1984-SDEE_Seismic Response of End-bearing Single Piles

7/29/2019 1984-SDEE_Seismic Response of End-bearing Single Piles

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Se i smic response o f end-bear ing s ing le p i l e sG E O R G E G A Z E T A SDep artmen t o f Ovi l Engineering, Rensselaer Polytechnic Inst i tu te, Troy, New York 12181, USA

A n u m e r i c a l s t u d y i s p r e s e n t e d o f t h e d y n a m i c r e s p o n s e o f e n d - b e a r i n g p i le s e m b e d d e d i n a n u m b e ro f i d e a li z e d s o i l d e p o s i t s a n d s u b j e c t e d t o v e r t ic a l ly p r o p a g a t i n g h a r m o n i c S - w a v e s. R e s u l t s, f o r b o t h' k i n e m a t i c ' a n d ' i n e r t i a l ' i n t e r a c t i o n , a r e o f f e r e d i n t h e f o r m o f d i m e n s i o n l e s s g r a p h s a n d f o r m u l a ec o v e r i n g a w i d e r a n g e o f e x c i t a t i o n f r e q u e n c i e s a n d o f c r u c i a l m a t e r i a l a n d g e o m e t r i c p a r a m e t e r s .P r a c t ic a l a s p e c t s o f t h e e v a l u a t i o n o f t h e i n f l u e n c e o f p il e s o n t h e e f f e c t i v e s e is m i c e x c i t a ti o n o f as t ruc ture a re d i s cus sed and a case h i s to ry i l lus t ra te s the use fu lnes s o f the p resen ted re su l t s .Key Words : dynam ic re sponse , end-bea r ing p i l e s , so i l depos i t s , ha rm onic S -waves , s e i sm ic exc i t a t ion .

I N T R O D U C T I O NT h e r e s p o n s e o f p i l e s t o v e r t i ca l l y p r o p a g a t i n g S - w a v es h a sb e e n s t u d i e d b y s e v e ra l a u t h o r s . H 2 D e s p i t e t h e s i g ni f ic a n tprogres s in unders tand in g the s e i sm ic behav ior o f singlep il es , s eve ra l ques t ion s rem a in unan swere d , e spec ia l ly wi threga rd to th e p rac t i ca l a s ses sm ent o f the in f luence o f p i l e son the s e i sm ic exc i t a t ion o f a s t ruc tu r e . F or ins tance , it hasbeen sugges ted tha t , s ince f l ex ib le p i l e s fo l low the g roundm o t i o n , t h e r e i s n o n e e d t o m o d i f y t h e i n p u t e x c i t a t i o n .C o n s e q u e n t l y , a n a l y si s p r o c e d u r e s o f t e n m a k e u s e o f t h esam e des ign re sponse spec t ra a s fo r s t ruc tu res on sha l lowfounda t ions . 3 , l a, 14 The v a l id i ty o f th i s a pp rox im a t io n , how -eve r , has no t ye t been a deq ua te ly ve r i f i ed . In fac t , the re su l t sof th i s s tudy show tha t , in ce r ta in cases , even re la t ive ly longp i le s m a y a p p r e c i a b l y m o d i f y t h e b a s e e x c i t a t i o n o f a s u p -p o r t e d s t r u c t u r e .F u r t h e r m o r e , o n l y a v e r y l i m i t e d n u m b e r o f r e s u lt s a r ea v a i la b l e in t h e f o r m o f d i m e n s io n l e s s g r a p h s a n d f o r m u l a e .S uch re su l t s would be use fu l no t o n ly fo r deve lop ing ani m p r o v e d u n d e r s t a n d i n g o f t h e m e c h a n i c s o f t h e p r o b l e ma n d c h e c k i n g t h e a c c u r a c y o f s o p h i s t i c a te d s o l u t i o n s , b u ta l so fo r m ak ing pre l im ina ry des ign e s t im a tes in p rac t i ce .By con t ra s t , s eve ra l pa ram et r i c s tud ies have been publ i shedfor p i l e s sub jec ted a t the i r head to hor izon ta l s t a t i c ' * ' l s -2or dy na mi c 4 ' v , H, 19-25 forces .

T h i s p a p e r p r e s e n t s a n e x t e n s i v e p a r a m e t e r s t u d y , c o n -d u c t e d w i t h t h e e f fi c i e n t f i n i te - e l e m e n t f o r m u l a t i o ndeve lo ped by Roesse t and h i s co-worke rs . 4 '24 Resu l t s a r ep r e s e n t e d i n t h e f o r m o f d i m e n s io n l e s s s o i l -p i l e i n t e r a c t i o nand am pl i f i ca t ion fac to rs , a s we l l a s p i l e -head im pedancef u n c t io n s . A m o n g t h e g r o u p s o f p r o b l e m p a r a m e t e r si n f l u e n c in g t h e r e s p o n s e m o s t i m p o r t a n t ( f o r e ac h p a rt i c u l a rso i l p rof i l e ) have been foun d to be : the s t i ffnes s ra t ioEp/E s of the p i l e Youn g ' s m od ulus ove r a cha rac te r i s t i cY o u n g ' s m o d u l u s o f t h e s o i l d e p o s i t ; t h e s l e n d e r n e s s r a t i oL id of the l eng th ove r t i l e d iam e te r o f the p i l e ; the f re -q u e n c y r a t i o f/f~ o f t h e e x c i t a t i o n f r e q u e n c y o v e r t h ef u n d a m e n t a l n a t u r a l f r e q u e n c y o f t h e u n p e r t u r b e d s o i ldepos i t in ve r t i ca l S -waves : and th e re la t ive f requen cyf a c t o r fst/fx, w h e r e f s t i s t h e f u n d a m e n t a l f r e q u e n c y o f t h ep i l e - s u p p o r t e d s u p e r s t r u c t u r e , C o n c l u s i o n s a r e d ra w n o nPaper received September 1983. Discussion closes June 1984.

the va l id i ty o f c ur ren t s e i sm ic design prac t i ces and a caseh i s t o r y i s p r e s e n t e d i n v o l v i n g a c t u a l e a r t h q u a k e r e c o r d s o n ,a n d n e a r b y , a p il e d f o u n d a t i o n . T h e u s e f u l n e ss o f t h e r e s u lt so f f e r e d i n t h e p a p e r i s i l l u s tr a t e d b y c o m p a r i n g t h e o r e t i c a la n d r e c o r d e d t r a n s f e r f u n c t i o n s .

S T A T E M E N T O F T H E P R O B L E MT h e s y s t e m s t u d i e d r e f e r s t o a n e n d - b e a r in g p i le s u p p o r t i n ga b l o c k o f m a s s M ( s u p e r - s t r u c t u r e ) a n d b e i n g e m b e d d e d i na so i l s t ra tu m of th icknes s L (F ig . l a ) . Ver t i ca l ly inc iden th a r m o n i c S - w a v e s c o n s t i t u t e t h e b a s e e x c i t a t i o n , w h i c h i sd e s c r i b e d t h r o u g h t h e d i s p l a c e m e n t : ug (t) = Ug. exp(i21rft).The p i l e i s a l inea r ly e la s t i c f l exura l beam wi th a c i rcu la rc r o s s - se c t i o n o f d i a m e t e r d , Y o u n g ' s m o d u l u s Ep a n d m a s sdens i ty pp . The so i l i s m ode led a s a l inea r ly hys te re t i c con-t i n u u m w i t h c o n s t a n t P o i s s o n 's r a t i o vs , m ass dens i ty P sa n d h y s t e r e t i c d a m p i n g r a t i o / 3 s , b u t w i t h Y o u n g ' s m o d u l u sE ( z ) w h i c h v a r i e s w i t h d e p t h f r o m t i l e g r o u n d s u r f a c e .

Three so i l m ode l s a re cons ide red , each wi th a d i f fe ren tv a r i a t i o n o f E ( z ) , as ske tched in F ig . 2 . In Mode l A, E ( z )i s p r o p o r t i o n a l t o d e p t h , r e p r e s e n t i n g u n i f o r m s o f t n o r m a l l y -conso l ida ted c lay depos i t s . In Mode l B , E ( z ) i s p r o p o r t i o n a lt o t il e s q u a re r o o t o f z - a n id e a l i z at i o n a p p r o p r i a t e f o run i f o rm depo s i t s o f cohes ion les s so il s. F ina l ly , Mode l C hasa m o d u l u s E ( z ) = E s, c o n s t a n t w i t h d e p t h - t y p i c a l o f s t i f fo v e r c o n s o l i d a t e d c l a y d e p o s i t s . T h e s e t h r e e m o d e l s m a ya d e q u a t e l y r e p r e s e n t t h e d y n a m i c c h a r a c t e r i s ti c s o f a f a i rl yw i d e r a n g e o f s o il p r o f i le s e n c o u n t e r e d i n n a t u r e .I t i s c o n c e p t u a l l y a t t r a c t i v e a n d c o m p u t a t i o n a l l y c o n -v e n i e n t t o e x p r e s s t h e r e s p o n s e o f t h e s y s t e m s h o w n inF ig . l a a s a supe rp os i t ion o f two eff ects : 3'14'26'27 ( 1 ) akinemafic interaction e f fec t , invo lv ing the re sponse to base -r o c k e x c i t a t i o n o f t h e s y s t e m s h o w n i n F i g. l b , w h i c hd i f fe r s f r o m t h e c o m p l e t e s y s t e m o f F i g . l a i n t h a t t h e m a s sof the su pe r -s t ruc tu re i s s e t equa l to ze ro : (2 ) an inertialinteraction e f f e c t , r e f e r r i n g t o t h e r e s p o n s e o f t h e c o m p l e t ep i l e - s o i l - s t r u c t u r e s y s t e m t o e x c i t a t i o n b y D ' A l e m b e r tfo rces , --Mi~k, as soc ia ted w i th the acce le ra t ion , /~k , o f thesupe r -s t ru c ture due to the k inem at ic in te rac t ion (F ig . l c ) .Th i s supe r pos i t ion i s exac t i f the an a lyses in bo th s t eps a re

r i g o r o u s ly p e r f o r m e d . T h e p o p u l a r i t y , h o w e v e r , o f t h ea p p r o a c h s t e m s f r o m a s u g g e s t e d a p p r o x i m a t i o n t o t h e0261-7277/84/020082-12 52 .008 2 Soi l Dyn amics and Ea rthquake Engineering, 1984, Vol . 3 , No. 2 1984 CML Publications


