 00  3 co  ? o  5 %  200  _l Ul  Q  V  200  Increases  Linearly  With  Depth  a.  ui a  300*  300  0  I  SHEAR KIP / FIQ. 2 0 a  Strength  ca  0. UJ  100  ac  CC  o  Uniform Strength  ENVELOPES FOR EL  300  FT. L A Y E R  CENTRO  FIG.20C  O F MAX. S H E A R  STRESS  SUBJECTED  FIG.  20b  TO  FORCE-DISPLACEMENT  RELATIONSHIPS.  3  SQ. FT.  SHEAR  STRENGTH  ASSUMPTIONS 300  EARTHQUAKE.  2  STRENGTH  FOR  FT. L A Y E R .  u  where  =  a  / m  (TA)  (6.1)  T  =  the shear s t r e s s at the base of t h e . t o p mass  A  =  the a r e a on w h i c h x  m  =  the top mass  T h e r e f o r e , from Eq.  6.1,  directly proportional  acts  the maximum a c c e l e r a t i o n of the top mass i s  to the maximum shear s t r e s s .  v i s c o p l a s t i c system w i t h no v i s c o u s i s the y i e l d s t r e s s ( p r o v i d e d  I n an  elastic  damping, the maximum shear s t r e s s  y i e l d i n g o c c u r s ) and hence the maximum  a b s o l u t e a c c e l e r a t i o n i s independent of the base a c c e l e r a t i o n . ever, i f viscous  damping i s i n c l u d e d  t h a n the y i e l d s t r e s s .  How-  then the dynamic s t r e s s i s h i g h e r  From F i g . 20a  i t may  be seen t h a t the dynamic  s t r e s s at the base of the top mass i s t w i c e the y i e l d s t r e s s when 5% c r i t i c a l damping i s i n c l u d e d . the top mass i s i n c r e a s e d  Thus the maximum a b s o l u t e a c c e l e r a t i o n  by a f a c t o r o f two  by the i n c l u s i o n o f  of  5%  damping. I n the p r e v i o u s a n a l y s i s i t was v a r i e s l i n e a r l y w i t h depth. be assumed.  I n F i g . 20b,  l i n e a r l y varying  However, any v a r i a t i o n of y i e l d s t r e s s  a u n i f o r m y i e l d s t r e s s e q u a l to t h a t of  case a t the c e n t r e o f the l a y e r i s shown.  the y i e l d s t r e s s may Fig.  assumed t h a t the y i e l d s t r e s s .  be c o n s i d e r e d a . f u n c t i o n  the  In a d d i t i o n ,  of s t r a i n as shown i n  20c where the y i e l d s t r e s s d e c r e a s e s by a f a c t o r of 10 when the  d i m e n s i o n a l d i s p l a c e m e n t , U, i n c r e a s e s softening  may  from 2 to 3, d e p i c t i n g a  non-  strain  material. The  responses at the s u r f a c e  f o r the c o n d i t i o n s  of the 300  of c o n s t a n t s t r e n g t h  and  foot h o r i z o n t a l layer  l i n e a r l y varying  strength  are  1.0  co co  f \  <  Str ength  In e r e a s e s  u.  ^"Uniform  a.  ?  u. o  Linearly  With D e p t h  Strength  (  ( Fig.20 b )  Fig.  20b  —*  1  UJ  y  /  <  "\ s.  \  OT  \  m  \  s  V  )  s  / s —  r-  z O UJ  I-  UJ  uj  z O  >  *5 a ar -  -1.0  UJ  ^Strength -2.0 3  1  I  4  5  TIME  F I Q . 21a  EFFECT  OF  HORIZONTAL  z o  OF  2 % _l UJ  2  I  VARIOUS LAYER  EL CENTRO,  STRENGTH OF  1940  SOIL  Decreases  IN  Displacement  ( Fig. 20c)  1  I 6  7  1 8  10  S E C O N D S .  ASSUMPTIONS  WHEN  With  ON T H E  SUBJECTED  TO  SURFACE  THE  N-S  DISPLACEMENTS  OF A  300  FT.  COMPONENT  EARTHQUAKE.  I  o o o>  <2 UJ  2  o O K _J  co  ao u.  < O  -10  3 TIME  FIG.21b  EFFECT  OF  HORIZONTAL OF  EL  VARIOUS LAYER  CENTRO,  STRENGTH OF SOIL  1940  4  5  IN  SECONDS  ASSUMPTIONS  WHEN  SUBJECTED  EARTHQUAKE.  ON T H E TO  THE  SURFACE N-S  ACCELERATIONS  COMPONENT  OF  A  300  FT.  - 95 compared i n F i g s . 21a and b.  For u n i f o r m s t r e n g t h the r e l a t i v e  dis-  placements a r e somewhat s m a l l e r w h i l e the a b s o l u t e a c c e l e r a t i o n s a r e l a r g e r t h a n f o r the case o f l i n e a r l y v a r y i n g s t r e n g t h .  The  absolute  a c c e l e r a t i o n curve i s more peaked f o r the case of u n i f o r m s t r e n g t h .  This  i s because no p l a s t i c a c t i o n i s o c c u r r i n g i n the top h a l f o f the l a y e r f o r t h i s c o n d i t i o n as w i l l be shown The  subsequently.  s u r f a c e responses f o r a s t r a i n s o f t e n i n g m a t e r i a l as  d e p i c t e d by curve "B" o f F i g . 19c are a l s o shown i n F i g s . 21a and The y i e l d s t r e s s was  assumed to v a r y l i n e a r l y w i t h depth.  b.  Comparing the  responses w i t h those o b t a i n e d from the assumption o f l i n e a r l y v a r y i n g y i e l d s t r e s s but w i t h e l a s t i c p e r f e c t l y p l a s t i c m a t e r i a l i t may that surface displacements ening m a t e r i a l .  The  be seen  a r e c o n s i d e r a b l y g r e a t e r f o r the s t r a i n  soft-  s u r f a c e a c c e l e r a t i o n s on the o t h e r hand a r e s m a l l e r  f o r the s t r a i n s o f t e n i n g m a t e r i a l .  However, the maximum s u r f a c e a c c e l e r -  a t i o n w h i c h o c c u r s a t a time o f 2.2  seconds i s the same f o r b o t h  T h i s i s because a t a time of 2.2  seconds the n o n - d i m e n s i o n a l  o f the top mass, U ^ Q , i s o n l y 1.7. u n t i l U^Q i s g r e a t e r than 2.0, a c c e l e r a t i o n i s not reduced.  displacement  S i n c e the y i e l d s t r e s s does not  the s t r e n g t h and  reduce  t h e r e f o r e the maximum  The n o n - d i m e n s i o n a l d i s p l a c e m e n t ,  g r e a t e r than 3 a t a time of 2.4  cases.  seconds and s u b s e q u e n t l y  the  U^Q, i s  surface  a c c e l e r a t i o n s a r e l e s s than one h a l f . t h o s e o b t a i n e d f o r the  elastic  perfectly plastic material.  considered,  I f no v i s c o u s damping had been  t h e n r e d u c i n g the y i e l d s t r e s s by a f a c t o r o f t e n would a l s o reduce the a c c e l e r a t i o n s by a f a c t o r o f t e n .  So t h a t v i s c o u s damping i s  r e s p o n s i b l e f o r the magnitude o f the a c c e l e r a t i o n s a f t e r 2.4  largely, seconds.  - 96 The  n o n - d i m e n s i o n a l d i s p l a c e m e n t , U, i s the r a t i o o f t h e  t o t a l i n t e r - s t o r e y d i s p l a c e m e n t t o the e l a s t i c i n t e r - s t o r e y t h a t would j u s t cause y i e l d i n g .  The maximum v a l u e o f U i s i n f a c t  i d e n t i c a l t o t h e d u c t i l i t y f a c t o r , u, as d e f i n e d (1960).  displacement  by V e l e t s o s and Newmark  The d u c t i l i t y f a c t o r g i v e s a measure o f t h e amount o f p l a s t i c  deformation which occurs.  D u c t i l i t y f a c t o r s l e s s than o r e q u a l t o  u n i t y imply e l a s t i c a c t i o n , w h i l e d u c t i l i t y f a c t o r s g r e a t e r than unity imply p l a s t i c a c t i o n .  The h i g h e r t h e d u c t i l i t y f a c t o r t h e g r e a t e r t h e  r a t i o o f p l a s t i c a c t i o n compared t o e l a s t i c a c t i o n . Relationships Fig.  22.  between d u c t i l i t y f a c t o r and depth a r e shown i n  For t h e assumption o f u n i f o r m s t r e n g t h ,  the d u c t i l i t y  factor  i s h i g h e s t a t t h e b a s e , a p p r o x i m a t e l y e q u a l t o e i g h t , and l e s s than u n i t y f o r the top h a l f o f the l a y e r .  This implies  t h a t most o f t h e  y i e l d i n g i s t a k i n g p l a c e a t o r near t h e base and no y i e l d i n g i s o c c u r r i n g i n t h e upper h a l f o f t h e l a y e r .  F i n n (1966) c o n s i d e r i n g  v i s c o p l a s t i c material obtained a s i m i l a r r e s u l t . varying  strength,  s o i l t o be a  F o r the case of l i n e a r l y  the d u c t i l i t y f a c t o r i s h i g h e s t a t t h e s u r f a c e ,  being  about 6.5 a t the top mass, d e c r e a s i n g t o 2.5 at a depth o f 100 f t . and 1.8 a t the base o f the l a y e r . p l a c e near the s u r f a c e , levels. 20c  a l t h o u g h some y i e l d i n g does t a k e p l a c e a t a l l  When t h e s t r e n g t h  (curve " B " ) ,  T h e r e f o r e , most o f t h e y i e l d i n g i s t a k i n g  d e c r e a s e s w i t h d i s p l a c e m e n t as shown i n F i g .  then e s s e n t i a l l y a l l y i e l d i n g takes p l a c e i n the top 100  f t . , w i t h maximum y i e l d i n g t a k i n g p l a c e a t t h e s u r f a c e . Newmark • (1965) s u g g e s t s t h a t a deep c l a y l a y e r w i l l respond to earthquake e x c i t a t i o n m a i n l y i n i t s fundamental mode.  An a n a l y s i s was  -  97 -  5 %  50  100  150  UJ  o < u.  oc  OT 2 0 0 o _l UJ  m a. UJ  250  a  300 26  FIG.  22  RELATIONSHIP  BETWEEN  LAYER  SUBJECTED  OF  CENTRO  EL  TO  DUCTILITY 0-10  EARTHQUAKE.  SECS.  FACTOR OF T H E  AND N-S  DEPTH  FOR  COMPONENT  28  300  30  FT.  - 98 performed t o v e r i f y t h i s and determine t h e e f f e c t o f p l a s t i c a c t i o n on t h e f r e q u e n c y component o f the s u r f a c e  accelerations.  A power s p e c t r a l d e n s i t y a n a l y s i s o f E l Centro earthquake ( F i g . 23) i n d i c a t e s t h a t most o f t h e energy i s b e i n g t r a n s m i t t e d of 2 c.p.s.  a t a frequency  The fundamental f r e q u e n c y o f t h e c l a y l a y e r , assuming  v i s c o p l a s t i c m a t e r i a l , i s a 1/2 c.p.s.  A.power s p e c t r a l d e n s i t y  analysis  of t h e s u r f a c e a c c e l e r a t i o n s from t h e v i s c o e l a s t i c a n a l y s i s , F i g . 23, i n d i c a t e s t h a t most o f t h e energy i s b e i n g t r a n s m i t t e d frequency.  a t t h e fundamental  T h i s tends t o v e r i f y Newmark's (1965) s u g g e s t i o n .  s p e c t r a l density a n a l y s i s of the surface a c c e l e r a t i o n s  A power  from t h e e l a s t i c .  v i s c o p l a s t i c a n a l y s i s ( F i g . 23) i n d i c a t e s t h a t most o f the energy i s a l s o being transmitted  a t the fundamental f r e q u e n c y .  Thus p l a s t i c a c t i o n does  not seem t o a l t e r the f r e q u e n c y a t w h i c h most o f t h e energy i s t r a n s m i t t e d . I t c o u l d be c o n c l u d e d from t h i s , t h a t power s p e c t r a l d e n s i t y a n a l y s e s on earthquakes w h i c h have been r e c o r d e d on deep c l a y l a y e r s , w i l l g i v e the fundamental f r e q u e n c y o f t h e s e l a y e r s .  However, d a t a r e v i e w e d  by Wiggins (1964) s u g g e s t s t h a t t h i s i s n o t n e c e s s a r i l y the case.  The  t r a n s m i t t a l o f shock waves appears t o be a more complex problem than i s u s u a l l y assumed f o r a n a l y s i s p u r p o s e s . SLOPING LAYERS OF SOIL The response o f a 50 f t . s l o p i n g l a y e r o f s o i l s u b j e c t  t o a base  m o t i o n c o r r e s p o n d i n g t o t h e f i r s t 10 seconds of t h e N-S component o f E l Centro earthquake i s next examined.  