Three Dimentional Soil Structure Response to Earthquakes

7/29/2019 Three Dimentional Soil Structure Response to Earthquakes

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B u l l e ti n o f t h e S e is m o l o g i c a l S o c i e t y o f A m e r i c a , V o l . 6 3 , N o . 3 , p p. 1 0 4 1 - 1 0 5 6 . J u n e 1 9 73

THREE-DIMENSIONAL SOIL-STRUCTURE RESPONSE TO EARTHQUAKESBY WILLIAMWEAVER, JR., GREGG E. BRANDOW, AND KAARE HOEG

A B S T R A C TCalculations for the dynamic response of muitistory buildings to earthquake

accelerations of bedrock should include the effects of the soil and the structuralfoundation. For this purpose a three-dimensional analytical model is developed,consisting of the following parts. The superstructure is modeled as a tier building(with rigid floor diaphragms, space frame members, bracing, and setbacks); thefoundation is approximated by a rigid block in combination with piles (prismaticmembers with pinned ends), and the soil is idealized by a finite-element mesh(three-dimensional rectangular prisms) with special boundary conditions (viscousdamping at lateral boundaries and a rigid boundary at bedrock). This analyticalmodel is incorporated into a computer program, which performs response analysesfor specified earthquakes. Sample problems are included to demonstrate the capa-bilities of both the analytical model and the computer program. The results showthat the combination of a three-dimensional analytical model and the presence ofunderlying soil has important influences on the calculated responses of mnltistorybuildings to earthquakes.

[ N T R O D U C T I O NThe response of a building to earthquake ground motions depends not only upon its

own characteristics and the nature of the excitation, but also upon the properties ofunderlying soil. With the aid of digital computers, structural engineers are able to analyzebuilding frames in a highly refined manner, and soil engineers have progressed in theirunderstanding of free-field ground surface motions from a knowledge of local soilconditions. However, insufficient attention has been given to the combined soil-foundation-structural system and its response to earthquake excitations.

The importance of local soil conditions upon ground surface motions during earth-quakes has been shown analytically and experimentally (Gutenberg, 1957; Kanai e t a l . ,1959; Seed e t a l . , 1969; Tsai, 1967; Wiggins, 1964). Analysts initially utilized rathersimple analytical models (Biot, 1943; Hashiba and Whitman, 1968; Jacobsen, 1938;1958; Parmelee, 1967; 1968 ; Whitman, 1969) to represent what is actually a very complexproblem. Since the advent of digital computers, however, investigators have developeddiscretized analytical models with many degrees of freedom (Lycan and Newmark, 196i ;Minami e t a l . , 1969; 1970; Penzien e t a l . , 1964).

The finite-element method has been applied to the soil-structure problem by Khanna(1969) and Wilson (1969), who analyzed two-dimensional frameworks subjected to onehorizontal component of ground motion. In both of these studies rigid boundarieswere assumed to exist at bedrock and along vertical planes at arbitrary distances fromthe structures. Unfortunately, in dynamic analysis, the side boundaries cause artificiallyreflected waves that obscure the true soil-structure response. However, Lysmer andKuhlemeyer (1969) devised a method for synthesizing nonreflective boundaries by usingviscous dampers at boundary nodes.Most of the previous investigations have yielded some insight into the soil-structureinteraction problem, but the analytical models apply at best only to two-dimensional

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1042 W I L L IA M W E A V E R , J R . , G R E G G E . B R A N D O W , A N D K A A R E H O EGs i tu a t io n s . T h e p r i m a r y o b j e c t iv e o f th i s p a p e r i s t o p r e s e n t a t h r e e - d i m e n s i o n a l a n a l y t i c a lm o d e l f o r s o i l - f o u n d a t i o n - s t r u c t u r e i n t e r a c t i o n t h a t r e p r e s e n t s a c t u a l c o n d i t i o n s m o r er e a l i s t i c a l l y t h a n i t s p r e d e c e s s o r s . T h i s m o d e l a n d a m e t h o d f o r c a l c u l a t i n g d y n a m i cr e s p o n s e t o e a r t h q u a k e g r o u n d m o t i o n a r e d e s c r i b e d h e r e i n . S e l e c t e d e x a m p l e s a r ei n c l u d e d t o d e m o n s t r a t e t h e n a t u r e o f r e s u l t s o b t a i n a b l e f r o m a s p e c i a l - p u r p o s e c o m -p u t e r p r o g r a m d e v e l o p e d i n t h is p r o j e c t .

A N A L Y T I C A L M O D E LThree-dimensional assemblage. F i g u r e 1 s h o w s t h e t h r e e - d i m e n s i o n a l a n a l y t i c a l m o d e l

s e l e c t e d f o r t h i s s t u d y . I t c o n s i s ts o f a mu l t i s t o r y ( t i e r) b u i l d i n g p r o j e c t i n g a b o v e g r o u n dl e v el , a r ig i d s t r u c t u r a l f o u n d a t i o n b e l o w g r o u n d l e v el , a n d a n e t w o r k o f f i n it e e l e m e n t sr e p r e s e n t i n g t h e s o il a b o v e b e d r o c k . V e r t ic a l o r i n c l in e d p i le s d r i v e n t o b e d r o c k m a y b ei n c l u d e d , a n d d a m p e d b o u n d a r y c o n d i t i o n s o n t h e f i n i t e e l e m e n t s s i m u l a t e a s o i l o fi n f i n i t e e x t e n t i n h o r i z o n t a l d i r e c t i o n s . A l l ma t e r i a l s i n t h e a n a l y t i c a l mo d e l a r e a s s u me dt o b e l i n e a r l y e l a s ti c , a n d d i s p l a c e m e n t s r e l a t i v e t o b e d r o c k a r e a s s u me d t o b e s u f f ic i e n t lys m a l l t h a t t h e o r i g i n a l g e o m e t r y m a y b e u s e d t h r o u g h o u t t h e a n a l y s i s . A c t i o n s a n d

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THREE-DIMENSIONAL SOIL-STRUCTURE RESPONSE TO EARTHQUAKES 1043d i s p l a c e m e n t s a r e t a k e n t o b e p o s i t i v e w h e n i n th e p o s i t i v e d i r e c t i o n s o f o r t h o g o n a l( x y z ) r e f e r e n c e a x e s.

