UBC Theses and Dissertations

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UBC Theses and Dissertations

Seismic response of retaining structures Salgado, Francisco Manuel Goncalves Alves 1981

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SEISMIC RESPONSE OF RETAINING STRUCTURES B . S c , The T e c h n i c a l U n i v e r s i t y o f L i s b o n ( I . S . T . ) , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA APRIL 1981 C J FRANCISCO MANUEL GONCALVES ALVES SALGADO, 1981 BY FRANCISCO MANUEL GONCALVES ALVES SALGADO i n In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of ^ % U J ^ L ^ ^ -^ 7 The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 - ABSTRACT -A s i m p l e method of a n a l y s e s which a l l o w s both the earthquake i n d u c e d f o r c e s and d i s p l a c e m e n t s of r e t a i n i n g s t r u c t u r e s t o be computed i s p r e -s e n t e d . The method c o n s i d e r s both the weight of the w a l l and the f l e x i -b i l i t y and s t r e n g t h o f both the b a c k f i l l and f o u n d a t i o n s o i l . A s i n g l e degree o f freedom p e r f e c t e l a s t i c - p l a s t i c model i s used and the e q u a t i o n o f motion i s i n t e g r a t e d t o y i e l d the time h i s t o r i e s o f w a l l f o r c e and d i s p l a c e m e n t . The method i s a p p l i e d t o a g r a v i t y r e t a i n i n g w a l l s t r u c t u r e s u b j e c t e d t o t h r e e d i f f e r e n t a c c e l e r a t i o n time h i s t o r i e s . The r e s u l t s i n d i c a t e t h a t : (1) the dynamic d i s p l a c e m e n t s w i l l be s m a l l f o r w a l l s h a v i n g the u s u a l f a c t o r of s a f e t y a g a i n s t s l i d i n g > 1.5; (2) the maximum dynamic f o r c e on the w a l l i n c r e a s e s as the f a c t o r of s a f e t y a g a i n s t s l i d -i n g i n c r e a s e s and can be g r e a t e r than the Mononobe-Okabe v a l u e when s l i d i n g i s p r e v e n t e d from o c c u r r i n g . - i i -TABLE OF CONTENTS ABSTRACT 1 . INTRODUCTION 2. METHOD OF ANALYSIS 2.1 G e n e r a l 2.2 I n c r e m e n t a l E q u a t i o n o f Motion 2.3 G o v e r n i n g E q u a t i o n s 2.3.1 G e n e r a l 2.3.2 I n c r e m e n t a l E q u a t i o n o f Motion 2.3.3 Step by Step I n t e g r a t i o n 2.3.4 Summary of Procedure 3. MODEL PARAMETERS AND RETAINING STRUCTURE USED IN THE ANALYSES 3.1 G e n e r a l 3.2 C h o i c e o f Mass f o r E q u i v a l e n t Lumped System 3.3 C h o i c e o f S p r i n g C o n s t a n t s , U s i n g Measured S t r e s s - S t r a i n R e l a t i o n s and E m p i r i c a l E q u a t i o n s . 3.3.1 L a t e r a l S p r i n g 3.3.1.1 S t r a i n s R equired t o Reach A c t i v e and P a s s i v e S t a t e s . V a l u e s o f K^ and Kp Used i n the A n a l y s e s For D i f f e r e n t S o i l B a c k f i l l s . 3.3.2 Base S p r i n g . E s t i m a t i o n o f Foun d a t i o n S t i f f n e s s , Kg 3.3.2.1 P o i s s o n ' s R a t i o , u 3.3.2.2 Modulus o f E l a s t i c i t y i n Shear, G 3.3.2.3 V a l u e s of Foundation S t i f f n e s s f o r D i f f e r e n t F o u n d a t i o n Types and L/B R a t i o s 3.4 C h o i c e of Damping f o r E q u i v a l e n t Lumped System 3.4.1 V a l u e s o f E f f e c t i v e Damping Used i n t h e A n a l y s i s , C B ( t ) 3.5 C h o i c e o f Time Increment, At, used i n the a n a l y s e s . 3.6 R e t a i n i n g W a l l Dimensions 4. EARTHQUAKE DATA USED IN THE ANALYSES - i i i -5. RESULTS 5.1 I n t r o d u c t i o n 5.2 Earthquake Induced D i s p l a c e m e n t s and Dynamic L a t e r a l F o r c e s as a F u n c t i o n of Time. 5.3 Maximum Di s p l a c e m e n t s Versus S t a t i c F a c t o r o f S a f e t y A g a i n s t S l i d i n g . 5.3.1 E f f e c t of Foundation and Earthquake C o n d i t i o n s 5.3.2 Comparison w i t h Newmark's A n a l y s i s 5.4 Maximum Dynamic E a r t h C o e f f i c i e n t , K d y m a x V e r s u s S t a t i c F a c t o r of S a f e t y A g a i n s t S l i d i n g . 5.4.1 E f f e c t o f Foundation and Earthquake C o n d i t i o n s 5.4.2 E f f e c t o f S o i l B a c k f i l l S t a t e o f Compaction 5.4.3 Comparison w i t h Mononobe-Okabe A n a l y s i s . 6. CONCLUSIONS BIBLIOGRAPHY APPENDIX - PROGRAMME WALLQUAKE - i v -LIS ! OF TABLES Ta b l e I - S o i l B a c k f i l l Parameters. T a b l e I I - S p r i n g C o n s t a n t s f o r R i g i d Base R e s t i n g on E l a s t i c H a l f -Space T a b l e I I I - P o i s s o n ' s R a t i o V a l u e s T a b l e IV - T e s t Procedures f o r Measuring Modulus and Damping C h a r a c t e r i s t i c s . T a b l e V - Ranges of G m a x T a b l e VI - F o u n d a t i o n Parameters V a l u e s t o Assess G. T a b l e V I I - S p r i n g Constant V a l u e s , Kg. T a b l e V I I I - E f f e c t i v e Damping V a l u e s , Cg. T a b l e IX - N a t u r a l P e r i o d , T, and Time Increment, At. T a b l e X - Earthquake Data. T a b l e XI - V a l u e s Correspondent t o the S o f t e r F o u n d a t i o n C o n d i t i o n s . - v -FIGURE INDEX F i g u r e 1 -F i g u r e 2 F i g u r e 3 F i g u r e 4 F i g u r e 5 F i g u r e 6 -F i g u r e 7 F i g u r e 8 F i g u r e 9 F i g u r e 10 -F i g u r e 1 1 -F i g u r e 12 -F i g u r e 13 -F i g u r e 14 -F i g u r e 15 -F i g u r e 16 -F i g u r e 17 -F i g u r e 18 -F i g u r e 19 -F i g u r e 20 -F i g u r e 21 -C a n t i l e v e r R e t a i n i n g W a l l Model L a t e r a l S p r i n g S t r a i n s Required t o Reach A c t i v e and P a s s i v e S t a t e s i n a Dense Sand Base s p r i n g R e l a t i o n s h i p between Base and L a t e r a l S p r i n g s , and W a l l Movement. S i n g l e Degree o f Freedom (S.D.O.F.) Ge n e r a l C h a r a c t e r i s t i c s o f Damping F o r c e , D(t) and L o a d i n g f o r c e , E ( t ) . Motion of System D u r i n g Time Increment E a r t h C o e f f i c i e n t s v s . S t r a i n . S p r i n g Constant C o e f f i c i e n t s f o r R e c t a n g u l a r F o u n d a t i o n s I l l u s t r a t i o n of Shear Modulus L o c a t i o n Assessment C a n t i l e v e r R e t a i n i n g W a l l Used i n the A n a l y s i s . Approximate R e l a t i o n s h i p s Between Maximum A c c e l e r a t i o n s on Rock and Other L o c a l S i t e C o n d i t i o n s . Earthquake Induced Displacements as a F u n c t i o n of Time. Dynamic L a t e r a l Force as a F u n c t i o n o f Time. Maximum Displacements v s . S t a t i c F a c t o r of S a f e t y A g a i n s t S l i d i n g . Maximum Displacements v s . S t a t i c F a c t o r of S a f e t y A g a i n s t S l i d i n g E f f e c t o f L/B R a t i o . Comparison w i t h Newmark D i s p l a c e m e n t s . Maximum Dynamic E a r t h C o e f f i c i e n t , K, v s . S t a t i c F a c t o r dymax of S a f e t y A g a i n s t S l i d i n g . Maximum Dynamic E a r t h C o e f f i c i e n t , K d y m a x v s . A c c e l e r a t i o n L e v e l . E f f e c t o f S o i l B a c k f i l l Compaction. - v i -F i g u r e 22 - Maximum Dynamic E a r t h C o e f f i c i e n t , K d y , a ;, v s . S t a t i c F a c t o r i n g Comparison With Mononobe-Okabe F i g u r e 23 - Maximum Dynamic E a r t h C o e f f i c i e n t , K d y m a x v s . A c c e l e r a t i o n L e v e l . E f f e c t o f S o i l B a c k f i l l Compaction. Comparison w i t h Mononobe-Okabe R e s u l t s . - v i i -ACKNOWLEDGEMENT The w r i t e r wishes t o acknowledge h i s a p p r e c i a t i o n t o Dr. P e t e r Byrne f o r h i s s u g g e s t i o n s and the 100% support shown d u r i n g the inumerous c o n s u l t a t i o n s throughout the development o f the p r e s e n t s t u d y . The w r i t e a l s o wishes t o acknowledge h i s a p p r e c i a t i o n f o r the f i n a n c i a l s u p p o r t o f f e r e d by the N a t u r a l S c i e n c e and E n g i n e e r i n g C o u n c i l of Canada (Grant P. 5109). - v i i i -JOAO and MARTA 1. INTRODUCTION The earthquake i n d u c e d f o r c e s on r e t a i n i n g w a l l s t r u c t u r e s a r e commonly computed from an e x t e n s i o n of the Coulomb-Rankine s l i d i n g wedge t h e o r y i n which the t r a n s i e n t earthquake f o r c e s on the s o i l b a c k f i l l are r e p r e s e n t e d by an e q u i v a l e n t s t a t i c f o r c e d e s i g n a t e d by a s e i s m i c c o e f f i c i e n t . T h i s method was d e v e l o p e d f o r dry c o h e s i o n l e s s m a t e r i a l s by Okabe (1926) and Mononobe and Matsuo (1929), and assumes the w a l l y i e l d s s u f f i c i e n t l y t o produce minimum a c t i v e p r e s s u r e s . T h e i r e q u a t i o n f o r computing the earthquake in d u c e d f o r c e s on the w a l l i s g e n e r a l l y r e f e r r e d t o as the Mononobe-Okabe e q u a t i o n . These a d d i t i o n a l f o r c e s may cause f a i l u r e o f the s t r u c t u r a l components o f the w a l l and t h i s must be c o n s i d e r e d i n i t s d e s i g n . Prakash (1977) d i d an e x t e n s i v e r e v i e w of t e h e x i s t i n g methods t o compute e a r t h p r e s s u r e s on r e t a i n i n g w a l l s due t o s e i s m i c e x c i t a t i o n and c o n c l u d e s t h a t the above p s e u d o - s t a t i c method i s m a i n l y used f o r computing the dynamic increment of e a r t h p r e s s u r e s . However d i s t r e s s due t o d i s p l a c e m e n t caused by s l i d i n g a t the base o f the s t r u c t u r e s must a l s o be c o n s i d e r e d . The movements away from the b a c k f i l l can take p l a c e e a s i l y whereas the r e s i s t a n c e t o i t s movement towards the b a c k f i l l i s c o n s i d e r a b l y l a r g e r . T h e r e f o r e , a f t e r a number o f c y c l e s o f motion, the w a l l moves out and assumes a d i f f e r e n t p o s i t i o n . Such movements ahve been observed d u r i n g earthquakes i n C h i l e 1960, A l a s k a 1964 and N i i g a t a 1964 (Seed and Whitman, 1970). t h u s d i s p l a c e m e n t of the w a l l must be c o n s i d e r e d i n the a n a l y s i s s i n c e the maximum e a r t h p r e s s u r e d e c r e a s e s w i t h i n c r e a s i n g magnitude o f the w a l l d i s p l a c e m e n t , as shown by the e x p e r i m e n t a l s t u d i e s of Matsuo and Ohara (1960). 2. Newmark (1965) p r e s e n t e d a s i m p l e method f o r p r e d i c t i n g t h e earthquake i n d u c e d d i s p l a c e m e n t o f a s o i l mass, which a l t h o u g h developed f o r e a r t h s l o p e s , i s a l s o a p p r o p r i a t e f o r r e t a i n i n g s t r u c t u r e s . H i s method e s t i m a t e s the amount o f d i s p l a c e m e n t o f a r i g i d b l o c k s u b j e c t e d t o s e i s m i c e x c i t a t i o n s and r e s i s t e d by Coulomb f r i c t i o n a t the b l o c k -f o u n d a t i o n i n t e r f a c e . Numerous d e s i g n e r s used h i s simple r i g i d p l a s t i c approach: a) Slopes and Dams - Goodman and Seed (1966), Sharma (1975), M i n e i r o (1975), b) N u c l e a r Powerplants - K a u s e l e t a l (1979), c) R e t a i n i n g W a l l s - R i c h a r d s and Helms (1979), e t c . In t h i s method the d i s p l a c e m e n t o f the w a l l due t o s l i d i n g a l o n g i t s base i s computed from the time h i s t o r y of a c c e l e r a t i o n s , c o n s i d e r i n g the w a l l and a d j a c e n t s o i l t o respond as a s i n g l e degree o f freedom r i g i d p l a s t i c system. The f o r c e s on the w a l l , however, cannot be computed from t h i s method. The Mononobe-Okabe e q u a t i o n a l l o w s the s e i s m i c f o r c e s on the w a l l t o be computed but not the d i s p l a c e m e n t s . The Newmark approach a l l o w s t h e d i s p l a c e m e n t s t o be computed but not the w a l l f o r c e s . I t would be d e s i r a b l e t o have a s i n g l e method o f a n a l y s i s which would a l l o w both the f o r c e s and d i s p l a c e m e n t s o f the w a l l t o be computed. A r a t i o n a l method o f a n a l y s i s r e q u i r e s t h a t both the f l e x i b i l i t y and s t r e n g t h o f the s o i l s u r r o u n d i n g the w a l l ( s o i l f o u n d a t i o n and s o i l b a c k f i l l ) be c o n s i d e r e d t o g e t h e r w i t h the weight and s t r u c t u r a l s t i f f n e s s o f the w a l l i t s e l f . A r i g o r o u s a n a l y s i s would r e q u i r e a f i n i t e element d i s c r e t i z a t i o n o f t h e domain t o g e t h e r w i t h a time s t e p i n t e g r a t i o n o f t h e r e s u l t i n g e q u a t i o n s o f motion. Such an a n a l y s i s i s complex, and h e r e i n a s i m p l e r model based on a s i n g l e degree o f freedom system i s p r e s e n t e d . 