The l a y e r i s shown i n F i g . 24a.  m a t e r i a l i s assumed t o have a u n i t w e i g h t y  =  100 p . c . f . ,  The  u= 0.45 and a  99 NOTE:  4.0  DUCTILITY FACTOR EQUALS 6.5 FOR ELASTIC - VISCOPLASTIC CASE.  3.5  ACCELERATIONS  3.0  AT  SURFACE  OF 300 FT.  CLAY  LAYER ,  VISCOELASTIC , X = 5 %  2.5  UJ *  2.0  UJ  <  /•  ACCELERATIONS  AT  SURFACE  OF 300 FT CLAY  It  S Q.  LAYER,  1.5  ELASTIC-VISCOPLASTIC,  X  =5%  in  oc  UJ S  o a.  1.0  0-10  SECS.  EL CENTRO EARTHQUAKE  (ACCELERATIONS)  0.5  2  FREQUENCY, FIG. 23  4  3  RESULTS  OF  CYCLEs/sEC.  POWER  SPECTRAL  DENSITY  ANALYSES  -  2  F I G . 24a  SLOPING  SOIL  LAYER.'  FIG  24b  4  3  Young's  L3h C,o  m I—vvV^ 10  -y,  NO.  - i  [  m  2  mi X.  c,  7777777777777777777 FIG.24c  SHEAR  MODEL  AND  PROPERTIES  OF  VARIATION  E,  OF  p. S. i .  MODULUS  E,  LAYER.  Mass Stiffness C r i t i c a l m k Damping k i p sec/ft k i p / f t . * kip. sec / f t . Cc  1  1.71  100,000  10,7 3 9  2  1.71  47,300  5, 10 1  3  1.71  43,000  4,832  4  1.71  42,SOO  4,564  5  1.71  40,000  4,293  6  1.71  3 7,300  4 , 0 27  7  1.71  35,000  3,758  S  1.71  30,000  3,22 1  9  1.71  23,000  2,685  10  l.7l  18,000  1,933  50  -  5 x 10  Modulus  WITHIN  100  FT.  SOIL  Stc t i c  LAYER,  Stress  X=  lasti  lasfo - ^ ^ P l a s l ic,  FlO.S.  2.0  =0  2 Shear On  Stress  Planes  and  3  Parallel  FIG?drfSTRESSES  4  Shear to  AND  Strength  0 Shear  1 Stress  3 Planes  4  5  Parallel  6  7  to  the  8 Surface  kp. / s q . ft.  Surface  STRENGTH  2 on  FIG.24e  STATIC  AND  COMBINED  STATIC  PLUS  - 101 Young's modulus v a r i a t i o n as shown.  The l a y e r i s m o d e l l e d by 10 masses,  s p r i n g s and dashpots as i n d i c a t e d by F i g . 24c. The shear s p r i n g s  will  have an i n i t i a l f o r c e c o r r e s p o n d i n g t o t h a t o f s t a t i c shear s t r e s s e s . S t u d i e s p r e s e n t e d by Newmark (1965) and Goodman and Seed (1966) for a single-degree  o f freedom r i g i d p l a s t i c system d i d n o t c o n s i d e r  a v i s c o u s damping term.  R e s u l t s w i l l f i r s t be examined where v i s c o u s  damping i s z e r o , l a t e r t h e e f f e c t o f c o n s i d e r i n g damping w i l l be i n vestigated.  I t w i l l be assumed t h a t t h e earthquake m o t i o n i s p a r a l l e l  t o t h e base o f t h e l a y e r r a t h e r than h o r i z o n t a l . was made by Newmark (1965).  T h i s same assumption  S i n c e t h e v e r t i c a l component o f the e a r t h -  quake i s u s u a l l y n e g l e c t e d when t h e base i s h o r i z o n t a l , i t seems no more unreasonable t o neglect  the normal component when c o n s i d e r i n g a s l o p i n g  base problem. The' s t a t i c shear s t r e s s on p l a n e s p a r a l l e l t o t h e s u r f a c e i s shown i n F i g . 24d t o g e t h e r w i t h v a r i o u s assumed s t r e n g t h s which a r e e x p r e s s e d i n terms o f s t a t i c f a c t o r s o f s a f e t y .  I f the strength varies  l i n e a r l y w i t h depth and would p r o j e c t t h r o u g h t h e o r i g i n , of s a f e t y i s t h e same on every p l a n e .  I f t h e s t r e n g t h i s u n i f o r m , then  the f a c t o r o f s a f e t y i s l e a s t a t t h e base. shown i n F i g . 24e.  Maximum dynamic s t r e s s e s a r e  The maximum s t r e s s o c c u r s a t t h e base and f o r e l a s t i c  2 response i s 7.6 k p / f t . . viscous  then t h e f a c t o r  For e l a s t i c - v i s c o p l a s t i c  damping i s c o n s i d e r e d  to t h e y i e l d s t r e s s e s .  r e s p o n s e , s i n c e no  t h e maximum dynamic s t r e s s e s a r e i d e n t i c a l  The s t a t i c s t r e s s a t t h e base o f t h e l a y e r i s  2 2.2 k p . / f t . , thus f o r a s t a t i c f a c t o r o f s a f e t y o f 2.0 t h e y i e l d 2 would be 4.4 k p . / f t . , as shown i n F i g . 24d.  stress  - 102 -  The d i s p l a c e m e n t s o f t h e t o p s u r f a c e f o r t h e v a r i o u s assumptions i s shown i n F i g . 25.  strength  I t i s seen t h a t t h e magnitude o f t h e d i s -  placement i s v e r y s e n s i t i v e t o t h e y i e l d s t r e s s e x p r e s s e d i n terms o f s t a t i c factor of safety.  The maximum d i s p l a c e m e n t t h e r e f o r e , depends on  the s t a t i c f a c t o r o f s a f e t y and t h i s i s shown i n F i g . 26a. I t may be seen t h a t f o r t h e g i v e n s l o p e and e a r t h q u a k e , i f t h e s t a t i c f a c t o r o f s a f e t y i s g r e a t e r than 2, s u r f a c e d i s p l a c e m e n t s w i l l , be s m a l l .  The s u r f a c e  dis-  placements i n c r e a s e r a p i d l y as t h e f a c t o r o f s a f e t y drops below 2 and become v e r y l a r g e as t h e f a c t o r o f s a f e t y approaches u n i t y . The r e l a t i o n s h i p between d u c t i l i t y f a c t o r and depth i s shown i n F i g . 26b f o r some o f t h e above s t r e n g t h assumptions.  F o r t h e case o f  u n i f o r m s t r e n g t h and a f a c t o r o f s a f e t y = 1.25 a t t h e base, t h e d u c t i l i t y f a c t o r has a maximum v a l u e o f 250 a t t h e bottom o f t h e l a y e r and d e c r e a s e s r a p i d l y , so t h a t a l l y i e l d i n g i s e s s e n t i a l l y c o n f i n e d  t o t h e base.  Where t h e s t r e n g t h i s assumed t o be l i n e a r w i t h depth t h e d u c t i l i t y f a c t o r i s highest  a t t h e s u r f a c e a l t h o u g h y i e l d i n g does take p l a c e throughout t h e  depth o f t h e l a y e r . The e f f e c t o f v i s c o u s shown i n F i g . 27.  damping on t h e s u r f a c e d i s p l a c e m e n t s i s  F o r a f a c t o r o f s a f e t y o f 1.5 t h e i n t r o d u c t i o n o f  v i s c o u s damping c o r r e s p o n d i n g t o X = 5% reduces t h e maximum d i s p l a c e m e n t from 0.64 f t . t o 0 . 3 . f t . , and would have a p p r o x i m a t e l y t h e same e f f e c t on d i s p l a c e m e n t as i n c r e a s i n g t h e f a c t o r o f s a f e t y t o 1.75. ( F i g . 2 6 a ) . For a s t r a i n s o f t e n i n g m a t e r i a l t h e y i e l d s t r e s s i s a f u n c t i o n o f s t r a i n and t h e f o r c e d e f o r m a t i o n curve may be as shown i n t h e i n s e r t o f F i g . 28.  I f t h e peak y i e l d f o r c e , Q , c o r r e s p o n d s t o a s t a t i c f a c t o r o f  INCLINED  LAYER OF S O I L WHEN  S U B J E C T E D TO T H E N - S  COMPONENT  OF  EL CENTRO  EARTHQUAKE  '  - 104 -  F.O.S.  0  0.2  0.4  MAX.  FIG. 2 6 a  DISPLACEMENT  RELATIONSHIP SLOPING EL  0.6  CENTRO  0.8  OF  BETWEEN  LAYER  = 1.25  TOP  TO  Base,  SURFACE  IN  0-10  SECS.  Uniform  Strength  K4  1.2  1.0  FACTOR OF S A F E T Y  SUBJECTED  at  FT.  AND OF  DISPLACEMENT N-S  FOR  COMPONENT  50  FT.  OF  EARTHQUAKE.  0  100  F I G . 26b  RELATIONSHIPS LAYER EL  FACTOR  BETWEEN  DUCTILITY  SUBJECTED  CENTRO  200  DUCTILITY  TO  0-10  EARTHQUAKE.  SECS.  300  /x FACTOR OF  THE  AND N-S  DEPTH  FOR  COMPONENT  50 OF  FT.  SLOPING  -  SOT  -  - 106 s a f e t y o f 1.5 t h e n an u l t i m a t e y i e l d f o r c e , to a f a c t o r o f s a f e t y o f 1.25.  = .8330^ w i l l c o r r e s p o n d  Thus, as t h e n o n - d i m e n s i o n a l d i s p l a c e m e n t ,  U, i n c r e a s e s from 5 t o 10, t h e s t a t i c f a c t o r o f s a f e t y drops from 1.5 to 1.25.  The d i s p l a c e m e n t s o f t h e t o p s u r f a c e f o r such an assumption a r e  shown i n F i g . 28.  As e x p e c t e d , they range between those o b t a i n e d f o r  e l a s t i c p e r f e c t l y p l a s t i c m a t e r i a l and f a c t o r s o f s a f e t y o f 1.25 and 1.5. A comparison o f v i s c o e l a s t i c and e l a s t i c - v i s c o p l a s t i c  responses  o f t h e h o r i z o n t a l and s l o p i n g l a y e r s o f s o i l examined i n d i c a t e s t h e following points: 1.  I n c r e a s e d v i s c o u s damping causes reduced d i s p l a c e m e n t s whereas p l a s t i c y i e l d i n g causes i n c r e a s e d d i s p l a c e m e n t s .  2.  F o r t h e case o f v i s c o e l a s t i c r e s p o n s e i n c r e a s e d v i s c o u s damping causes a c c e l e r a t i o n s t o be reduced.  However, f o r  e l a s t i c - v i s c o p l a s t i c r e s p o n s e , i n c r e a s e d v i s c o u s damping causes i n c r e a s e d a c c e l e r a t i o n s . 3.  A l o w e r y i e l d s t r e s s causes i n c r e a s e d d i s p l a c e m e n t s and reduced a c c e l e r a t i o n s .  The above p o i n t s suggest t h a t v i s c o u s damping and p l a s t i c a c t i o n may have o p p o s i t e e f f e c t s on t h e dynamic response o f e a r t h s t r u c t u r e s . Therefore,  t h e concept o f u s i n g i n c r e a s e d v i s c o u s damping t o a l l o w f o r  the energy d i s s i p a t e d i n p l a s t i c y i e l d i n g as suggested by Ambraseys (1960) and Seed (1966) may g i v e erroneous i n e r t i a f o r c e r e s u l t s . 6.3  DYNAMIC RESPONSE OF AN EARTH DAM The v a r i o u s t y p e s o f m o t i o n t o be e x p e c t e d i n an e a r t h o r r o c k -  0  I  2  3  5  4  6  7  8  9  TIME IN SECONDS FIG.28  EFFECT  ON  SURFACE  DISPLACEMENTS  CAUSED  BY STRENGTH  10 ,  VARYING  WITH  DISPLACEMENT.  M  o SOFT.  INCLINED  LAYER  SUBJECTED  TO T H E N - S C O M P O N E N T  OF E L CENTRO  EARTHQAKE.  ,  - 108 f i l l dam subjected to earthquake motion were discussed i n Chapter 3. I t was suggested by Newmark (1965) that the character of the motion depends on the type of m a t e r i a l .  I f y i e l d i n g i s confined to a s i n g l e  plane or curved surface, then block type movement occurs.  Whereas, i f  displacements occur throughout the dam, a general deformation w i l l exist.  Results presented  condition  from the above studies on h o r i z o n t a l and  sloping layers of s o i l i n d i c a t e that the strength parameters d i c t a t e where movement of y i e l d i n g takes place.  The method used herein f o r e s t -  imating the dynamic displacements of earth dams i s an extension of the p r i n c i p l e s used f o r h o r i z o n t a l and sloping layers and consequently does not require an assumption with regard to the character of motion. A 285 f t . high e a r t h - f i l l  dam with side slopes 2.1 and over-,  l y i n g bedrock i s shown i n F i g . 29a. The m a t e r i a l i s assumed to have a u n i t weight, y = 130 p.c.f., a Poisson's r a t i o , u = 0.4 and a v a r i a t i o n i n modulus, E, as shown.  The dam i s modelled by 11 masses to the l e f t .  