T ie r b u i ld in g m o d e l . T h e t h r e e - d i m e n s i o n a l a n a l y t i c a l m o d e l f o r s t a t i c a n d d y n a m i ca n a l y s e s o f t i e r b u i l d i n g s h a s b e e n d e s c r i b e d i n p r e v i o u s p a p e r s ( W e a v e r e t a l . , 1 9 6 6 ;! 9 6 8 ). T h e c a p a b i l i t y o f in c l u d i n g t h r e e- d i m e n s i o n a l s h e a r c o r e s w as a d d e d b y M a n n i n g( 1 9 70 ) , w h i l e v a r i o u s p o s s i b il it ie s f o r b r a c i n g a n d s e t b a c k s w e r e c o n t r i b u t e d b y B r a n d o w( 1 97 0 ). F i g u r e 2 i l lu s t r at e s a ll o f t h e s e f e a tu r e s , a l t h o u g h s h e a r w a ll s w e r e n o t a c t u a l l yu t i l i z e d i n th i s p r o j e c t .

T h e p r i m a r y f r a m i n g m e m b e r s i n t h e ti e r b u il d in g m o d e l c o n s i st o f p r i s m a t ic b e a m sa n d c o l u m n s a r r a n g e d i n a r e c t a n g u l a r p a t t e r n , a s i n d i c a t e d i n F i g u r e 2 . C o n n e c t i o n s

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b e t w e e n b e a m s a n d c o l u m n s a r e a s s u m e d t o b e r i gi d , w h e r e a s t h o s e a t th e e n d s o f b r a c e sa r e t a k e n a s p i n n e d . F l o o r a n d r o o f d i a p h r a g m s a r e id e a l i ze d a s l a m i n a e h a v i n g i n f in i ter i g i d i t i e s i n t h e i r o w n p l a n e s a n d f i n i t e r i g i d i t i e s n o r m a l t o t h e i r p l a n e s , i n t h e f o r m o fb e a m s t i f f n e s s e s . E a c h l a m i n a h a s t h r e e r i g i d - b o d y m o t i o n s w i t h w h i c h i n e r t i a a c t i o n sa r e a s s o c i a t e d : t r a n s l a t i o n s i n t h e x a n d y d i r e c t i o n s a n d r o t a t i o n i n t h e z s e n s e . J o i n td i s p l a c e m e n t s w i t h w h i c h n o i n e r t i a a c t i o n s a r e a s s o c i a t e d ( x a n d y r o t a t i o n s a n d zt r a n s l a t i o n ) a r e e l i m i n a t e d i n t h e c o n d e n s a t i o n p r o c e s s l e a d i n g t o t h e s t i f f n e s s m a t r i xf o r s t o r y d i s p l a c e m e n t s .

M a s s e s o f t h e t i e r b u i l d i n g t r i b u t a r y t o t h e f r a m i n g l e v el s a r e a s s u m e d t o b e u n i f o r m l yd i s t r i b u t e d o v e r th e l a m i n a e . U n d e r t h i s a s s u m p t i o n , t h e c e n t e r o f m a s s a t e a c h l ev e l


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1044 WILLIAM WEAVER, JR., GREGG E. BRANDOW, AND KAARE HOEGc o i n c i d e s w i t h t h e c e n t r o i d o f t h e a r e a . T h e r e s u l t i n g m a s s - i n e r t i a m a t r i x r e f e r r e d t o t h ec e n t r o i d o f t h e a r e a f o r t h e k th f r a m i n g l e v e l m a y b e w r i t te n a s[ :11 0M k = m k 0 10 0 r k 2 ( 1 )w her e m k i s the m ass a t the k th l eve l and r k is t h e r a d i u s o f g y r a t i o n w i t h r e s p e c t t o t h ec e n t r o i d . T h e m a s s m a t r i x f o r t h e e n t i r e t i e r b u i l d i n g i s d i a g o n a l w h e n c e n t r o i d s o fl a m i n a e a r e c h o s e n a s r e fe r e n c e p o i n t s .

F i n i t e e l e m e n t s f o r t h e s oi l. T h e t h r e e - d i m e n s i o n a l s o li d r e c t a n g u l a r e le m e n t o f M e l o s h( 1 9 6 3 ) w a s c h o s e n t o r e p r e s e n t t h e s o i l a b o v e b e d r o c k . T h i s e l e m e n t h a s e i g h t c o r n e rn o d e s a n d t h r e e t y p e s o f g e n e r i c a n d n o d a l d i s p l a c e m e n t s , a s in d i c a t e d i n F i g u r e 3 .

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f o r i = x , y , z a n d = x / a , q = y / b , ~ = z / c . T h i s f o r m u l a t i o n l e a d s t o a 2 4 2 4 s ti f fn e s sm a t r i x f o r t h e e l e m e n t , w h i c h m a y b e p a r t i t i o n e d t o s e g r e g a t e t e r m s a s s o c i a t e d w i t h e a c ho f t h e e i g h t n o d e s , a s f o l l o w s


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THREE-DIMENSIONAL SOIL-STRUCTURE RESPONSE TO EARTHQUAKES 1045

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B e c a u s e o f th e l a y e r e d n a t u r e o f s e d i m e n t a r y s o il d e p o s i ts , o r t h o t r o p i c s t re s s - st r a inr e l a t io n s h i p s a r e c o n s i d e r e d a n e s s en t ia l f e a t u r e o f th i s e l e m e n t i n t h e p r e s e n t a p p l i c a t io n .F o r x , y , a n d z a s p r in c i p a l d i r e c t i o n s o f o r t h o t r o p y , t h e s e r e l a t i o n s h i p s a re

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_ay~_l L 0 0 0 0 0 E66 _~yz[i n wh i c h t h e s y m b o l s o a n d e d e n o t e s t re s s a n d s t r a in , r e s p e c t i v e l y . N i n e i n d e p e n d e n tc o n s t a n t s E u a r e r e q u i r e d t o r e p r e s e n t a n o r t h o t r o p i c m a t e r i a l i n t h r e e d i m e n s i o n s( E j i = E l i ) .i n a p r e l i m i n a r y v i b r a t i o n a l s t u d y , n a t u r a l f r e q u e n c i e s f o r a r e p r e s e n t a t i v e p r o b l e mu s i n g t h e c o n s i s te n t - m a s s m a t r i x ( A r c h e r , 1 96 3) o f th e e l e m e n t w e r e c o m p a r e d a g a i n s tt h o s e u s i n g a l u m p e d - m a s s m a t r i x o f t h e f o l lo w i n g f o r m