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k i n e m a t i c i n t e r a c t i o n effe cts: 3'14'26 s i n c e i n m a n y c a s e sp i le s t e n d t o f o l l o w th e g r o u n d , s o i l -p i l e in t e r a c t i o n i si g n o r e d a n d t h e f r e e - f i e l d m o t i o n i s u s e d a s i n p u t i n t h ei n e r t ia l i n t e r a c t i o n s t e p . O n e o f t h e p u r p o s e s o f t h i s p a p e ris t o c o m m e n t o n t h e r a n g e o f v a l id i ty o f t h is a p p r o x i m a -t i o n .A n o t h e r p o t e n t i a l l y s i g n i f i c a n t s i m p l i f i c a t i o n m a y a l s ob e s u g g e st e d i n t h e c o n t e x t o f t h e k i n e m a t i c - i n e r t i a ld e c o m p o s i t i o n . 14 Firs t, reca ll 16' ~s, ~9, 22 th at , in th e m aj or it yo f a c t u a l c a s e s , p i l e d e f o r m a t i o n s d u e t o l a t e r a l e x c i t a t i o n

. . . . . . " I l l

Y G Uf ll I~ NU IU l I I I )L Pclston's ~olio %

// / / / / 7 / / , / , , . . . : H/ / / , H:I / : 'H/H/HH'/ , /~ /~ M/HHI' e / / /O l l t ) . u l t I ~ t

K I N E M A T I C I N T E R A C T I O N I~olsless*~e,s.vcvurer---.-~

I r

uolt), uoe

[ , . ~ . ., . L , . . E . . c . , o . I

I - l t u . l q I

/ N(c )~. , , ]= = ",- r

t

. . ~ / ~ , .

Figure 1. (a) Ge om etry o f soil-pile-structure interactionproblem; (b) decomposit ion into kinem atie and inertialinteraction problems; (e) two-step a na lysis o f inertialinteraction

Seismic respo nse o f end -beating single piles." G. Gazetast r a n s m i t t e d f r o m t h e s u p e r - s t r u c t u r e a t t e n u a t e v e r y r a p i d l yw i t h d e p t h ( t y p i c a l l y w i t h i n 1 0 - 1 5 d i a m e t e r s f r o m t h eg r o u n d s u r f a c e ) . T h e r e f o r e , s h e a r st r a in s i n d u c e d i n t h e s o ild u e t o i n e r t i a l i n t e r a c t i o n m a y b e s i g n i f i c a n t o n l y n e a r t h egrou nd sur face . 6 '28 By co n t ra s t , ve r t i ca l S -waves induce int h e f r e e -f i e ld s h e a r s t r a i n s t h a t a r e l i k e l y t o b e i m p o r t a n ton ly a t re la t ive ly dee p e leva t ions . T hus , s ince so il s tra ins a rec o n t r o l l e d b y i n e r t i a l e f f e c t s n e a r t h e g r o u n d s u r f a c e a n db y k i n e m a t i c e f f e c t s a t g r e a t e r d e p t h s , t h e s u p e r p o s i t i o nm a y b e a p p r o x i m a t e l y v a l i d e v e n i f n o n l i n e a r s o i l b e h a v i o ri s e x p e c t e d , d u r i n g a s t r o n g b a s e e x c i t a t i o n .N o t e a l s o t h a t , a s o r i g i n al l y p r o p o s e d b y K a u s e l a n dR o e s s e t 27 f o r e m b e d d e d f o u n d a t i o n s , i n e r ti a l i n t e r a c t i o na n a l y s e s i n t h e f r e q u e n c y d o m a i n c a n b e c o n v e n i e n t l yp e r f o r m e d i n t w o s t e p s , s k e t c h e d i n F i g . l c . D e t e r m i n a t i o no f t h e d y n a m i c i m p e d a n c e s , fffH H, :ZtrMMa n d :Z('HM,w h i c he x p r e s s d y n a m i c f o r c e - d i s p l a c e m e n t r a t i o s a t t h e h e a d o ft h e p i l e , is a c e n t r a l t a s k o f t h i s a p p r o a c h .F i n a ll y , k i n e m a t i c - i n e r t i a l d e c o m p o s i t i o n i s p a r t i c u l a r l ys u i t a b l e f o r p a r a m e t r i c s t u d i e s , a n d p r o v i d e s c o n s i d e r a b l ei n s ig h t i n t o t h e m e c h a n i c s o f p i l e - s o i l - st r u c t u r e i n t e r a c t io n .P A R A M E T R I C R E S U L T S : K I N E M A T I C I N T E R A C T I O NI n t h e a b s e n c e o f a p i l e , a v e r t i c a l ly i n c id e n t S - w a v e w o u l di n d u c e o n l y h o r i z o n t a l d i s p l a c e m e n t s in t h e f r e e - fi e l d s o il .F o r a b a s e r o c k m o t i o n ug(t) = ug expff2rrft) a n d a h o m o -g e n e o u s s o i l s t r a t u m , t h e o n e - d i m e n s i o n a l ' a m p l i f i c a t i o n 'the ory 29 wo uld g ive fo r the s t eady -s ta te f ree - f i e ld d i sp lace -m e n t a t g r o u n d s u r f a c e l e v el :

Uo (t) = Uo exp(i27 rft) ( l a )Uo 2

- - = ( lb )ug exp( iqL ) + exp( - - iq L )( L i s t h e s t r a t u m t h i c k n e s s a n d q2= 4rr2f: /[V2(1 + 2i/3s)],wh ere V and /~s a re the S -wave ve loc i ty and the in te rna l(hy s te re t i c ) dam ping in the so i l : i 2 = - - 1 .) I f /~ s = 0 , equa -t i o n ( l b ) y i e l d s Uo/Ug = 1/cos(2~rfL/Vs) w h i c h t e n d s t oi n f i n i t y ( r e s o n a n c e ) a t t h e n a t u r a l s h e a r f r e q u e n c i e s o f t h es t r a t u m , f n = ( 2 n - - 1) Vs/4L, n = l , 2 , 3 . . . .A cy l indr ica l p i l e d i f f rac t s the inc iden t and re f l ec tedo n e - d i m e n s i o n a l v e r t i c a l S - w a v e s , t h e r e b y m o d i f y i n g t h e' f ree ' wave f i e ld . As a re su l t , the hor izo n ta l d i sp lacem enta t o p t h e p i l e , up (t) = Up exp(i2rrft) di f fe rs f rom uo( t ) o fe q u a t i o n ( 1 ) . I n a d d i t i o n , t h e p i le t o p e x p e r i e n c e s a r o t a -t ion , Cp( t ) = Cp exp(i27rft). I t i s c o n v e n i e n t t o p o r t r a y t h ee f f e c t s o f k i n e m a t i c i n t e r a c t i o n b y i n tr o d u c i n g t h e d i s-p l a c e m e n t a n d r o t a t i o n kinematic interaction factors

_ Up (pproI u - - - a n d I ~ - ( 2 )U 0 U O

0 E ( z ) 0,fZ = dt

SOIL SOILMODEL MODELA B

Figure 2. The three soil models studied

E ( z )

, Z SOILMODELC0 E ( z )

NSoil Dynam ics and Earthquake Engineering, 1984, Iiol. 3, No. 2 8 3


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Seismic response o end-be arin g s ingle pi les: G. Gazetasa n d a l s o , t h e d i s p l a c e m e n t a n d r o t a t i o n kinematic ampli f i -cat ion factors

_ Up q)proA u - - - an d A ~ = ~ ( 3 )Ug ugw h e r e r 0 = d /2 i s t he r ad i us o f t he p i l e , whi l e Up and Uo a r et h e a m p l i t u d e s o f h o r i z o n t a l d i s p l a c e m e n t o f t h e p il e t o pa n d t h e f r e e - f i e l d g r o u n d s u r f a c e , r e s p e c t i v e l y , r e l a t i v e t ot h e b a s e ; t h e t o t a l r e s p e c t i v e d i s p l a c e m e n t s a r e up( t ) + Ug( t)a n d Uo(t) + us(t ) . W i t h o u t k i n e m a t i c i n t e r a c t io n Up w o u l d b eequa l t o Uo, and (pp equa l t o ze r o . T h en , lu = 1, I 0 = A O = 0a n d A u w o u l d b e g i v e n b y e q u a t i o n ( l b ) . I n t h e s e q u e l,k i n e m a t i c i n t e r a c t i o n e f f e c t s a r e s t u d i e d i n t e rm s o f t h e s ef o u r in t e r a c t i o n a n d a m p l i f i c a t i o n f a c t o r s .