2. METHOD OF ANALYSIS 2.1 G e n e r a l A n a l y s i s of s o i l s t r u c t u r e i n t e r a c t i o n u s u a l l y r e q u i r e s the d e t e r m i n a t i o n o f a s e t o f f o u n d a t i o n s t i f f n e s s ( o r compliance) f u n c t i o n s . These f u n c t i o n s are r e q u i r e d t o e s t a b l i s h a r e l a t i o n s h i p between f o r c e s a t t h e s t r u c t u r e f o u n d a t i o n and the r e l a t i v e motions of the f o u n d a t i o n and, t h e ground a t a l o c a t i o n w e l l removed from the s t r u c t u r e . However, i n g e n e r a l , these f u n c t i o n s are f r e q u e n c y dependent which p r e c l u d e s a n a l y s i s i n the time domain (Whitman (1 9 7 2 ) ) . In p r a c t i c e t h ese f r e q u e n c y dependent f u n c t i o n s are u s u a l l y r e p l a c e d by e q u i v a l e n t masses, d a s h p l o t s and s p r i n g s . By doing t h i s , methods such as Modal S u p e r p o s i t i o n and Step by Step i n t e g r a t i o n can be f o l l o w e d . The s t e p by Step i n t e g r a t i o n method was used i n the p r e s e n t a n a l y s i s . 2.2 A n a l y t i c a l Model The c a n t i l e v e r r e t a i n i n g w a l l shown i n F i g u r e 1 i s m o d e l l e d as a s i n g l e lumped mass, connected t o the f r e e f i e l d w i t h two e l a s t i c - p l a s t i c s p r i n g s as shown on F i g u r e 2. The base s p r i n g w i t h s t i f f n e s s Kg i s c o u p l e d w i t h a dashpot with e f f e c t i v e damping C . I t i s assumed t h a t th e s o i l c o n t a i n e d i n t h e domain above the base and shown as the shaded zone i n F i g u r e 1, moves w i t h the w a l l and hence the e q u i v a l e n t s i n g l e mass i s t h e mass of the w a l l p l u s t h e mass of the s o i l i n the shaded zone. The l i n e A-B above the h e e l of the w a l l r e p r e s e n t s the e f f e c t i v e f a c e o f the w a l l . L a t e r (Chapter 3.2) some c o n s i d e r a t i o n s are p r e s e n t e d r e g a r d i n g t h e e q u i v a l e n t o r e f f e c t i v e mass chosen above. The f o r c e i n the l a t e r a l s p r i n g r e p r e s e n t s the l a t e r a l e a r t h f o r c e d e v e l o p e d a t the e f f e c t i v e f a c e o f the w a l l d u r i n g the s e i s m i c a c t i o n . The f o r c e - d e f l e c t i o n c h a r a c t e r i s t i c s o f t h i s s p r i n g are shown on F i g u r e R O C K FIG.2 M O D E L . FIG.I CANTILEVER RETAINING WALL. W A L L M O V E M E N T , X FIG.3 L A T E R A L SPR I N G . 5. T h i s l a t e r a l s p r i n g has two d i s t i n c t l i m i t i n g f o r c e s o r y i e l d p o i n t s : t h e upper r e p r e s e n t s the P a s s i v e f o r c e ( P p ) c o r r e s p o n d e n t t o the s o i l b a c k f i l l c h a r a c t e r i s t i c s and t h e lower, the A c t i v e f o r c e (P&). The s t i f f n e s s (K ) of the s p r i n g i n the p l a s t i c range i s based on T e r z a g h i L (1934) and Lambe and Whitman (1969). On F i g u r e 4 i s shown a t y p i c a l r e l a t i o n s h i p between the c o e f f i c i e n t o f e a r t h p r e s s u r e v e r s u s h o r i z o n t a l s t r a i n , i n p e r c e n t , f o r the case o f a dense sand base on L a b o r a t o r y d a t a ( a f t e r Lambe and Whitman, 1969). 7i r f -«—Ko - 1 0 - 5 0 + 5 + 1 0 Horizontal strain (%) Fig.4 STRAINS REQUIRED TO REACH ACTIVE AND PASSIVE STATES IN A DENSE SAND (after Lambe and Whitman, 1969) Three d i f f e r e n t r e l a t i v e d e n s i t y c o n d i t i o n s of the s o i l b a c k f i l l were c o n s i d e r e d i n the a n a l y s i s , c o r r e s p o n d e n t r e s p e c t i v e l y t o dense, medium dense and l o o s e c o n d i t i o n s . The r e l a t i o n s h i p used i n the a n a l y s i s 6. between e a r t h c o e f f i c i e n t s and h o r i z o n t a l s t r a i n f o r these t h r e e cases a r e p r e s e n t e d i n Chapter 3 (Model P a r a m e t e r s ) . I n i t i a l l y , the f o r c e i n t h i s s p r i n g i s the s t a t i c v a l u e P , and as the w a l l moves away from the s t b a c k f i l l d u r i n g the earthquake, the f o r c e drops t o the a c t i v e v a l u e . Should the w a l l move towards t h e b a c k f i l l the f o r c e i n c r e a s e s towards the p a s s i v e v a l u e as shown on F i g u r e 3. The i n i t i a l s t a t i c f o r c e v a l u e , P s t , can be c o n s i d e r e d e i t h e r as t h e f o r c e c o r r e s p o n d e n t t o , an " a t r e s t " c o n d i t i o n o r t o an " a c t i v e . c o n d i t i o n " . These two s t a r t i n g c o n d i t i o n s were c o n s i d e r e d i n the a n a l y s i s . The lower o r base s p r i n g r e p r e s e n t s the compliance o f the f o u n d a t i o n s o i l r e l a t i v e t o the f r e e f i e l d and i t s y i e l d l i m i t r e p r e s e n t s t h e l i m i t i n g f r i c t i o n a l r e s i s t a n c e t h a t can be m o b i l i z e d a t the base. The f o r c e d e f l e c t i o n c h a r a c t e r i s t i c s o f t h i s s p r i n g are shown on F i g u r e 5. WALL M O V E M E N T , X FIG.5 B A S E SPRING. Under the p r e - e a r t h q u a k e s t a t i c c o n d i t i o n the f o r c e i n t h i s s p r i n g w i l l be Q o p p o s i n g the s t a t i c f o r c e from the l a t e r a l s p r i n g , i e . Q = s t s t P . As the w a l l moves away from the b a c k f i l l , the f o r c e i n t h i s s p r i n g s t may i n c r e a s e t o the y i e l d v a l u e a t which time base s l i p o c c u r s . I f the w a l l moves towards the b a c k f i l l the f o r c e drops and may change s i g n and y i e l d on the n e g a t i v e s i d e as shown on F i g u r e 5. The s p r i n g l i m i t i n g f o r c e i s e x p r e s s e d i n terms o f the s t a t i c f a c t o r o f s a f e t y a g a i n s t s l i d i n g (F ), which i s d e f i n e d as f o l l o w s , s R e s i s t a n c e " a " F o r c e s D r i v i n g F o r c e s (1) t h e r e f o r e from e q u a t i o n (1) the base s p r i n g l i m i t i n g f o r c e ( r e s i s t a n c e f o r c e ) can be d e r i v e d : Base S p r i n g L i m i t i n g F o r c e (Q y) = F s - S t a t i c L a t e r a l F o r c e ( 2 ) t h e s t a t i c l a t e r a l f o r c e can be taken as an i n i t i a l ' a c t i v e 1 or 'at r e s t ' v a l u e . Assuming an a c t i v e v a l u e , the above e q u a t i o n can be r e w r i t t e n as: °-v = F s P A (3) In the a n a l y s i s performed, d i f f e r e n t v a l u e s of F g were c o n s i d e r e d , which c o r r e s p o n d t o d i f f e r e n t c o n d i t i o n s of the s t r u c t u r e f o u n d a t i o n . A h i g h r e p r e s e n t s a r e t a i n i n g w a l l where no movement i n r e l a t i o n t o i t s base i s expe c t e d , such as the case o f a r e t a i n i n g w a l l founded on p i l e s . 8. In o r d e r t o v i s u a l i z e the r e l a t i o n s h i p between the f o r c e s on the two s p r i n g s and the r e l a t i v e w a l l d i s p l a c e m e n t a schematic diagram i s p r e s e n t e d on f i g u r e 6. P o i n t s 1,2,3 and 4 r e p r e s e n t p o s s i b l e s t a g e s d u r i n g the motion. Fig. 6 RELATIONSHIP BETWEEN BASE AND LATERAL SPRINGS AND WALL MOVEMENT 2.3. G o v e r n i n g E q u a t i o n s  2.3.1 G e n e r a l As mentioned p r e v i o u s l y a s t e p - b y - s t e p i n t e g r a t i o n method was used i n the p r e s e n t a n a l y s i s . The method chosen uses the l i n e a r a c c e l e r a t i o n i n c r e m e n t a l p r o c e d u r e o u t l i n e d by Clough and P e n z i e n (1975), which i s a good approach f o r n o n - l i n e a r a n a l y s i s . In t h i s approach, the response i s e v a l u a t e d f o r a s e r i e s o f s h o r t time i n c r e m e n t s At, g e n e r a l l y taken o f e q u a l l e n g t h f o r c o m p u t a t i o n a l c o n v e n i e n c e . In the computer programme de v e l o p e d by the w r i t e r t h i s time increment At i s , when needed, d i v i d e d by 2, 4, 6, 2n, i n o r d e r t o a c h i e v e a h i g h degree of p r e c i s i o n a t " c r i t i c a l p o i n t s " such as change from e l a s t i c t o p l a s t i c or changes i n s i g n o f v e l o c i t y . The degree o f p r e c i s i o n i s c o n t r o l l e d by an i n p u t e d parameter. The dynamic e q u i l i b r i u m c o n d i t i o n i s s a t i s f i e d a t the b e g i n n i n g and end o f each time i n t e r v a l . The n o n l i n e a r n a t u r e of the system i s e s t a b l i s h e d by c a l c u l a t i n g new parameters a t the b e g i n n i n g of each time i n c r e m e n t . In t h p r e s e n t s t u d y t h r e e parameters behave n o n - l i n e a r l y : the s t i f f n e s s of t e h base and l a t e r a l s p r i n g s ( n o n - l i n e a r i t y o c c u r s a t changes from e l a s t i c t o p l a s t i c range) and the damping a s s o c i a t e d w i t h t h e base s p r i n g (C i s e q u a l t o an i n i t i a l g i v e n v a l u e w h i l e the base s p r i n g remains e l a s t i c , and assumes a z e r o v a l u e i n the p l a s t i c range o f the s p r i n g . ) The complete response i s o b t a i n e d by u s i n g the v e l o c i t y and d i s p l a c e m e n t a t the end of one time i n t e r a l as the i n i t i a l c o n d i t i o n s f o r th e n ext i n t e r v a l , and the p r o c e s s i s t o be c o n t i n u e d s t e p - b y - s t e p from th e time z e r o t o any d e s i r e d time. The s t r e n g t h and d e f o r m a t i o n p r o p e r t i e s o f both t h e b a c k f i l l and 10. f o u n d a t i o n s o i l may degrade d u r i n g the s h a k i n g , p a r t i c u l a r l y so i f t h e y comprise o f s a t u r a t e d l o o s e t o medium dense g r a n u l a r m a t e r i a l . T h i s c o u l d be accounted f o r i n the a n a l y s i s by changing the s p r i n g p r o p e r t i e s as the s h a k i n g p r o c e e d s . However, i n the a n a l y s i s p r e s e n t e d h e r e i n i t i s assumed t h a t the s o i l s do not degrade and the p r o p e r t i e s o f the s p r i n g s a r e k e pt c o n s t a n t w i t h time ( w i t h the e x e p t i o n when they change from e l a s t i c t o p l a s t i c o r v i c e v e r s a ) . 2.3.2 I n c r e m e n t a l E q u a t i o n o f Motion The s i n g l e degree of freedom system used i n the a n a l y s i s i s p r e s e n t e d i n F i g u r e 7(a) and the f o r c e s a c t i n g on the mass o f the system a r e i n d i c a t e d i n F i g u r e 7 ( b ) . The g e n e r a l c h a r a c t e r i s t i c s o f the damping and l o a d i n g f o r c e s are shown i n F i g u r e 8. The c h a r a c t e r i s t i c s o f both l a t e r a l and base s p r i n g f o r c e s were a l r e a d y d e s c r i b e d (see F i g u r e 3 and 5 ) . / / / _T£_ B / KB /—JULQMMJULUi^-\ / 9 ^ (a) D(t) <-Q(t) <• 1(0 < > <— w • < •> P(t) > E(t) (b) Fig. 7 SINGLE DEGREE OF FREEDOM (S.D.O.F) (a) BASIC S.D.0.F f M F O R C E EQUILBRIUM (b) Fig. 8 GENERAL CHARACTERISTICS OF (a) DAMPING FORCE, D ( t ) (b) LOADING FORCE, E (t) C = Dashpot w i t h damping C B B K r = Base s p r i n g w i t h s t i f f n e s s 12. X ( t ) = R e l a t i v e v e l o c i t y K L = L a t e r a l s p r i n g w i t h s t i f f n e s s E ( t ) = L o a d i n g f o r c e = earthquake f o r c e = -m ^ ( t ) m = Mass of the system = Earthquake a c c e l e r a t i o n D(t) = Damping f o r c e Q(t) = R e s i s t a n c e f o r c e I ( t ) = I n e r t i a f o r c e P ( t ) = E a r t h L a t e r a l F o r c e At any i n s t a n t o f time t , the e q u i l i b r i u m o f f o r c e s a c t i n g on the mass m r e q u i r e s : I ( t ) + D(t) + Q(t) = E ( t ) + P ( t ) '(4) w h i l e a s h o r t time l a t e r the e q u a t i o n would be: I(t + A t ) + D(t+At) + Q(t+At) = E(t+At) + P(t+At) (5) s u b t r a c t i n g e q u a t i o n (5) from e q u a t i o n (4) g i v e s the i n c r e m e n t a l form o f t h e e q u a t i o n o f motion f o r the time i n t e r v a l t , which i s as f o l l o w s : A l ( t ) + AD(t) + AQ(t) = AF(t) + AP(t) The i n c r e m e n t a l f o r c e s i n t h i s e q u a t i o n may be e x p r e s s e d as f o l l o w s : A l ( t ) = I(t+AT) - I ( t ) = m Ax'(t), 13. AD( t ) = A(t+At) - D(t) = C _ ( t ) A X ( t ) , B AQ(t) = Q(t+At) - Q ( t ) = K ( t ) A X ( t ) , (7) B A F ( t ) = F(t+At) - F ( t ) = -m A X ^ t ) , and, AP(t) = P(t+At) - P ( t ) = K ( t ) AX(t) In the above e q u a t i o n s , m = mass o f the system as d e f i n e i n F i g u r e 1 and A x ( t ) , A x ( t ) , and AX(t) a r e r e s p e c t i v e l y the r e l a t i v e d i s p l a c e m e n t increment, the r e l a t i v e v e l o c i t y increment and the r e l a t i v e a c c e l e r a t i o n increment of the w a l l r e f e r r e d t o the ground f o u n d a t i o n , a t time t , and Ax ^ ( t ) i s the ground a c c e l e r a t i o n increment a t the same time t . Cg^t) r e p r e s e n t s the damping p r o p e r t i e s c o r r e s p o n d e n t t o the v e l o c i t y e x i s t i n g d u r i n g the time i n t e r v a l as i n d i c a t e d i n f i g u r e 8 ( b ) . In t h e a n a l y s i s performed C ( t ) was assumed t o have a c o n s t a n t v a l u e C w h i l e t h e base B B s p r i n g remains e l a s t i c , and a z e r o v a l u e as soon as i t re a c h e s the p l a s t i c range. K ( t ) and K ( t ) r e p r e s e n t the s t i f f n e s s p r o p e r t i e s o f t h e base and B L l a t e r a l s p r i n g s r e s p e c t i v e l y c o r r e s p o n d e n t t o the w a l l movement X ( t ) as shown i n F i g u r e s 3 and 5. Both w i l l have a c o n s t a n t v a l u e and e q u a l t o K and K r e s p e c t i v e l y w h i l e i n t h e e l a s t i c range and a s p e c i f i e d v a l u e B L w h i l e i n the p l a s t i c range. T h i s l a t t e r w i l l be e x p l a i n e d i n d e t a i l i n the f o l l o w i n g c h a p t e r . S u b s t i t u t i n g the f o r c e e x p r e s s i o n s o f e q u a t i o n s (7) i n t o e q u a t i o n s (6) l e a d s t o the f i n a l form of t h e i n c r e m e n t a l e q u i l i b r i u m e q u a t i o n s f o r time t : mAX ( t ) + C B ( t ) A x ( t ) + V t ) ^ ( t ) - -"tobM + * L ( t ) to(t) (8) 14. 