M e = p V I / 8 (5 )i n wh i c h p is t h e m a s s d e n s i t y o f t h e m a t e r i a l , V ( = 8 a b e ) d e n o t e s t h e v o l u m e o f t h ee l e m e n t , a n d I i s t h e i d e n t i t y m a t r i x ( o f o r d e r 2 4 x 2 4 ). T h e s e c o m p a r i s o n s i n d i c a te d t h a te i t h e r t y p e o f m a s s r e p r e s e n t a t i o n w o u l d s uf fi ce , a n d t h a t g i v en b y e q u a t i o n ( 5) w a sc h o s e n b e c a u s e o f it s g r e a t e r s i m p l i c i t y . F o r l a y e r e d s o i ls th e v a r i a t i o n o f m a s s d e n s i t ya n d s o il p r o p e r t i e s w i t h d e p t h m a y b e e a s il y i n c l u d e d i n t h e a n a l y s is .

F o u n d a t i o n a n d p i le s . T h e p o r t i o n o f a m u l t i s t o r y b u i l d i n g b e l o w g r o u n d l ev e l ist y p i c a l l y v e r y s t i f f a n d m a s s i v e in c o m p a r i s o n w i t h t h e s u p e r s t r u c t u r e a n d is c h a r a c t e r i z e db y t h i c k b a s e m e n t w a l l s , h e a v y f l o o r s l a b s , a n d l a r g e f o o t i n g s o r m a t f o u n d a t i o n s .T h e r e f o r e , t h is p o r t i o n o f t h e s t r u c t u r e i s a s s u m e d t o b e a t h r e e - d i m e n s i o n a l s o l id r i g idb o d y h a v i n g s ix d e g r ee s o f f r e e d o m ( t h r e e c o m p o n e n t s o f t r a n s l a t io n a n d t h r e e o fr o t a t i o n ) w i t h w h i c h i n e r t ia a c t i o n s a r e a s s o c ia t e d . T h e m a s s - i n e r t ia m a t r i x f o r p r i n c i p a la x e s t h r o u g h t h e c e n t e r o f m a s s f o r s u c h a f o u n d a t i o n b l o c k i s a 6 x 6 d i a g o n a l a r r a ywi t h t h e t e r m s MB , M , , M B * , l xx , l ry , a n d I ~ i n d i a g o n a l p o s i t i o n s . He r e t h e s y m b o l MBr e p r e s e n t s th e m a s s o f t h e b l o c k ( a s s o c i a t e d wi t h x a n d y t r a n s l a t i o n a l a c c e l e r a t i o n s ) ,w h i l e M B * is t h e s a m e m a s s a u g m e n t e d b y a p o r t i o n o f t h e m a s s o f th e s u p e r s t r u c t u r e( a s s o c i a t e d wi t h z t r a n s l a t i o n a l a c c e l e r a t i o n s ) . T h e t e r m s 1 ~ , l y y , a n d I= a r e t h e m o m e n t so f i n e r t ia o f t h e b l o c k a b o u t p r i n c i p a l b o d y a x e s . M a s s es a s s o c i a t e d w i th n o d e s o f f in i tee l e m e n t s a n d s t r u c t u r a l m e m b e r s a t t a c h e d t o t h e r i g i d b o d y m u s t b e t a k e n i n t o a c c o u n tw h e n c a l c u l a ti n g t h e l o c a t i o n o f it s c e n t e r o f m a s s a n d i ts m a s s - i n e r t ia p r o p e r t i e s .F u r t h e r m o r e , a n a x i s - r o t a t i o n t r a n s f o r m a t i o n i s r e q u i r e d i n c a s e s w h e r e t h e p r i n c i p a la x e s o f t h e f o u n d a t i o n b l o c k a r e n o t p a r a ll e l t o g l o b a l r e f e r e n c e a x e s.


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1046 WI L L IAM W E AVE R, JR . , GRE GG E . BR ANDOW , AND KAARE HOE GP i le s , p i e r s , o r c a i s s o n s o f t e n s u p p o r t t h e f o u n d a t i o n s o f ta l l b u i l d i n g s i n s o f t s o il s,

a n d a x i a l - fo r c e p r i s m a t i c m e m b e r s a r e i n c l u d e d i n t h e a n a l y t ic a l m o d e l t o r e p r e s e n t s u c he n t it ie s d r iv e n t o b e d r o c k . F i g u r e 4 s h o w s a p i le m e m b e r o f g e n e r a l o r i e n t a t i o n , f o r w h i c h

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t h e s t i ff n es s m a t r i x is a 6 x 6 a r r a y c o m m o n l y a s s o c i a t e d w i t h t h e s p a c e t ru s s t y p e o ff r a m i n g m e m b e r ( W e a v e r , 1 96 7). T h e s ti ff n es s m a t r i x f o r a p il e m a y b e w r i tt e n a s

S p = L S k j S k k J (6 )w h i c h c o n s i st s o f 3 x 3 su b m a t r i c e s p e r t a i n i n g t o d i s p l a c e m e n t s a t t h e j a n d k e n d s o f t h em e m b e r .