of the d i v i s i o n and 11 masses to the r i g h t of the d i v i s i o n connected by i n t e r - s t o r e y springs and dashpots and also by h o r i z o n t a l springs as shown i n F i g . 29b. I t w i l l be i n i t i a l l y assumed that l e f t and r i g h t hand masses at any l e v e l are equal, and values f o r the masses, spring constants and c r i t i c a l damping factors are tabulated f o r t h i s condition (Table I ) . This assumption corresponds to considering the dam as divided down the centre l i n e and held apart by h o r i z o n t a l springs which w i l l be i n i t i a l l y i n compression to represent the s t a t i c s t r e s s condition.  An estimate  of the h o r i z o n t a l spring s t i f f n e s s may be made by assuming l e f t and r i g h t hand masses f o r any h o r i z o n t a l s l i c e are separated by a distance L/2  a ro  TJ  O  o  o  ro co o  x  X  m H X  N O O o  > r co TJ X  •n ©  o o -o  I  X  ro  t-  m  I  r  CD  z -I > o co — zz H —  -n -n  —-  J-  ro  or  73,  CO  o m  o -n  \ ro  2  O o m  m J> x  r  o  ©  z  > 2  m > x H X  m  to co  D  Tl  Shear  Stress  a  Plane  in  P. S.  on I.  Horizontal  > 2  285  Height  2  in  FT.  ooo  Ol  o  O  ro ro CD O oi  -110-  TABLE I PHYSICAL PROPERTIES ASSOCIATED WITH THE EARTH DAM MODEL of FIG 29b. Values associated with masses on the left-hand side Shear Stiffness k kip/ft.  Horizontal Stiffness kip/ft.  Critical damping kip, sec/ft.  Average Static shear stress kip/ft-*  No.  Mass m kip/sec /ft  1  58.0  201,500  542  49,892  - 2.81  2  53.0  98,500  591  24,389  - 2.52  3  48.0  85,500  560  21,170  -2.44  4  42.8  72,000  622  19,065  - 2.01  5  37.8  61,000  632  15,104  - 1.56  6  32.8  53,500  722  13,247  -  7  27.7  40,500  737  10,028  - 0.94  8  22.7  34,000  884  8,418  - 0.89  9  17.6  23,000  965  5,695  - 0.58  10  12.6  17,100  1287  4,234  - 0.50  11  6.5  5,500  1235  1,362  - 0.34  2  1.29  - Ill ( F i g . 29c) and connected by an a x i a l f o r c e member o f Length L/2 and a r e a d whose s t i f f n e s s ik  where  E  =  h  =  — 2  d  i s t h a t o f t h e s o i l , namely /ft o\ (6.2)  E  Young's modulus o f t h e s o i l .  Other assumptions w i l l l a t e r be made w i t h r e g a r d t o t h e s t i f f n e s s o f t h e h o r i z o n t a l s p r i n g s t o determine t h e range o f i t s i n f l u e n c e on d i s p l a c e ments . The s t a t i c shear s t r e s s e s a t t h e base o f each h o r i z o n t a l  slice  were determined from t h e f i n i t e element a n a l y s i s u s i n g t h e same v a l u e s o f E and u as used f o r t h e d y n a m i c . a n a l y s i s . The shear s t r e s s u t i o n on a t y p i c a l h o r i z o n t a l p l a n e i s shown.in F i g . 29d.-  distrib-  The average  shear s t r e s s a t t h e base o f each h o r i z o n t a l s l i c e t o t h e l e f t o f t h e c e n t r e l i n e was c a l c u l a t e d and i s shown t a b u l a t e d i n T a b l e I . The i n t e r - s t o r e y s p r i n g f o r c e i s t h e average  shear s t r e s s m u l t i p l i e d by t h e  a r e a o v e r w h i c h i t a c t s and t h e h o r i z o n t a l s p r i n g f o r c e i s t h e d i f f e r ence between t h e shear f o r c e a t t h e top and bottom o f a s l i c e ( F i g . 3 0 ) . The i n t e r - s t o r e y damping c o e f f i c i e n t s , C^, a r e determined from v i s c o e l a s t i c c o n d i t i o n by assuming t h a t l e f t and r i g h t masses may o s c i l l a t e i n d e p e n d e n t l y as s i m p l e shear s t r u c t u r e s .  F i v e p e r cent o f  c r i t i c a l damping was used f o r a l l e l a s t i c v i s c o p l a s t i c a n a l y s e s . The m a t e r i a l was c o n s i d e r e d t o have a s t r e n g t h c o r r e s p o n d i n g t o one o f t h e f o l l o w i n g 1.  assumptions:  The shear s t r e n g t h i s u n i f o r m throughout t h e dam. corresponds t o purely cohesive m a t e r i a l .  This  - 112 -  LEFT  RIGHT  Qi+I  QI+I  }  (  Qh-H  Qi  FIG.  30  STATIC  FORCES  4  YIELD 285  ON T Y P I C A L  8  YIELD  31  tt„-Q  Equilibrium  1  +  r  Q,  Qi  0  FIG.  From  STRESS  OF S A F E T Y  12  STRESS  FT. HIGH  ELEMENTS.  IN  VERSUS  EARTH  = 1.5 .  16  20  K.S.F.  DEPTH  DAM  RELATIONSHIP  HAVING  A  STATIC  FOR FACTOR  - 113  2.  -  The shear s t r e n g t h depends on the normal s t r e s s and c o r r e s p o n d s to p u r e l y f r i c t i o n a l m a t e r i a l .  The shear s t r e n g t h o r y i e l d s t r e s s i s assumed t o be independent o f r e l a t i v e d i s p l a c e m e n t and number o f s t r e s s c y c l e s and t o c o r r e s p o n d t o a s t a t i c f a c t o r of safety of.1.5.  For purely cohesive m a t e r i a l the 2  s t r e n g t h r e q u i r e d f o r a f a c t o r o f s a f e t y o f 1.5 i s 6.8 k i p / f t . (1948), and f o r a f r i c t i o n a l m a t e r i a l t h e s t r e n g t h r e q u i r e d t o t a n <j> = .75.  Taylor  corresponds  Average assumed s t r e n g t h v e r s u s depth r e l a t i o n s h i p s  are shown i n F i g . 31. The r e l a t i v e d i s p l a c e m e n t s o f t h e c r e s t o f the dam t o t h e l e f t and r i g h t o f the c e n t r e l i n e due t o a base a c c e l e r a t i o n c o r r e s p o n d i n g to t h e f i r s t 10 seconds o f t h e N-S component o f E l Centro earthquake are shown i n F i g . 32a f o r the p u r e l y c o h e s i v e c o n d i t i o n .  I t may be seen  t h a t from 0 - 2  together.  s e e s , t h e l e f t and r i g h t hand s i d e s move  However, a f t e r 2 s e e s , the movement o f t h e masses no l o n g e r c o i n c i d e due to the occurrence of p l a s t i c y i e l d .  W h i l e the masses move t o g e t h e r t h e  f o r c e i n t h e top h o r i z o n t a l s p r i n g remains c o n s t a n t  (Fig.32b).  As  p l a s t i c y i e l d o c c u r s the h o r i z o n t a l s p r i n g f o r c e i s reduced and a t 2.8 s e e s , the r e l a t i v e d i s p l a c e m e n t has been s u f f i c i e n t t o cause the top s p r i n g f o r c e t o drop t o z e r o .  The c o n d i t i o n o f zero a l l o w a b l e  tension  was used, c o n s e q u e n t l y the h o r i z o n t a l f o r c e cannot drop below z e r o .  Where  the d i s p l a c e m e n t s o f the top masses c o i n c i d e o r n e a r l y c o i n c i d e the f o r c e r e t u r n s to t h e h o r i z o n t a l s p r i n g .  A r e l a t i v e d i s p l a c e m e n t between the  top masses s u f f i c i e n t . t o  cause t h e h o r i z o n t a l s p r i n g f o r c e t o drop t o  zero c o u l d be c o n s i d e r e d  as a c r a c k i f the m a t e r i a l cannot t a k e t e n s i o n .  UJ CO  < ffi UJ  >  < _1  Ul  cc  CO CO  <  o. o  -  u  H  o  4 0. CO 3 TIME 32a  FIG.  DISPLACEMENTS  © z  N-S  COMPONENT  RESULTING OF  E L  4 IN  AT CREST OF  CENTRO  5  SECONDS 285  FT  EARTHQUAKE.  HIGH  DAM  WHEN THE  SYMMETRICAL  BASE  DIVISION AND  IS  SUBJECTED  COHESIVE  TO  THE  MATERIAL.  cc  a. co  HORIZONTAL  N  a. o  STIFFNESS  AS  PER  TABLE  I  60  cc o X  SPRING  c o a  40  20  UJ o ac o u.  FIG  3 TIME 32b  F O R C E IN T O P H O R I Z O N T A L  SPRING  IN AS  10  4 SECONDS A FUNCTION OF  TIME.  - 115 A f t e r 4.1 s e e s , t h e f o r c e i n the top 5 s p r i n g s was i n d i c a t i n g a c r a c k to a depth of 135 f e e t . at  found to remain z e r o  The w i d t h of r e s i d u a l c r a c k  the c r e s t a t the end o f 10 s e e s , o f s h a k i n g would be a p p r o x i m a t e l y  1/2 i n . S i n c e the s t r e n g t h of the m a t e r i a l was  assumed c o n s t a n t f o r the  above a n a l y s i s and s i n c e both s t a t i c and d y n a m i c - s t r e s s e s can be  expected  to be h i g h e r near the base, i t would be r e a s o n a b l e to assume t h a t most o f the y i e l d i n g would t a k e p l a c e a t o r towards  the base o f the dam.  33 shows t h a t h i g h e r d u c t i l i t y f a c t o r s o c c u r a t the base,  Fig.  indicating  t h a t i n d e e d most o f the y i e l d i n g takes p l a c e a t the base of the dam.  It  might t h e n be e x p e c t e d t h a t c r a c k s s h o u l d develop from the base o f the dam  r a t h e r t h a n from the t o p .  However, d i s p l a c e m e n t s a r e o b t a i n e d by  summation o f a l l the r e l a t i v e d i s p l a c e m e n t s from the base upward, hence the d i s p l a c e m e n t s w i l l always be l a r g e r towards  the top and t h e r e f o r e ,  c r a c k s w i l l o c c u r a t the top even though y i e l d i n g i s o c c u r r i n g a t the base. The d i s p l a c e m e n t s o f the top o f the dam i a l a r e shown i n F i g . 34a. placement  I t may  f o r a f r i c t i o n a l mater-  be seen t h a t the d i f f e r e n t i a l  dis-  between the top masses i s not as g r e a t as f o r c o h e s i v e m a t e r i a l .  The f o r c e i n the top h o r i z o n t a l s p r i n g i s shown i n F i g . 34b. drops to z e r o as y i e l d i n g p r o c e e d s .  The f o r c e s i n the top two  I t again horizontal  s p r i n g s were found to drop t o z e r o i n d i c a t i n g a c r a c k e x t e n d i n g to a depth of 60 f e e t .  The w i d t h of the c r a c k a t the c r e s t a f t e r 10 s e e s , o f  s h a k i n g would be 1/3 i n c h e s . I t appears r e a s o n a b l e to have c r a c k s i n c o h e s i v e m a t e r i a l , a l though p o s s i b l y some t e n s i l e s t r e n g t h s h o u l d then be a l l o w e d .  However,  - 116 -  FIG.  33  RELATIONSHIP 285 N-S  FT.  HIGH  BETWEEN EARTH  COMPONENT  OF  DUCTILITY  DAM  FACTOR  SUBJECTED  EL CENTRO  1940  TO  AND 10 S E C  DEPTH OF  EARTHQAKE.  FOR  THE  3 TIME FIG.  DISPLACEMENTS  34o  N-S  COMPT.  RESULTING  OF EL  4 IN  5  SECONDS  AT C R E S T OF 285  CENTRO  F T HIGH  DAM  EARTHQUAKE. MATERIAL  WHEN  THE  ASSUMED  BASE  TO B E  IS  SUBJECTED  TO  THE  FRICTIONAL.  e>  ~z  80  cc  0. CO IM  5  o  I  HORIZONTAL  SPRING  STIFFNESSES  AS  PER  TABLE  I  60 £  40  0. o  E  20  tu o o  FIG.  0  34b  FORCE  I  IN  2  TOP HORIZONTAL  3 TIME SPRING  IN AS  4 SECONDS A  5  6  FUNCTION OF TIME .  10  - 118 i n p u r e l y f r i c t i o n a l m a t e r i a l v e r t i c a l c r a c k s would n o t be p e r m i s s a b l e . I n a c t u a l f a c t , t h e h o r i z o n t a l f o r c e on a v e r t i c a l p l a n e , a l t h o u g h i t might drop t o zero f o r a s h o r t i n s t a n t o f t i m e , would not remain z e r o . Hence i n g r a n u l a r m a t e r i a l , c r a c k s which develop d u r i n g earthquake be e x p e c t e d t o c l o s e up a f t e r t h e earthquake motion ceases. d u r i n g an e a r t h q u a k e ,  codld  Even  downward motion o f m a t e r i a l may tend t o " f i l l  c r a c k s " and p r e v e n t t h e h o r i z o n t a l f o r c e d r o p p i n g t o z e r o , o r a l l o w i t t o drop t o z e r o o n l y f o r a v e r y s h o r t i n s t a n t o f time. I f t h e dam i s d i v i d e d a l o n g a l i n e w i t h a 1:1 s l o p e as shown i n F i g . 