2.3.3 Step by Step I n t e g t r a t i o n The b a s i c assumption o f the s o l u t i o n method i s t h a t the a c c e l e r a t i o n v a r i e s l i n e a r l y d u r i n g each time increment w h i l e the p r o p e r t i e s o f t h e system remain c o n s t a n t d u r i n g t h i s i n t e r v a l . The motion o f t h e mass d u r i n g the time i n t e r v a l i s i n d i c a t e d i n g r a p h i c a l form i n F i g u r e 9, t o g e t h e r w i t h e q u a t i o n s f o r the assumed l i n e a r v a r i a t i o n o f t h e a c c e l e r a t i o n and the c o r r e s p o n d i n g q u a d r a t i c and c u b i c v a r i a t i o n s o f t h e v e l o c i t y and d i s p l a c e m e n t , r e s p e c t i v e l y . deceleration (linear) x'(7) • 'x(t) • -%TZ Ax (t) velocity (quodratic) x It) X Ax(t) displacement (cubic) x(t) Ax(t) • 7 t t + At MOTION OF SYSTEM DURING TIME INCREMENT (based on linear acceleration) (after Wilson and Clough,l975) 15. E v a l u a t i n g t h e s e l a t t e r e x p r e s s i o n s a t the end of the i n t e r v a l (£=At) l e a d s t o the f o l l o w i n g e q u a t i o n s f o r the increment of v e l o c i t y and d i s p l a c e m e n t : (9.a) (9.b) At AX(t) = X ( t ) A t + AX(t) — ", . A t 2 A f AX(t) = X ( t ) A t + X ( t ) — + A X ( t ) — Now u s i n g the i n c r e m e n t a l d i s p l a c e m e n t as the b a s i c v a r i a b l e o f t h e a n a l y s i s , from e q u a t i o n 9a the i n c r e m e n t a l a c c e l e r a t i o n i s o b t a i n e d AX(t) = (AX (t) - X ( t ) A t ) - ^ t S u b s t i t u t i n g t h i s v a l u e i n e q u a t i o n (9.b) we o b t a i n AX(t) = X ( t ) A t + X ( t ) ^ + [(AX(t) - X ( t ) A t ) 2 ^ ] - ^ -At At' AX(t) = X ( t ) A t + X ( t ) — + A X ( t ) — - X ( t ) — From e q u a t i o n (11) A X ( t ) . i s o b t a i n e d At ", AX(t) = T ^ A X f t ) - 3X(t) - X ( t ) (10) (11) (12) and s u b s t i t u t i n g e q u a t i o n (12) i n t o e q u a t i o n (10) y i e l d s t o : * At- *" 2 AX(t) = [ ("77" AX(t) - 3X(t) - — X ( t ) ) - X ( t ) At] — At o r Ax'(t) = -^2 Ax(t) - X(t) - 3 X(t) (13) 16. S u b s t i t u t i n g e q u a t i o n s (12) and (13) i n t o e q u a t i o n (8) l e a d s t o the f o l l o w i n g form o f the e q u a t i o n o f motion: m [ — AX(t) - 7- X ( t ) - 3X(t)] + C f t ) [ - 7 - AX(t) - 3X(t) - -^f X ( t ) ] + A t 2 A t B At 2 + K B ( t ) A x ( t ) = -m Axb(t) + K L ( t ) Ax(t) f i n a l l y t r a n s f e r r i n g a l l forms a s s o c i a t e d w i t h the known i n i t i a l c o n d i t i o n s t o t h e r i g h t hand s i d e g i v e s : (14) K ( t ) A x ( t ) = Ap(t) (15) where: K ( t ) = K B ( t ) + ~^m + C B ( t ) - K L ( t ) (16) and 6 At Ap(t) = -mAXb(t) + m [ — X ( t ) + 3 X ( t ) ] + C R ( t ) [ 3 X ( t ) + — X ( t ) ] (17) A t 2 E q u a t i o n (15) i s e q u i v a l e n t to a s t a t i c i n c r e m e n t a l e q u i l i b r i u m r e l a t i o n s h i p and may be s o l v e d f o r the i n c r e m e n t a l d i s p l a c e m e n t by d i v i d i n g the i n c r e m e n t a l l o a d by the s t i f f n e s s . The dynamic b e h a v i o u r i s accounted f o r by the i n c l u s i o n o f i n e r t i a l and damping e f f e c t s i n the e f f e c t i v e l o a d and s t i f f n e s s terms. A f t e r the e q u a t i o n (15) i s s o l v e d f o r t h e d i s p l a c e m e n t increment, t h i s v a l u e i s s u b s t i t u t e d i n t o e q u a t i o n (12) t o o b t a i n the i n c r e m e n t a l v e l o c i t y . 17. T h i s n u m e r i c a l a n a l y s i s p rocedure i n c l u d e s two s i g n i f i c a n t a p p r o x i m a t i o n s : (1) t h a t the a c c e l e r a t i o n v a r i e s l i n e a r l y and (2) t h a t the damping and s t i f f n e s s p r o p e r t i e s remain c o n s t a n t d u r i n g the time s t e p . In g e n e r a l , n e i t h e r of the assumptions i s e n t i r e l y c o r r e c t even though t h e r r o r s are s m a l l i f the time s t e p i s s h o r t . In the p r e s e n t study s i n c e s t i f f n e s s and damping are assumed t o be c o n s t a n t ( e i t h e r i n the e l a s t i c o r p l a s t i c r a n g e s ) , those e r r o r s are m i n i m i z e d . 2.3.4 Summary o f the Procedure For any g i v e n time increment, the a n a l y s i s p r o c e d u r e c o n s i s t s of the f o l l o w i n g s t e p s : 1. 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 v a l u e s X ( t ) and X ( t ) a r e known, e i t h e r from v a l u e s a t the end o f the p r e c e d i n g increment or as i n i t i a l c o n d i t i o n s o f the problem. 2. With these v a l u e s and the s p e c i f i e d p r o p e r t i e s of the s t r u c t u r e , the damping C ( t ) and the s t i f f n e s s K ( t ) and B B K ( t ) f o r the i n t e r v a l are found and c o n s e q u e n t l y the L damping and s p r i n g f o r c e s D(t) and Q(t) and P ( t ) . 3. The i n i t i a l a c c e l e r a t i o n i s g i v e n by X ( t ) = - [-mX, (t) - D(t) - Q(t) + P ( t ) (18) m Jo 18. 4. The e f f e c t i v e l o a d increment AP(t) and e f f e c t i v e s t i f f n e s s K ( t ) a r e computed from e q u a t i o n s (16), ( 1 7 ) . 5. The d i s p l a c e m e n t increment i s g i v e n by e q u a t i o n ( 1 5 ) , and w i t h i t the v e l o c i t y increment i s found from e q u a t i o n (12). 6. F i n a l l y the v e l o c i t y and d i s p l a c e m e n t a t the end of the increment a r e o b t a i n e d from: X(t+At) = X ( t ) + AX(t) (19) X(t+At) = X ( t ) + AX(t) when s t e p 6 has been completed, t h e a n a l y s i s f o r t h i s time increment i s f i n i s h e d , and the e n t i r e p r o c e s s may be r e p e a t e d f o r the next time i n t e r v a l . As w i t h any 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 s s , the a c c u r a c y of t h i s s t e p by s t e p method w i l l depend on th e l e n g t h o f the time increment At. Three f a c t o r s must be c o n s i d e r e d i n the s e l e c t i o n o f t h i s i n t e r v a l : (1) the r a t e o f v a r i a t i o n o f the a p p l i e d l o a d i n g E ( t ) , (2) the c o m p l e x i t y of t h e n o n - l i n e a r damping and s t i f f n e s s p r o p e r t i e s , and, (3) t h e p e r i o d T o f v i b r a t i o n of the s t r u c t u r e . In g e n e r a l , an increment of At/T < 1/10 i s a good r u l e o f thumb f o r o b t a i n i n g r e l i a b l e r e s u l t s . In the a n a l y s i s performed an i n i t i a l time increment of T/10 was c o n s i d e r e d , and t h i s time increment i s s u b d i v i d e d any time a s i g n i f i c a n t sudden change t a k e s p l a c e as i t o c c u r s i n the y i e l d i n g of the e l a s t o p l a s t i c s p r i n g s o r i n v e l o c i t y s i g n a l changes. T h i s w i l l be l a t t e r e x p l a i n e d w i t h more d e t a i l i n the Appendix. 19. 3. MODEL PARAMETERS AND RETAINING STRUCTURE USED IN THE ANALYSIS 3.1 G e n e r a l As mentioned p r e v i o u s l y the a n a l y s i s i s based on a lumped mass, s p r i n g s and dashpot system which i s a p p r o x i m a t e l y e q u i v a l e n t t o t h e a c t u a l f o u n d a t i o n - s o i l system. In such lumped system the mass r e p r e s e n t s a l l of t h e i n e r t i a p r e s e n t i n the a c t u a l system, w h i l e the s p r i n g s and dashpot r e s p e c t i v e l y r e p r e s e n t a l l of the f l e x i b i l i t y and damping p r e s e n t i n the same system. The key s t e p i s t h e e v a l u a t i o n o f the parameters o f t h e e q u i v a l e n t lumped system. Once t h i s has been done, the mathematical p r o c e d u r e d e s c r i b e i n Chapter 2 can be used t o e s t i m a t e the response of t h e a c t u a l system. The f o u r b a s i c parameters of the model a r e : 1) Mass, 2) L a t e r a l s p r i n g c o n s t a n t , 3) Base s p r i n g c o n s t a n t , and, 4) Damping. The Procedure f o r o b t a i n i n g these parameters i s p r e s e n t e d h e r e i n . 3.2 C h o i c e o f Mass f o r E q u i v a l e n t Lumped System The mass o f the e q u i v a l e n t lumped system s h o u l d a t l e a s t i n c l u d e the mass of the r e t a i n i n g w a l l s t r u c t u r e p l u s the s o i l b a c k f i l l c o n t a i n e d i n the domain above the base as shown i n F i g u r e 1. In t h e l i t e r a t u r e i t i s found t h a t some r e s e a r c h e r s use an " e f f e c t i v e mass" c o n c e p t . T h i s concept i s o n l y j u s t i f i e d when a mass l a r g e r than t h a t of the f o u n d a t i o n b l o c k p l u s the s t r u c t u r e i s needed t o make t h e r e s ponse c u r v e of the lumped mass f i t the response c u r v e o f the a c t u a l system. Upon Whitman and R i c h a r t (1967) recommendations, the s i m p l e s t 20. assumption t h a t can be made i s s i m p l y t o take t h i s mass e q u a l t o t h a t of t h e f o u n d a t i o n b l o c k p l u s s t r u c t u r e and t o i g n o r e any " e f f e c t i v e mass" c o n c e p t . Whitman and R i c h a r t have shown i n t h e i r paper t h a t v a r i o u s r e s e a r c h e r s i n the p a s t have attempted t o e s t i m a t e the e f f e c t i v e mass on i n t u i t i v e grounds o r by f i t t i n g r e s p o n s i v e c u r v e s f o r lumped systems t o the response c u r v e s o f a c t u a l f o u n d a t i o n s [ C r o c k e t t e t a l (1949), Lorenz (1953) and Heukelom ( 1 9 5 9 ) ] . However, the r e s u l t s o b t a i n e d were w i d e l y s c a t t e r e d and sometimes u n r e a s o n a b l e . Based on the above, Whitman and R i c h a r t (1967) and Whitman (1972) recommend t h a t the f i c t i t i o u s " e f f e c t i v e mass" i s s m a l l and i t r e a l l y i s b e s t t o i g n o r e i t c o m p l e t e l y . T h e r e f o r e i n the p r e s e n t a n a l y s i s no " e f f e c c i v e mass" was c o n s i d e r e d . Next i t i s p r e s e n t e d the r e t a i n i n g w a l l dimensions used i n the a n a l y s i s , which a l l o w s t o compute the c o r r e s p o n d e n t mass. 21. 3.3 C h o i c e o f S p r i n g C o n s t a n t s , U s i n g Measured S t r e s s - S t r a i n R e l a t i o n s  and E m p i r i c a l E q u a t i o n s 3.3.1 L a t e r a l S p r i n g 3.3.1.1 S t r a i n s R e q u i r e d t o Reach A c t i v e and P a s s i v e S t a t e s As mentioned p r e v i o u s l y the s t i f f n e s s and y i e l d i n g p o i n t s o f t h e l a t e r a l s p r i n g were based on T e r z a g h i (1934) and Lambe and Whitman (1969). The f o l l o w i n g d e s c r i b e s t h e i r main c o n c l u s i o n s d e r i v e d from t r i a x i a l l a b o r a t o r y d a t a of t e s t s performed on dense sand samples: 1) Very l i t t l e h o r i z o n t a l s t r a i n , l e s s than -0.5% i s r e q u i r e d t o r e a c h the a c t i v e s t a t e . 