B e c a u se o f t h e p r e s e n c e o f b o t h r a d i a t i o n ( d u e t o d i f-f r a c t i o n ) a n d m a t e r i a l d a m p i n g i n t h e s y s t e m , t h e v a r i o u sd i s p la c e m e n t a n d r o t a t i o n c o m p o n e n t s a re n o t i n -p h a s ew i t h t h e e x c i t a t i o n . I t h a s b e e n c u s t o m a r y in t h e so i ld y n a m i c s l i t e r a tu r e t o u s e c o m p l e x n o t a t i o n i n o r d e r t oexpr es s d i f f e r enc es i n phase . I n t h i s s ense , t he f ou r ampl i f i -c a t i o n a n d i n t e r a c t i o n f a c t o r s a r e c o m p l e x f u n c t i o n s o ff r e q u e n c y . O n l y t h e ir a b s o lu t e v a l ue s ( a m p l i t u d e s ) a r es t ud i ed he r e i n : t h i s i s usua l l y su f f i c i en t f o r p r ac t i ca l app l i -ca t i ons , i s

F i g u r e s 3 - 5 p r e s e n t p a r a m e t r i c r e s u lt s f o r th e v a r i a t i o no f k i n e m a t i c a m p l i f i c a t io n a n d i n t e r a c t i o n f a c t o r s v e rs u s

Table 1. Expressions for natural shear requencies o] the three soildepositsSoil model f , /2J'Jl

A 1.21 Is/ / / 2.33B 0.56 IstH 2.66C 0.25 IslH 3.00

l ' = S-wave velocity at depth : = d below the ground surface

t h e f r e q u e n c y r a t i o f / f~. f j i s t h e f u n d a m e n t a l s h e a r f r e -q u e n c y ( in H z ) o f e a c h s t r at u m , c o m p u t e d f r o m t h ee x p r e s s i o n s o f T a b l e 1 f o r e a c h o f t h e t h r e e s o il p r o f i le s inFig. 2. 30Effect o f s t i f fness rat io EplEs

T h e i n f l u e n c e o f E p / E s i s por t r ayed i n F i g . 3 f o r a p i l eh a v i n g L id = 4 0 , Pp/Ps = 1 .60 ( . typ i ca l o f conc r e t e p i l e s )a n d b e i n g e m b e d d e d i n s o il M o d e l A w i t h Vs = 0 . 4 0 a n d13 = 0 . 0 5 . T h e f o l l o w i n g t r e n d s a r e w o r t h y o f n o t e i n F i g 3 .

1 . T h e p r e s e n c e o f t h e p i l e a n d t h e v a l u e o f Ep/Es d on o t h a v e a n y i n f l u e n c e o n t h e f i r s t r e s o n a n t f r e q u e n c y ,w h i c h p r a c t i c a l l y c o i n c i d e s w i t h f~ , t h e f u n d a m e n t a ls h e ar f r e q u e n c y o f th e u n p e r t u r b e d s o il s t r at u m .T h e r e f o r e , i n t h e s e q u e l , f ] is u s e d t o d e n o t e b o t hf r e q u e n c i e s , i n d i s c r i m i n a t e l y .

L ' d = 4 0

A u

2 0

10 > , 4 . : o o ) ' o . . . .' o o , . . . .. . . ., . :~ ' . ~ . ~ i

2 4

I I'X/ \A / \I / A ' , i / I '

2 4

I u

o s

F r e e F ~ e l d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o ,

o o ~ .. 1 0Ep/Es='~% ~ o . o s

Figure 3.

ii L I , , I L I , , * L , ` , , I0 2 4

fJ///////

//

/// . ~ " ~ .

2 4

Kin ema tic in teract ion ef fects: in f luence o f Ep/Es (L id = 40; soil Mod el A; ~Js = 0 .05 , vs = 0 .40 , Pp/Ps = 1.60 )

8 4 Soil Dy nam ics and Earthqu ake Engineering, 1984, Vol. 3 , No. 2


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2 . U p t o a f r e q u e n c y o f a b o u t 1 .5 0 f ] , p i le s o f a ll re l a ti v es t i f f n e s s e s a p p e a r t o e s s e n t i a l l y f o l l o w t h e m o v e m e n to f t h e g r o u n d ; h e n c e , t h e i r p r e s e n c e h a s n o p r a c t i c a le f f e c t o n t h e s e i sm i c m o t i o n a t g r o u n d - s u r f a c e l e v e l.3 . O n t h e o t h e r h a n d , a t h i g h e r f r e q u e n c i e s e v e n p r a c t i-ca l ly f l ex ib le p i l e s (Ep/Es a s l o w a s 2 9 0 ) m a y n o t b ea b l e t o f o l l o w t h e w a v y m o v e m e n t s o f t h e f r e e -f i e lda n d m a y t h e r e b y e x p e r i e n c e c o n s i d e r a b ly r e d u c e dd e f o r m a t i o n s . T h i s f i l t e r i n g e f f e c t i s s u b s t a n t i a l f o rs t i f fe r p i l e s (Ep /Es > 2 0 0 0 0 ) , t o t h e p o i n t t h a t a f t e rt h e s e c o n d n a t u r a l f r e q u e n c y , t h e p i le s e e m s t orem a in e s sen t i a l ly s t i l l, whi le the f ree - f i e ld so i l m assm o v e s c o n s i d e r a b l y . T h i s i s i n a g r e e m e n t w i t h t h ea c t u a l e a r t h q u a k e o b s e r v a t i o n s o f t h e r e s p o n s e o f ap i le f o u n d a t i o n , a s r e p o r t e d b y O t h a et al. 31 a n dTa j im i . 9

4 . A s a l re a d y m e n t i o n e d , a ro t a t i o n a l c o m p o n e n t o fm o t i o n d e v e l o p e d a t t h e h e a d o f a p i le i n a d d i t i o n t ot h e t r a n s l a t io n a l o n e ( t h i s c o m p o n e n t i s n o t p r e s e n ti n t h e f r e e - f ie l d s u rf a c e m o t i o n ) . T h e r o t a t i o n e x h i b i t ss e v e ra l p e a k s a t t h e n a t u r a l s h e a r f r e q u e n c i e s o f th edepos i t . Note tha t fo r re la t ive ly sof t p i le s the peakso f r o t a t i o n i n c r ea s e s u b s t a n t ia l l y a t t h e h i g h e r n a t u r a lf r e q u e n c i e s ; t h e o p p o s i t e i s t r u e f o r r e l a t i v e ly s t i f fp i l e s , in accord wi th the i r sm a l le r d i sp lacem ents .

T h e e f f e c t o f Ep/Es o n t h e k i n e m a t i c r e s p o n s e o f p i le se m b e d d e d i n t h e o t h e r t w o s o il m o d e l s , B a n d C , i s o f as im i la r na tu re .Ef fec t o f so i l p ro f i le

T y p i c a l s i m i l a r it i es a n d d i f f e r e n c e s i n k i n e m a t i c i n te r a c -t i o n d u e t o d i f f e r e n c e s in t h e t y p e o f so i l p r o f i l e m a y b e

Seismic response o f end-beating s ingle pi les: G. Gazetasseen in F ig . 4 . Th i s f igure com pare s the va r ia t ion o f thea m p l i f i c a t i o n a n d i n t e r a c t i o n f a c t o r s v e r s u s f/f~, fo r a p i l ee m b e d d e d i n e a c h o f t h e t h r e e s o i l M o d e l s A , B a n d C . I nal l three cases : L i d = 40 , t~ = 0.40 a nd /3 = 0.05 . Ho w-ever , E p / E s i s e q u a l t o 1 4 5 0 0 0 i n t h e t w o i n h o m o g e n e o u sM o d e l s, A a n d B , a n d e q u a l t o 5 0 0 0 0 i n t h e h o m o g e n e o u sM o d e l C . T h i s c h o i c e o f m o d u l i w a s m a d e i n o r d e r t o c o m -pare p i l e -so i l sys tem s wi th s im i la r ove ra l l s t i f fnes s . Thethree curves in each graph of F ig . 4 s e rve to qua l i t a t ive ly ,r a t h e r t h a n q u a n t i t a t iv e l y , h i g h li g h t t h e f o l l o w i n g t r e n d sa s s o c i a t e d w i t h e a c h m o d e l :

1 . In a l l ca ses , the s econd re sonance occurs a t a f re -q u e n c y a p p r o x i m a t e l y e q u a l t o f2 , t h e s e c o n d n a t u r als h e a r f r e q u e n c y o f t h e r e s p e c ti v e u n p e r t u r b e d s t r a t u m .F i g u r e 4 , in a g r e e m e n t w i t h T a b l e 1 , s h o w s t h a t f 2g e t s c l o s e r t o f ~ a s th e d e g r e e o f s o il i n h o m o g e n e i t yi n c r e as e s ( f r o m C to A ) .2 . A s u b s t a n t i a l in c r e a se i s o b s e r v e d in t h e p e a k r o t a t i o na t f = f l a s t h e d e g r e e o f s o i l i n h o m o g e n e i t y i n c r e a s es .B y c o n t r a s t , t h e t w o r e s o n a n t p e a k s o f h o r i z o n t a l d i s-p l a c e m e n t a s w e l l a s t h e s e c o n d r e s o n a n t p e a k o fro ta t ion a re l e s s s ensi t ive to d i f fe rences in the so i lp rof i l e .