G e o m e t r i c t r a n s f o r m a t i o n s . N o d e s o f f i n it e e l e m e n t s l p i le s , a n d f r a m i n g m e m b e r sa t t a c h e d t o t h e f o u n d a t i o n b l o c k m u s t d i s p l a c e i n a p a t t e r n d e t e r m i n e d b y t h e r i g i d -b o d y m o t i o n s o f t h e b l o ck . T h u s , t h e t r a n s l a ti o n s o f n o d e j in F i g u r e 4 m a y b e e x p r e ss e di n t e r m s o f t h e s ix d i s p l a c e m e n t s o f r e f e r e n c e p o i n t R o n t h e r ig i d b o d y a s

D ~ = T j R D R (7 )i n w h i c h , D j = {D j l , D j2 , D j 3 } ; D R = { D R 1 , D n 2 . . . . . DR6}; and

T jR = 1 0 - - ZR j 0 R j (8 )0 1 Y R j - - X R i

w h e r e t h e d i s t a n c e s x t ~ j, y g j , a n d Z R j a r e g i v e n i n F i g u r e 4 .S t if fn e s se s f o r a p i le m e m b e r m a y b e r e f e r r e d t o t h e p o i n t R i n F i g u r e 4 b y f o r m i n g

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THREE-DIMENSIONAL SOIL-STRUC TURE RESPONSE TO EARTHQUAKES 1047= I S R R S R k J

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o Ia n d T ~ R i s t h e t r a n s p o s e o f T p R . T h e a n a l o g o u s t r a n s f o r m a t i o n f o r a f i n i t e e l e m e n t i sm o r e c o m p l e x , h o w e v e r , b e c a u s e m o r e t h a n o n e n o d e o f a n e l e m e n t m a y b e a t t a c h e d t ot h e r i g i d b o d y . A n o p e r a t i o n t h a t c a n b e u s e d f o r a l l f in i te e l e m e n t s , r e g a r d l e s s o f th e i rc o n n e c t i v i t y w i th t h e f o u n d a t i o n b l o c k , c o n s is t s o f t r a n s f o r m i n g t h e 2 4 x 2 4 m a t r i x S ef r o m e q u a t i o n ( 3) t o t h e 3 0 x 3 0 m a t r i x SERa s f o l l o w s

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8 8 l S s 2 . . S s s S S RS R 1 S R 2 . . . S g 8 SRR

(11)

T h e u p p e r l e f t - h a n d p o r t i o n o f SER i s o f t h e f o r m g i v e n in e q u a t i o n ( 3 ); b u t c e r t a i ns u b m a t r i c e s w i ll b e n u l l d u e t o t h e n a t u r e o f th e 2 4 3 0 o p e r a t o r T e R , w h i c h i s c o m p o s e da s

I 1 R 0 . . . 0 T I R 1I 2R . . . O T2RTE R = (1 2)

L O . . . I8n TsR_[wh e r e I jR i s a 3 3 i d e n t i t y m a t r i x a n d T jR i s g iv e n b y e q u a t io n ( 8 ). T h e f o r m e r s u b -m a t r i x is n u ll i f n o d e j is a t t a c h e d t o t h e f o u n d a t i o n b l o c k .

T h e 1 2 12 m e m b e r s t if f ne s s m a t r i x S c f o r a f i r s t -s t o r y c o l u m n m u s t b e t r a n s f o r m e dt o a r e f e r e n c e p o i n t F o n t h e f i r s t - s t o r y l a m i n a a s w e l l a s t h e r e f e r e n c e p o i n t R o n t h ef o u n d a t i o n . I f t h e n o d a l d i s p l a c e m e n t s a r e t a k e n in t h e s e q u e n c e s h o w n i n F i g u r e 5,t h e d e s i r e d t r a n s f o r m a t i o n b e c o m e s

SCFR = T ~ R S cT F R =

i n wh ic h

w h e r e

TFR -~-

TjF =

I jj SjF S jR iISFj SFF SFRSRj SRF SRR[ i o o 1O T j F0 0 TkR

0 - - .FFj1| XF j lo l j

( 1 3 )

(14)

( 1 5 )


8/16

1048 WILLIAM WEAVER, JR . , GREG G E . BRANDOW , AND KAA RE Ht~EG~)

W

(b ) Z R C o l u m n -- -- ~ .,,k" 7 / / / / / / / / / / / /

: D R D k l / D k zI D R

Ro m, ,~ /, / D / R 1 D R 2 D R 5 IZ R k / /

~ D R 4 l / / / X R kIY R k J / /X R

R i g i d B l o c k . ~

D k 3; D k 6

vR

FIG. 5. C o l u m n a t t a c h e d t o f o u n d a t i o n : ( a ) f i r s t s t o r y l e v e l ; ( b ) g r o u n d l e v e l .

a n d

TkR

- 1 0 0 0 2 R k - - Y R k -0 1 0 - - Z R k 0 X R k0 0 1 YR k -- XR k 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

( 1 6 )

T r a n s f o r m a t i o n s f o r b r a c e s a t th e f ir st st o r y a r e si m i la r t o t h a t f o r a c o l u m n b u t s i m p l e ri n t h e ir d e t a i ls ( B r a n d o w , 1 97 1 ) .

Boundary condi t ions fo r the so i l . T h e v i s c o u s b o u n d a r y c o n d i t io n s d e v e l o p e d b y L y s m e ra n d K u h l e m e y e r ( 1 9 6 9 ) s e r v e t o r e p r e s e n t a s o il o f in f in i te e x t e n t h o r i z o n t a l l y . I n t h e ira p p r o a c h , t h e f o l l o w i n g c o n d i t i o n s a r e s p e ci fi ed a t t h e b o u n d a r i e s

aB = pVp~ a n d z B = p V s~ ( 1 7 )i n w h i c h a n a n d z s a r e t h e n o r m a l a n d s h e a r s tr e s se s a t a b o u n d a r y s u r f a ce , fi a n d ~ 'a r e t h e n o r m a l a n d t a n g e n t i a l v e l o c i t i e s o f t h e so i l a t s u c h a s u r f a c e , a n d V p a n d V a r et h e v e l o c i ti e s o f n o r m a l a n d s h e a r i n g s tr e ss w a v e s d e f i n e d b y


9/16

THREE-DIMENSIONAL SOIL-STRUCTURE RESPONSE TO EARTHQUAKES 1049Vp = (G/p) ' /2 /s a n d V = (G /p) '12 (18 )

in w h i c h G = s h e a r m o d u l u s o f s o il , p = m a s s d e n s i ty , a n d s = a n e l a s ti c it y c o n s t a n tg i v e n b y

s = [(1 - 2v)/2(1 - v) ] ' /2 (19)w h e r e v = P o i s s o n ' s r a t io . W i t h o r t h o t r o p i c m a t e r ia l s th e q u a n t i t ie s v a n d G i n e q u a t i o n s( 1 8 ) a n d ( 1 9 ) m u s t b e c a l c u l a t e d a s a v e r a g e v a l u e s.