2) L i t t l e h o r i z o n t a l compression, about +0.5% i s r e q u i r e d t o r e a c h o n e - h a l f of the maximum p a s s i v e r e s i s t a n c e , and, 3) Much more h o r i z o n t a l compression, about +2.0% i s r e q u i r e d t o r e a c h the f u l l maximum p a s s i v e r e s i s t a n c e . The above r e s u l t s are t y p i c a l f o r most dense sands. For l o o s e sands the f i r s t 2 c o n c l u s i o n s remain v a l i d , but the h o r i z o n t a l compression r e q u i r e d t o r e a c h f u l l p a s s i v e r e s i s t a n c e may be as l a r g e as 15%. The above r e s u l t s a p p l y when the i n i t i a l c o n d i t i o n i s a c o n d i t i o n ( a t r e s t c o n d i t i o n ) . I f i n i t i a l l y a / o * K then somewhat d i f f e r e n t h v o s t r a i n s w i l l be r e q u i r e d t o r e a c h the l i m i t i n g c o n d i t i o n s . Two i n i t i a l oh/ov c o n d i t i o n s were c o n s i d e r e d i n the a n a l y s i s : an a c t i v e c o n d i t i o n ( K A ) and an " a t r e s t " c o n d i t i o n (K ). The above l i m i t i n g c o n d i t i o n s were o assumed t o a p p l y a l s o f o r the i n i t i a l a c t i v e c o n d i t i o n (K ) and t h e r e f o r e the s t r a i n r e q u i r e d t o r e a c h o n e - h a l f of the maximum p a s s i v e r e s i s t a n c e was assumed t o be -1.0% as shown i n F i g u r e 10. T h i s s i m p l e l i n e a r approach i s c o m p a t i b l e w i t h the o v e r a l l s i m p l e t y p e o f a n a l y s i s . F o r more a c c u r a t e s o l u t i o n s , r e l a t i o n s h i p s between h o r i z o n t a l s t r a i n and e a r t h c o e f f i c i e n t based on l a b o r a t o r y data (as shown i n F i g u r e 4) s h o u l d be used i n s t e a d . The two d i f f e r e n t i n i t i a l d e p a r t u r e c o n d i t i o n s are e x p l a i n e d below: 1) I n i t i a l c o n d i t i o n - assumes t h a t the w a l l y i e l d s s u f f i c i e n t l y t o produce minimum a c t i v e p r e s s u r e s w h i l e i n s t a t i c , i . e . , the pre-dynamic e a r t h c o e f f i c i e n t c o r r e s p o n d s t o the a c t i v e c o n d i t i o n (K ). A 2) I n i t i a l K c o n d i t i o n - assumes t h a t a K c o n d i t i o n remains w h i l e o o i n s t a t i c i . e . , the pre-dynamic e a r t h c o e f f i c i e n t c o r r e s p o n d s t o the " a t r e s t " c o n d i t i o n (K ). o 3.3.1.2 Rankine V a l u e s o f K f tand Kp Used i n The A n a l y s i s f o r D i f f e r e n t S o i l B a c k f i l l s ~ ~ ~ ' ~ The Rankine v a l u e s o f K abd K were e s t i m a t e d u s i n g the s t a n d a r d A P e q u a t i o n s found i n the l i t e r a t u r e (Lambe and whitman, 1969): K - ^ f ^ l (20) A 1+Sinji and K = - i - ( 2 D P K A The d i f f e r e n t r e l a t i v e d e n s i t y c o n d i t i o n s were c o n s i d e r e d i n the a n a l y s i s c o r r e s p o n d e n t t o dense, medium dense and l o o s e s t a t e of compaction. In T a b l e I i t i s p r e s e n t e d the s o i l b a c k f i l l parameter v a l u e s , as w e l l as the mass o f t h e w a l l p l u s s o i l b a c k f i l l r e q u i r e d t o d e v e l o p a s t a t i c f a c t o r o f s a f e t y a g a i n s t s l i d i n g , F =1.5 f o r the t h r e e d i f f e r e n t s t a t e s o f compaction. These v a l u e s w i l l be used l a t e r i n the a n a l y s i s (see Chapter 5.3.2.) 23. TABLE I S o i l B a c k f i l l Parameters S o i l B a c k f i l l No. F r i c t i o n A n g l e <J> A c t i v e Coef. A P a s s i v e C o e f . K P U n i t Weight, Y S t a t e o f Compaction Mass o f w a l l (+) b a c k f i l l f o r F =1.5 s 1 40 0.217 4.60 135 l b / c f Dense 502 s l u g 2 35 0.271 3.69 130 l b / c f M.Dense 604 3 30 0.333 3.00 125 l b / c f Loose 714 Based on the above T a b l e I and on the s t r a i n s r e q u i r e d t o re a c h a c t i v e and p a s s i v e s t a t e s the f o l l o w i n g F i g u r e 10 show the r e l a t i o n s h i p between e a r t h c o e f f i c i e n t and s t r a i n f o r the t h r e e s t a t e s of compaction. In t h i s f i g u r e i t i s o n l y shown the i n i t i a l -1% s t r a i n and the c o r r e s -pondent e a r t h c o e f f i c i e n t v a l u e s f o r the t h r e e s t a t e s o f compaction, s i n c e i t was found i n t h e a n a l y s i s t h a t the r e t a i n i n g w a l l s t r a i n s towards t h e b a c k f i l l never exceed t h i s - 1 % s t r a i n v a l u e f o r the s t a n d a r d F =1.5. s 24. Fig. 10 EARTH COEFFICIENTS VS. STRAIN 25. 3.3.2 Base S p r i n g . E v a l u a t i o n o f Foundation S t i f f n e s s , For the e v a l u a t i o n of the base s p r i n g s t i f f n e s s , K the e q u a t i o n B d e v e l o p e d by Barkan (1962) was used. Barkan, assuming t h a t the s o i l i s i s o t r o p i c and homogeneous, t h a t a u n i f o r m d i s t r i b u t i o n of shear s t r e s s e x i s t s over the c o n t a c t a r e a , and computing the average h o r i z o n t a l d i s p l a c e m e n t of the c o n t a c t a r e a , d e r i v e d the f o l l o w i n g e q u a t i o n to e s t i m a t e the s p r i n g c o n s t a n t f o r a r i g i d r e c t a n g u l a r base r e s t i n g on e l a s t i c h a l f - s p a c e : K = 2(l+y) GB B x /El, (23) where: u = p o i s s o n ' s r a t i o G = shear modulus B = w i d t h o f f o u n d a t i o n L = l e n g t h of f o u n d a t i o n and, 3 = s p r i n g c o n s t a n t c o e f f i c i e n t ( g i v e n i n F i g u r e 2 6 . Fig. II SPRING CONSTANT COEFFICIENTS FOR RECTANGULAR FOUNDATIONS (after Lambe and Whitman , 1969) As a p o i n t o f i n t e r e s t the r e a d e r can f i n d s i m i l a r e q u a t i o n s f o r r i g i d c i r c u l a r and r e c t a n g u l a r b a s e s , r e s t i n g on e l a s t i c h a l f - s p a c e f o r d i f f e r e n t types o f motion on the f o l l o w i n g T a b l e I I d e r i v e d from Whitman and R i c h a r t (1967), and Whitman (1972). 27. T a b l e I I S p r i n g C o n s t a n t s f o r R i g i d Base R e s t i n g on E l a t i c H a l f - S p a c e MOTION SPRING CONSTANT REFERENCE CIRCULAR BASE V e r t i c a l v - 4 G R  K z 1-u Timoshenko and God d i e r (1951) H o r i z o n t a l v _ 32(1 - U)GR K x 7-8 p v - 8 R G B y c r o f t (1956) Whitman (1972) Rocking 8FR3  K<|> 3 ( l - n ) Borowicka (1943) T o r s i o n K 0 = i | GR 3 R e i s s n e r and S a g o c i (1944) RECTANGULAR BASE V e r t i c a l K Z = l V z / B L Barkan (1962) H o r i z o n t a l K = 2 ( 1 + U ) G B / B L •i it Rocking K<t> = T V * b l 2 Go rbunov-Po ss a dov (1961) Note: V a l u e s f o r 3 , 3 and B a r e g i v e n i n F i g u r e 11. z x <p In o r d e r t o compare the p r e v i o u s e q u a t i o n ( 2 3 ) f o r r e c t a n g u l a r base (Barkan 1 9 6 2 ) w i t h the e q u a t i o n s d e r i v e d from B y c r o f t ( 1 9 5 6 ) and Whitman ( 1 9 7 2 ) f o r c i r c u l a r base, f o r the case o f h o r i z o n t a l motion, a square base 8 . 5 x 8 . 5 f t 2 i s assumed r e s t i n g on a s o i l f o u n d a t i o n w i t h p o i s s o n ' s r a t i o u = 0 . 3 . The v a l u e s o b t a i n e d a r e as f o l l o w s : K = 2 ( l + u ) G3 /BL = 2 2 . 1 G Barkan ( 1 9 6 2 ) x x 8RG K = —— , = 2 2 . 5 G Whitman ( 1 9 7 2 ) x 2 1-u) 32(1-U)GR 28. K x = —7-8 V = 2 3 " 3 G B y c r o f t (1956) The above shows t h a t the t h r e e e q u a t i o n s g i v e c o n s i s t a n t v a l u e s . T h e r e f o r e e q u a t i o n (23) was chosen i n the p r e s e n t a n a l y s i s s i n c e i t c o r r e s p o n d s t o the r i g h t type of f o u n d a t i o n and a l l o w s t o manipulate i t f o r d i f f e r e n t L/B v a l u e s . However, c a r e s h o u l d be taken when a p p l y i n g t h i s e q u a t i o n t o the p a r t i c u l a r case of r e t a i n i n g w a l l s s i n c e r e t a i n i n g w a l l d e s i g n i s g e n e r a l l y based on a u n i t l e n g t h o f w a l l . Hence e q u a t i o n (23) was m o d i f i e d as f o l l o w s : Kg = 2(1+U) GB X ( i n f o r c e / l e n g t h / l e n g t h (24) The f o l l o w i n g p r e s e n t s the e s t i m a t e d v a l u e s f o r V and G used i n the a n a l y s i s f o r d i f f e r e n t s o i l and rock f o u n d a t i o n c h a r a c t e r i s t i c s . 3.3.21 P o i s s o n ' s R a t i o , V 29. I t i s p o s s i b l e t o compute P o i s s o n ' s r a t i o f o r s o i l s from measured v a l u e s o f the compression-wave and shear-wave v e l o c i t i e s t hrough the s o i l . However, t h e s e computations i n v o l v e s m a l l d i f f e r e n c e s of r a t h e r l a r g e number, and s i g n i f i c a n t e r r o r s are p o s s i b l e . G e n e r a l l y , i t has been found t h a t P o i s s o n ' s r a t i o v a r i e s from about 0.25 t o 0.35 f o r c o h e s i o n l e s s s o i l s and from about 0.35 t o 0.45 f o r c o h e s i v e s o i l s which are c a p a b l e of s u p p o r t i n g b l o c k - t y p e f o u n d a t i o n s . ( R i c h a r t e t a l . 1970). F o r l i n e a r l y e l a s t i c and i s o t r o p i c r o c k s , u i s i n the range 0.0 t o 0.5 and i s o f t e n assumed t o be 0.25 (Goodman 1979). Con s e q u e n t l y , f o r d e s i g n purposes l i t t l e e r r o r i s i n t r o d u c e d i f P o i s s o n ' s r a t i o i s assumed as shown on T a b l e I I I , f o r the case of d i f f e r e n t f o u n d a t i o n c o n d i t i o n s c o n s i d e r e d i n the a n a l y s i s . TABLE I I I P o i s s o n ' s R a t i o V a l u e s F o u n d a t i o n C o n d i t i o n P o i s s o n ' s R a t i o , V Rock 0.25 C o h e s i o n l e s s S o i l 0.30 S o f t C l a y 0.40 30. 3.3.2.2 Modulus o f E l a s t i c i t y i n Shear G b . l S o i l F o u n d a t i o n The modulus o f e l a s t i c i t y i n Shear, G i s the most i m p o r t a n t o f a l l parameters i n v o l v e d i n a lumped system due t o i t s e f f e c t s on t h e computation o f the s p r i n g c o n s t a n t v a l u e (see e q u a t i o n ( 2 3 ) ) . S i n c e the main o b j e c t i v e o f t h i s study i s the development o f a s i m p l e method of a n a l y s e s t o determine d i s p l a c e m e n t s and l a t e r a l h o r i z o n t a l f o r c e s on the w a l l , the w r i t e r i s not g o i n g t o extend t o o much on t h e shear modulus e v a l u a t i o n but simply emphasize i t s importance and g i v e a b r i e f r e v i e w o f how i t can be o b t a i n e d . F o r more d e t a i l s r e g a r d i n g G e v a l u a t i o n the r e a d e r can f i n d i t i n the f o l l o w i n g r e f e r e n c e s : Whitman and R i c h a r t (1967), H a r d i n and B l a c k (1968 and 1969), Seed and I d r i s s (1970), H a r d i n and D r n e v i c h (1970) R i c h a r t e t a l (1970), Whitman (1972 and 1976), Ohsaki and Iwasaki (1973) and, Anderson e t a l (1978). There a r e two main approaches t o determine G: D i r e c t methods ( f i e l d and lab) and e m p i r i c a l methods. C r i t i c a l s o i l a n a l y s i s g e n e r a l l y r e q u i r e d e t e r m i n a t i o n o f G by g e o p h y s i c a l methods, such as w i t h t h e c r o s s h o l e o r downhole t e c h n i q u e . However, i n c e r t a i n s i t u a t i o n s i t i s n e c e s s a r y t o r e l y on l a b o r a t o r y t e s t i n g methods o r e m p i r i c a l e q u a t i o n s e i t h e r t o supplement g e o p h y s i c a l t e s t i n g o r t o v e r i f y the c r e d i b i l i t y o f moduli determined by g e o p h y s i c a l methods. 