3 . T h e h o r i z o n t a l i n t e r a c t i o n f u n c t i o n s I u = l u ( f / f 0r e v e a l t h a t , i n t h e f r e q u e n c y r a n g e s t u d i e d ( f / f l~


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Seismic response o end-b ear ing s ingle pi les: G. Gazetas

Ep/Es=]45 OO [ ~

20

Au ~c

2 4

A@ 0,5

AI I) \

2 4

I u0 .5

Figure 5.

L / d F r e e f i e l d

_ ' ;o / \

"\ , , . / / \, , , t l l l l [ l , l , , i , , , l l2 4

f / f )

oJL/d=1O~-.../ \ f -.,..I , / \ / \

o.o, I ,o \ / \/ / jL -o 2 I

f / f ~K inema tic in teract ion ef fects: in f luence o f Li d (E p/Es = 145 000; soi l Mod el B; (is = 0 .05, us = 0 .40, Pp/Ps = 1.60)

Effect o f s lenderness rat io LidF i g u re 5 p o r t r a y s t h e e f f e c t o f L i d o n t h e k i n e m a t i c

i n t e r a c t i o n / a m p l i f i c a t i o n f o r p i le s w i t h Ep/Es = 145 00 0 i ns o il M o d e l B ( m o d u l u s p r o p o r t i o n a l t o U z ) . T h r e e v a l u es o fL id a r e e x a m i n e d . 1 0 , 2 0 a n d 4 0 , w h i c h c o v e r a f a i rl y w i d er ange o f s l ende r nes s r a t i os o f ac t ua l p i l e s .

I t i s ev i den t t ha t L i d h a s a p r o f o u n d e f f e c t o n p i l e - h e a dr o t a t i on , a t a l l f r equenc i es . On t he o t he r hand , i ts i n f l uenceo n d i s p l a c e m e n t s b e c o m e s a p p r e c i a b l e o n l y a t f r e q u e n c i e sg r e a t e r t h a n a b o u t 1 . 50 f ] , w h e n t h e s h o r t e r p i l es p r o d u c es t r onge r f i l t e r i ng e f f ec t s .

T h e r o t a t i o n a m p l i t u d e s a t f r e q u e n c i e s f < j q a r e u n d e r -s t a n d a b l y h i g h e r f o r t h e s h o r t e r p i le s w h i c h e x p e r i e n c ea b o u t t h e s a m e r e l at i v e d i sp l a c e m e n t a s t h e l o n g e r p i le s b u to v e r a s h o r t e r l e n g t h . H o w e v e r , a t f ~ f 2 shor t p i l e s expe r i -ence r e l a t i ve l y sma l l r o t a t i ons , cons i s t en t w i t h t he i r sma l l( due t o f i l t e r i ng) d i sp l acement s .Syn thes is o f resul ts fo r 1 ,

T h e v a l u e o f 1, f r om a l l ca ses s t ud i ed a r e r ep l o t t ed i nF i g . 6 , f o r each o f t he t h r ee so i l p r o f i l e s . F i gur e 6a app l i e st o P r of i l e A and por t r ays t he va r i a t i on o f lu ver sus t hed i m e n s i o n l e s s f r e q u e n c y p a r a m e t e r

f (E p] O'10 (Z ) "O'40"

for Prof i le B, F ig. 6b plots Iu a s a f u n c t i o n o ff " g p ' '1 6 ( L 1 - ' 3 5

(4 )

( 5 )

F i na l l y , f o r P r of i l e C , F i g . 6c shows lu a s a f u n c t i o n o fF# oFC = -~1" \-E-# " d ) (6 )

De t e r mi n ed by t r i a l and e r r o r so t ha t t he r e su lt s f a l lw i t h i n a r e l a ti v e l y n a r r o w b a n d , e a c h o f t h e s e d i m e n s i o n l e s sp a r a m e t e r s e n c o m p a s s e s t h e t h r e e k e y n o r m a l i z e d p r o b l e mp a r a m e t e r s , Ep/Es , L id an d f / fF T h e s c a t t e r o f e a c h s e t o f' d a t a p o i n t s ' a r o u n d t h e r e s p e c t iv e ' a v e r a g e ' c u rv e i s v e r ys m a l l f o r p r a c t ic a l a p p l i c a t i o n s . T h e r e f o r e , t h e v a l u e o f t h ek i n e m a t i c d i s p l a c e m e n t i n t e r a c t i o n f a c t o r , lu , m a y b e e s t i -m a t e d r e a d i l y a n d w i t h e n g i n e e r i n g a c c u r a c y f r o m t h e' a v e r a g e ' c u r v e o f t h e p e r t i n e n t s o il m o d e l .

I t i s n o t e d t h a t t h e ' a v e r a g e ' lu cur ves o f F i g . 6 a r e o f as i m i l a r n a t u r e t o t h o s e p r o p o s e d b y E l s a b e e et al. 32 f o r

8 6 Soil Dy nam ics and Earthq uake Engineering, 1984, Vol. 3 , No. 2


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1

I u

Seismic response of end-bearing single piles: G. Gazetas%/%

2 9 0 4 0-'( 2 9 0 0 0 4 0/ ~ 1 4 5 0 0 0 4 0

1 4 5 0 0 0 ! 00 1 4 5 0 0 0 2 0

- - " a v e r a g e " c u r v e

xu & x , ~ 0 x J( 0

1 I I1 2 3

1

I u

o A oo 9

o 0 oxx

I I I I I2 4 6 8 1 0%

1

I u

Figure 6.H)-(6))

E p / E s L / dA S O 4 0 5 0 0 4 0* ~ a ~ t ~ ~ so lo ,oo 5 0 0 0 0 1 S

" .

~' A ~ n ....o

0 A

I I I 2 0 4 0

%K inemat ic in te ract ion fac tors, lu , in te rms o f the d imens ionless f requen cy parameters FA, F B and F c ( equat ions

Soil Dyn am ics and Earth quak e Engineering, 1984, Vol. 3, No. 2 8 7


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S e i s m i c r e s p o n s e o e n d - b e a r i n g s i n g le p i l es : G . G a z e t a sembedded c i rcu la r founda t ions . The prac t i ca l s ign i f i canceof such curves i s appa ren t : by mu l t ip ly ing a g iven f ree -f i e ld des ign re sponse spec t rum wi th the appropr i a t e i n t e r -ac t ion curve , one ma y de r ive the des ign re spon se spec t rumtha t must be input a t t he base of a s t ruc ture o n p i le sfounda t ion . A t the same t ime , however , t he ro t a t iona lc o m p o n e n t o f t h e i n p u t m o t i o n , e x p r e s sed th r o u g h I0 ,sh o u l d n o t b e n e g l e c te d .P A R A M E T R I C R E S U L T S : D Y N A M I C I M P E D A N C E SThe s t eady-s t a t e re sponse of p i l e s t o l a t e ra l dynamic forcest ransmi t t ed f rom the supe r - s t ruc ture can be eas i ly com-p u t e d o n c e t h e t h r e e d y n a m i c i m p e d a n c e s , g (H H , ~MMan dJtrHm, assoc ia t ed wi th swaying , rock ing and coupled sway-ing- rock ing osc i l l a t ions , r e spec t ive ly , have been de r ived .These imp edances a re ra t ios be twee n exc i t i ng force (ormoment ) and re su l t i ng p i l e -head d i sp lacement (o r ro t a t ion) ,a n d t h e y a r e c o m p l e x f u n c t i o n s o f t h e f r e q u e n c y 03 = 2 rr f.In th is pape r we express each imped ance a s, ~r = K ( k + 2 i D ) (7 )in which K i s t he s t a t i c s t i f fness in the pa r t i cu la r mode ,k = k(03) the dyna mic s t i f fness coe f f i c i en t , and D = D(03)the ' e f fec t ive ' damping ra t io o f t he sy s t em. D wi ll i n gene ra lc o n s i s t o f t w o c o m p o n e n t s : o n e , w h i c h i s f r e q u e n c y -indepe nden t and a ri se s f rom the presence of hys t e re t i cdamping in the so i l, and ano the r one , wh ich inc reases w i thf r e q u e n c y a n d e x p r es se s t h e a m o u n t o f r a d ia t io n d a m p i n gin the sys t em.In prac t i ce , t he ma jor i ty o f p i l e s sub jec t ed to l a t e ra lh e a d l o a d i n g are f l e x i b l e , i n t he sense tha t t hey do no tde fo rm over the i r en t i re l eng th . Ins t ead , p il e de forma t ionsand st resses reduce to negl igible proport ions wi thin a dis-t ance l a ( o n t h e o r d e r o f 1 0 t o 15 d i a m e t e r s ) f r o m t h e g r o u n dsur face . We name la: th e ac t i ve pi le length. With f lexiblepiles, L > la , the exact pi le length L is an i rre levant para-me te r , hav ing no in f luence on the re sponse .