F o r t h e d i s c r e t i z e d a n a l y t i c a l m o d e l i n F i g u r e 1 , t h e d a m p i n g m a t r i x B , p e r t a i n i n g t on o d e s o n v i s c o u s b o u n d a r i e s , c o n t a i n s c o e f f i ci e n ts o n t h e d i a g o n a l f o r h y p o t h e t i c a ld a s h p o t s . T h e c o n t r i b u t i o n o f o n e f in i te e le m e n t t o s u c h a c o e f fi c ie n t f o r t h e i t" b o u n d a r yd i s p l a c em e n t is c o m p u t e d a s

B u = A p V v /4 o r B i t = A p V s / 4 (20 )d e p e n d i n g u p o n w h e t h e r t h e d i s p l a c e m e n t i s n o r m a l o r t a n g e n t i a l t o t h e b o u n d a r y .T h e s y m b o l A i n th e s e e x p r e s s i o n s r e p r e s e n t s t h e s u r f a c e a r e a o f t h e f a c e o f t h e f in i tee l e m e n t o n t h e b o u n d a r y .

METHOD OF SOLUTIONEquations o f motion . T h e e q u a t i o n s o f m o t i o n f o r t h e d i sc r e ti z ed a n a l y t ic a l m o d e l

s u b j ec t ed t o e a r t h q u a k e g r o u n d a c c e le r a ti o n s m a y b e w r i tt e n a sM 6 + c o + s o = A ( 0 ( 2 1 )

i n wh i c h t h e s y mb o l s D , 1 ) , a n d D r e p r e s e n t d i s p l a c e me n t s , v e l o c i t i e s , a n d a c c e l e r a t i o n so f n o d e s ( o r r e f e r e n c e p o i n t s) r e l a ti v e to b e d r o c k . T h e i d e n t if i er s M , C , a n d S d e n o t em a s s , d a m p i n g , a n d s ti ff n es s m a t r ic e s f o r t h e d e g r e e s o f f r e e d o m , a n d A ( t ) is a v e c t o r o fe q u i v a l e n t a c t i o n s ( o r l o a d s ) t h a t v a r y w i t h t i m e .

I n t h e i r f in a l f o r m t h e s t if f n e ss a n d ma s s m a t r i c e s i n e q u a t i o n ( 21 ) a r e- L s : o 1= S SRR SRsSSR SSS

MTT 0 0 JM = 0 M n n 0

0 0 M s s

( 2 2 )

( 2 3 )

I n t h e s e e x p r e s s i o n s t h e s u b s c r i p t s T , R , a n d S p e r t a i n t o t h e t i e r b u i l d i n g , t h e r i g i df o u n d a t i o n b l o c k , a n d t h e s o i l . S u b ma t r i c e s i n t h e s t i f f n e s s ma t r i x a r e f i l l e d , b u t t h o s e i nt h e m a s s m a t r i x a r e d i a g o n a l. T h e n u m b e r s o f d e g re e s o f f r e e d o m w i th w h i c h m a s se s( o r m a s s m o m e n t s o f in e r t ia ) a r e a s s o c ia t e d a r e a s f o l lo w s :

T i e r b u i l d i n g :R i gi d f o u n d a t i o n b l o c k :S o i l :

3 t im e s n u m b e r o f s to r ie s62 ti m e s n u m b e r o f u n c o n s t r a in e d n o d e s .

V e r t ic a l d i s p l a c e m e n t s o f u n c o n s t r a i n e d n o d e s i n t h e s oi l n e t w o r k a r e e li m i n a t e d i n am a t r i x c o n d e n s a t i o n p r o c e d u r e l e a d i n g t o t h e f in a l fo r m o f t h e s ti ff n es s m a t r i x ,


10/16

1 0 5 0 W I L LI A M W E A V E R , J R . , G R E G G E . B R A N D O W , A N D K A A R E H O E G

T h e d a m p i n g m a t r i x C i n e q u a t i o n ( 2 1 ) i s b e s t o b t a i n e d a sC = B + M ~ ( 2 ) , i p i 4 ~ ' ) M (2 4)i=1

i n w h i c h B i s a d i a g o n a l d a m p i n g m a t r i x w i t h t e r m s d u e t o d a s h p o t s o n v i s c o u s b o u n d -a r ie s . T h e l a s t p a r t o f e q u a t i o n ( 2 4 ) is d e ri v e d f r o m m o d a l a n a ly s i s, w h e r e i = m o d ei n d e x , n = n u m b e r o f n a t u r a l m o d e s o f i n t er e s t, y~ = r a t i o o f d a m p i n g t o c r i ti c a l d a m p i n gi n m o d e i , p~ = u n d a m p e d a n g u l a r f r e q u e n c y o f m o d e i, a n d ~ = e i g e n v e c t o r f o r m o d e in o r m a l i z e d w i t h r e s p e c t t o M .

B e c a u s e m u l t i s t o r y b u il d i n g s a r e m u c h m o r e s e n s it iv e t o h o r i z o n t a l g r o u n d m o t i o n st h a n t o v e r t i c a l m o t i o n s , t h e l o a d v e c t o r i s r e s t r i c t e d t o c o n t a i n t e r m s d u e o n l y t o t h ef o r m e r . E a r t h q u a k e r e c o r d s a r e u su a l ly i n t h e f o r m o f a c c e l e r o g r a m s ; s o t h e l o a d v e c t o ri s c o n s t i t u t e d a s f o l lo w s

A ( t ) i = - a ~ ( t ) M , ( i f i d e n o t e s x t r a n s l a t i o n ) ( 2 5 a )A ( t ) i = - a y ( t ) M ~ ( i f i d e n o t e s y t r a n s l a t i o n ) ( 2 5 b )A ( t ) i = 0 (o therw ise ) . (25c)

T h e t e r m s a ~ ( t ) a n d a y ( t ) r e p r e s e n t e a r t h q u a k e a c c e l e r a t io n s o f b e d r o c k i n t h e x a n d yd i r e c t i o n s , r e s p e c t i v e l y , w h i c h m a y b e t r e a t e d a s p i e c e w i s e - l i n e a r f o r c i n g f u n c t i o n s( W e a v e r e t a l . , 1968).