1) D i r e c t Methods The f o l l o w i n g T a b l e IV, d e r i v e d from Seed and I d r i s s (1970) p r e s e n t s the t e s t p r o c e d u r e s f o r measuring modulus and damping c h a r a c t e r i s t i c s . TABLE IV T e s t P r o c e d u r e s f o r Measuring Modulus and Damping C h a r a c t e r i s t i c s ( A f t e r Seed and I d r i s s , 1970) G e n e r a l Procedure T e x t C o n d i t i o n Approximate S t r a i n Range P r o p e r t i e s Determined D e t e r m i n a t i o n o f h y s t e r e t i c 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 T r i a x i a l compression Simple Shear T o r s i o n a l Shear 10" 10" 10" 2 2 2 t o t o t o 5% 5% 5% Modulus; Modulus; Modulus; damping damping damping F o r c e d V i b r a t i o n L o n g i t u d i n a l v i b r a t i o n s T o r s i o n a l v i b r a t i o n s Shear v i b r a t i o n s - l a b Shear v i b r a t i o n s - f i e l d 10" 10" 10" 4 it i+ t o t o t o 10 -10" 10 -• 2% -2% -2% Modulus; Modulus; Modulus; Modulus damping damping damping Free v i b r a t i o n t e s t s L o n g i t u d i n a l v i b r a t i o n s T o r s i o n a l v i b r a t i o n s Shear v i b r a t i o n s - l a b Shear v i b r a t i o n s - f i e l d 10" 10" 10" 10" 3 3 3 3 t o t o t o to 1% 1% 1% 1% Modulus Modulus Modulus F i e l d S e i s m i c Response Measurement of motions i n d i f f e r e n t l e v e l s i n d e p o s i t Modulus, damping 2) E m p i r i c a l Methods A comprehensive su r v e y o f the f a c t o r s a f f e c t i n g the shear moduli o f s o i l s and e x p r e s s i o n s f o r d e t e r m i n i n g t h i s p r o p e r t y have been p r e s e n t e d by Ha r d i n and D r n e v i c h (1970). In t h e i r s t u d y , an e m p i r i c a l e q u a t i o n was p r e s e n t e d t o determine the v a l u e s of maximum shear modulus G ( a t e s s e n t i a l l y z e r o - s t r a i n ) . T h e i r e q u a t i o n max i s as f o l l o w s : r - i47fin v (2.973 - e)' Gmax " 1 4 7 6 0 x m .a,-. ,1/2 (O.C.R) a( a'm) (25) where G = maximum shear modulus i n p s f . max e = v o i d r a t i o O.C.R. = o v e r c o n s o l i d a t i o n r a t i o a = parameter t h a t depends on the p l a s t i c i t y index of the s o i l , and, a' = mean p r i n c i p a l e f f e c t i v e s t r e s s i n p s f . m Seed and I d r i s s (1970) d e v e l o p e d a r e l a t i o n s h i p which i s v a l i d o n l y f o r sands and i s as f o l l o w s : G = 1000 K 0 m = > ( a') 1 / 2 max 2 max m (26) where G - Shear modulus max K - parameter t h a t r e f l e c t s the i n f l u e n c e of v o i d r a t i o and s t r a i n 2 max a m p l i t u d e , and a' = mean p r i n c i p a l e f f e c t i v e s t r e s s , m F o r c l a y s , Seed and I d r i s s used an e q u a t i o n of the form (27) G -2M = c o n s t a n t Su where Su i s t h e u n d r a i n e d s h e a r i n g s t r e n g t h o f the c l a y . L a b o r a t o r y and i n s i t u t e s t d a t a o b t a i n e d by s e v e r a l r e s e a r c h e r s were used t o e s t a b l i s h t h e c o n s t a n t . I t s v a l u e v a r i e d from 1000 t o 3000. Ohsaki and Iwasaki (1973) d e r i v e d an e q u a t i o n f o r sands and c l a y s , by e m p i r i c a l l y c o r r e l a t i n g c r o s s h o l e v e l o c i t y data t o SPT N - v a l u e s . The e q u a t i o n i n terms of G i s as f o l l o w s : " — I (28) G = 1200 N 0 , 8 max where N i s t h e N-value o b t a i n e d d u r i n g the SPT. Anderson e t a l (1978) d i d a study r e g a r d i n g the e s t i m a t i o n o f i n s i t u s h e a r moduli a t competent s i t e s , where i n s i t u , l a b o r a t o r y and e m p i r i c a l methods were compared. T h e i r main c o n c l u s i o n s a r e as f o l l o w s : 1) The H a r d i n - B l a c k (or H a r d i n - D r n e v i c h ) method t y p i c a l l y o v e r e s t i -mates the l a b o r a t o r y G a t competent s i t e s by a f a c t o r o f 1.1 max t o 1.5 and u n d e r e s t i m a t e s f i e l d v a l u e s of G by a f a c t o r o f max 1.3 t o 2.5. 34. 2) The S e e d - I d r i s s method f o r sands p r o v i d e s a c l o s e r a p p r o x i m a t i o n of the f i e l d G where K i s s e l e c t e d on the b a s i s of f i e l d max 2 max d a t a . 3) The O h s a k i - I w a s a k i method o v e r p r e d i c t s the f i e l d G by 25 max p e r c e n t or more a t competent s i t e s . 4) For important s i t e s i t i s e s s e n t i a l t h a t the i n s i t u g e o p h y s i c a l t e s t s ( p r e f e r a b l y c r o s s h o l e ) be performed t o e s t a b l i s h modulus. E m p i r i c a l and l a b o r a t o r y methods i n v o l v e v a r i a t i o n s which are too l a r g e t o j u s t i f y t h e i r use f o r o t h e r than a p p r o x i m a t e l y G max o r e s t a b l i s h i n g the g e n e r a l c r e d i b i l i t y o f G max E q u a t i o n (25) was used i n the a n a l y s i s t o e s t i m a t e the maximum shear modulus of the s o i l f o u n d a t i o n . I t i s known t h a t G depends on c o n f i n i n g s t r e s s - d e p t h and s t r a i n l e v e l . In the p r e s e n t a n a l y s i s the v a r i a t i o n of G w i t h s t r a i n was not taken i n t o account d i r e c t l y a t the end o f each c o m p u t a t i o n a l s t e p as c o u l d have been done by u s i n g the r e l a t i o n s h i p between G and s t r a i n suggested by Seed and I d r i s s (1970). I n s t e a d the f o l l o w i n g s t e p s based on Whitman (1972 and 1976) were used t o determine G: 1 - F o r h o r i z o n t a l t r a n s l a t i o n i t i s g e n e r a l l y r e a s o n a b l e t o use the average modulus f o r a depth e q u a l t o the r a d i u s of the f o u n d a t i o n . (In t h e a n a l y s i s h a l f o f the w i d t h , B/2, was c o n s i d e r e d ) . Whitman recommends t h a t judgement must be e x e r c i s e d i n the a p p l i c a t i o n of t h e s e r e s u l t s . In p a r t i c u l a r , i t may be n e c e s s a r y t o reduce the modulus i f t h e r e i s a s o f t l a y e r a t the s u r f a c e . 2 - A more d i f f i c u l t problem i s t o account f o r the r e d u c t i o n of G w i t h s t r a i n . Whitman sugges t s the use of an e m p i r i c a l r u l e t o determine t h e modulus, by s i m p l y a d o p t i n g a range o f v a l u e s f o r shear modulus and assuming that any value within t h i s range i s possible. If i s the shear modulus determined using r e l i a b l e data, then the following ranges are proposed (see Table V). TABLE V Ranges of G m a x (After Whitman (1972) MOTION MACHINE FOUNDATION LARGE EARTHQUAKE VERTICAL AND HORIZONTAL °'7 Gmax t o Gmax °- 5 Gmax t o Gmax ROCKING 1 °' 5 Gmax t o Gmax ° ' 3 3 Gmax t o Gmax 1 A shear modulus value = 1/2 G was adopted i n the present analysis max to account f o r i t s change with s t r a i n . The only step not followed by the writer to Whitman's suggestions was in the c a l c u l a t i o n of mean p r i n c i p a l e f f e c t i v e stress (o' ). Whitman m recommends to include the e f f e c t of the weight of the structure when estimating a' . However i n the present a n l y s i s the weight of the m ret a i n i n q wall and b a c k f i l l were not taken into account f o r the a' 3 m evaluation. The reasoning followed i s that although the column of s o i l 36. d i r e c t l y beneath the b l o c k f o u n d a t i o n becomes " s t i f f e r " due t o the weight o f the r e t a i n i n g w a l l and s o i l b a c k f i l l , the s u r r o u n d i n g f o u n d a t i o n s o i l t o t h a t column o f s o i l does not get a f f e c t e d and i t i s t h i s s u r r o u n d i n g s o i l t h a t w i l l r e a c t t o any developed h o r i z o n t a l f o r c e . The f o l l o w i n g F i g u r e 12 i l l u s t r a t e s the above. 'soil R-stiff pile (b) Fig. 12 ILLUSTRATION OF SHEAR MODULUS LOCATION ASSESSMENT F i g u r e 12(b) i l l u s t r a t e s i n a more r e a l i s t i c manner the above c o n s i d e r a t i o n s , where the column o f s o i l w i t h shear modulus, G shown i n F i g u r e 12(a), can be v i s u a l i z e d as s t i f f p i l e when compared w i t h the s u r r o u n d i n q s o i l w i t h shear modulus, G . The d e f o r m a t i o n o f the s t i f f s o i l p i l e t o any developed h o r i z o n t a l f o r c e F w i l l be c o n t r o l l e d by the shear modulus o f the s u r r o u n d i n g s o i l G ... s o i l 37. b.2 - Rock Foundation To e s t i m a t e the s p r i n g c o n s t n a t c o r r e s p o n d e n t t o a rock type f o u n d a t i o n the v a l u e s of G enocuntered i n the l i t e r a t u r e were used (Goodman 1979). G v a l u e s f o r rock range from 10 x 1 0 6 t o 400 x 1 0 6 p s f (or h i g h e r ) . A t y p i c a l v a l u e o f G = 16 x 1 0 6 p s f c o r r e s p o n d e n t t o s i l s t o n e was chosen. 3.3.2.3 V a l u e s o f G and K Used i n the A n a l y s e s max 13 For the d i f f e r e n t type of f o u n d a t i o n c o n d i t i o n s used i n the a n a l y s i s , T a b l e VI p r e s e n t s the assumed s o i l parameters r e q u i r e d t o a s s e s s v a l u e s f o r G u s i n g e q u a t i o n (26) and t h e c o r r e s p o n d e n t v a l u e s o b t a i n e d , max TABLE VI Fo u n d a t i o n Parameters V a l u e s t o A s s e s s G FOUNATION CONDITION VOID RATIO, e OVER CONSOLIDATION RATIO (O.C.R.) UNIT WEIGHT Y( L b / c f ) SHEAR MODULUS G m a x < L b / s f ) C o h e s i o n l e s s S o i l 0.43 1 135 5.84 x 1 0 5 S o f t C l a y 2.00 1 98 0.38 x 10 5 Rock (from b.2) 160.00 x 1 0 5 38. U s i n g e q u a t i o n (24) and the shear modulus v a l u e s p r e s e n t e d i n T a b l e V I , v a l u e s f o r K (base s p r i n g c o n s t a n t ) c o r r e s p o n d i n g t o d i f f e r e n t base B l e n g t h and width r a t i o s (L/B) were o b t a i n e d . These v a l u e s a r e p r e s e n t e d on the f o l l o w i n g t a b l e . TABLE V I I S p r i n g Constant V a l u e s , K R ( L b / f t / f t ) x 1 0 6 L/B C o h e s i o n l e s s S o i l S o f t C l a y Rock 1 1.50 0.10 43.2 2 1.00 0.08 30.2 4 0.75 0.06 21.6 6 0.67 0.05 19.4 8 0.61 0.04 17.6 10 0.57 0.038 16.2 3.4 C h o i c e of Damping f o r E q u i v a l e n t Lumped System The dashpot of t h e lumped system r e p r e s e n t s the damping of the s o i l . T h e r e a r e two t y p e s of damping - the l o s s o f energy through p r o p a g a t i o n of waves away from th e immediate v i c i n i t y o f the f o o t i n g , and the i n t e r n a l energy l o s s w i t h i n the s o i l due t o h y s t e r e t i c and v i s c o u s e f f e c t s . L i n e a r dashpots a r e used t o d e r i v e s i m p l e u s e f u l mathematical e x p r e s s i o n s f o r the response o f t h e lumped system. In g e n e r a l the damping, C, o f a dashpot element i s e x p r e s s e d i n terms o f t h e damping r a t i o , X, which r e l a t e s C t o t h e " c r i t i c a l damping", C (X = C/C ). c c Most o f the c o n s i d e r a t i o n s and r e f e r e n c e s d e s c r i b e d above f o r the shear modulus a l s o a p p l y f o r t h e e v a l u a t i o n of the damping r a t i o X. The 39. damping r a t i o A v a r i e s w i t h , number o f c y c l e s , N (and t h e r e f o r e w i t h s t r a i n ) , mean e f f e c t i v e s t r e s s , a' and w i t h P o i s s o n ' s r a t i o u. No m c o r r e c t i o n o f the damping r a t i o w i t h the above f a c t o r s was taken i n t o a ccount a t the end of each computation s t e p and i n s t e a d an e m p i r i c a l r u l e was used: a 5% v a l u e f o r damping r a t i o was c o n s i d e r e d t o r e p r e s e n t r a d i a i t o n and i n t e r n a l damping e f f e c t s . T h i s v a l u e i s c o n s i d e r e d t o be c o n s e r v a t i v e and t h e r e f o r e on the s a f e s i d e . Whitman (1976) uses as h i s r u l e o f thumbs the f o l l o w i n g : I f the d e s i g n earthquake has a peak a c c e l e r a t i o n o f 0.25g then 15% i n t e r n a l damping i s r e a s o n a b l e , and i f i t has a peak a c c e l e r a t i o n o f < O.lg, 5% i n t e r n a l damping i s a c c e p t a b l e . As mentioned p r e v i o u s l y damping was o n l y c o n s i d e r e d i n the e l a s t i c range of the base s p r i n g of the i d e a l i z e d model. No damping was c o n s i d e r e d i n t h e p l a s t i c range, s i n c e i n t h i s range the f o r c e r e q u i r e d t o m a i n t a i n the mass i n motion i s c o n s t a n t , which i m p l i e s t h a t the r e s i s t a n t f o r c e i s a l s o c o n s t a n t and o n l y a f u n c t i o n of t h e f r i c t i o n a l r e s i s t a n c e d e v e l o p e d between f o u n d a t i o n b l o c k and f o u n d a t i o n s o i l . R e gardin the l a t e r a l s p r i n g , no damping was c o n s i d e r d , s i n c e t h e i n c r e a s e o r d e c r e a s e of the f o r c e i n t h i s s p r i n g i s o n l y dependent on the r e l a t i v e movement of the b l o c k f o u n d a t i o n to the ground s u r f a c e , which i s c o n t r o l l e d by the movements a l l o w e d by th e base s p r i n g . 3.4.1 V a l u e s of E f f e c t i v e Damping Used i n the A n a l y s i s , C r The damping f o r c e D(t) (see e q u a t i o n (4) and F i g u r e 8 ( a ) ) i s e x p r e s s e d by: 40. D(t) = a, x(t) (29) where: C = e f f e c t i v e damping B X ( t ) = r e l a t i v e v e l o c i t y The e f f e c t i v e damping C can be e x p r e s s e d as f o l l o w s : C B = 2m X % = 2mX ( ^ ) ^ 2 (30) where: m = mass of the w a l l and s o i l b a c k f i l l = 502 s l u g s X = damping r a t i o = 5% K = f o u n d a t i o n s t i f f n e s s B Based on e q u a t i o n (30) and K v a l u e s from T a b l e IV the f o l l o w i n g B v a l u e s f o r C were o b t a i n e d as shown i n T a b l e V I I . B TABLE V I I I E f f e c t i v e Damping V a l u e s , C B ( L b / f t / s e c ) L/B S t i f f S o i l C o h e s i o n l e s s S o i l S o f t C l a y Rock 1 3250 2700 743 14,700 2 2740 2240 634 12,300 4 2600 1940 550 10,400 6 2470 1800 500 9,900 8 2400 1750 450 9,400 10 2000 1700 440 9,000 41. 