T o k e e p t h e n u m b e r o f in d e p e n d e n t p r o b l e m p a r a m e t e r ssma ll (w i tho ut se r ious ly re s t r i c ting the range of p rac t ica lva l id i ty o f t he re su l t s ) on ly f l ex ib l e p i le s a re s tud ied in th i ssec t ion . L i s kep t cons t an t , equa l t o 40d . Most o f t he pre -sented resul ts , however, wi l l a lso be appl icable wi th largerand even wi th somew ha t sma l l er va lues of L , a s l ong asL ~>ta.Table 2 presen t s s imple express ions for e s t ima t ing theact ive length of pi les in each of the three soi l profi les. Theseexpress ions were de r ived f rom the re su l t s o f t he f in i t e -e l ement ana lyses and a re appl i cab le w i th reasonable accur -acy for a fa i r ly w ide range of f requenc ies . A t dep ths be low

z = l a a head- loaded p il e would exper i enc e de for ma t io nswhich a re w i th in 5% of those at t he top ; fu r the rm ore ,removing th i s l ower ( id ling) pa r t o f t he p i le wou ld on lyins ign i f ican t ly a f fec t i ts h ead impedances .St a t i c s t i f f nes ses

Table 3 g ives s imple express ions for t he ( l eng th- inde -pendent ) s t a t i c s t i f fnesses K H H , K M M a n d K H M assoc ia t edwi th each of t he th ree s tud ied so i l Model s. These expres-s ions were ob ta ined by f i t t i ng the s t a t i c f in i t e -e l ementresu l ts and the i r exc e l l en t accuracy has been ve r i fi ed byc o m p a r i n g w i th t h e e x p r e s s io n s o f B l a n e y et al . 4 for t hecase of hom ogen eous so il and of Rand olph 16 for so i l w i thm o d u l u s p r o p o r t i o n a l t o d e p t h . T h e s i m p l i c i ty o f t h eseformulae makes them pa r t i cu la r ly a t t r ac t ive for des ignc o m p u t a t i o n s .

Table 2. Active length o f pile under lateral dynamic loadblg at theto pActive length Ia

Soil mod el Expression Typical range

I p] 1'6A 3.2d \Tss ] 6a - 15d(E = Esz/d)B 3 2d lE-P '~ . . . 6 a -1 7 d

C [ E \ ~/s 8 d - 20 d(E = Es) 3.3d ~ s )

Table 3. Expressions jbr static st iJJhess o f f lexible piles em bed dedin three soil profilesK H H K M M K H MSoil model dEs c ~ ~ s c

A 0.6 0 (Eff_~).. . / E \ . . . . [Ep,O.oo( E = E sz / d ) , ~, 0.14 [-~-sP -- 0.17 ~- --~t E \0.58 I E \o.s~B 0.79~_~p i ' E \o.7~( E = E s x / ~ 7 " ~ , _ ~ , 0"15 ~ s P ~ -- 0'24 ~'~-sP

C I E \ . .. . [ E \o.,s [Ep~O.SO(E = E s, 1"08 ~-~sP 0.16 ~..~.sP -- 0.22~--~-~~

D y n a m i c s t if f n e ss c o e f f ic i e n t s a n d e f f e c t i v e d a m p i n g r a t io sFigures 7 , 8 and 9 por t ray the va r i a t ion wi th f requencyof the th ree pa i rs o f s ti f fness coe f f i c i en t s and dam ping

ra t ios , namely ( k H i 4 , D H H ) , ( k M M , D M M ) an d (kHM, DI4M),for each of t he th ree so i l Mode ls , A , B and C. Seve ra l va lueso f E p / E s cover ing a ra the r ex t rem e range of p rac t i ca l s i tua -t ions and a hys t e re t i c damping ra t io in the so i l equa l t o0 .05 a re cons ide red . The re su l t s a re p lo t t ed ve rsus f / f bw h e r e f l = t h e f u n d a m e n t a l sh e a r f r e q u e n c y o f t h e u n p e r -tu rbed so il dep os i t , g iven b y th e express ions of Table 1 forthe con s ide red prof i l e s .Aga in , t he choice o f f l a s a normal i z ing pa ramete r i shard ly arbi t rary. Even in this case of a head -loade d pi le ,F igs . 7 -9 revea l t ha t r e sonance pheno me na occur a lmostprecisely a t f = f l , fo r a l l soi l profi les and a l l E p / E s ra t ios.A t re sona nce , t he d ynam ic s t i ffness coe f f i c i en t s expe r i en cea dip , w hich is especia l ly sharp fo r st i f f pi les ( i .e . wi th largeE p / E s ra t ios) . Moreover , be low f l t he e f fec t ive dampingra t ios a tt a in sma ll and f requ ency - indep ende nt va lues whichre f l ec t t he ma te r i a l (hys t e re t i c ) damping in the sys t em. A tsuch low f requenc ies , no rad ia t ion damping i s p resen t s incene i the r sur face n or bo dy rad ia l ly -propaga t ing waves can bephys ica l ly c rea t ed in the so il s t ra tum. But a s soon as fexceeds f~ , damping ra t ios s t a r t inc reas ing wi th fd u e to thedeve lop ing rad ia t ion dam ping .I t i s a lso wo r th no t ing a few in t e res t ing t rends in F igs .7 - 9 .

1 . The va r i a t ion of t he dynamic s t i f fness coe f f i c i en t swi th f requ ency is no t d ram at i c , exce pt pe rhaps fork t t H of re la t ively soft pi les ( E p / E s ~ 1 5 0 0 ) i n t h einhom ogen eous so il p rof i l e s A and B. Pa r t i cu la r ly

8 8 S o i l D y n a m i c s a n d E a r t h q u a k e E n g i n e e r in g , 1 9 8 4 , V o l. 3 , N o . 2


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~HH

L > ~ ' a

E o / E s

- - _ D

E p / e s /- // e

/ 4,p

< ~=.. . . . ~ ' , , ,

d /.... j / /]]/ /o~ / , /] / / /

1 ' /s /Yo . . . . ~ . . . . , ~ ,

k ~ D ~ / 7o u

o., J y. . . . L . . . . . ~ * . . . . ~ ' ' ' " ' -|/II f/fl

Figure 7 . Dynam ic s t i ffness coe f f ic ien ts and dam pingratios for f l exib le piles em bed ded in soil Mo del A ([Js = O.0 5;u = 0.40, Pp/Ps = 1.60)

i n s e n si t iv e t o v a r i a t i o n s in f r e q u e n c y a r e t h e r o c k i n gc o e f f i c i e n t s , kMM , rega rd le s s o f so i l p rof i l e . Assum ingc o n s t a n t ( f r e q u e n c y - i n d e p e n d e n t ) d y n a m i c s ti f fn e s se sf o r d e s ig n p u r p o s e s w i ll i n t r o d u c e o n l y s m a l l e r r o r s inm o s t a c t u a l s i t u a t i o n s .2 . T h e e f f e c t i v e d a m p i n g r a t i o s a re c o n s i d e r a b l y l o w e ri n t h e r o c k i n g t h a n t h e s w a y i n g m o d e o f v i b r a t i o n .This i s t rue fo r a l l f requenc ies and i s cons i s ten t wi ththe w e l l kn ow n resu l t s fo r sha l low r ig id fou nda t ions , a33 . T h e t y p e o f v a r i a ti o n o f s o i l m o d u l u s w i t h d e p t h h a sa p r o f o u n d e f f e c t o n t h e d a m p i n g r a ti o s , es p e c i a ll y a th i g h f r e q u e n c i e s . F o r i n s t a n c e , c o m p a r e t h e r e s u l tsf o r M o d e l s A a n d C . I n A , a s E p /E s inc reases , the ra teo f i n c r e a s e o f DHH w i t h f r e q u e n c y d r o p s c o n s i d e r -a b l y ; t h u s , t h e s o f t e s t p i le s e x h i b i t t h e h i g h e s t DHHva lues a t f / f] ~ : 5 . The oppos i t e i s t rue wi th Mode l C .

The d im ens ion les s g raphs in F igs . 7 -9 a long wi th thef o r m u l a e i n T a b l e s 1 - 3 m a k e i t p o s s i b le t o r e a d i l y e v a l u a t es t a t i c / d y n a m i c s t if f n e s se s a n d d a m p i n g r a t i o s o f p i le s in av a r i e t y o f a c tu a l s i t u a ti o n s , w i t h o u t t h e n e e d o f a c o m p u t e r .C O M B I N E D K I N E M A T I C - I N E R T I A L I N T E R A C T I O N

A n e x a m p l e o f a c o m p l e t e s o i l - p i l e - s tr u c t u r e i n t e r a c t i o na n a l y si s , in w h i c h t h e e f f e c t s o f b o t h k i n e m a t i c a n d i n e r ti a lin te rac t ion a re com bined , i s p resen ted in F ig . 10 . Thee x a m p l e r e f e r s t o t h e s t e a d y - s t a t e r es p o n s e a t o p a p i le

Seismic response o end -bea ring single piles: G. Gazetase m b e d d e d i n a l i n e a r l y i n h o m o g e n e o u s d e p o s i t ( M o d e l A )a n d s u p p o r t i n g a m a s s m , l o c a t e d j u s t a b o v e t h e g r o u n dsur face . Th i s is the s im ples t pos s ib le m od e l o f a supe r -s t r u c t u r e , w h i c h m a y n e v e r t h e l e s s h e lp h i g h l i g h t k e y a s p e c tso f t h e r e s p o n s e . T h e n a t u r a l f r e q u e n c y o f s u c h a s u p e r -s t ruc ture i s g iven by

i n w h i c h K h i s t h e s t a t i c f o r c e - d i s p l a c e m e n t r a t i o o f a p i l es u b j e c t e d t o a s o le h o r i z o n t a l f o r c e a n d f r e e to r o t a t e . K h isre la ted to the s t a t i c s t i f fnes ses o f Tab le 3 :