35a, t h e n t h e masses to t h e l e f t and r i g h t a t any l e v e l w i l l not be e q u a l .  The s p r i n g s t i f f n e s s and damping c o e f f i c i e n t s w i l l a l s o be  d i f f e r e n t , but these a r e r e a d i l y c a l c u l a t e d and a r e shown i n Table I I . A g a i n y i e l d s t r e n g t h s c o r r e s p o n d i n g to a s t a t i c f a c t o r o f s a f e t y o f 1.5 f o r b o t h a p u r e l y c o h e s i v e and p u r e l y f r i c t i o n a l m a t e r i a l were chosen and these a r e shown i n F i g . 35b.  I t i s seen t h a t f o r a f r i c t i o n a l m a t e r i a l ,  s i n c e t h e weight t o a r e a r a t i o i s not t h e same f o r an  unsymmetrical  d i v i s i o n , t h e s t r e n g t h s l e f t and r i g h t w i l l not be t h e same.  The s t a t i c  s h e a r s t r e s s e s c a l c u l a t e d from t h e f i n i t e element a n a l y s i s a r e n o t u n i f o r m as was shown i n F i g . 29d„  Hence t h e average s t a t i c shear s t r e s s e s w i l l  not be t h e same l e f t and r i g h t f o r an unsymmetrical  division.  In Fig.  35c i t may be seen t h a t t h e average s t r e s s t o t h e l e f t of t h e d i v i s i o n i s a p p r o x i m a t e l y t h r e e times t h a t to the r i g h t . The dynamic d i s p l a c e m e n t s of t h e c r e s t of t h e dam f o r t h e assumption  of c o h e s i v e m a t e r i a l and an unsymmetrical  d i v i s i o n were found  to be v e r y s i m i l a r to those o b t a i n e d from the s y m m e t r i c a l  division.  - 119 -  •*—  U_ 2 8 5  c  <v in 2 0 0 a  m  a> > o n  <  T  Material Right Sides  Cohesive Left and  -Right Side  100  Left Side-  .C  Frictional  '55  Material  o>  I  0  4  Yield  8  12  16  20  Stresses on Horizontal Planes, K.S. F  Fig 35b Yield Stresses for Unsymmetrical Division of Dam  fr  285  w 200 CD 0)  Left S ide  > o  .O <  100  V Right Side 1. 0  1  Static  2  3  4  5  Shear Stresses on Horizontal Planes, K.S.F  Fig35c Static Shear Stresses for Unsymmetrical  Division of Dam  -  120 -  TABLE II PHYSICAL PROPERTIES A S S O C I A T E D WITH THE U N S Y M M E T R I C A L D I V I S I O N O F D A M (Fig. 35a)  No.  Mass Shear Stiffness m k kip/sec^/ft kip/ft.  Horizontal stiffness kip/ft. .  Critical ddmping kip.sec.ft.  Average Static shear stress kip/ft 2  1  29.0  101,000  541  •24,900  - 3.60  2  26.7  49,300  590  12,200  - 3.23  3  24.0  42,800  560  10,600  - 2.88  4  21.5  38,500  622  9,520  - 2.42  5  18.9  30,500  632  7,550  - 1.93  6  16.3  26,700  722  6,600  - 1.63  7  13.9  20,300  736  5,000  - 1.53  8  11.3  17,000  884  4,220  - 1.15  9  8.8  11,500  965  2,850  - 1.00  10  6.3  8,540  1287  2,110  - 0.82  11  3.3  2,750  1235  680  - 0.65  N O T E : Above values of mass, shear stiffness, c r i t i c a l damping and average static shear stress are associated with left side. Values for the right side are three times those for the left side. '  - 121 Thus c a l c u l a t e d d i s p l a c e m e n t a t t h e c r e s t a r e e s s e n t i a l l y independent o f the d i v i d i n g l i n e chosen.  S i n c e observed f i s s u r e s i n dams have g e n e r a l l y  been v e r t i c a l , i t would appear more r e a l i s t i c to have a v e r t i c a l o f t h e dam.  division  A n a l y s i s i n d i c a t e s t h a t f o r homogeneous c o h e s i v e m a t e r i a l ,  no one p a r t i c u l a r v e r t i c a l p l a n e i s more l i a b l e to c r a c k i n g than any other.  Thus, a number o f f i s s u r e s r a t h e r t h a n a s i n g l e c r a c k w i l l  likely  occur i n p r a c t i c e . The dynamic d i s p l a c e m e n t s a t t h e c r e s t o f t h e dam f o r f r i c t i o n a l m a t e r i a l a r e shown i n F i g . 36.  I t may be seen t h a t t h e d i f f e r e n t i a l  dis-  placement a t t h e end o f 10 seconds o f s h a k i n g i s a p p r o x i m a t e l y t e n times h i g h e r than f o r t h e case o f a s y m m e t r i c a l d i v i s i o n ( F i g . 3 4 a ) . A f t e r a time o f 1.8 seconds t h e top s u r f a c e t o t h e l e f t and r i g h t o f t h e d i v i s i o n no l o n g e r move t o g e t h e r .  Y i e l d i n g causes t h e dam t o spread and t h e  h o r i z o n t a l f o r c e s i n t h e top 6 h o r i z o n t a l s p r i n g s t o drop t o z e r o i n t h e 10 second p e r i o d . 160'.  T h i s c o u l d be i n t e r p r e t e d as a c r a c k t o t h e depth o f  The w i d t h o f t h e c r a c k a t t h e s u r f a c e a f t e r t h e 10 second  shaking  p e r i o d would be 4". I t was mentioned e a r l i e r t h a t a c r a c k o r f i s s u r e i s not compatible with ideal f r i c t i o n a l material. i n t r o d u c t i o n of a t e n s i o n l i m i t .  The concept o f a c r a c k a r i s e s by t h e  However, i f t h e h o r i z o n t a l s p r i n g s t i f f -  ness i s made v e r y l o w , then l a r g e d i f f e r e n t i a l d i s p l a c e m e n t s w i l l o n l y have a v e r y s m a l l e f f e c t on t h e f o r c e i n t h e s p r i n g .  The s p r i n g f o r c e ,  Q^, a t any t i m e , t , i s g i v e n by Q, = Q + k6 h ^s x  where  Q  =  the s t a t i c f o r c e i n the s p r i n g  (6.3)  UNSYMMETRICAL  DIVISION  OF DAM  0.5  UJ  CO  <  CD  UJ  >  < _l  Id  tr  co co < OL  o  u. o UJ  z  UJ  o < _l  0. CO -0.6 4  5  TIME  FIG. 36  DISPLACEMENTS N - S  RESULTING  COMPONENT  AT C R E S T  OF E L C E N T R O  OF  to  6  IN S E C O N D S  285  EARTHQUAKE.  FT. HIGH  DAM WHEN  UNSYMMETRICAL  BAS E S UB JE CT E D DIVISION  AND  TO  T H E  FR ICTIONAL  MATERIAL.  r-o I  6  =  the d i f f e r e n t i a l d i s p l a c e m e n t between l e f t  and  r i g h t masses k  =  the h o r i z o n t a l s p r i n g  stiffness  I f the h o r i z o n t a l s p r i n g s t i f f n e s s i s put e q u a l to z e r o , then the a t a l l times remains c o n s t a n t  and e q u a l to the s t a t i c f o r c e .  would be a s e v e r e c o n d i t i o n and  force  This  c o u l d be.expected to produce l a r g e  displacements. The  dynamic d i s p l a c e m e n t s f o r the assumptions of zero  t a l s p r i n g s t i f f n e s s are shown i n F i g . 37. and  r i g h t s i d e are i d e n t i c a l up t o 1.8  displacements diverge.  The  d i s p l a c e m e n t s of  seconds, a f t e r w h i c h time  At the end of the 10 second s h a k i n g  r e s i d u a l d i s p l a c e m e n t at the top of the l e f t s i d e i s 1.32 r i g h t s i d e i s zero.  downward and p r e s e r v e s a c o n s t a n t The  the  dropped  as a  crack.  l a t e r a l force.  r e l a t i o n s h i p between d u c t i l i t y f a c t o r and I t may  the y i e l d i n g t a k e s p l a c e near the top of the dam. c o n s i d e r a b l y more y i e l d i n g t a k e s p l a c e i f the dam The  the  s p r e a d s , m a t e r i a l a c t u a l l y moves  f r i c t i o n a l m a t e r i a l i s shown i n F i g . 38,  rically.  the  f t . and a t  to zero t h i s d i f f e r e n t i a l d i s p l a c e m e n t cannot be c o n s i d e r e d t h a t as the dam  left  period,  S i n c e the h o r i z o n t a l s p r i n g f o r c e has not  I t c o u l d be s p e c u l a t e d  horizon-  depth f o r  be seen t h a t most of A l s o , as  expected,  i s d i v i d e d unsymmet-  a n a l y s i s suggests t h a t the c l o s e r the d i v i s i o n i s brought  t o e i t h e r the upstream or downstream f a c e , the more s e v e r e the c o n d i t i o n . T h i s i n d i c a t e s t h a t f o r a f r i c t i o n a l m a t e r i a l , a s u r f a c e or near s u r f a c e s l o p e movement i s most l i k e l y .  T h i s i s i n agreement w i t h  mode of f a i l u r e o u t l i n e d by Newmark f o r g r a n u l a r m a t e r i a l .  the  However, the  UNSYMMETRICAL  DIVISION FRICTIONAL  TIME  FIG.  37  DISPLACEMENTS N-S  COMPONENT  RESULTING OF  AT CREST OF  EL CENTRO  MATERIAL  IN S E C O N D S  285 FT  E A R T H Q U A K E .  HIGH  DAM WHEN  UNSYMMETRICAL  T H E BASE DIVISION  SUBJECTED AND  TO T H E  FRICTIONAL  MATERIAL. I  - 125 FRICTIONAL STATIC  FIG.  38  RELATIONSHIP 285  F T . HIGH  THE  N - S  F.O.S.  BETWEEN EARTH  =  DUCTILITY  DAM  COMPONENT  MATERIAL  OF  1.5.  FACTOR  SUBJECTED EL  CENTRO  TO  10  AND  DEPTH  SECONDS  EARTHQUAKE.  OF  FOR  - 126 -  theory presented here i s r e s t r i c t e d  to horizontal  displacements,  whereas t h e a c t u a l d i s p l a c e m e n t s w i l l a l s o have downslope components. The maximum d i f f e r e n t i a l d i s p l a c e m e n t s between l e f t and r i g h t masses f o r a l l cases examined a r e shown p l o t t e d v e r s u s h e i g h t i n F i g . 39. I t may be seen t h a t t h e d i f f e r e n t i a l d i s p l a c e m e n t s symmetrical d i v i s i o n  and f o r c o h e s i v e m a t e r i a l .  are small for a  F o r an  unsymmetrical  s u b d i v i s i o n and f r i c t i o n a l m a t e r i a l t h e d i f f e r e n t i a l d i s p l a c e m e n t s a r e zero i f t h e h o r i z o n t a l s p r i n g s a r e assumed t o be r i g i d and a maximum o f 1.32 f t . i f t h e h o r i z o n t a l s p r i n g s a r e assumed t o be v e r y  flexible.  I n t h e f o r e g o i n g examples, a l t h o u g h the m a t e r i a l was c o n s i d e r e d to v a r y w i t h depth, i t was assumed t o be homogeneous on h o r i z o n t a l planes.  However, t h e method o f a n a l y s i s i s a l s o thought  to be.applic-  a b l e t o t h e dynamic a n a l y s i s o f e a r t h dams w i t h c l a y c o r e s .  F o r such a  dam t h e s t a t i c shear s t r e s s e s on h o r i z o n t a l p l a n e s c o u l d a g a i n be c a l c u l a t e d u s i n g t h e f i n i t e element method o f a n a l y s i s w i t h a p p r o p r i a t e s o i l parameters.  F o r a s o f t c o r e t h e s t a t i c shear s t r e s s e s would be expected  to be h i g h e r than f o r t h e c o n d i t i o n o f no c o r e .  The average shear  s t r e n g t h on h o r i z o n t a l p l a n e s would be l e s s due t o t h e presence s o f t core.  The h o r i z o n t a l s p r i n g s t i f f n e s s  the f l e x i b i l i t y o f t h e s o f t c o r e ,  c o u l d be chosen to a l l o w f o r  A spring stiffness  e q u a l t o zero  should give a severe c o n d i t i o n f o r a n t i c i p a t e d displacements given  of the  due t o a  earthquake. The y i e l d s t r e n g t h o f t h e m a t e r i a l was c o n s i d e r e d to be i n d e -  pendent o f d i s p l a c e m e n t . i n t h e examples o f t h i s s e c t i o n .  