3.5 Ch o i c e o f Time Increment, At, Used i n t h e A n a l y s i s In o r d e r t o determine the minimum time i n t e r v a l , At, t o use i n the a n a l y s i s , the n a t u r a l p e r i o d , T o f the s t r u c t u r e has t o be e s t i m a t e d . In g e n e r a l , an increment o f At/T < 10, i s a good r u l e f o r o b t a i n i n g r e l i a b l e r e s u l t s . The p e r i o d T i s r e l a t e d w i t h the s p r i n g s t i f f n e s s c o n s t a n t , K , and B the mass of the system, m, as f o l l o w s : 2TT T = — (29) W n K but, W = • — (30) n m 2 Trv'm hence, T = — (31) where W = n a t u r a l f r e q u e n c y o f the system, m The e f f e c t o f the l a t e r a l s p r i n g c o n s t a n t K was n e g l e c t e d s i n c e i t i s much s m a l l e r than the base s p r i n g c o n s t a n t K . B Then u s i n g the v a l u e s o f K from T a b l e V I I , the At/T < 10 c o n d i t i o n , B adn e q u a t i o n (31) the c o r r e s p o n d e n t v a l u e s f o r At were o b t a i n e d as shown i n T a b l e IX. 42. TABLE IX N a t u r a l P e r i o d , T, and Time Increment, At, i n Seconds L/B COHESIONLESS SOIL SOFT CLAY RC XTK T At T At T At 1 0.12 0.01 0.45 0.04 0.03 0.002 2 0.14 0.01 0.50 0.05 0.04 0.004 4 0.16 0.02 0.58 0.06 0.04 0.004 6 0.17 0.02 0.63 0.06 0.05 0.005 8 0.18 0.02 0.71 0.07 0.05 0.005 10 0.19 0.02 0.72 0.07 0.05 0.005 3.6 R e t a i n i n g W a l l Dimensions The g r a v i t y r e t a i n i n g s t r u c t u r e used i n the a n a l y s i s i s 20 f e e t h i g h w i t h a base o f 8.5 f e e t as shown i n F i g u r e 13. T h i s r e t a i n i n g c a n t i l e v e r w a l l was d e s i g n e d f o r a s t a t i c o f s a f e t y a g a i n s t s l i d i n g F = 1.5, c o n s i d e r i n g a s o i l b a c k f i l l i n a dense s t a t e of s compaction (cj>=40° and y=135 l b / c f t ) and a f r i c t i o n f a c t o r o f 0.55 a t t h e i n t e r f a c e o f b l o c k f o u n d a t i o n and s o i l f o u n d a t i o n . The above dimensions were e s t i m a t e d f o l l o w i n g the s t a n d a r d p r o c e d u r e s used i n p r a c t i c e as o u t l i n e d by Peck, Hanson and Thornborn (1973). Below i s shown t h e b a s i c dimension d e s i g n r u l e s : a) BASE - The r a t i o o f the width o f the base t o the o v e r a l l h e i g h t of the w a l l commonly v a r i e s from 0.40 t o 0.65. The s m a l l e r r a t i o i s a p p r o p r i a t e i f the base i s s u p p o r t e d by f i r m s o i l and i f t h e b a c k f i l l has a h o r i z o n t a l s u r f a c e and c o n s i s t s o f c l e a n sand o r g r a v e l . The r a t i o used i n d e s i g n was 0.43. 43. The t h i c k n e s s o f the base commonly l i e s i n the range o f 1/12 t o 1/8 o f the h e i g h t o f the w a l l . The r a t i o used i n d e s i g n was 1/10. b) STEM - The t h i c k n e s s o f the stem must be s u f f i c i e n t t o r e s i s t the shea r and moments due t o the e a r t h p r e s s u r e a g a i n s t the back o f the w a l l . A u n i f o r m stem 1 f o o t t h i c k was used i n the a n a l y s i s and s t r u c t u r a l c o n s i d e r a t i o n s were not taken i n t o account s i n c e are i r r e l e v a n t f o r the p r e s e n t a n a l y s i s . 4> 0) O o CM <0 0 o <\J FIG.I3 C A N T I L E V E R RETAINING WALL USED IN THE A N A L Y S I S . 44. 4. EARTHQUAKE DATA USED IN THE ANALYSES Three e a r t h a c c e l e r a t i o n r e c o r d s were c o n s i d e r e d c o r r e s p o n d i n g t o ; 1) a Rock f o u n d a t i o n , 2) a deep c o h e s i o n l e s s s o i l f o u n d a t i o n and 3) a s o f t c l a y f o u n d a t i o n . In T a b l e X i t i s p r e s e n t e d d a t a r e f e r r e d t o the t h r e e e a r t h q u a k e s . TABLE X Earthquake Data FOUNDATION CONDITIONS EARTHQUAKE RECORD YEAR STATION MAXIMUM ACCELERATION SCALLING FACTOR Rock San Fernando 1971 S t a t i o n 12 Lake Hughes 0.35g 0.50 Deep C o h e s i o n l e s s S o i l E l C e ntro 1940 117 0.35g 0.33 S o f t C l a y Alameda Park 1962 Mexico C i t y 0.09g 0.27 The San Fernando r e c o r d was o b t a i n e d on r o c k and i s an a p p r o p r i a t e f r e e f i e l d motion t o use when the base o f the r e t a i n i n g w a l l r e s t s on r o c k . The E l Centro r e c o r d i s a p p r o p r i a t e where deep c o h e s i o n l e s s d e p o s i t s are p r e s e n t and the Alameda Park, Mexico C i t y r e c o r d i s a p p r o p r i a t e f o r s i t e s u n d e r l a i n by deep s o f t c o m p r e s s i b l e c l a y d e p o s i t s . The t h r e e earthquake a c c e l e r a t i o n r e c o r d s were s c a l e d t o r e p r e s e n t a peak a c c e l e r a t i o n o f 0.5g on rock and hence the San Fernando r e c o r d was s c a l e d t o 0.5g f o r rock c o n d i t i o n s . F i e l d e v i d e n c e s u g g e s t s t h a t peak a c c e l e r a t i o n s a s s o c i a t e d w i t h s t r o n g ground s h a k i n g a r e d e a m p l i f i e d as the y pass through s o i l d e p o s i t s , and t h e r e f o r e lower a c c e l e r a t i o n l e v e l s e t a l (1976) and shown i n F i g u r e 14. 45. « *> u u < E E K o 5 0 .6 0 .5 0.4 0.3 0.2 0. I 0 . 0 Note - Relotionships shown above 0 3 g ore based on extrapolation of data base . Rock / i S . F e r n o n d o I97I) Stiff soil conditions Deep cohesionless 'soils . (E l Centro I940H Soft to medium » t i f f clay ond sand (Alameda 1962) 0.1 0 . 2 0.3 0.4 0.5 0.6 Maximum Acceleration in Rock 0.7 FIG 14 APPROXIMATE RELATIONSHIPS BETWEEN MAXIMUM ACCELERATIONS ON ROCK AND OTHER LOCAL SITE CONDITIONS. ( Af ter S e e d et a l . 1976) 4 6 . 5. RESULTS 5.1 INTRODUCTION The method was applied to a number of s o i l structure systems, using the model parameters data described in Chapter 3, for the three d i f f e r e n t earthquake e x c i t a t i o n s of the San Fernando, E l Centro and Alameda Park. In general a duration of 10 seconds of motion was analysed. The accumulated p l a s t i c displacements and the maximum developed horizontal pressures computed. The following presents the maximum displacements and maximum developed horizontal pressures versus the s t a t i c factor of safety against s l i d i n g and the e f f e c t of foundation and earthquake conditions on the response of the system. Comparisons were made with the standard methods of Newmark (displacements) and Mononobe-Okabe (horizontal pressures). The e f f e c t of d i f f e r e n t state of compactions of the s o i l b a c k f i l l was also considered. 5.2 Earthquake Induced Displacements and Dynamic L a t e r a l Forces as a  Function of Time. Typi c a l time h i s t o r i e s of the earhquake induced displacements of a wall having a s t a t i c factor of safety against s l i d i n g , F = 1.5 are shown in Figure 15. O.I4p 0.12 P s r p o Time in seconds FIG.15 E A R T H Q U A K E INDUCED DISPLACEMENTS AS A FUNCTION OF T I M E . The earthquake r e c o r d used was the E l C e n t r o (1940) s c a l e d t o 0.33g and the e q u i v a l e n t f o u n d a t i o n c h a r a c t e r i s t i c s were: K = 1 x 10^ L b / f t and B C^ = 2240 L b / f t / s e c . A dense s o i l b a c k f i l l was c o n s i d e r e d . As expected, the d i s p l a c e m e n t accumulates with time, and maximum v a l u e s o c c u r s a t the end of the s h a k i n g p e r i o d . I t may be seen t h a t somewhat l a r g e r d i s p l a c e m e n t s o c c u r when the pre-earthguake s t a t i c f o r c e on the w a l l i s the " a t r e s t " v a l u e P , r a t h e r than the a c t i v e v a l u e P . o A The l a t e r a l dynamic f o r c e s on the w a l l are shown i n F i g u r e 16. The l a t e r a l dynamic f o r c e , P , i s e x p r e s s e d i n terms of the dynamic e a r t h dy c o e f f i c i e n t K , g i v e n by: dy K d y " few2 | (32) where: Y = u n i t weight of s o i l b a c k f i l l H = h e i g h t o f the w a l l ^ 0 . 3 4 7 r 5 . 0.304 U J "E o & 0.260 2 ? 0.217 o ° ° 0.000. E L C E N T R O R E C O R D 10 Time in seconds FIG.16 DYNAMIC L A T E R A L FORCE AS A FUNCTION OF T IME. 48. I t may be seen t h a t o s c i l l a t e s between 0.217(1.0 K^) and about 0.304(1.4K ) and t h a t t h e maximum v a l u e may oc c u r a t any time d u r i n g the s h a k i n g ( g e n e r a l l y i n phase w i t h peak a c c e l e r a t i o n ) . A p r e - e a r t h q u a k e l a t e r a l f o r c e e q u a l t o the " a t r e s t " v a l u e , P , r a t h e r than the a c t i v e o v a l u e , P , causes h i g h e r dynamic f o r c e s i n the i n i t i a l p e r i o d o f s h a k i n g (but never h i g h e r than the i n i t i a l P ). However, once s l i d i n g o c c u r s , the o dynamic f o r c e s are s i m i l a r f o r both c a s e s . In the r e s u l t s which f o l l o w o n l y the maximum d i s p l a c e m e n t s and dynamic e a r t h c o e f f i c i e n t , K , are shown. In a d d i t i o n , i t i s assumed dy max t h a t t h e pre-earthquake s t a t i c f o r c e e q u a l s the a c t i v e f o r c e and the s o i l b a c k f i l l i s i n a dense s t a t e o f compaction. 5.3 Maximum Di s p l a c e m e n t s V e r s u s S t a t i c F a c t o r o f S a f e t y A g a i n s t S l i d i n g 5.3.1 E f f e c t o f Fo u n d a t i o n and Earthquake C o n d i t i o n s The maximum d i s p l a c e m e n t s o f the w a l l as a f u n c t i o n o f the s t a t i c f a c t o r o f s a f e t y a g a i n s t s l i d i n g , F , i s shown i n F i g u r e 17. A r a t i o s L/B = 10 was c o n s i d e r e d , which c o r r e s p o n d t o the more f l e x i b l e c o n d i t i o n s t u d i e d . 49. 10.0 8.0 c a> E a> o a to E E x o LEGEND © — • S.Fernando, 1971 Rock Foundation scaled to 0.5g x — x El Centro Deep Cohesionless Foundation scaled to 0.33g o-—oAlameda Deep Soft Foundation scaled to 0.27g L/B= 10 1.0 1.2 1.4 1.5 1.6 IB Static Factor of Safety Against Sliding 2.0 Fig. 17 MAXIMUM DISPLACEMENTS VS. STATIC FACTOR OF SAFETY AGAINST SLIDING (MOTION DURATION =IOsec.) For a l l t h r e e r e c o r d s i t may be seen t h a t the d i s p l a c e m e n t s are s m a l l f o r F > 1.5. The maximum d i s p l a c e m e n t f o r F =1.5 was e q u a l t o 0.49 f e e t s s and was o b t a i n e d u s i n g the earthquake r e c o r d c o r r e s p o n d e n t t o the deep s o f t c l a y f o u n d a t i o n c o n d i t i o n s . For F < 1.3 the d i s p l a c e m e n t s i n c r e a s e s markedly and are c o n s i d e r a b l y l a r g e r f o r the l a r g e r predominant p e r i o d o f th e Alameda Park r e c o r d . In o r d e r t o i l l u s t r a t e the i n f l u e n c e o f L/B r a t i o i n F i g u r e 18 below i t i s shown the r e s u l t s o b t a i n e d u s i n g the Alameda r e c o r d f o r d i f f e r e n t L/B r a t i o s . Static Factor of Safety Against Sliding Fig. 18 MAXIMUM DISPLACEMENTS VS. STATIC FACTOR OF SAFETY AGAINST SLIDING (MOTION DURATION =IOsec.) EFFECT OF L / B RATIO As i t may be seen the response t o L/B e q u a l t o 1 and 10 i s almost the same. S i m i l a r r e s u l t s were o b t a i n e d f o r the r e m a i n i n g earthquake r e c o r d s and f o u n d a t i o n c o n d i t i o n s . 51. 5.3.2 Comparison With Newmark's A n a l y s i s The maximum d i s p l a c e m e n t s a r e compared w i t h these o b t a i n e d by Newmark (1965) i n F i g u r e 19, and i t may be seen t h a t the a n a l y s i s p r e s e n t e d h e r e i n p r e d i c t s d i s p l a c e m e n t s t h a t are i n good agreement w i t h the s i m p l e r Newmark r i g i d p l a s t i c model. 