S t e p - b y - s t e p s o l u ti o n . T h e p r e s e n c e o f m a t r i x B i n e q u a t i o n ( 2 4 )~ p r ec lu d e s t h e p o s s i b i l it yo f a n a l y zi n g th e p r e s e n t p r o b l e m b y t h e n o r m a l - m o d e m e t h o d . T h e r e f o r e , t h e s t ep - b y -s t e p m e t h o d o f W i l s o n ( 1 9 68 ) w a s u t i li z e d, w h i c h i n v o l v e s t h e a p p r o x i m a t i o n t h a ta c c e l e r a t i o n s v a r y l i n e a r l y w i t h i n a ti m e i n t e r v a l A t.

I n t h e s t e p - b y - s te p p r o c e d u r e t h e t i m e o f i n t e re s t is d i v i d e d i n t o v e r y s h o r t i n te r v a ls .D u r i n g e a c h i n t e r v a l , th e f o r c i n g f u n c t i o n s a ~ ( t ) a n d a y ( t ) a r e a ss u m e d t o b e c o n s t a n t a n da r e a s s i g n ed t h e i r v a l u e s a t t h e m i d p o i n t o f th e t i m e i n t e rv a l 2 A t.

C o m p u t e r p r o g r a m . A c o m p u t e r p r o g r a m n a m e d S O I L T I E R w a s w r i t t e n i nF O R T R A N I V ( H ) f o r t h e I B M 3 6 0/ 67 a t S t a n f o r d U n iv e r s it y a n d i s d o c u m e n t e d i nB r a n d o w ( 1 9 7 1 ) . I t c a n a n a l y z e t i e r b u i l d i n g s ( w i t h o r w i t h o u t b r a c i n g a n d s e t b a c k s )t h a t a r e e i t h e r s u p p o r t e d b y a f o u n d a t i o n e m b e d d e d i n s o i l ( w i t h o r w i t h o u t p i l e s )o r a r e re s ti n g d ir e c tl y u p o n b e d r o c k . T h e o u t p u t p r o d u c e d b y th e p r o g r a m m a y b ee i t h e r fr e q u e n c i e s a n d m o d e s h a p e s f o r th e u n d a m p e d s y s t em ( w i t h o u t v is c o u s b o u n d a r i e s )o r t h e r e s u lt s f r o m n u m e r i c a l i n t e g r a t i o n o f th e e q u a t i o n s o f m o t i o n f o r t h e s y s te m s u b -j e c t e d t o h o r i z o n t a l a c c e l e r a t i o n s o f b e d r o c k .

EXAMPLEST h e f o l l o w i n g e x a m p l e s i n d i c a te t h e c a p a b i l it i es o f t h e m e t h o d d e s c r i b e d h e r e in a n d

a r e n o t i n t e n d e d t o r e p r e s e n t t y p i c a l c a se s .1 . T w o - s t o r y b u i l d i n g . T o d e m o n s t r a t e h o w c r u d e t h e f i n it e - e le m e n t m e s h c a n b e , a

v e r y s i m p l e s t r u c t u r e w a s a n a l y z e d f o r s e v e r a l s o il n e t w o r k s . F i g u r e 6 s h o w s a t w o -s t o r y b u i ld i n g s u p p o r t e d o n a f o u n d a t i o n b l o c k t h a t is 6 f t d e e p a n d is e m b e d d e d w i t h in9 9 f i n it e e l e m e n t s ( 5 x 5 x 4 - 1 ) r e p r e s e n t i n g t h e s o il . C o l u m n s i n t h e b u i l d i n g a r e a s s u m e dt o h a v e c r o s s - s e c t i o n a l a r e a s o f 9 i n z , p r i n c i p a l m o m e n t s o f i n e r t i a o f 1 10 i n 4 , a n dt o r s i o n c o n s t a n t s o f 2 0 in '~ ; b e a m s h a v e m o m e n t s o f i n e r t i a o f 2 0 0 i n 4 a n d t o r s i o nc o n s t a n t s o f 2 0 i n 4 . Y o u n g ' s m o d u l u s f o r a l l m e m b e r s i s E - - 3 0 ,0 0 0 k s i. T h e t r i b u t a r yw e i g h t a t e a c h s t o r y is t a k e n a s 2 1 6 l b / f t 2 , a n d t h e f o u n d a t i o n b l o c k h a s a u n i f o r m l y -


11/16

THREE-DIMENSIONALSOIL-STRUCTURERESPONSE TO EARTHQUAKES 1051

S t o r y 2S t o r y 1 ~ \

i ~ - B e d r o c k. 5 @ 2 4 ' - 0 " : 1 2 0 ' - 0 "

~ I ~z''

_ ~ 8 ' 0 " / ' 6 0 ' - 0 "

No te : V i scous boundary no ts h o w n in f i g u r eFIG. 6. Example 1 Two-story building.

0 .3(1 Xg

0

- 0 . 3

" " 8

X - D i r e c t i o n

Time (sec)

0 .3a yg

0

- 0 . 3

Y - D i r e c t io n

FIG. 7. Bedrock accelerations for example 1.

distributed total weight of 260 kips. The soil is assumed to be homogeneous and i sotropicwith You ng 's modulus E = l0 ksi, Poi sson 's ratio v = 0.45, and unit weight of 120lb/ft 3 (V e ~ 1200 fps and V s ~ 365 fps).

The response of this system to the bedrock accelerations shown in Figure 7 was calcu-lated for a time of 2 sec using 100 steps of 0.02 sec. Fo r this purpose, damping was assumedto be 10 per cent of critical damping in the fundamental mode of vibration with rigidboundaries. Similar analyses were also conducted using cruder meshes for the same


12/16

1052 WIL L IAM WE AVE R, JR . , GRE G G E. BRANDOW , AND KA ARE HOE Gv o l u m e o f so il , a s f o ll o w s : 1 8 e l e m e n ts ( 3 x 3 2 ) , a n d 9 e le m e n t s ( 3 3 x 1 ). F o r t h el a t t e r a n a l y s e s t h e d e p t h o f t h e f o u n d a t i o n b l o c k w a s r e d u c e d t o z e r o . F i g u r e 8 sh o w sm a x i m u m x - d i s p l a c e m e n t s i n t h e s o i l , w h i c h o c c u r r e d a t a p p r o x i m a t e l y t h e s a m e t i m ef o r a l l a n a l y s e s . A t g r o u n d l e v el , t h e s e m a x i m a a r e v e r y c l o s e ; t h e r e s u l t f o r 1 8 e l e m e n t sd i f f e r s f r o m t h a t f o r 9 9 e l e m e n t s b y o n l y a b o u t 6 p e r c e n t w h i l e t h a t f o r 9 e l e m e n t s i se v e n c l o s e r .