However, s t r a i n  s o f t e n i n g m a t e r i a l can be i n c o r p o r a t e d i n e x a c t l y t h e same manner as was  - 127 -  STATIC  / S y mmet r i c a 1 D i v i s i o n ,  r /  I \  /t-*^~ /  1  n met Unsyr rial. Mate  F. 0 . S.  Cohesive  r i ca1 K  as  as  Calc  U n s y rn m e t r i c a l  Division,  Mate r i a l ,  Fricti onal  ^^^^^^^^^^  K  =  n  /  1 /  0.2  0.4  DIFFERENTIAL LEFT  39  h  Frictional  •~.  1/ n llii 1i  FIG.  <  /  1/  i  1  Calc.  \\ //  /  1.5  M a t e r i a 1.  Division, h  =  AND  DIVISION  FOR T H E  OF  A  285  N - S  1.0  BETWEEN  OF T H E DIVISION. —  DISPLACEMENTS  SECS.  0.8  DISPLACEMENT  RIGHT  HORIZONTAL  0.6  FT. HIGH  DAM OF  MASSES  1.4  TO T H E  FT-  TO T H E L E F T  COMPONENT  1.2  AND RIGHT OF T H E  SUBJECTED EL  CENTRO  TO  0 - 1 0  EARTHQUAKE..  d o n e , f o r h o r i z o n t a l and s l o p i n g l a y e r s o f s o i l  (Chapter  6.2).  In  a d d i t i o n , t h e r e would be l i t t l e problem i n i n c o r p o r a t i n g a p r o c e d u r e t o a l l o w t h e y i e l d s t r e s s t o v a r y w i t h the number o f c y c l e s o f s t r e s s o r strain.  6.4  AVERAGE DYNAMIC SEISMIC COEFFICIENTS  Seed (1966) proposed a method f o r e s t i m a t i n g t h e earthquake s t a b i l i t y o f an e a r t h s t r u c t u r e u s i n g average dynamic s e i s m i c i e n t s and t h i s was d i s c u s s e d i n Chapter 3. presented  Seed and M a r t i n  coeffic(1966),  a method f o r o b t a i n i n g these average d y n a m i c , s e i s m i c c o e f f i c -  i e n t s based on v i s c o e l a s t i c response t h e o r y .  Since considerable  plastic  a c t i o n a l s o o c c u r s d u r i n g a s t r o n g m o t i o n earthquake (Ambraseys 1960), i t was f e l t t h a t i t would be o f c o n s i d e r a b l e i n t e r e s t t o i n v e s t i g a t e t h e e f f e c t o f p l a s t i c a c t i o n on average dynamic s e i s m i c Using  coefficients.  t h e dam d e s c r i b e d i n S e c t i o n 6.3 average dynamic s e i s m i c  c o e f f i c i e n t s were c a l c u l a t e d f o r b o t h v i s c o e l a s t i c and e l a s t i c - v i s c o p l a s t i c conditions.  The dam was assumed t o have a s t a t i c f a c t o r o f  s a f e t y o f 1.5, and b o t h c o h e s i v e and f r i c t i o n a l m a t e r i a l s were A base a c c e l e r a t i o n c o r r e s p o n d i n g 1940  considered.  t o t h e N-S component o f E l C e n t r o ,  earthquake was used f o r a l l cases  analyzed.  Average dynamic s e i s m i c c o e f f i c i e n t s f o r v i s c o e l a s t i c response corresponding  t o A = 5 and 20% a r e shown i n F i g . 40.  F o r t h e top 1/5 o f  the dam and A = 5%, t h e maximum v a l u e o f t h e average dynamic s e i s m i c c o e f f i c i e n t i s 1.0.  F o r d e s i g n purposes t h e e f f e c t o f t h e earthquake i n  the f i r s t 10 s e e s , c o u l d be r e p r e s e n t e d  by 12 c y c l e s o f a "maximum  - 130 -  equivalent  seismic c o e f f i c i e n t " equal  t o 0.5 a t a p e r i o d o f 0.75 s e e s .  F o r the top 1/5 o f t h e dam and X = 20% t h e maximum v a l u e o f t h e average dynamic s e i s m i c c o e f f i c i e n t during  the f i r s t  i s 0.64 and the e f f e c t o f t h e earthquake  10 seconds c o u l d be r e p r e s e n t e d  a "maximum e q u i v a l e n t s e i s m i c c o e f f i c i e n t " e q u a l 0.75  by about 6 p u l s e s o f to 0.3 a t a p e r i o d o f  sees. Average dynamic s e i s m i c c o e f f i c i e n t s  response corresponding  for elastic-viscoplastic  to c o h e s i v e m a t e r i a l a r e shown i n F i g . 41. I t  may be seen t h a t the average dynamic s e i s m i c c o e f f i c i e n t s 1/5 o f t h e dam a r e almost i d e n t i c a l to those o b t a i n e d  f o r t h e top  for viscoelastic  response and X = 5% ( F i g . 4 0 ) . F o r the f u l l depth o f the dam, p l a s t i c a c t i o n reduces the maximum v a l u e o f the average dynamic s e i s m i c c o efficient  from 0.38 ( v i s c o e l a s t i c )  t o 0.3.  The s e i s m i c  coefficients  shown i n F i g . 41 were f o r an unsymmetric d i v i s i o n o f the dam and f o r horizontal spring stiffnesses  as used p r e v i o u s l y ( T a b l e I I ) .  Average dynamic s e i s m i c c o e f f i c i e n t s response c o r r e s p o n d i n g 43.  for elastic-viscoplastic  to f r i c t i o n a l m a t e r i a l a r e shown i n F i g s . 42 and  The u n s y m m e t r i c a l d i v i s i o n o f t h e dam was c o n s i d e r e d  of v a r y i n g  and t h e e f f e c t  the h o r i z o n t a l s p r i n g s t i f f n e s s was i n v e s t i g a t e d .  dynamic s e i s m i c c o e f f i c i e n t s  f o r the c o n d i t i o n of i n f i n i t e l y s t i f f o r  r i g i d h o r i z o n t a l s p r i n g s a r e shown i n F i g . 42. viscoelastic  response  Average  When compared  with  ( F i g . 40) i t may be seen t h a t f o r t h e top 1/5 o f  the dam the e f f e c t o f p l a s t i c a c t i o n has been t o reduce t h e maximum v a l u e o f the average dynamic s e i s m i c c o e f f i c i e n t  from 1.0 t o 0.87.  However, t h e e f f e c t o f t h e earthquake i n terms o f the number o f  - ZGT-  - 133s i g n i f i c a n t c y c l e s o f a "maximum e q u i v a l e n t s e i s m i c c o e f f i c i e n t " i s e s s e n t i a l l y t h e same f o r b o t h v i s c o e l a s t i c and e l a s t i c - v i s c o p l a s t i c assumptions. Average dynamic s e i s m i c c o e f f i c i e n t s f o r t h e c o n d i t i o n o f infinitely  f l e x i b l e h o r i z o n t a l s p r i n g s a r e shown i n F i g . 43.  shown a p p l y t o masses t o t h e l e f t o f t h e d i v i s i o n . c o r r e s p o n d t o outward movement o f t h e masses. values lead to i n s t a b i l i t y  The v a l u e s  The n e g a t i v e v a l u e s  Hence, o n l y t h e n e g a t i v e  o f t h e s o i l mass t o t h e l e f t o f t h e d i v i s i o n  and t h e r e f o r e o n l y these v a l u e s s h o u l d be c o n s i d e r e d .  T h i s was a l s o  the case f o r p r e v i o u s cases o f u n s y m m e t r i c , s u b d i v i s i o n b u t t h e e f f e c t was  small.  The maximum v a l u e o f t h e average dynamic s e i s m i c c o e f f i c i e n t  for  t h e t o p 1/5 o f the dam i s 0.35 and t h e e f f e c t o f the earthquake  c o u l d be r e p r e s e n t e d by 12 c y c l e s o f a "maximum e q u i v a l e n t s e i s m i c c o e f f i c i e n t " e q u a l t o 0.3 a t a p e r i o d o f 0.75 sees. infinitely  The e f f e c t o f t h e  f l e x i b l e h o r i z o n t a l springs i n contrast to r i g i d springs  t h e r e f o r e has been t o c o n s i d e r a b l y reduce b o t h t h e maximum v a l u e o f t h e "average dynamic s e i s m i c c o e f f i c i e n t " and t h e "maximum e q u i v a l e n t s e i s m i c coefficient". The s i g n i f i c a n t f e a t u r e s from F i g s . 40 t o 43 i n c l u s i v e a r e shown i n T a b l e I I I .  I t may be seen t h a t f o r b o t h t h e top 1/5 o f t h e  dam and t h e whole dam where each i s c o n s i d e r e d s e p a r a t e l y , dynamic seismic c o e f f i c i e n t s are approximately  the same f o r v i s c o e l a s t i c and  e l a s t i c - v i s c o p l a s t i c responses.corresponding  to both cohesive m a t e r i a l  and f r i c t i o n a l m a t e r i a l w i t h r i g i d h o r i z o n t a l s p r i n g s (A = 5% a l l c a s e s ) . For f r i c t i o n a l m a t e r i a l and h o r i z o n t a l s p r i n g s t i f f n e s s e s e q u a l t o z e r o ,  TIME 3  IN  SECONDS 4  5  z UJ u.  0.4  UI o o o  2 to UJ CO  0.0  -0.2  Ui CO  <  or  UJ > <  1.  Material  Assumed  2. U n s y m m e t r i c a l -0.8  u." u.  Ui  o u o  as  0.4  Shown  3.  Horizontal  4.  X  --  1  wV  0.2  >  CO 0 . 0 Ui CO UJ - 0 . 2  V  o  < ar•  V  5  to be  Division  in F i g . Spring  Frictional. of  Dam  35a Stiffness,  K =0 k  %  1  WHOLE  DAM  V  v  Ui - 0 . 4  >  I  < FIG  2  3  4 TIME  43  AVERAGE N-S  DYNAMIC  COMPONENT  HORIZONTAL  SEISMIC OF  5  6  8  7  10  SECONDS  COEFFICIENTS  EL CENTRO,  SPRINGS.  IN  FOR  285  194 0 E A R T H Q U A K E .  FT. HIGH  EARTH  FRICTIONAL  DAM  MATERIAL  SUBJECTED AND  TO  FLEXIBLE  THE OJ  TABLE III SEISMIC C O E F F I C I E N T S FROM A N A L Y S I S OF 285 FT. N-S COMPONENT  Portion of Dam  Condition  H I G H EARTH D A M SUBJECT T O 0-10 SECS O F THE  O F EL C E N T R O  EARTHQUAKE  M a x . A v e r . Equivalent Seismic M a x . Aver. Coefficient Coefficient  N o . of Significant Stress Pulses  Period of Stress Pulses-Sees.  Viscoelastic, X = 5%  Top 1/5 Full Height  1.0 0.38  0.5 0.2  12 10  0.75 0.75  Viscoelastic, ?v = 20%  Top 1 /5 Full Height  0.64 0.2  0.3 0.2  6 5  0.75 0.75  E l a s t i c - V i s c o p l a s t i c , Cohesive M a t e r i a l , X = 5%, as calculated  Top 1/5  1.0  0.5  12  0.75  Full Height  0.3  0.2  10  0.75  E l a s t i c - V i s c o p l a s t i c , Frictional Material  Top 1/5  0.88  0.5  12  0.75  Full Height  0.32  0.2  10  0.75  Top 1/5  0.35  0.3  12  0.75  Full Height  0.2  0.2  6  0.75  X= 5%, k, =  OO  . h E l a s t i c - V i s c o p l a s t i c , Frictional Material, X = 5%, k, =0 n  - 136 the s e i s m i c c o e f f i c i e n t s a r e c o n s i d e r a b l y s m a l l e r t h a n above.  For the  top 1/5 o f t h e dam t h e "maximum average s e i s m i c c o e f f i c i e n t " i s the same as t h a t o b t a i n e d by i n c r e a s i n g t h e damping c o e f f i c i e n t s a n a l y s i s to correspond  i n viscoelastic  t o X = 20%. However, t h e number o f s i g n i f i c a n t  p u l s e s i s 12 f o r t h e e l a s t i c - v i s c o p l a s t i c a n a l y s i s v e r s u s e l a s t i c response.  6 for visco-  F o r t h e f u l l h e i g h t o f the dam t h e e l a s t i c - v i s c o -  p l a s t i c and t h e v i s c o e l a s t i c a n a l y s i s w i t h X = 20% g i v e e s s e n t i a l l y t h e same s e i s m i c  coefficients.  The v a r i a t i o n o f t h e maximum average s e i s m i c c o e f f i c i e n t f o r potential  s l i d i n g blocks extending  t o v a r i o u s depths i s shown i n F i g . 44.  For a l l c o n d i t i o n s o f b o t h v i s c o e l a s t i c and e l a s t i c - v i s c o p l a s t i c r e s p o n s e , t h e maximum v a l u e d e c r e a s e s w i t h depth below t h e c r e s t .  