0.05 O.I VALUES OF = A M A X R E S I S T A N C E C Q E F F . M A X . E A R T H Q U A K E A C C . FIG.I9 COMPARISON WITH NEWMARK DISPLACEMENTS. For t h i s comparison the E l Ce n t r o (1940) r e c o r d was s c a l e d t o a maximum a c c e l e r a t i o n o f 0.5g and the time s c a l e a l t e r e d t o produce a maximum v e l o c i t y o f 30 i n c h e s / s e c o n d . A t o t a l d u r a t i o n o f motion o f 30 seconds was a l s o c o n s i d e r e d and zero damping assumed. The y i e l d a c c e l e r a t i o n , N was computed as f o l l o w s : 52. "N" was d e f i n e d by Newmark as b e i n g a c o e f f i c i e n t t h a t when m u l t i p l i e d by g ( a c c e l e r a t i o n o f g r a v i t y ) wil£> produce the minimum a c c e l e r a t i o n v a l u e , a c t i n g i n the p r o p e r d i r e c t i o n t h a t would j u s t overcome the r e s i s t a n c e t o s l i d i n g of the element, hence: P A F S = P A + N ^ M (33) where P„ = a c t i v e f o r c e A F = s t a t i c f a c t o r of s a f e t y a g a i n s t s l i d i n g s N = Newmark's c o e f f i c i e n t g = g r a v i t y a c c e l e r a t i o n M - mass e q u a t i o n (30) can be r e w r i t t e n as f o l l o w s : N W = P A ( F S - D (34) where w = weight N = 4 < F s - X ) (35) 53. 5.4 Maximum L'ynamic E a r t h C o e f f i c i e n t , K, V e r s u s S t a t i c F a c t o r o f dymax S a f e t y 5.4.1 E f f e c t o f Found a t i o n and Earthquake C o n d i t i o n s . The maximum e a r t h c o e f f i c i e n t , K, , ve r s u s the s t a t i c f a c t o r o f dymax s a f e t y a g a i n s t s l i d i n g , F s , i s shown i n F i g u r e 20 f o r a l l 3 earthquake r e c o r d s . .736 I.5I9 X o £ "O 0) 1.302 5= I.085 o o O o IxJ u E o c >» Q E 3 E X o 0.868 0.65I 0.434 0.2I7 LEGEND • -S .Fernando, I97I 0.5g - El Centro , I940 0.22g o - Alameda, I962 0.2lg L / B = 10 L / B = I.O 1.5 ^0 3.0 4.0 5.0 Static Factor of Safety Against Sliding I0.0 T A B L E I X K bxl0 6 lb/ft. 0g Ib.At/sec. 0.038 440 0.I00 743 0.570 I700 1.500 2700 16.200 9000 43.200 I4700 Fig. 20 MAXIMUM DYNAMIC EARTH COEFFIC IENT,K d y m Q X VS. STATIC FACTOR OF SAFETY AGAINST SLIDING Two d i f f e r e n t f o u n d a t i o n c o n d i t i o n s (L/B = 1 and L/B = 10) were a n a l y s e d f o r a l l 3 r e c o r d s as shown i n above F i g u r e 20. As i t may be seen K, i n c r e a s e s w i t h F and v a r i e s c o n s i d e r a b l y w i t h earhtquake r e c o r d dymax s and f o u n d a t i o n c h a r a c t e r i s t i c s . In o r d e r t o have a b e t t e r u n d e r s t a n d i n g of the i n t e r p r e t a t i o n o f the r e s u l t s shown i n F i g u r e 20, i n t h i s same F i g u r e i t i s p r e s e n t e d i n T a b l e IX the f o u n d a t i o n parameters ( s p r i n g c o n s t a n t ) and ( e f f e c t i v e damping) c o r r e s p o n d e n t to each c u r v e . These v a l u e s were d e r i v e d from T a b l e s VI and V I I (Chapter 3 ) . F o l l o w i n g i s p r e s e n t e d a b r i e f d i s c u s s i o n o f the r e s u l t s : 1 - San Fernando (1971) - Rock F o u n d a t i o n - F o r t h i s p a r t i c u l a r r e c o r d the wide range of f o u n d a t i o n c h a r a c t e r i s t i c s used (K v a r i e s from B 16.2 x 1 0 6 t o 43.2 x 1 0 6 ( L b / f t ) and C v a r i e s from 9000 t o 14700 B L b / f t / s e c ) d i d not a f f e c t the maximum develop e d K, v a l u e s which were dymax 0.239(1.1 K ) and 0.224(1.03 K ) f o r the h i g h e r and lower f l e x i b l e A A f o u n d a t i o n c o n d i t i o n s . T h i s r e f l e c t s t h a t when a r e t a i n i n g w a l l i s found on a v e r y s t i f f f o u n d a t i o n i t s r e l a t i v e motion towards the s o i l b a c k f i l l i s almost i n e g l i q i b l e . These maximum K v a l u e s d e v e l o p e d a t or v e r y 3 dymax c l o s e t o F = 1.0. s 2 - E l C e n t r o (1940) - Deep C o h e s i o n l e s s F o u n d a t i o n - F o r the case of the E l C e n t r o r e c o r d the i n f l u e n c e of f o u n d a t i o n c h a r a c t e r i s t i c s i s n o t i c a b l e . A maximum v a l u e of K = 0.582(2.68 K ) o c c u r s f o r F > 3.0 dy A s and L/B=10. For the s t i f f e r case o f L/B=l a maximum K = 0.312(1.44 K ) dy A was o b t a i n e d . T h i s shows t h a t i t i s important t o s e l e c t the r i g h t f o u n d a t i o n d e s i g n parameters, r e g a r d i n g the s t r u c t u r a l d e s i g n o f the r e t a i n i n g s t r u c t u r e . 3 - Alameda (1962) - Deep S o f t C l a y F o u n d a t i o n - For t h i s case a tremendous i n f l u e n c e o f the f o u n d a t i o n c h a r a c t e r i s t i c s on the maximum K dy v a l u e s o b t a i n e d was o b t a i n e d . In g e n e r a l the maximum K, v a l u e s occur f o r dy F >3 as f o r t h e E l C e n t r o r e c o r d . A maximum v a l u e of K, =1.187(5.47 K ) s dy A o c c u r s f o r the case L/B=10 where the c o r r e s p o n d e n t f o u n d a t i o n parameters are K =0.038 x 1 0 6 L b / f t and C = 440 L b / f t / s e c . Again the importance of B B c h o o s i n g the a p p r o p r i a t e f o u n d a t i o n parameters i s emphasized. The above K maximum v a l u e s , the K v a l u e s f o r F =1.5 and dymax dy s t h e i r o r d e r o f magnitude compared w i t h t h e p a s s i v e e a r t h p r e s s u r e c o e f f i c i e n t s , K a r e shown i n T a b l e XI. P TABLE XI K, V a l u e s Correspondent to the S o f t e r F o u n d a t i o n C o n d i t i o n s dy EARTHQUAKE RECORD FOUNDATION CONDITION K d y < F s = 1 - 5 ) ^ ( F s = 1 . 5 ) dymax dymax K p San Fernando (1971) Rock 0.23g 0.05 0.23g 0.05 E l C e n t r o (1940) Deep C o h e s i o n l e s s S o i l 0.0356 0.08 0.582 0.13 Alameda Park (1962) Deep S o f t C l a y 0.540 0.12 1.187 0.26 T a b l e X shows the e f f e c t o f f o u n d a t i o n c o n d i t i o n s on the response o f d e v e l o p e d K v a l u e s . F o r a F = 1.5 K, v a l u e s range from 0.239 (roc k ) dy s dy t o 0.543 ( s o f t c l a y ) c o r r e s p o n d e n t t o 0.05 t o 0.12 of the maximum p a s s i v e v a l u e . Reqarding maximum K v a l u e s the range i s wider and v a r i e s from dy 0.239 (rock) t o 1.187 ( s o f t c l a y ) o r from 0.05 t o 0.26 i n terms o f maximum p a s s i v e v a l u e s (K ). I t i s a l s o seen t h a t the maximum h o r i z o n t a l f o r c e P d e v e l o p e d o c c u r s f o r the s o f t c l a y type o f f o u n d a t i o n and eq u a l s a p p r o x i m a t e l y a 1/4 of K^. 5 7. 5.4.2 E f f e c t o f S o i l B a c k f i l l S t a t e o f Compaction So f a r a dense s o i l b a c k f i l l w i t h a z e r o s l o p e angle has been con-s i d e r e d . To study the e f f e c t o f the s o i l b a c k f i l l s t a t e o f compaction on t h e maximum developed l a t e r a l f o r c e the t h r e e s o i l b a c k f i l l s p r e s e n t e d i n T a b l e I (page 23) were used i n the a n a l y s i s t o g e t h e r w i t h the E l C e n t r o (1940) and Alameda (1962) r e c o r d s . These two r e c o r d s were a p p l i e d f o r f o u n d a t i o n c o n d i t i o n s c o r r e s p o n d i n g to a deep c o h e s i o n l e s s s o i l and a deep s o f t c l a y r e s p e c t i v e l y and a f o u n d a t i o n f l e x i b i l i t y o f L/B = 1 0 . A f a c t o r of s a f e t y a g a i n s t s l i d i n g , F = 1 . 5 was c o n s i d e r e d f o r a l l cases which s means t h a t t h r e e d i f f e r e n t masses were used f o r the t h r e e s o i l b a c k f i l l s t a t e s o f compaction, as shown i n T a b l e I . In F i g u r e 21 i s shown the v a r i a t i o n o f the maximum e a r t h c o e f f i c i e n t , K v e r s u s a c c e l e r a t i o n l e v e l f o r the two f o u n d a t i o n c o n d i t i o n s and t h r e e dy s o i l b a c k f i l l s t a t e s o f compaction. x o 0 \ I . 1 1 1 0.0 O.I 0.2 0.3 0.4 0.5 x Gravity Fig. 21 MAXIMUM DYNAMIC EARTH COEFFICIENT, Kdymax VS. ACCELERATION LEVEL EFFECT OF SOIL BACKFILL COMPACTION 59. As e x p e c t e d the maximum developed h o r i z o n t a l f o r c e s i n c r e a s e w i t h i n c r e a s i n g a c c e l e r a t i o n l e v e l and d e c r e a s i n g r e l a t i v e d e n s i t y o f s o i l b a c k f i l l . 6 0 . 5.4.3 Comparison With Mononobe-Okabe A n a l y s i s The maximum l a t e r a l f o r c e s p r e d i c t e d from the Mononobe-Okabe e q u a t i o n f o r maximum a c c e l e r a t i o n s o f 0.33g and 0.27g are shown with arrows i n F i g u r e 22. 1.736 .519 X o E S 1.302 c O o O o LU o E o c >s Q E 3 E 'x o 1.085 0.868 0.651 0.434 0.217 LEGEND • - S.Fernando,l97l 0.5g - E I Centro, 1940 0.22g O - Alameda, 1962 0.2lg L / B = 10 - L / B = Mononobe Okabe 0.33g Mononobe Okabe 0.27g -* c — 1.0 1.5 2.0 3.0 4.0 5.0 Static Factor of Safety Against Sliding 10.0 Fig. 22 MAXIMUM DYNAMIC EARTH COEFFIC IENT,K d y m Q X VS. STATIC FACTOR OF SAFETY AGAINST SLIDING COMPARISON WITH MONONOBE OKABE RESULTS They s h o u l d be compared w i t h the E l Centro and Alameda Park r e c o r d r e s u l t s . The Mononobe-Okabe e q u a t i o n does not d i r e c t l y c o n s i d e r the e f f e c t o f base s l i d i n g . The assumption i n v o l v e d i s t h a t s u f f i c i e n t base s l i d i n g or r o t a t i o n o c c u r s t o m o b i l i z e the a c t i v e c o n d i t i o n s . In F i g u r e 22 i s shown t h a t the l a t e r a l f o r c e s p r e d i c t e d from the Mononobe-Okabe e q u a t i o n l i e w i t h i n the range p r e d i c t e d i n the p r e s e n t a n a l y s i s f o r the s o i l f o u n d a t i o n c o n d i t i o n s c o r r e s p o n d i n g t o the E l C e n t r o r e c o r d . However s i g n i f i c a n t d i f f e r e n t i n r e s u l t s i s o b t a i n e d when comparing the l a t e r a l f o r c e s p r e d i c t e d from the Mononobe-Okabe e q u a t i o n f o r 0.27g maximum h o r i z o n t a l a c c e l e r a t i o n w i t h the r e s u l t s o b t a i n e d from the p r e s e n t a n a l y s i s f o r t h e s o i l f o u n d a t i o n c o n d i t i o n s f o the Alameda Park r e c o r d . To i l l u s t r a t e the above F i g u r e s 21 i s re-shown i n F i g u r e 23 and p r e s e n t s the Mononobe-Okabe r e s u l t s u s i n g t h e same i n p u t d a t a . The Mononobe-Okabe r e s u l t s agree f a i r l y w e l l w i t h t h e p r e s e n t d a t a , f o r the case o f the E l Ce n t r o r e c o r d , a l t h o u g h s l i g h t l y lower, but f o r t h e case o f the Alameda r e c o r d the Mononobe-Okabe r e s u l t s a r e c o n s i d e r a b l y lower, f o l l o w i n g t h e 0.1g a c c e l e r a t i o n l e v e l ( F i g u r e 23) the Mononobe-Okabe v a l u e s d i f f e r a p p r o x i m a t e l y of a f a c t o r r a n g i n g from 1.7 t o 2.0. The f a c t t h a t the l a t e r a l dynamic p r e s s u r e s s h o u l d v a r y w i t h the amount o f base s l i d i n g , as shown i n F i g u r e s 20 and 22, i s i n agreement w i t h Whitman (1978). 62. Loose backfill (0= 30 ° ) L / B = 10 • 1 1 1 i I 0.0 0.1 0.2 0.3 0.4 0.5 x Gravity Fig. 23 MAXIMUM DYNAMIC EARTH COEFFICIENT, Kdymax VS. ACCELERATION LEVEL EFFECT OF SOIL BACKFILL COMPACTION COMPARISON WITH MONONOBE OKABE RESULTS 63. who c o n s i d e r e d t h a t the dynamic p r e s s u r e s on w a l l s t h a t moved r i g i d l y w i t h the u n d e r l y i n g s o i l s h o u l d be c o n s i d e r a b l y g r e a t e r than the p r e s s u r e s p r e d i c t e d by the Mononobe-Okabe e q u a t i o n . In a d d i t i o n , Rowland and Elms (1979) e s t i m a t e d t h a t the dynamic l a t e r a l f o r c e s on damaged b r i d g e abutments i n New Zeal a n d were about 1.4 to 1.8 times the v a l u e s p r e d i c t e d from the Mononobe-Okabe e q u a t i o n . 64. 7. CONCLUSIONS A simple method o f a n a l y s i s which a l l o w s both the earthquake i n d u c e d f o r c e s and d i s p l a c e m e n t s o f r e t a i n i n g s t r u c t u r e s t o be computed i s p r e s e n t e d . The method c o n s i d e r s both the weight o f the w a l l and t h e f l e x i b i l i t y and s t r e n g t h o f both the b a c k f i l l and f o u n d a t i o n s o i l . However, the p o s s i b i l i t y of s t r e n g t h l o s s i s not c o n s i d e r e d . The method o f a n a l y s i s was a p p l i e d t o a 20 f e e t h i g h c a n t i l e v e r w a l l s u b j e c t e d t o t h r e e d i f f e r e n t earthquake e x c i t a t i o n s r e p r e s e n t i n g s o f t t o hard f o u n d a t i o n s o i l c o n d i t i o n s . The r e s u l t s i n d i c a t e the f o l l o w i n g : 1 ) The dynamic d i s p l a c e m e n t s o f t h e w a l l d e c r e a s e w i t h i n c r e a s i n g s t a t i c f a c t o r o f s a f e t y a g a i n s t s l i d i n g , F s , and w i l l be low f o r t h e c o n v e n t i o n a l w a l l t h a t has Fs > 1.5. 2) The Newmark method g i v e s a good e s t i m a t e o f earthquake induced w a l l d i s p l a c e m e n t s . 3) The maximum dynamic h o r i z o n t a l f o r c e i n c r e a s e s w i t h t h e s t a t i c f a c t o r of. s a f e t y a g a i n s t s l i d i n g , F s, and may be g r e a t e r than t h e v a l u e s p r e d i c t e d from the Mononobe-Okabe e q u a t i o n f o r w a l l s t h a t a r e p r e v e n t e d from s l i d i n g . 