K~MK h = K H H - - - - ( 9 )K M MT h e a p p r o x i m a t i o n i n e q u a t i o n ( 8 ) c o n s i s t s i n t h a t t h es t a ti c i n s t e a d o f t h e d y n a m i c f o r c e - d i s p l a c e m e n t r a t i o i sused . Th i s i s jus t i f i ed in v iew of the fa c t tha t kHH, kHM a n dkM M a re re la t ive ly insens i t ive to f requency (F igs . 7 -9) .M o r e o v e r , i n F i g . 1 0 , f s t f r o m e q u a t i o n ( 8 ) i s m e r e l y u s e da s a c o n v e n i e n t n o r m a l i z i n g p a r a m e t e r ; i t w a s n o t u s e d i nt h e a n a l y s e s , w h i c h a r e ' e x a c t ' .F i g u r e 1 0 i ll u s t ra t e s t h e e f f e c t o f t h e r e l a t iv e f r e q u e n c yf a c t o r f s t / f ] , f o r a p i l e w i t h L / d = 4 0 a n d E p /E s = 2 9 0 0 0 .A f e w t r e nd s a r e w o r t h y o f n o t e .

T h e r e s p o n s e o f t h e s o i l - p i l e - s t r u c t u r e s y s t e m e x h i b i t sr e s o n a n t p e a k s a t t w o d i f f e r e n t s e t s o f f r e q u e n c i e s : th e

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Soil Dyn am ics and Ear thqua ke Engineering, 1 984, VoL 3, No. 2 8 9


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f ir s t s e t c o r r e s p o n d s t o t h e n a t u r a l f r e q u e n c i e s o f th e s u p e r -s t r uc t u r e . T h i s i s i n agr eeme nt w i t h t he r e su lt s o f o t he rau t hor s .Z , 9 M o r eover , t he h i ghes t peak s o f t he d i sp l acem enti n t e r a c t i o n c u r v e s , Iu ( f / f l : f sT / f l ) , a l ways o ccur a t f ~ f s t,t h e n a t u r a l f r e q u e n c y o f t h e s u p e r - s t ru c t u r e . H e n c e , a s t h em a s s m i n c r e a se s , t h e h i g h e s t o f t h e p e a k s t e n d s t o o c c u ra t l o w e r f r e q u e n c i e s o f e x c i t a t i o n w h e r e r a d i a t i o n d a m p i n gm a y b e p a r t i c u l a r l y s m a l l ( Fi g s. 7 - 9 ) : s u c h p e a k s m a y t h u sbe qu i t e sha r p . On t he con t r a r y , a r e l a t i ve l y sma l l mass( large f s r / f l ) m a y b e a s s o c i a t e d w i t h v e r y l a r g e d a m p i n g a n dv e r y f la t r e s o n a n t p e a k s .

T he r e su l t s f o r o t he r p i l e s and so i l p r o f i l e s (. no t shownh e r e f o r l a c k o f s p a c e ) a re i n q u a l it a t iv e a g r e e m e n t w i t ht hose i n F i g . 10 .

A N A L Y S I S O F A C A S E - H I ST O R YS in c e p l a ce a n d t i m e o f e a r t h q u a k e o c c u r r e n c e c a n n o t b ea c c u r a t e l y p r e d e t e r m i n e d , t h e r e i s g e n e r a l l y o n l y a l i m i t e da m o u n t o f r e c o r d e d f i e ld i n f o r m a t i o n w i t h w h i c h t h ea d e q u a c y o f d e v e l o p e d a n a l y t i c a l m e t h o d s c a n b e j u d g e d .M o r e o v e r , t h e s e is m i c r e s p o n s e o f s t r u c t u r e s f o u n d e d o np i l e s e m b e d d e d i n s o f t s u b s o i l i s a c o m p l i c a t e d p r o b l e m a n di t is d i f f i cu l t t o c l ea r l y de l i nea t e t he r o l e o f p i l e - so i l i n t e r -a c t i o n o n t h e r e s p o n s e . F o r t h i s re a s o n , a w e l l d o c u m e n t e dc a s e h i s t o r y p r e s e n t e d b y O h t a et al . 31 i s o f s i gn i f i cance a ndi s s t ud i ed he r e i n .

T h e v i b r a t i o n o f a n l 1 - s to r e y a p a r t m e n t b u i l d in g s u p -p o r t e d o n p i le s w a s r e c o r d e d d u r i n g s e v e n e a r t h q u a k e s .T h e b u i l d in g c o n s is t s o f r e i n f o r c e d c o n c r e t e - a n d - s te e lc o m p o s i t e f r a m e a n d r is e s n e a r l y 3 1 m a b o v e th e g r o u n d .T h e g r o u n d f l o o r i s o f t h e ' p y l o t i s ' t y p e a n d t h e r e i s n ob a s e m e n t . T h e p l a n a n d t w o c r o s s - s ec t i o n s o f th e b u i l d in ga r e shown i n F i g . 11 .

T he bu i l d i ng i s f ou nde d on cas t - i n - p l ace con c r e t e p i l es o fabo u t 25 m i n l eng t h and 1 .4 m i n d i am e t e r . T w o 2 .3 m-h i gh f o o t i n g b e a m s r u n n i n g i n t h e l o n g i t u d in a l d i r e c t io n

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t r a n s f e r t h e l o a d s o f t h e s u p e r - s t r u c t u r e o n t h e p i l e s . T h eso i l depos i t i s o f a l luv ia l o r ig in and cons i s t s o f a l t e rna t ingl a y e r s o f sa n d a n d s il t. B e l o w a b o u t 2 5 m f r o m t h e g r o u n dsur face the so i l beco m e s dense g rave l ly -sand and s t i f f c lay .P r i o r t o r e c o r d i n g t h e r e p o r t e d g r o u n d m o t i o n s s e v er a l f i e ldt e s t s h a d b e e n c a r r i e d o u t t o o b t a i n t h e d y n a m i c so i lp r o p e r t i e s a t v a r i o u s d e p t h s . T h e s e t e s t s i n c l u d e d S H w a v ev e l o c i t y m e a s u r e m e n t s w i t h t h e s o - ca l le d W e l l - S h o o t i n gm eth od and fo rced v ibra t io n t e s t s on p i l e s. 31 F igure 11p o r t r a y s t h e S H - w a v e v e l o c i t y p r o f i l e .Charac te r i s t i c s o f the s even recor ded s e i sm ic even t s a reg iven in Tab le 4 . These even t s can be roughly c la s s i f i ed in totw o ca tegor ie s : sm a l l -m agni tud e , nea r -d i s tan t even t s (ML ~< 5a n d R < 4 0 k m ) ; a n d m o d e r a t e a n d l a rg e - m a g n it u d e f a r-d i s t a n t e v e n ts ( M / > 5 .5 a n d R > 6 5 k m ) .

T h e r e s p o n s e o f t h e s o i l - p il e - s t r u c t u r e s y s t e m d u r i n gt h e se g r o u n d s h o c k s w a s m o n i t o r e d b y m e a n s o f 27 a c c e le r o -m e t e r s , e i g h t d i s p l a c e m e n t - m e t e r s , f o u r e a r t h - p r e s s u r eg a u g e s a n d t w o p o r e w a t e r p r e s s u r e g a u g e s . A s i n d i c a t e d i nF i g . 11 , t h e a c c e l e r o m e t e r s w e r e p l a c e d a l o n g t h r e e d i f f e r e n tve r t i ca l axes : on the p i l e -bu i ld ing ax i s ; on the ax i s 5 m aw ayf r o m t h e p i l e ( t o b e c a l l e d ' n e a r b y - s o i l ' a x i s ) ; a n d o n a n a x i s35 m away f rom the p i l e (which e s sen t i a l ly i s a ' f ree - f i e ld 'a x i s ). A s u m m a r y o f t h e r e c o r d e d r e s u l t s is p r e s e n t e d b e l o w .