I t may  be seen t h a t f o r c o h e s i v e m a t e r i a l the p l a s t i c a c t i o n i n v o l v e d has l i t t l e e f f e c t on t h e maximum v a l u e a t t h e t o p o f the dam.  Actually,  no p l a s t i c  a c t i o n o c c u r s i n t h e top h a l f o f t h e dam ( F i g . 3 3 ) . Towards t h e base the p l a s t i c a c t i o n i n v o l v e d w i t h c o h e s i v e m a t e r i a l does reduce t h e maximum v a l u e i n comparison t o t h a t o b t a i n e d f o r v i s c o e l a s t i c response and X = 5%. For f r i c t i o n a l m a t e r i a l a c o n s i d e r a b l e range i n t h e maximum v a l u e s o f the s e i s m i c c o e f f i c i e n t o c c u r s depending on t h e s t i f f n e s s o f the h o r i z o n t a l springs.  The maximum v a l u e s f o r t h e assumption o f v i s c o e l a s t i c  response and A = 20% l i e s as an approximate median l i n e w i t h i n t h e above range. S e i s m i c c o e f f i c i e n t s o b t a i n e d from e l a s t i c - v i s c o p l a s t i c analyses  f o r a g i v e n earthquake a r e r e l a t e d  to displacements.  Where d i s -  placements a r e . l a r g e , s e i s m i c c o e f f i c i e n t s w i l l be s m a l l and v i c e v e r s a .  - 137 -  r-  (/) UJ  or o  * o -i  UJ  m •x. o o _l  CD O  <  o 0.  X  r-  0. UJ  a  0.2  0.4  MAXIMUM  FIG.  44  VARIATION DEPTH EL  0.6  AVERAGE  SEISMIC  IN M A X I M U M  FOR A  CENTRO,  0.8  COEFFICIENT  AVERAGE  2 8 5 F T . HIGH 1940  1.0  SEISMIC  DAM  EARTHQUAKE.  COEFFICIENT  SUBJECTED  TO  WITH  - 138  The  i n t r o d u c t i o n of i n c r e a s e d v i s c o u s damping to account f o r n o n - e l a s t i c  behaviour  r e d u c e s . b o t h the d i s p l a c e m e n t s  and the s e i s m i c  coefficients.  I t appears t h a t i t would be p o s s i b l e u s i n g a v i s c o e l a s t i c a n a l y s i s to s e l e c t a v i s c o u s damping c o e f f i c i e n t w h i c h would g i v e "maximum e q u i v a l e n t seismic c o e f f i c i e n t s " corresponding e l a s t i c - v i s c o p l a s t i c response.  to those o b t a i n e d from an  However, the number of  significant  c y c l e s i s l i k e l y to be l e s s f o r the v i s c o e l a s t i c a n a l y s i s . a p p r o p r i a t e damping c o e f f i c i e n t i s r e l a t e d  Since  t o the s t r e s s - s t r a i n  the  charact-  e r i s t i c s o f the m a t e r i a l and the d e s i g n e a r t h q u a k e , i t would seem more reasonable  to a p p l y an e l a s t i c v i s c o p l a s t i c a n a l y s i s than to a r b i t r a r i l y  assume some v a l u e o f the damping c o e f f i c i e n t , such as 20%, and  perform  a v i s c o e l a s t i c a n a l y s i s as has been done by Ambraseys (1960) and and M a r t i n  (1966).  Seed  -  - 139 CHAPTER 7  SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH  7.1  SUMMARY Review o f earthquake d e s i g n a n a l y s e s suggest t h a t t h e  r a t i o n a l d e s i g n o f an e a r t h s t r u c t u r e s u b j e c t t o earthquake f o r c e s s h o u l d be based on t h e e s t i m a t e d magnitude o f d i s p l a c e m e n t s  produced  r a t h e r than on t h e f a c t o r o f s a f e t y o f a p o t e n t i a l s l i d i n g b l o c k o f t h e material.  I n a t t e m p t i n g t o o b t a i n a measure o f t h e s e d i s p l a c e m e n t s t h e  f o l l o w i n g two b a s i c approaches 1.  have been proposed:  Newmark (1965) proposed a d i r e c t method o f e s t i m a t i n g d i s placements.  He i d e a l i z e d a p o t e n t i a l s l i d i n g b l o c k o f t h e  s t r u c t u r e by a r i g i d - p l a s t i c s i n g l e - d e g r e e o f freedom Knowing t h e y i e l d a c c e l e r a t i o n o f t h e system,  system.  displacements  can be c a l c u l a t e d f o r any assumed e x c i t a t i o n . 2.  Seed (1966) proposed an i n d i r e c t method f o r e s t i m a t i n g d i s placements.  He f i r s t assumes t h e s t r u c t u r e responds  as a  m u l t i - d e g r e e o f freedom v i s c o e l a s t i c system, thus a l l o w i n g i n e r t i a f o r c e s t o be c a l c u l a t e d f o r a g i v e n e x c i t a t i o n . These f o r c e s a r e then i n c o r p o r a t e d i n a s l i p c i r c l e a n a l y s i s a l l o w i n g s t r e s s e s t o be e s t i m a t e d .  He suggests t h a t these  s t r e s s e s can t h e n be a p p l i e d to a sample o f t h e m a t e r i a l a l l o w i n g s t r a i n s t o be determined and from t h e s e s t r a i n s a measure o f d i s p l a c e m e n t s can be o b t a i n e d .  - 140 -  The method proposed by Newmark i s d i r e c t b u t o v e r s i m p l i f i e s the problem.  F i e l d o b s e r v a t i o n on t h e b e h a v i o u r o f e a r t h s t r u c t u r e s s u b j e c t  to  e x c i t a t i o n i n d i c a t e t h a t they respond e s s e n t i a l l y as a m u l t i - d e g r e e  of  freedom v i s c o e l a s t i c system f o r s m a l l a m p l i t u d e v i b r a t i o n , w h i l e f o r  l a r g e amplitude v i b r a t i o n associated w i t h high i n t e n s i t y earthquakes, considerable p l a s t i c y i e l d i n g occurs. the  The method proposed by Seed has  u n d e s i r a b l e f e a t u r e t h a t a v i s c o e l a s t i c d i s p l a c e m e n t f i e l d i s assumed  when c a l c u l a t i n g i n e r t i a f o r c e s w h i c h a r e then used t o e s t i m a t e a new displacement f i e l d . the  Thus two d i s p l a c e m e n t f i e l d s a r e a s s o c i a t e d w i t h  same f o r c e system. The e l a s t i c - v i s c o p l a s t i c method o f a n a l y s i s proposed i n t h i s  t h e s i s a l l o w s d i r e c t c a l c u l a t i o n o f d i s p l a c e m e n t s t o be made.  The  s t r u c t u r e i s m o d e l l e d by a m u l t i - d e g r e e o f freedom system w h i c h responds i n a v i s c o e l a s t i c manner t o s m a l l a m p l i t u d e v i b r a t i o n s w h i l e t h e i n t r o d u c t i o n o f a y i e l d s t r e s s causes p l a s t i c a c t i o n t o o c c u r d u r i n g l a r g e amplitude v i b r a t i o n s .  The model t h e r e f o r e c l o s e l y d u p l i c a t e s t h e f i e l d  b e h a v i o u r d e s c r i b e d i n t h e p r e v i o u s paragraph and hence t h e method p r o posed would appear t o be an advance on c u r r e n t methods o f a n a l y s i s . The a n a l y s i s does, however, have a d i s a d v a n t a g e i n t h a t d i s placements a r e r e s t r i c t e d t o a o n e - d i m e n s i o n a l f i e l d whereas i n .plane s t r a i n problems, t h e d i s p l a c e m e n t f i e l d w i l l i n g e n e r a l be twodimensional.  The i n c o r p o r a t i o n o f p l a s t i c a c t i o n i n t o a f i n i t e  element  a n a l y s i s , w h i c h would a l l o w a t w o - d i m e n s i o n a l d i s p l a c e m e n t f i e l d , be more d e s i r a b l e .  However, t h i s i s a complex problem and no such  a n a l y s i s has as y e t (1967) been p r e s e n t e d .  would  - 141 The e q u a t i o n s o f m o t i o n o f t h e proposed  i d e a l i z e d system were  presented together w i t h a step-by-step s o l u t i o n procedure.  Computer  programs were developed w h i c h a l l o w t h e d i s p l a c e m e n t s , v e l o c i t i e s and a c c e l e r a t i o n s w i t h i n t h e system acceleration.  t o be c a l c u l a t e d f o r any assumed base  T y p i c a l e a r t h s t r u c t u r e s were a n a l y z e d u s i n g a base  a c c e l e r a t i o n c o r r e s p o n d i n g t o t h e 0 - 1 0 s e e s , o f t h e N-S component o f E l Centro earthquake  7.2  and r e s u l t s were d i s c u s s e d i n c o n s i d e r a b l e d e t a i l .  CONCLUSIONS The d a t a p r e s e n t e d i n t h i s t h e s i s l e a d s t o t h e f o l l o w i n g  conclusions: 1.  The magnitude of earthquake  induced d i s p l a c e m e n t s depend  on t h e dynamic s t r e s s - s t r a i n r e l a t i o n s h i p s o f t h e m a t e r i a l , t h e geometry o f t h e s t r u c t u r e and t h e base accelerations. 2.  Approximate dynamic.displacements  o f an e a r t h s t r u c t u r e ,  w h e r e i n t h e m a t e r i a l behaves i n an e l a s t i c - v i s c o p l a s t i c manner, can be c a l c u l a t e d where d i s p l a c e m e n t s  occur  e s s e n t i a l l y p a r a l l e l t o t h e base o f t h e s t r u c t u r e . 3.  The type o f damping p r e s e n t i n a s t r u c t u r e has a l a r g e e f f e c t on d i s p l a c e m e n t s .  The i n t r o d u c t i o n o f v i s c o u s  damping w i l l reduce d i s p l a c e m e n t s , whereas t h e i n t r o duction of p l a s t i c y i e l d i n g w i l l lead to increased displacements. 4.  F o r a v i s c o e l a s t i c system, i n c r e a s e d v i s c o u s damping  - 142 leads t o lower a c c e l e r a t i o n s seismic  coefficients.  and hence lower  However, f o r an e l a s t i c -  v i s c o p l a s t i c system, t h e i n t r o d u c t i o n o f v i s c o u s damping causes i n c r e a s e d increased  seismic  accelerations  coefficients  and hence  (provided  plastic  a c t i o n does o c c u r ) . 5.  Considerable magnification  of surface  accelerations  of s o i l l a y e r s can be expected f o r s m a l l base accelerations. accelerations 6.  F o r l a r g e base a c c e l e r a t i o n s ,  surface  a r e l i m i t e d by p l a s t i c a c t i o n .  E s t i m a t e s o f i n e r t i a f o r c e s from v i s c o e l a s t i c response t h e o r y a r e u n l i k e l y t o be c o r r e c t .  7.  <  I n e a r t h dam a n a l y s i s , t h e s t i p u l a t i o n t h a t  displace-  ments must be h o r i z o n t a l may i n some cases be i n t o l e r able.  A response a n a l y s i s w h i c h would a l l o w  vertical  d i s p l a c e m e n t s i n a d d i t i o n to h o r i z o n t a l d i s p l a c e m e n t s would t h e r e f o r e be v e r y 7.3  desirable.  SUGGESTIONS FOR FURTHER RESEARCH The  w r i t e r considers  that useful information  c o u l d be o b t a i n e d  1.  I n v e s t i g a t i o n o f t h e type o f damping p r e s e n t i n e a r t h  from:  structures. 2.  Development o f an a n a l y s i s t o model t h e performance o f an e a r t h dam w h e r e i n each h o r i z o n t a l s l i c e i s c o n s i d e r e d t o be  - 143 -  comprised o f a number o f p a r t s r a t h e r than two as assumed i n t h i s t h e s i s . (  Damping i n a d d i t i o n to  h o r i z o n t a l s p r i n g s c o u l d be i n c o r p o r a t e d between t h e s e elements.  3.  I n v e s t i g a t i o n i n t o the f e a s i b i l i t y  of u s i n g t h e  f i n i t e element method f o r n o n - l i n e a r dynamic response a n a l y s e s of e a r t h s t r u c t u r e s .  