D i s p l a c e m e n t s in t h e x d i re c t i o n o f t h e s e c o n d s t o r y a n d t h e f o u n d a t i o n s la b a r ep l o t t e d a g a i n s t t i m e i n F i g u r e 9 . F o r p r a c t i c a l p u r p o s e s , t h e r e s u l t s f r o m a l l t h r e e m e s h e sa p p e a r t o b e s a t i s f a c to r y . T h u s , e v e n t h e v e r y c r u d e 9 - e l e m e n t m e s h w o u l d s u ff ic e f o rt h i s e x a m p l e .

2. Ten-story building. A t e n - s t o r y e x a m p l e is p r e s e n t e d t o s h o w s o m e o f t h e e ff e c ts o fv a r y i n g t h e d e p t h o f s o il u n d e r a t al l b u i l d in g . F i g u r e 1 0 g i v es t h e l a y o u t o f a h y p o -t h e t i c a l b u i l d i n g t h a t h a s a s t e e l f r a m e a n d f o u r b r a c e d b a y s a b o v e t h e f i r s t s t o r y l e v e l .T h e t o t a l w e i g h t t r i b u t a r y t o e a c h f r a m i n g l e v e l i s a s s u m e d t o b e 2 8 3 k i p s . M a s s a n di n e r t ia p r o p e r t i e s o f t h e f o u n d a t i o n w e r e c o m p u t e d a s i f i t w e r e a so l id c o n c r e t e m a tw i t h a t h i c k n e s s o f 5 f t , b u t i t s t h i c k n e s s i n t h e a n a l y t i c a l m o d e l w a s r e d u c e d t o z e r o f o rp u r p o s e s o f t h i s p a r a m e t e r s t u d y . T h e f i n i t e- e l e m e n t m e s h ( n o t s h o w n in F i g u r e 1 0) w a st a k e n a s 3 sp a c e s a t 1 00 ft e a c h w a y h o r i z o n t a l l y t o f o r m n e t w o r k s o f 9 e l e m e n t s ( 3 x 3 x1), 18 e leme nts (3 x 3 x 2 ) , o r 27 e lem ents (3 x 3 x 3 ), d ep end ing up on the de p th o f thes o il . P r o p e r t i e s o f t h e s o i l a r e t h e s am e a s i n e x a m p l e 1, a n d d a m p i n g w a s d e t e r m i n e d o nt h e b a si s o f 1 0 p e r c e n t o f c r it ic a l d a m p i n g in t h e f u n d a m e n t a l m o d e .

T h e f o r c i n g f u n c t i o n s u s e d in t h is e x a m p l e c o n s i s t o f e a r t h q u a k e a c c e l e r o g r a m s f o r1 2 s ec o f th e E u r e k a ( C a l i f o r n i a ) e a r t h q u a k e o f D e c e m b e r 2 1, 1 95 4. F i g u r e 11 c o n t a i n sc o m p u t e r p l o t s o f d i g it i ze d a c c e l e r a ti o n s f o r t h e n o r t h - s o u t h a n d e a s t - w e s t c o m p o n e n t so f m o t i o n . T h e m a x i m u m c o m p o n e n t o f a c c e l e r a ti o n w a s 0 . 28 t i m e s g r av i t y in t h e e a s t-w e s t d i re c t i o n , a n d t h e m a g n i t u d e o f th i s e a r t h q u a k e w a s 6 .7 o n t h e R i c h t e r s c al e.R e s p o n s e c a l c u l a t i o n s w e r e c o n d u c t e d f o r 1 2 s e c u s i n g 6 0 0 s t e p s o f 0 . 0 2 s e c.

M a x i m u m s h e a r f o r c es a n d m o m e n t s p r o d u c e d in t h e fi rs t s t o r y o f t h is b u i l d i n g b yt h e E u r e k a e a r t h q u a k e a r e p l o t t e d a g a i n s t d e p t h o f so il in F i g u r e 12 . T h e v a r i a t i o n o ft h e s e q u a n t i t i e s f o r d e p t h s f r o m z e r o ( n o s o i l - - b e d r o c k a t s u r f a c e ) to 2 0 0 ft is c o n s i d e r -a b l e . F o r e x a m p l e , t h e x - s h e a r f o r a so i l d e p t h o f 6 0 f t is a p p r o x i m a t e l y 1 50 p e r c e n t o ft h a t f o r z e r o d e p t h . A m o r e c o m p l e t e i l l u s t r a t i o n o f s t o r y sh e a r s i s g i v e n in F i g u r e s 1 3aa n d b , w h e r e b o t h x - a n d y - s h e a r s f o r a s o il d e p t h o f 6 0 f t a r e c o m p a r e d w i t h c o r r e s p o n d -i n g v a l u e s f o r b e d r o c k a t t h e s u r fa c e .

CONCL USIONST h e a n a l y t i c a l m o d e l d e s c r i b e d h e r e i n c o n s t i t u t e s a u s e fu l t o o l f o r i n v e s ti g a ti n g t h r e e-

d i m e n s i o n a l s o i l - f o u n d a t i o n - s t r u c t u r e i n t e r a c t i o n f o r m u l t i s t o r y b u i l d i n g s s u b j e c t e d t oe a r t h q u a k e s . T h e u s e o f f in i te e l e m e n t s to r e p r e s e n t th e s o i l a p p e a r s p r o m i s i n g b e c a u s et h is t e c h n i q u e is v e r sa t il e , s t r a i g h t f o r w a r d , a n d a m e n a b l e t o c o m p u t e r p r o g r a m m i n g .