F i g u r e 1 2 a p o r t r a y s t h e d i s t r i b u t i o n o f th e p e a k a b s o l u t ev a l ue s o f a c c e l e r a t i o n s r e c o r d e d a l o n g t h e t h r e e a x e s d u r i n g

Seismic response of end-bearing single piles: G. Gazetast h e s e v e n e a r t h q u a k e s . O f p a r t i c u l a r i n t e r e s t i n t h i s p a p e ri s t h e r e l a t i o n s h i p b e t w e e n t h e ' f r e e - f i e ld ' g r o u n d s u r f a c ea c c e l e r a t i o n , / ~ o , a n d t h e a c c e l e r a t i o n a t t h e h e a d o f t h ep i l e , / ~ p . A n y d i f f e r e n c e b e t w e e n / ~ o a n d / ~ t , i s t h e r e s u l t o fp i l e -so i l - s t ruc tu re in te rac t ion , a s d i s cus sed in p reced ings e c t io n s . F i g u r e 1 2 b p l o t s t h e r a t i o o f t h e r e s p e c t iv e p e a kv a l u e s , m a x / ~ p / m a x / ~ o , f o r e a c h e a r t h q u a k e . I t i s e v i d e n tt h a t f o r n e a r b y e v e n t s :

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Figure 12 . C ase h is tory : (a) d is t r ibu t ion w i th z o f the re -corded peak accelerations; (b) variation with epicentrald is tance o f the ra t io o f peak acce lerat ion a top th e p i le andat the f ree -f ieM ground sur face

Table 4. Ca sehistory: recorded earthquakes3~

EarthquakeNo. Year Name

MaximumEpicentral free-fieldd i s t a n c e F o c a l d e p t h acceleration(kin) (kin) Magnitude (gad)

1 1975 Central Chiba2 1976 Eastern Tok yo3 1976 Eastern Saitana4 1 9 7 6 E a s t e r n Y a m o n a s h i5 1978 Near lzu Ohshima6 1978 Of f Miyagi7 1978 Off Miyagi

45 70 4.6 10.20 40 4.2 28.640 70 4.8 22.065 20 5.5 27.4110 0 7.0 21.3400 50 6.7 13.4350 40 7.4 57.4

Soil Dyn am ics and Earth quak e Engineering, 1984, Vol. 3, No. 2 91


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Seismic response o end-bearing single piles: G. GazetasT hese va l ues a r e i n qua l i t a t i ve agr eement w i t h t he r e su l t so f t h is p a p e r . I n d e e d , a s a n t i c ip a t e d , t h e th r e e n e a r b ym o t i o n s ( n o t s h o w n h e r e ) a r e v e r y ri c h in h i g h - f r e q u e n c yc o m p o n e n t s ; 3 ] s u c h c o m p o n e n t s c o n t r i b u t e t h e m o s t t op e a k a c c e l e r a t io n . S i n c e k i n e m a t i c i n t e r a c t i o n f i lt e r s t h eh i g h f r e q u e n c y c o m p o n e n t s o f m o t i o n ( r e c a l l F i g . 6 ) , i t i sn a t u r a l t o h a v e p e a k a c c e l e r a t i o n s a t t h e p i l e h e a d s m a l l e rt han t hose a t t he f r ee f ie l d . Conver se l y , d i s t an t r ec or ds a r er i c h i n l o w a n d m e d i u m f r e q u e n c y c o m p o n e n t s w h i c h a r en o t a s m u c h i n f l u e n c e d b y t h e p r e s e n c e o f p i le s ; h e n c e , t h ep e a k a c c e l e r a t i o n s a t t h e f r e e f i el d a n d t h e p i le h e a d a r e o fa b o u t t h e s am e m a g n i t u d e .

F u r t he r m or e , F i g . 13 p l o t s t h e r a t i o { )'p ff .) 'o f t he F our i e rA m p l i t u d e S p e c t r a o f t h e p i l e- h e a d a n d t h e f r e e - fi e ld a c -c e l e r a t io n r e c o r d s . T h e p l o t s h o w s th a t , i n d e e d , ( 1 ) h i g hf r e q u e n c y c o m p o n e n t s o f t h e s e is m i c m o t i o n a re f i lt e re do u t b y t h e p i le - s o i l - s tr u c t u r e i n t e r a c t i o n ; (2 ) l o w f r e q u e n c yc o m p o n e n t s a re n o t a f f e c t e d b y t h e pi le o r s t r u c t u r e ; a n d( 3 ) c o m p o n e n t s i n t h e f r e q u e n c y r a n ge b e t w e e n t h e f u n d a -m e n t a l f r e q u e n c y o f t h e s o il s t r a t u m , f~ , a n d t h e f u n d a -m e n t a l f r e q u e n c y o f t h e s u p e r - s t r u c t u r e , l e t , a r e s u b s t a n t i a l lya m p l i f i e d d u e t o p i l e - s o i l - s t r u c t u r e i n t e r a c t i o n .

A l s o p l o t t e d i n F ig . 1 3 is a n a p p r o x i m a t e t h e o r e t i c a lcur ve de r i ved i n a s i mp l i f ied wa y on t he bas i s o f t he r e su l t so f t h is p a p e r , a s f o l l o w s : f i rs t , t h e s o il is a p p r o x i m a t e d a sa h o m o g e n e o u s 2 0 - m t h i c k s t r a t u m w i t h V s = 1 1 0 m / s ,vs = 0 . 4 0 , Ps = 1 .60 t / m 3 , E s ~- 5 4 M N / m 2 , a n d a f u n d a -m e n t a l s h e ar f r e q u e n c y f l = 1 1 0 / ( 4 x 2 0 ) ~ 1 .3 8 H z . F o rt he p i l e , Ep ~-22 00 0 M N / m 2 , d = 1 .4 m, and , e f f ec t i ve l y ,L = 2 0 m . T h e r e f o r e :

E p ~ L- - - 4 0 7 a nd - - - 1 4 . 3 ( 11 )E~ oS i nce t he spac i n g b e t w een p i l e s i s f a i r l y la r ge , s - - 8 .35 m ~ -6 d , o n e m a y e x p e c t o n l y s m a l l e r r o r s i n t h e a n a l y s i s d u e t op i l e - so i l - p i l e i n t e r ac t i o n e f f ec t s . ] ' u

T o a c c e s s t h e k i n e m a t i c i n t e r a c t i o n e f f e c t s , t h e d i m e n -s i o n l e s s f r e q u e n c y p a r a m e t e r F i s o b t a i n e d f r o m e q u a t i o n( 6) :

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Figure 13. Casehistory." comparison of actually recordedand approximately predicted ratio of Fourier-Amplitudespectra atop the pile and at the free-field ground surface

a n d 1u i s s c a l l e d f r o m F i g . 6 c . F o r t h e f r e q u e n c y r a n g e o fi n t e r e s t, 0 < f < 5 . 3 H z , F c < 6 . 2 a n d , a s a fi rs t a p p r o x i m a -t i o n , Iu ~ 1.

T h e a c t i v e p il e le n g t h i s o b t a i n e d f r o m T a b l e 2 :I a = 3 .3 x 1 .4 x ( 40 7) 1 '5 ~ - 15 .4 m ( 13)

wh i ch is l es s t ha n L = 20 m: henc e , t he p i le i s f l ex i b l e ,d e s p i te i ts v e r y l a rg e d i a m e t e r . T a b l e 3 m a y t h e n b e c o n -s u l t e d t o o b t a i n t h e s t a t i c s t i f f n e s s e s . F o r e x a m p l e :

KHH = 1 .08 X 1 .4 X 54 X ( 40 7) '21 ~ 288 M N / m 1 ,14)S i mi l a r l y :

KMM ~- 2 1 4 8 M N . m / ra d , KHM ~--470 M N / r a da n d , f r o m e q u a t i o n ( 9 ),

K h -- 1 8 5 M N / m .A v e r y s i m p l e m o d e l , a p p r o p r i a t e o n l y f o r p r e l i m i n a r y

d e s ig n c o m p u t a t i o n s , i s a d o p t e d f o r t h e s u p e r - s t r u c t u r e ,w h i c h i s a s s u m e d t o v i b r a te o n l y i n i ts f ir s t n a t u r a l m o d e . 34T h e c o r r e s p o n d i n g f ir s t n a t u r a l f r e q u e n c y , f s t , i s o b t a i n e da s f o l l o w s : f i rs t , w e e s t i m a t e r o u g h l y t h e f i x e d - b a se s m a l l-d e f o r m a t i o n n a t u r a l f r e q u e n c y , f * , u s in g a n em p i r ic a lf o r m u l a f r o m t h e l i t e r a tu r e : 3 s'3 6

f s~ ~ 24 H- S 4 ~ 24( 31 ) - 3 /4 _ 1 .83 Hz ( I 5 )B e c a u s e o f t h e f l e x i b i li t y o f th e s u p p o r t ( c o n s i s t i n g o f t h ep i l e - so i l s y s t e m ) f s t w i ll b e i n fe r i o r t o f ~ . U s i n g D u n k e r l e y ' srule 34

1 1 47r 2 1 4rr 2- - ~ - - - -- 0 .3 4 (16 )f ~ f * ~ ~ K . /me 1 . 8 3 ~ 1 8 5 0 0 0 / 1 9 5f r o m w h i c h : fs t ~ 1 .7 1 H z . I n e q u a t i o n ( 1 6 ) M e = 1 9 5 t o n sis t h e ' e f f e c t i v e ' m a s s o f b u i ld i n g p a r t i c i p a t i n g i n t h e f i rs tm o d e a n d c o r r e s p o n d i n g t o e a c h p i l e : 3 4

11 2

Me . . . . ( 1 7 )11X M ;4i = 1

i n w h i c h M i a n d ~b ar e t h e m a s s a n d m o d a l d i s p l a c e m e n to f t h e i t h f l o o r . A s s u m i n g a l i n ea r m o d e s h a p e a n d a u n i f o r md i s t r i b u t i o n o f m a s s a l o n g t h e h e i g h t l e a d s t o M e = 0 . 7 5 M ,34w h e r e M - - - 2 6 0 t o n s i s t h e t o t a l m a s s o f th e b u i l d i n g c a r r ie db y e a c h p i l e .