LITERATURE CITED  AMBRASEYS, N. N., 1960, "The S e i s m i c S t a b i l i t y o f E a r t h Dams", P r o c . , 2nd World Conference on Earthquake Engrg., Japan, 1960, V o l . I I . BISHOP, A. W., 1955, "The Use o f S l i p C i r c l e i n t h e S t a b i l i t y A n a l y s i s of Slopes", Geotechnique, V o l . 5, No. 1. CHOPRA, A. K., 1967, "Earthquake Response o f E a r t h Dams". J o u r n a l o f t h e S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , ASCE, V o l 93, No. SM2. CLOUGH, R. W., 1967. " A n a l y s i s o f Embankment S t r e s s e s and D e f o r m a t i o n s " . J o u r n a l o f t h e S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , ASCE, V o l . 93, SM4. CLOUGH, R. W., and CHOPRA, A.K., 1966. "Earthquake S t r e s s A n a l y s i s i n E a r t h Dams". J o u r n a l o f t h e E n g i n e e r i n g Mechanics D i v i s i o n , ASCE, V o l . 92, No. EM2. DUKE, C. M., 1960. "Foundations and E a r t h S t r u c t u r e s i n E a r t h q u a k e s " . P r o c e e d i n g s o f t h e 2nd World Conference on Earthquake Engrg., J a p a n , 1960, V o l . I I FINN, W. D., 1 9 6 6 a . " S t a t i c and S e i s m i c B e h a v i o u r o f an E a r t h Dam". Canadian G e o t e c h n i c a l J o u r n a l , V o l I V , No. 1. FINN, W. D., 1966b."Earthquake S t a b i l i t y o f Cohesive S l o p e s " . J o u r n a l o f t h e S o i l Mechanics and Founda t i o n s D i v i s i o n , ASCE, V o l . 92, No. SMI. GOODMAN, R. E. and SEED, H. B o l t o n , 1966. "Earthquake Induced D i s p l a c e m e n t s i n Sand Embankments". J o u r n a l of t h e S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , ASCE, V o l . 92, No. SM2. HALL, J . R., 1963. " D i s s i p a t i o n o f E l a s t i c Wave Energy i n Granular S o i l s " . J o u r n a l o f t h e S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , ASCE, V o l . 89, No. SM6. LOWE, J . , and KARAFIATH, L., 1959. " S t a b i l i t y o f E a r t h Dams Upon Drawdown". P r o c e e d i n g s , 1 s t Pan American Conf. on S o i l Mechanics and F o u n d a t i o n Engrg., Mexico C i t y , M e x i c o , 1959.  MELIN, J . W. , 1958. " N u m e r i c a l I n t e g r a t i o n by the B e t a Method". P r o c e e d i n g s , ASCE Conference on E l e c t r o n i c Computation, Kansas C i t y , November, 1958. MONONOBE, N., TAKATA, A., and MATUMURA, M., 1936. " S e i s m i c S t a b i l i t y o f the E a r t h Dam". P r o c e e d i n g s , 2nd Congress on Large Dams, Washington, D. C. 1936, Vol. IV. MORGENSTERN, N. R. and PRICE, V. E., 1965. "The A n a l y s i s of t h e S t a b i l i t y o f G e n e r a l S l i p S u r f a c e s " . G e o t e c h n i q u e , V o l . XV, No. 1. NEWMARK, N. M., 1959. "A Method o f Computation f o r S t r u c t u r a l Dynamics". J o u r n a l o f the E n g i n e e r i n g Mechanics D i v i s i o n , P r o c e e d i n g s ASCE, V o l . 85, No. EM3, J u l y 1959. NEWMARK, N.M., 1963. "Earthquake E f f e c t s on Dams and Embankments". P r e s e n t e d a t the ASCE S t r u c t u r a l Energy Conf., San F r a n c i s c o , C a l i f . October 1963. NEWMARK, N.M., 1965. " E f f e c t s o f Earthquake on Dams and Embankments". Geotechnique, V o l . XV, No. 2. NIELSEN, N.N., 1964. "Dynamic Response o f M u l t i s t o r e y B u i l d i n g s " . Phd. T h e s i s , Earthquake E n g i n e e r i n g R e s e a r c h , C a l i f o r n i a I n s t i t u t e o f Technology, Pasadena, C a l i f . , June 1964. NIELSEN, N.N., 1966. " V i b r a t i o n T e s t s o f a N i n e - S t o r e y S t e e l Frame B u i l d i n g " . J o u r n a l o f the E n g i n e e r i n g Mechanics D i v i s i o n , V o l . 92, No. EMI. PEACOCK, W. H. and SEED, H. B o l t o n , 1968." "Sand L i q u e f a c t i o n under C y c l i c L o a d i n g Simple Shear C o n d i t i o n s " . J o u r n a l o f t h e S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , ASCE, V o l . 94, SM3. PENZIEN, J . , 1960. " E l a s t o - P l a s t i c Response o f I d e a l i z e d M u l t i - S t o r e y Structures Subjected to a Strong Motion Earthquake". P r o c , 2nd World Conf. on Earthquake Engrg. , Japan, 1960, V o l . I I . RASHID, Y. R., 1966. "Dynamic Response o f E a r t h Dams t o Earthquake". Graduate Student Research R e p o r t , U n i v . of C a l i f o r n i a , B e r k e l e y , C a l i f . , 1961.  SEED, H. B o l t o n , 1966. "A Method f o r Earthquake R e s i s t a n t D e s i g n o f E a r t h Dams". J o u r n a l o f t h e S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , V o l . 92, No. SMI. SEED, H. B o l t o n , and MARTIN, 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 Design". J o u r n a l o f the S o i l Mechanics and F o u n d a t i o n s D i v i s i o n , V o l . 92, No. SM3. SEED, H. B o l t o n , and LEE, K.K., 1966. " L i q u e f a c t i o n o f S a t u r a t e d Sands D u r i n g C y c l i c L o a d i n g " . Journal of the S o i l Mechanics and Foundations- D i v i s i o n , V o l . 92, No. SM6. VELETSOS, A. S., and NEWMARK, N. M., 1960. " E f f e c t o f I n e l a s t i c B e h a v i o u r on t h e Response o f Simple Systems to Earthquake M o t i o n s " . P r o c . 2nd World Conf. on Earthquake Engrg., Japan, 1960, V o l . I I . WIGGINS, J.H., 1964. " E f f e c t o f S i t e C o n d i t i o n s on E a r t h quake I n t e n s i t y . " J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n . V o l . 90, ST2 . WILSON, E.L. and CLOUGH, R.W., 1962. "Dynamic Response by Step-by-Step M a t r i x A n a l y s i s " . Symposium on t h e Use of Computers i n C i v i l E n g i n e e r i n g , L i s b o n , P o r t u g a l , O c t o b e r , 1962. WILSON, S.D., and DIETRICH, R. J . , 1960. " E f f e c t o f C o n s o l i d a t i o n on E l a s t i c and S t r e n g t h P r o p e r t i e s o f C l a y " . Research Conference on S h e a r - S t r e n g t h o f C o h e s i v e S o i l s , B o u l d e r , C o l o . , 1960.  - 147 'APPENDIX I DERIVATION OF THE EXPRESSION FOR AVERAGE DYNAMIC SEISMIC COEFFICIENTS OF A WEDGE OF AN EARTH STRUCTURE  The e x p r e s s i o n f o r t h e average dynamic s e i s m i c c o e f f i c i e n t s of a p o t e n t i a l s l i d i n g wedge o f a dam s u b j e c t e d to e a r t h q u a k e f o r c e s was g i v e n by Eq. 4.23.  av  k ( t )  • i  T h i s e q u a t i o n was d e r i v e d as f o l l o w s :  n=°°  3u  I / n=l  jt^  2  ( A 1 )  where W = t h e w e i g h t o f t h e wedge and t h e r e m a i n i n g  3 u 2 at  3u = " c T~~ 31  2  Now  symbols have been d e f i n e d i n Chapter 4.  &  j and sxnce  u  - u> u n n  J (3 o n h  2  = -  w  S  V  •  N  3u  (A2)  2  /l-X  OJ  T  (t)  _v  (/AA J ;  6 Ji(6 )  2  n  n  2 J (6 o nh  n  n  /1-X  n/  1  n  V (t) n  /  A  /  B Ji(3) n n  2  n  1  S u b s t i t u t i n g Eqs. A3 and A4 i n Eq. A2 g i v e s  3 u  2 J (B  2  _ § H 3t d  =  °  n  h  T io / 1 - X n/ n 2  B J n 1  r  (B ) n  V(t)+V(t)} n n  S u b s t i t u t i n g Eq. A5 i n Eq. A l and i n t e g r a t i n g y i e l d s  v  (A4)  (A5)  n=°° 2payh J i (g X ) ± J 1 n=l o ) / B Ji(B) 1  k(t)  =  n  n  Now  and  V (t) + c V ( t ) }  t  n  n  1 o -^pay g  W  =  c  =  co n  =  2A co n n 7-^-  h  V  s  S u b s t i t u t i n g f o r W, c and co i n Eq. A6 y i e l d s n  k  (  t  )  = a  v  4 V s  J i ( 0 ±) _ah _ o  i2  A/ i  T  { v  / o \  g y B ' 1-A^ J i ( B )  +  _n  n  2A w  .  } n  n  n  I f A i s assumed s m a l l such t h a t / 1-A = 1, then n / n. ' 2  4 V av  g y ° J  J ( B £)  2A  X  3n  Ji(S) • 1  1  n  co n  n  v  '  J  - 149 -  APPENDIX I I COMPUTER PROGRAMS AND TEST PROBLEMS  Computer Programs  Four computer programs were developed d u r i n g t h e r e s e a r c h period. Program 1 i s used t o c a l c u l a t e t h e n a t u r a l f r e q u e n c i e s , mode shapes and c r i t i c a l damping f a c t o r s o f a v i s c o e l a s t i c Determination o f the n a t u r a l frequencies i s e s s e n t i a l l y  structure. an e i g e n -  v a l u e problem and a s u b r o u t i n e was used f o r t h i s purpose.  The  c r i t i c a l damping f a c t o r s a r e determined from t h e n a t u r a l f r e q u e n c i e s as d i s c u s s e d i n Chapter 5.7. A b l o c k diagram f o r t h i s program i s shown i n F i g . 45. Program 2 determines t h e e l a s t i c - v i s c o p l a s t i c response o f a s i m p l e shear type s t r u c t u r e t o earthquake m o t i o n .  The program  i n t e g r a t e s t h e c o u p l e d e q u a t i o n s o f m o t i o n by the method suggested by P e n z i e n (1960), t h e t h e o r y f o r w h i c h i s p r e s e n t e d i n d e t a i l i n Chapter 5.2. P e n z i e n s u g g e s t s t h a t a time i n t e r v a l e q u a l t o a p p r o x i m a t e l y o n e - t e n t h t h a t o f t h e l o w e s t n a t u r a l p e r i o d s h o u l d be used i n t h e i n t e g r a t i o n p r o c e d u r e . ory  T h i s was found t o be s a t i s f a c t -  p r o v i d e d t h e damped n a t u r a l p e r i o d was used.  A b l o c k diagram  f o r t h i s program i s shown i n F i g . 46. Program 3 i s used t o determine t h e e l a s t i c - v i s c o p l a s t i c response o f a s h e a r - t y p e s t r u c t u r e such as a dam, where t h e shear s t r e s s on p a r a l l e l p l a n e s i s n o t u n i f o r m .  The program i n t e g r a t e s  the  c o u p l e d e q u a t i o n s o f m o t i o n f o r the model p r e s e n t e d and d i s c u s s e d  in  Chapter 5.2. A b l o c k diagram f o r t h i s program i s shown i n F i g . 47.  PROGRAM 1 Determines Natural Frequencies, Mode Shapes and the C r i t i c a l Damping Factors of an Idealized Structure START  Read:  1. Masses 2. Inter-storey Spring Stiffness, k  C a l c u l a t e Stiffness M a t r i x [K]  Calculate -1/2  [M]  -1/2  [K][M]  _  =[K]  C a l l Subroutine for Eigen values and Eigen Vectors  C a l c , Mode Shapes  UI  -1/2  i  = [M]  [<f>]  C a l c . C r i t i c a l Damping factors _ 2 k  Print:  / a ) i  1. Frequency 2. Mode Shape 3. C r i t i c a l Damping Factors  i  END F i g . 45  PROGRAM 2  - 151 -  Determines the E l a s t i c - V i s c o p l a s t i c Response of an Idealized Structure (Uniform Stress on Parallel Planes) by Integration of Coupled Equations of Motion START Ttead:  1. N o . of Masses 2 . Per cent C r i t i c a l Damping 3 . Time Interval  Read:  1. 2. 3. 4. 5. 6. 7.  Masses Springs Yie|d Stresses C r i t i c a l Damping Factors Static Shear Stresses Cross-Sectional Area Ductility Factors  Read Earthquake Data  f  C a l c . N o n - D i m e n s i o n a l Dispis. & V e l s . at M i d - I n t e r v a l  Yes  Repeat Simi lar Process For End of Period  C a l c . S p r i n g Force  C a l c . Spring Force  V  C a l c . Non-Dimensional Accelerations  C a l c . Dispis. V e l s . , Aces. etc. Print:  Dispis., V e l s . , A c e s . , etc. END F i g . 