4) The i n i t i a l p r e - earthquake s t a t i c p r e s s u r e whether i t be " a t r e s t " o r a c t i v e c o n d i t i o n has o n l y a s m a l l e f f e c t on the maximum dynamic f o r c e on the w a l l . 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Newmark, N.M., 1965, " E f f e c t s of Earthquakes on Dams and Embankments", Geote c h n i q u e , London, England, V o l . XV, No. 2, 1926. Okabe, S., 1926, "General Theory of E a r t h P r e s s u r e , " J o u r n a l o f the Japan S o c i e t y o f C i v i l E n g i n e e r s , V o l . 12, No. 1, 1926. Ohsa k i , Y., and Iwasaki, R., 1973, "On Dynamic Shear M o d u l i and P o i s s o n ' s R a t i o o f S o i l D e p o s i t s " , S o i l s and Fo u n d a t i o n s ( J a p a n ) , V o l . 13, No. 4, pp. 61-73. Prakash, S., 1977, " S o i l Dynamics and i t s A p p l i c a t i o n t o F o u n d a t o i n E n g i n e e r i n g - S e i s m i c Response o f S o i l D e p o s i t s , Embankments, Dams and S t r u c t u r e s " , P r o c e e d i n g s 9th I n t e r n a t i o n a l C o n f e r e n c e on S o i l Mechanics and Fo u n d a t i o n E n g i n e e r i n g , Tokyo, 1977, pp. 624-630. Peck, R.B., Hanson, W.E., and Thornburn, T.H., 1974, "Fo u n d a t i o n E n g i n e e r i n g " . P u b l i s h e d by John W i l e y and Sons, Inc., pp. 415-416. R e i s s n e r , E., and S a g o r i , H.F., 1944, " F o r c e d T o r s i o n a l O s c i l l a t i o n s o f an E l a s t i c H a l f - S p a c e " , J o u r n a l o f A p p l i e d P h y s i c s , V o l . 15, pp. 652-662. R i c h a r d s , R., and Elms, D., 1979, " S e i s m i c B e h a v i o u r o f G r a v i t y R e t a i n i n g W a l l s , " J o u r n a l o f Geot. Eng. D i v . , ASCE, V o l . 105, No. GT4, A p r i l 1979. R i c h a r t , F.E., J r . , H a l l , J.R., J r . , and Woods, R.D., 1970, " V i b r a t i o n s of S o i l s and F o u n d a t i o n s , " P r e n t i c e - H a l l , I n c . , Englewood C l i f f s , N.J. 6 7 . Seed, H.B. and I d r i s s , I.M., 1970, " S o i l M oduli and Damping F a c t o r s f o r Dynamic Response A n a l y s e s " , Earthquake Eng. Res. Cen., U. of C a l . 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Timoshenko, S.P., and G o o d i e r , J.N., 1951, Theory o f E l a s t i c i t y , McGraw H i l l Book Co., I n c . , New York. Whitman, R.V., and R i c h a r t , J r . , F.E., 1967, "Design Procedures f o r D y n a m i c a l l y Loaded F o u n d a t i o n s " , J o u r n a l of the S o i l Mechanics and F o u d n a t i o n s D i v i s i o n , ASCE, Nov. 1967. Whitman, R.V., 1972, " A n a l y s i s of S o i l - s t r u c t u r e I n t e r a t i o n . A S t a t e - O f -T h e - A r t Review", School of Eng. M.I.T., P. 72-3, S o i l P u b l i c a t i o n No. 300. Whitman, R.V., 1976, " S o i l - P l a t f o r m I n t e r a t i o n " . I n t e r n a t i o n a l C o n f e r e n c e on the Review o f O f f s h o r e S t r u c t u r e s , 1976, V o l . 1. pp. 817-829. Whitman, R.V., 1978, "Response of G r a v i t y W a l l s t o Earthquake Ground M o t i o n " , P r o c e e d i n g s o f the ASCE G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n S p e c i a l t y C o n f e rence - E a r t h q u a k e . E n g i n e e r i n g and S o i l Dynamics, Pasadena, 1978. APPENDIX PROGRAMME WALLQUAKE 69. APPENDIX - PROGRAMME WALLQUAKE A. Input Data The programme wallquake reads 17 parameters, through 3 READ Input Statements, d e s c r i m i n i z e d below. 1 s t READ STATEMENT - U s i n g a FORMAT (7F 10.3) the parameters a r e : SAFETY, EKA, EKP, RH0, SK, WM, C where: SAFETY = S t a t i c f a c t o r of s a f e t y a g a i n s t s l i d i n g EKA = A c t i v e e a r t h c o e f f i c i e n t = ( 1 - S i n cf>)/(1+Sin $) (<j> = f r i c t i o n angle of s o i l b a c k f i l l ) EKP = P a s s i v e e a r t h c o e f f i c i e n t = 1/EKA RH.0 = 0.5yH 2, q u a n t i t y t h a t when m u l t i p l i e d by an e a r t h c o e f f i c i e n t , K, g i v e s the c o r r e s p o n d e n t e a r t h f o r c e . (Y = u n i t weight o f s o i l b a c k f i l l and H = h e i g h t of t h e w a l l ) . SK = Base s p r i n g s t i f f n e s s c o n s t a n t , K as d e f i n e d i n B (3.3.2.2). WM = Mass of the w a l l p l u s s o i l b a c k f i l l as d e f i n e d i n ( 3 . 2 ) . C = E f f e c t i v e damping, C , as d e f i n e d i n ( 3 . 4 ) . B 2nd READ STATEMENT - U s i n g a FORMAT (4F 10.3) the parameters a r e : T0, T I , SCALE 1, TURN. where: T0 = Time i n t e r v a l used i n the earthquake a c c e l e r a t i o n r e c o r d . Most o f the earthquake r e c o r d s , have i t s a c c e l e r a t i o n d a t a p r e s e n t e d i n 0.02 seconds time i n t e r v a l s , such as th e San Fernando and th e E l C e n t r o r e c o r d s . The Alameda Park r e c o r d as an e x c e p t i o n has i t s d a t a i n 0.01 seconds time i n t e r v a l s . 70. T h i s parameter i s used w i t h two o b j e c t i v e s : a) S u b d i v i d e the earthquake d a t a . I f TI = T0 no s u b d i v i s i o n o f earthquake a c c e l e r a t i o n data i s performed. I f TI = 0.5 T0 the double o f the earthquake data i s o b t a i n e d f o r the same r e c o r d d u r a t i o n and i s s t o r a g e d i n an a p p r o p r i a t e a r r a y f o r l a t e r use. b) To i n p u t the d e s i r e d i n i t i a l time increment, At. (see (3.5) ) . T h i s parameter i s used t o determine the "SCALE" used t o s c a l e the earthquake a c c e l e r a t i o n v a l u e s . I f SCALE 1 = 0.5 t h a t means t h a t the earthquake a c c e l e r a t i o n v a l u e s of the r e c o r d w i l l be s c a l e d t o 0.5g (g = a c c e l e r a t i o n o f g r a v i t y ) Example SCALE 1 g SCALE = MAX. ACCELERATION VALUE OF RECORD Fo r the Case of San Fernando Record (1971) the maximum a c c e l e r a t i o n v a l u e = 3462, assuming SCALE 1 = 0.5, and g = 32.2 f e e t / s e c SCALE = 9 = 0.004650 3462 : Parameter used t o l i m i t the number of s u b i t e r a t i o n s a t " c r i t i c a l p o i n t s " . 71. " C r i t i c a l p o i n t s " a r e p o i n t s such as p o i n t 1 (change from e l a s t i c t o p l a s t i c ) or p o i n t 2 (change i n s i g n o f v e l o c i t y ) . At these p o i n t s i t i s n e c e s s a r y t o d e c r e a s e the i n i t i a l i n p u t e d time i n t e r v a l , At, (or TI) i n o r d e r t o o b t a i n a b e t t e r a c c u r a c y when t u r n i n g those " p o i n t s " . The r u l e s used i n the programme are as f o l l o w s : P o i n t 1 - The programme i s o n l y p e r m i t t e d t o t u r n p o i n t 1 (or s i m i l a r y i e l d p o i n t s ) when |X -X |<0.001. A 1 U n t i l t h i s c o n d i t i o n i s not s a t i s f i e d the i n i -t i a l time increment, T I , i s s u b d i v i d e d i n smal-l e r increments and the programme s u b i t e r a t e d . P o i n t 2 - The programme i s o n l y p e r m i t t e d t o t u r n p o i n t 2 (or s i m i l a r ) when |X |<0.001 (remember t h a t o f B p o i n t 2, X 2=0.0). S i n c e t o s a t i s f y the above two r u l e s the i n i t i a l time increment, TI was, a t t i m e s , g r e a t l y reduced, i t was found t h a t by u s i n g a l i m i t f o r the r e d u c t i o n o f T I , good a c c u r a c y was a l s o o b t a i n e d . T h i s l i m i t i s e x p r e s s e d by the parameter "TURN". I f TURN=10 t h a t i m p l i e s t h a t the programme t u r n s any " C r i t i c a l P o i n t " e i t h e r by s a t i s f y i n g 72. one o f above r u l e s ( p o i n t 1 r u l e " o r p o i n t 2 r u l e " ) o r when a 'reduced' time increment = TI/10 i s o b t a i n e d . ( i f TI=0.01 s e c . then the minimum a l l o w a b l e i s 0.001 S e c ) . 3rd READ STATEMENT - U s i n g a FORMAT (516) the parameters a r e : NP, IFORM, IG, IE, IK0 where: NP = Number of a c c e l e r a t i o n v a l u e s t o be r e a d from the e a r t h -quake a c c e l e r a t i o n r e c o r d . IFORM = Is a f l a g which can assume the v a l u e s 0 or 1. IFORM = 0, reads the i n p u t e d earthquake a c c e l e r a t i o n . r e c o r d w i t h a 8F10.0 FORMAT. IFORM = 1, reads the i n p u t e d earthquake a c c e l e r a t i o n r e c o r d w i t h a 10F8.0 FORMAT. Most of the earthquake a c c e l e r a t i o n r e c o r d s are p r e s e n t e d w i t h a 8F10.0 FORMAT, such as the San Fernando and E l C e n t r o Records. The Alameda Park, however, i s p r e s e n t e d i n the o t h e r type of FORMAT. IG = Is a f l a g which can assume v a l u e s of 0 o r 1. IG = 0, means t h a t the u n i t system used i s : FOOT, POUND, and SECOND. IG = 1, the u n i t system i s : METER, NEWTON and SECOND. Note t h a t the above i n p u t parameters, RH0, SK, WM and C have t o agree w i t h the system of u n i t s choosen. IE = Is a f l a g t o c o n t r o l o u t p u t d a t a . IE = 0, o u t p u t s a t i t l e c o r r e s p o n d e n t t o the San Fernando r e c o r d . IE = 1, o u t p u t s a t i t l e c o r r e s p o n d e n t t o the E l C e n t r o r e c o r d . IE = 2, o u t p u t s a t i t l e c o r r e s p o n d e n t t o the Alameda Park r e c o r d . IKJZf = I s a f l a g t o c o n t r o l the i n i t i a l ACTIVE o r "AT REST" c o n d i t i o n . IKJ3 = 0, c o r r e s p o n d s t o an i n i t i a l ACTIVE c o n d i t i o n . IK? = 1, c o r r e s p o n d s t o an i n i t i a l "AT REST" (K ) o c o n d i t i o n . B. Output Data The programme WALLQUAKE ou t p u t s the f o l l o w i n g d a t a : MAX.. DISP. = Maximum d i s p l a c e m e n t i n f e e t (IG=0) or i n meters (IG=1). F. SAFETY = S t a t i c f a c t o r of s a f e t y a g a i n s t s l i d i n g used i n the a n a l y s i s . HOR. F.R. = Maximum h o r i z o n t a l f o r c e r a t i o o b t a i n e d , R . P (R = K, /K ) p dy A N. POINT = Number of earthquake a c c e l e r a t i o n v a l u e s used in' the a n a l y s i s . N o t e : l ) T h i s number of earthquake a c c e l e r a t i o n v a l u e s used i n the a n a l y s i s may o r may not be eq u a l t o the number of e a r t h a c c e l e r a t i o n v a l u e s r e a d from the earthquake f i l e . I f T0 = T I i t w i l l be t h e same I f T0 = 2TI th e n t h e number of earthquake a c c e l e r a t i o n v a l u e s used are double o f t h a t r e a d . . 2) A l s o the d u r a t i o n o f motion = N. POINT x TI PEAK ACC. = Maximum a c c e l e r a t i o n v a l u e from t h e r e c o r d . ACC. SCALE = S c a l e used t o s c a l e the r e c o r d a c c e l e r a t i o n v a l u e s t h i s d i m e n s i o n l e s s parameter w i l l have d i f f e r e n t v a l u e s f o r IG=0 o r IG=1 (see d e f i n i t i o n o f SCALE 1 ) . YIELD DISP. = Y i e l d d i s p l a c e m e n t of the base s p r i n g i n f e e t (IG=0) or i n meters (IG=1). T h i s y i e l d d i s p l a c e m e n t c o r r e s p o n d s t o the 1 s t t u r n i n g p o i n t or y i e l d p o i n t o f t h i s s p r i n g . LIMIT SPRING FORCE = Is the l i m i t i n g f o r c e o f the base s p r i n g which c o r r e s p o n d s t o the above y i e l d p o i n t . T h i s l i m i t i n g f o r c e = F 0.5 YH 2K as d e f i n e d p r e v i o u s l y , t i n g A C. Sample o f Input and Output Data F o l l o w i n g i s shown a copy of t y p i c a l i n p u t data u s i n g the 3 earthquake r e c o r d s used i n the a n a l y s i s , w i t h the c o r r e s p o n d e n t o u t p u t v a l u e s . Sample of Input and Output Data 1 3 0 2 0. 02 3 500 End of F i l e 1 1. SO 2 0. 02 3 500 End o f F i l e 1 1. 50 2 0. 01 3 1C00 End o f F i l e 1 2 + 3 End o f F i l e 1 End of F i l e 1 End of F i l e 0. 2170 0. 002 0 0 0. 217 0 01 0 0 0. 217 0. 01 1 O MAX. DISP 0. 000404 MAX DISP 0 1106G7 MAX DISP 0 474 7 77 4. 6 0 5 0 4 60 0 33 1 4. 60 0. 275 2 F S A F E T Y 3 00000 F SAFETY 1 50000 F S A F E T Y 1 50000 27000 0 10 0 ?7000 0 10. 0 ! 1600000. 1500000 0 57000 0 10. 0 38000. 00 502 64 502 6-1 504. 64 10400 .0 2700 0 440. 00 EARTHQUAKE RETAINING WALL STUDY-S FERNANDO(1971> HOR F P N. POINT PEACK ACC. ACC SCALE Y IELD DISP. 1 0312 5000 3 4 6 2 . 0 0 0 .004650 0 000814 E A R T H G U A K E - R E T A I N I N G - W A L L - S T U P Y - E L CENTRO(1940) HOR F R. N .POINT PEACK ACC. ACC. SCALE Y I E L D DISP 1 4998 1000 3417 00 0 . 0 0 J 1 1 0 0 . 0 0 5 8 5 9 EARTHQUAKE RETAINING WALL STUDY-ALAMEDA P. (1962) HOR F R N.POINT PEACK ACC ACC SCALE Y IELD DISP 2 4V13 1001 3 CO 2 951^66 0 231276 L IM IT SPRING FORCE 17577 .00 L IM IT SPRING FORCE 87B8. 50 L IMIT SPRING FORCE 8 7 8 8 . 5 0 In 

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