I n t h e i n t e re s t o f si m p l i c i t y a n d c o n s i s t e n c y w i t h t h eo v e r a ll a p p r o x i m a t i o n , t h e b u i l d i n g i s m o d e l e d a s a s in g l emass , m = 4rrZf~Ka, s u p p o r t e d o n a n u n r e s t r a i n e d - h e a dpi le wi th K h = 1 8 5 M N / m 2 , a n d l o c a t e d a b o v e t h e g r o u n ds u r f a c e . T h e a c t u a l h e i g h t o f t h e s u p e r - s t r u c t u r e , i m p o r t a n ta s i t i s f o r t he r e sponse o f i t s h i ghe r s t o r eys , has on l y am i n o r e f f e c t o n t h e r e s p o n s e a t t h e b a s e o f t h e s t r u c t u r e , 37w h i c h i s o f i n t e r e st h e r e . I n a d d i t i o n , t h e f r e q u e n c y v a r i a ti o no f d y n a m i c s t i f f n e s se s i s a l s o i g n o r e d .

T h u s , t h e p r o b l e m r e d u c e s t o c o m p u t i n g th e r e s p o n se o fa o n e - d e g r e e - o f - f r e e d o m s y s t e m h a v i n g m a s s m , s t i f fn e s s K aa n d f r e q u e n c y - d e p e n d e n t d a m p i n g ra t io D r , a n d b e in gs u b j e c t e d t o a h a r m o n i c f r e e - f i e l d b a s e m o t i o n ( s i n c eI u ~- 1) . D h is o b t a i n e d a t e a c h f r e q u e n c y , o n t h e b a si s o ft h e g r a p h s o f F i g . 9 , f o r Ep/Es = 4 0 7 . H o w e v e r , s i n c e t h ea c t u a l s o i l p r o f il e i s n o t u n d e r l a i n b y a r i gi d b e d r o c k b u tr a t h e r b y g r a v e l ly - s a n d a n d c l a y l a y e r s o f a n a v e r ag e v e l o c i t yV sr --- 4 0 0 m / s , r a d i a t i o n d a m p i n g w o u l d b e u n d e r p r e d i c t e d

9 2 Soil Dynamics and Earthquake Engineering, 1984, Vol. 3, No. 2


12/12

Seismic respon se o f end-bearing single piles: G. Gazetasfrom Fig. 9. Roesset 29 has suggested th at an a ddit ion alequivalent radiation dampin g

2 V 3"1 21 10 1. 38 0.24Dr . . . . (18)n V . f n40 0 f fshould be added to D h. At f = fst this additional dampingratio amounts to a substantial 14%.

The resulting approximate steady-state theoretical curvefor Up/Uo predicts very well the most impo rtan t trendsobserved in the ratio of the recorded Fourier AmplitudeSpectra. This gives confidence in the usefulness of theresults presented in the paper.

ACKNOWLEDGEMENTSThe author thanks Raman Krishnan and Alvaro Velez fortheir help in o btaining some of the n umerical results shownin the paper, and Ricardo Dobry for his stimul ating com-ments at the early stages of the presented work.

R EFER EN C ES1 Nair , K. Dynamic and earthquake forces on deep foundations,Performance of Deep Foundations ASTM STP 444, 1969,pp. 229-2632 Flores-Berrones, R. Behavior of end bearing piles under seismicforces, 9th Int. Conf. Soil Mech. Fdn. Engng., Proc. SpecialtySession 10, Tokyo 19773 Flores-Berrones, R. and Whitman, R. V. Seismic responses ofend-bearing piles, J. Geotech. Engng., ASCE 1982, 108 (GT4),5544 Blaney, G. W., Kausel, E. and Roesset, J. M. Dynamic stiffness

of piles, Proc. Second Int. Conf. Num. Meth. in Geomech.,ASCE Blickshurg, Virginia, 1976, pp. 1001-10125 Takemiya, H. and Yamada, Y. Layered soil-pile-structureinteraction, lnt. J. Earthq. Engng. Struet. Dyn. 1981, 9, 4376 Kagawa, T. and Kraft, L. M. Lateral load-deflection relation-ship of piles subjected to dynamic loadings, Soils and Founda-tions 1980, 20 (4), 197 Kagawa,T. and Kraft, L. M. Dynamic characteristics of literalload-deflection relationships of flexible piles, Int. J. Earthq.Engng. Struct. Dyn. 1981,9, 538 Harada, T., Kubo, K. and Katayama, T. Dynamic soil-structureinteraction by continuum formulation method, Report, b~st.Industrial Science, Tokyo 1981, 29 (5), 19 Tajimi, H. Seismic effects on piles, State-of-the-Art ReportNo. 2, Int. Conf . Soil Mech. Found. Engng., Prec. SpecialtySession 10 1977, pp. 15-2610 Wolf,J. P. and Von Arx, G. A. Travelling waves in a group ofpiles taking pile-soil-pile interaction into account, Prec. 7thWorld Conf. Earthq. Engng., lstanbul, Turkey 1980, pp. 467-47011 Kaynl i, A. M. Dynamic stiffness and seismic response of pilegroups, Research Report R82-03 Massachusetts Inst. of Tech.,198212 Nogami,T., Idriss, I. M., Power, M. and Chang, C. Y. Effect ofradiation damping on earthquake response of pile supportedoff-shore platforms, bit. J. o f Earthq. Engng. Struct. Dyn.1983, 11 (3)13 Prakash, S. Soil Dynamics, McGraw-Hill, 1981, Ch. 7

14 Dobry, R. and O'Rourke, M. J. Discussion on 'Seismic responseof end-bearing piles' by R. Flores-Bertones and R. V. Whitman,J. Geotech. Engng., ASCE 1983, 10915 Poulos, H. G. and Davis, E. H. Pile Foundation Analysis andDesign, John Wiley and Sons, 198016 Randolph, M. F. Response of flexible piles to literal loading,Geotechnique 1981,31 (2), 24717 Banerjee, P. K. and Davies, T. G. The linear behavior of axiallyand laterally loaded single piles embedded in nonhomogeneoussoils, Geotechnique 1978, 28 (3), 30918 Krishnan, R., Gazetas, G. and Velez, A. Static and dynamicliteral deflexion of piles in nonhomogeneous soil stratum,Geotechnique 1983, 33 (3)19 Velez, A., Gazetas, G. and Krishnan, R. Lateral dynamic re-sponse of constrained-head piles, J. Geotech. Engng., ASCE1983, 109 (8)20 Dobry, R., Vicente, E., O'Rourke, M. J. and Roesset, J. M.Horizontal stiffness and damping of single piles, J. Geotech.Engng., ASCE 1982, 108 (GT3), 43921 Novak, M. and El Sharnouby, B. Stiffness constants of singlepiles, J. Geotech. Engng., ASCE 1983, 109 (7), 96122 Kulemeyer, R. L. Static and dynamic literally loaded floatingpiles, J. Geotech. Engng., ASCE 1979, 105 (GT2), 28923 Kobori, T., Minai, R. and Babe, K. Dynamic behavior of aliterally loaded pile, Prec. Specialty Session 10, 9th lnt. Conf.Soil Mech. Found. Engng., Tokyo 197724 Roesset, J. M. and Angeildes, D. Dynamic stiffness of piles,Numerical Methods in Offshore Piling, Inst. of Civil Engineers,London, 22-23 May 197925 Nogami,T. and Novak, M. Resistance of soil to a horizontallyvibrating pile, Int. J. Earthq. Engng. Struct. D3,n. 1977, 5, 24926 Whitman, R. V. and Bielak, J. Foundations, Design of Earth-quake Resistant Structures, Rosenblueth, E., ed., John Wiley& Sons, 198027 Kausel , E. and Roesset, J. M. Soil-structure interact ion fornuclear containment structures, Prec. ASCE Power DivisionSpecialty Conf. Boulder, Colorado, 197428 Gazetas, G. and Dobry, R. Horizontal response of piles inlayered soils, J. Geotech. Engng., ASCE accepted for publi-

cation29 Roesset, J. M. Soi l amplification o f earthquakes, NumericalMethods #7 Geotech. Engng., Desai, C. S. and Christian, J. T.,eds., McGraw-Hill, 197730 Dobry, R., Oweis, I. and Urzua, A. Simplified procedures forestimating the fundamental period o f a soil profile, BulletinSeismological Soc. of Am. 1976, 66 (4), 129331 Ohta, T., Uchiyama, S., Niwa, M. and Ueno, K. Earthquakeresponse characteristics of structure with pile foundation onsoft subsoil layer and its simulation analysis, Prec. 7th WorldConf. on Earthq. Engng., lstanbul. Turkey 1980, pp. 467-47032 Elasbe, F. and Morray, J. P. Dynamic behavior of embeddedfoundations,Publication No. R 77-33, MIT 197733 Gazetas, G. Analysis of machine foundation vibrations: state ofthe art, hit. J. of Soil Dynamics and Earthq. Engng. 1983, 2(1),234 Veletsos, A. S. Dynamics of structure-foundation systems,Structural & Geotechnical Mech., Hall, W. J., ed., Prentice-Hall,197735 Tassios, T. P. and Gazetas, G. For a Predraft era New GreekAseismic Code, National Technical University of Athens,Greece, 197936 Tentative provisions for the development of seismic regulationsfor buildings, A TC-3-06 Report, Applied Technology Council,Pale Alto, California, 197837 Richart, F. E., Jr. and Chou, C.-S. Note on stiffness and damp-hag of pile systems, Prec. Specialty Session 10, 9th Int. Conf.Soil Mech. Found. Engng., Tokyo 1977

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1984-SDEE_Seismic Response of End-bearing Single Piles