46  |  PROGRAM 3 Determines the E l a s t i c - V i s c o p l a s t i c Response of an Idealized Structure for the Ease of N o n - U n i f o r m Stresses on Horizontal Planes by the Solution of the Coupled Equations .of M o t i o n START Read:  Read:  ]. 2. 3. 4. 5. 6. 7.  1. N o . of Left Masses 2. Per cent C r i t i c a l Damping 3. Time Interval Masses Left & right I Springs Left & Right Y i e l d Stresses Left & Right C r i t i c a l Damping Factors Left & Right Static Shear Stresses Left & Right Areas Left & Right D u c t i l i t y Factors Left & Right Read Earthquake Data  C a l c . Non-Dimensional Displs.& V e l s . at M i d Interval C a l . Spring Force  Repeat Similar Process for end of Period  Yes  Calc.  Spring Force  C a l f . Displs. Left and Right  C a l c . Non-Dimensional Accelerations  Calc.  Displs., V e l s . , etc.  Aces.,  4-  Print: D i s p l s . , V e l s . , & A c e s . END  F i g . 47  PROGRAM 4 Determines the Viscoelastic Response of an Idealized Structure by Integrating the Uncoupled Equations of M o t i o n Using the Method Described by Wilson and Clough (1962)  START  r Read:  1. 2. 3.  Read:  1. 2.  N o . of Modes Per cent C r i t i c a l Damping Time Interval  Frequencies Mode Shapes  READ E A R T H Q U A K E DATA  C a l c u l a t e M o d a l Dispis , V e l s . and A c e s .  1 C a l c u l a t e x D i s p i s . , V e l s . , and A c e s .  Print  x D i s p i s . , V e l s . , and Aces  END  F i g . 48  - 154 -  Program 4 d e t e r m i n e s the v i s c o e l a s t i c response o f a s t r u c t u r e s u b j e c t e d to earthquake m o t i o n .  T h i s program i n t e g r a t e s the un-  c o u p l e d e q u a t i o n s of m o t i o n by the method o u t l i n e d by W i l s o n and Clough (1962) and d i s c u s s e d i n Chapter 4.4 and 4.5.  A b l o c k diagram  f o r t h i s program i s . shown i n F i g . 48. Test Problems  When new  computer programs a r e b e i n g developed i t i s i m p o r t a n t  t h a t they be a d e q u a t e l y checked to i n s u r e t h a t they a r e p e r f o r m i n g the desired calculations.  The b e s t method f o r c h e c k i n g a program i s t o  use i t t o a n a l y z e problems f o r w h i c h t h e s o l u t i o n i s a l r e a d y known.and compare r e s u l t s .  U s u a l l y i t i s not p o s s i b l e t o f i n d a problem which  c o n t a i n s a l l the many f a c e t s c o n t a i n e d i n a program.  However, a  number of problems can be a n a l y z e d ,each of which-cheeks the v a r i o u s a s p e c t s o f the program, thus e n s u r i n g t h a t a l l s e c t i o n s a r e p e r f o r m i n g correctly. D e t a i l e d examples of the response o f m u l t i d e g r e e of freedom e l a s t i c - v i s c o p l a s t i c systems t o known f o r c i n g f u n c t i o n s a r e not r e a d i l y available.  However, such a response f o r a two degree o f freedom  system s u b j e c t t o a d i s p l a c e m e n t f u n c t i o n u^ = 0.5 S i n 4irt i n c h e s was  found (Biggs 1964).  I n t h i s example v i s c o u s damping i s not c o n s i d -  e r e d so t h a t the response i s e l a s t i c - p l a s t i c r a t h e r than e l a s t i c viscoplastic.  The system i s shown i n F i g . 49a.  S i n c e the programs  developed by the w r i t e r a r e d e s i g n e d to use ground a c c e l e r a t i o n s r a t h e r t h a n ground d i s p l a c e m e n t s , the ground d i s p l a c e m e n t , u , can  - 155 -  be- r e p l a c e d by the ground a c c e l e r a t i o n u  =. - 8 T r S i n 4 i r t i n / s e c . 2  2  A t t = 0 the' ground d i s p l a c e m e n t i s z e r o , however, the i n i t i a l v e l o c i t y i s not z e r o but e q u a l s 2TT i n / s e c .  T h i s can be c o n s i d e r e d as  -equivalent..to..a.fixed, base .but w i t h each of the masses h a v i n g an i n i t i a l - v e l o c i t y • = - 2n i n / s e c R e l a t i v e d i s p l a c e m e n t s c a l c u l a t e d u s i n g Program 2 and a time i n t e r v a l , At = 0.01  s e e s , a r e shown i n F i g . 49b  e x a c t l y w i t h those o b t a i n e d by B i g g s .  were found t o compare  T h i s two degree of freedom  system t h e r e f o r e checks t h e e l a s t i c - p l a s t i c p o r t i o n o f Program 2. Program 2 can a l s o be used t o s o l v e v i s c o e l a s t i c problems p r o v i d e d the y i e l d s t r e s s e s a r e chosen s u f f i c i e n t l y h i g h t h a t the dynamic s t r e s s e s do n o t e n t e r t h e p l a s t i c range.  The  viscoelastic  response of a 300 f t . h o r i z o n t a l l a y e r of s o i l . s h o w n i n F i g . 50a s u b j e c t e d to a base a c c e l e r a t i o n c o r r e s p o n d i n g to the N-S E l Centro earthquake was  and  component o f  determined u s i n g b o t h Programs 2 and  4.  The d i s p l a c e m e n t o f t h e . t o p mass as a f u n c t i o n o f time i s shown i n Fig.  50c f o r b o t h methods.  I t may  i d e n t i c a l f o r the two-methods. placement 4 was  be seen t h a t the d i s p l a c e m e n t s are  I n f a c t , a f t e r 10.seconds the d i s -  c a l c u l a t e d from Program 2 was  0.4806 f e e t and from Program  0.4807 f e e t . I t was  assumed i n the above c a l c u l a t i o n s t h a t damping was  the r e l a t i v e v i s c o u s type (Chapter 5.6), and t h a t damping i n the mode, X, - 5%. C10  T h i s a l l o w e d t h e damping c o e f f i c i e n t s C^,  i n Program 2 t o b e . c a l c u l a t e d by Eq. 5.44  shown i n F i g . 50b.  of first  C2  and these v a l u e s a r e .  I n Program 4 which i n t e g r a t e s the  uncoupled  U 2  ~ '  156  u  m  u —u  =0.177 kip. s e c . /in 2  2  k = 58.0 kip./in , Q y S 21 kips 2  m  = 0.294 kip. s e c / in 2  2  k, = 55.6 k i p / i n , Q = 29.5 k i p s y  ^ FIG 49 Q  >  U n =-8TT SIN4TTt 2  ELASTIC - PLASTIC  SYSTEMS,  AFTER  BIG6S  (1964)  0.8  co  a  si  o c  0.4 TIME  0.0  2  0 4  \  06  0 .8  IN SEC.  1.  CN  3  CO  cu a a  Si •H  TIME IN SEC.  FIG. 4 9 b  RESPONSE  OF  ELASTIC — PLASTIC SYSTEM  OF  FIG. 4 9 Q  - 157 -  CLAY  LAYER  y - 10 0 v = 0. 5  o o to  P. C . F.  300  =)ll(=)ll(=)IK=)IU=)IK=)ll(=)lll=)ll(=)IK=]ll(  0  20  FIG.  50Q  SECTION  AND  PROPERTIES  m io  3  1—wvv*—| m2  1—vv!&—j  -3—i C 3  •  c  2  m,  c,  1——j MODEL  OF  SOIL  FIG  AND  CLAY  395.5  2  9. 3 2  6060  7. 9 0 8  178.8  11.6  3  9.32  5440  1 2.559  160.8  18.5  141.7  25.3  4  3.389  4800  17.  9.3 2  4460  21.902  131.6  32.3  6  9.32  4000  26.848  1 18.0  39.5  200  7  9.32  3200  31.954  94.0  47.0  8  9.32  2400  37.416  70.8  55.0  9  9.32  1600  42.510  47. 2  62.5  10  9.32  800  50.116  23.6  72.7  PROPERTY  VALUES  VALUES  LAYER  And Are  Super  position  Coupled  Solutions  Identical E la sti c \ -  5 %  Response in  TIME  MOTION EL  TOP  SURFACE  CORRESPONDING  C E N T R O -  1940  5.0  9.32  Modal  OF  X  5  0.6  DISPLACEMENT  LAYER  13400  PROPERTY  MODEL OF  CLAY  r» 9. 3 2  MASS Kip  LAYER  50b  FOOT  Damping Coeff.C  1  m  300  60  K.S.I.  Natural Spring Stiffness  NO.  ClO  OF  40  IN  E  THE  VERSUS N-S  EARTHQUAKE.  TIME  FOR  COMPONENT  f'Mode IN  SECS.  BASE OF  of m o t i o n , v a l u e s o f the damping, A , i n each mode are r e -  equations quired.  These a r e o b t a i n e d from Eq. 5.41  X  i " 2  u  2A  w i t h a = 0, i . e .  i  , (A9)  X  where 3 =  from Eq, A9 w i t h i = 1 Oil  thus A.  5co.  =  with  l  to  1  Ai  = 5  1  V a l u e s of the n a t u r a l f r e q u e n c i e s , UK, and the p e r c e n t a g e o f c r i t i c a l damping i n each mode, A_^, The  a r e shown i n F i g .  r e s u l t s shown i n F i g , 50c show t h a t damping i s b e i n g  c o r r e c t l y i n c o r p o r a t e d i n the c o u p l e d 2.  50b.  i n t e g r a t i o n p r o c e d u r e o f Program  I t a l s o shows t h a t the m i d - a c c e l e r a t i o n i n t e g r a t i o n method i s q u i t e  reliable.  Many a d d i t i o n a l problems were a l s o s o l v e d u s i n g Program 2  and a l l i n d i c a t e t h a t t h i s program performs the. d e s i r e d c a l c u l a t i o n s . Program 3 i s a more complex v e r s i o n of Program 2 and s o l v e s  the  model shown i n F i g . 51 w h i c h i n c l u d e s h o r i z o n t a l s p r i n g s i n a d d i t i o n to  shear s p r i n g s .  I t can be checked i n a number o f ways by  r e s u l t s w i t h those o b t a i n e d by Program 2, o f t h r e e such s e p a r a t e 1)  The  comparing  f o l l o w i n g a r e examples  checks:  I f i t i s assumed t h a t a)  the h o r i z o n t a l s p r i n g s , k^, i.e,,  b)  (k.)  T  = ( k . ) ^ and IK  (m.)  T  XL  L and R denote l e f t and  = (ra.) 1  where the s u b s c r i p t s K  right  the l e f t and r i g h t y i e l d f o r c e s (Q)^and ( Q ^ ) same  stiffness,  l e f t and r i g h t masses move t o g e t h e r  IL'  c)  are of i n f i n i t e  a r e R  the  FIG.  51  MODEL  SOLVED  BY  PROGRAM  -  3  - 160 then t h e r e s p o n s e o f t h e system s h o u l d be i d e n t i c a l t o t h a t d e t e r m i n e d from Program 2 u s i n g k. = l / 2 ( k . ) 1  and m. = 1 / 2 l '  (m,) + 1 / 2 l L T  + l/2(k.) ,  1 L  I K  (m.)^. l R  T h i s was found t o be. s o . 2)  I f i t i s assumed t h a t t h e h o r i z o n t a l s p r i n g s t i f f n e s s e s a r e z e r o , t h e n - l e f t and r i g h t s i d e s respond i n d e p e n d e n t l y of each o t h e r and. each i s e q u i v a l e n t by Program.2.  to t h e system s o l v e d  The r e s p o n s e can t h e r e f o r e be checked  using  Program 2 and t h i s was done. 3.  I f t h e ' r a t i o o f t h e i n t e r - s t o r e y s p r i n g s t i f f n e s s , k^, t o the mass, nu, i s e q u a l f o r l e f t and r i g h t hand s i d e s , i . e . , (k./m.) = (k./m.)„ and i f t h e system remains v i s c o e l a s t i c , l l L l l R T  J  then no dynamic f o r c e s h o u l d springs.  occur i n the h o r i z o n t a l  T h i s i s so s i n c e l e f t and r i g h t hand s i d e s  have t h e same n a t u r a l f r e q u e n c i e s  will  and mode shapes, and  hence w i l l respond i n t h e same way t o the same f o r c i n g function.  Program 3 under t h e s e c o n d i t i o n s d i d c a l c u l a t e  zero dynamic f o r c e i n t h e h o r i z o n t a l s p r i n g s . The check t e s t s p e r f o r m e d i n d i c a t e t h a t t h e programs d e v e l o p e d are c o r r e c t l y c a r r y i n g out the d e s i r e d c a l c u l a t i o n s .  

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