S a m p l e p r o b l e m s i n d i c a t e t h a t a r a t h e r s m a l l n u m b e r o f fi n it e e l e m e n t s i s a p p a r e n t l ys u ff ic i en t f o r c a l c u l a t in g t h e d y n a m i c r e s p o n s e o f a s t r u c t u r e . O n t h e o t h e r h a n d ,d e t e r m i n a t i o n o f s tr e ss e s in t h e s o i l t y p i c a ll y r e q u ir e s a m o r e r e f i n ed n e t w o r k . T h e r e p o r tb y B r a n d o w ( 19 7 1) c o n t a i n s t h e r e s u lt s f o r a n u m b e r o f p a r a m e t e r s t u d ie s , i n c l u d in g t h ee f f e c ts o f v a r y i n g s o i l d e p t h t o b e d r o c k , Y o u n g ' s m o d u l u s f o r t h e s o i l, s ti f fn e s s es o f t h es t r u c t u r a l f r a m e w o r k , a n d f o u n d a t i o n c o n d i t i o n s . T h e s e s t u d i e s s h o w t h a t t h e i n f l u e n c e so f t h e s o il a n d f o u n d a t i o n u p o n t h e r e s p o n s e o f th e s u p e r s t r u c t u r e c a n s e l d o m b e i g n o re d .E x c e p t f o r t h e m a t t e r s o f c o m p u t e r t im e a n d s t o r a g e , t h e r e a r e n o g r e a t o b s t a cl e s f o re x t e n d i n g t h e m e t h o d t o i n c lu d e n o n l i n e a r s t r e s s- s t r a in - r e l a ti o n s h i p s in t h e s o il .


13/16

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16/16

1056 WILLIAM WEAVER, JR., GREGG E. BRANDOW~ AND KAARE HC)EGREFERENCES

Archer, J. S. (1963). Consistent mass matrix for distributed mass systems, J . S truct . Div . , ASCE, 89,161-178.Blot, M. A. (1943). Analytical and experimental methods in engineering seismology, Tra n s . AS C E, 108,365-385.Brandow, G. E. (1970). Computer programs for statics and dynamics of tier buildings, Technical ReportNo. 129 , Department of Civil Engineering, Stanford University.Brandow, G. E. (1971). Soil-foundation-structure interaction during earthquake excitations, TechnicalReport No. 145, Department of Civil Engineering, Stanford University.Gutenberg, B. (1957). The effect of ground on earthquake motion, Bull. Seism. Soc. Am. 47, 221-251.Hashiba, T. and R. V. Whitman (1960). Soil-structure interaction during earthquakes, Soils Found.,Japan, 8, 1-12.Jacobsen, L. S. (1939). Natural periods of uniform cantilever beams, Tra n s . AS C E , 104, 402-439.Jacobsen, L, S. and R. S. Ayre (1958). Engineering Vibrations, McGraw-Hill Book Co., Inc., New York,110-112.Kanai, K., T. Tanaka, and S. Yoshizawa (1959). Comparative studies of earthquake motions on theground and underground, Part 1, Bull . Earthquake Res. Inst . , Tokyo Univ. 37, 53-88.Khanna, J. (1969). Elastic soil-structure interaction, Proe. Worm Conf. Earthquake Engineering, 4tk,Chile, 3, 143-152.Lycan, D. L. and N, M. Newmark (1961). Effects of structure and foundation interaction, J. Eng.M eek . Div . , AS C E, 87, 1-31.Lysmer, J. and R. S. Kuhlemeyer (1969). Finite dynamic model for infinite media, J. Eng. Mech. Div.,A S C E , 95, 859-877.Manning, T. A. (1970). The analysis of tier buildings with shear walls, Technical Report No. 128, Depart-ment of Civil Engineering, Stanford University.Melosh, R. J. (1963). Structural Analysis of Solids, J. Struet. Div., ASCE, 89, 205-223.Minami, J. K. and J. Sajurai (1969). Some effects of substructure and adjacent soil interaction on theseismic response of buildings, Proe. W orm Conf. Earthquake Eng ., 4th, Chile 3, 71-86.

Minami, K., J. Sakurai, and T. Katayama (1970). Some effects of substructure proportions and adjacentsoil interaction on the seismic response of buildings, Proc. Japan Earthquake Eng. Symp., 3rd,To kyo 1-8.Parmelee, R. A. (1967). Building-foundation interaction effects, J. Eng. Mech . Div . , ASCE, 93, 131-152.Parmelee, R. A., D. S. Perelman, and S. L. Lee (1968). Seismic response of structure-foundation systems,J . Eng . Mech . Div ., ASC E, 94, 1295-1315.Penzien, J., C. F. Scheffey, and R. A. Parmelee (1964). Seismic analysis of bridges on long piles, J. Eng.Mech . Div . , ASCE, 90, 223-254.Seed, H. B., I. M. Idriss, and F. W. Kiefer (1969). Characteristics of rock motions during earthquakes,J . So i l Mech . Found. Div. , ASC E, 95, 1199-1218.Seed, H. B. and I. M. Idriss (1969). Influence of soil conditions on ground motions during earthquakes,J . So i l Mech . Found. Div ., A SCE , 95, 99-138.

Tsai, Nien-Chien (1967). Influence of local geology on earthquake ground motion, Earthquake Engineer-ing Research Laboratory, California Institute of Technology, Pasadena, California.Weaver, W., Jr. (1967). Computer Programs for Structural Analysis, D. Van Nostrand Co., Inc., Princeton,New Jersey.Weaver, W., Jr. and M. F. Nelson (1966). Three-Dimensional Analysis of Tier Buildings,J. Struct. Div.,A S C E , 92 , 385-404.Weaver, W., Jr., M. F. Nelson, and T. A. Manning (1968). Dynamics of tier buildings, J. Eng. Mech.Div . , ASCE, 94, 1455-1475.Whitman, R. V. (1969). The current status of soil dynamics, Appl. Mech. Rev. 22, 1-8.Wiggins, J. H. (1964). Effects of site conditions on earthquake intensity, J . S truct . Div . , ASCE, 90, 279-313.Wilson, E. L. (1969). A method of analysis for the evaluation of foundation-structure interaction, Proc.W orm Conf. Earthquake Eng., 4th, Chile, 3, 87-150.Wilson, E. L. (1968). A computer program for the dynamic stress analysis of underground structuresReport No . 68-1 , Structural Engineering Laboratory, University of California, Berkeley.

DEPARTMENT OF CIVIL ENGINEERING BRANDOW t~ JOHNSTONSTANFORD UNIVERSITY ASSOCIATESSTANFORD, CALIFORNIA (W.W. ANDK.H.) Los ANGELES, CALIFORNIA(G.E.B.)

Manuscript received November 20, 1972

Three Dimentional Soil Structure Response to Earthquakes

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