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E L S E V I E R 0 2 6 7 - 7 2 6 1 ( 9 4 ) 0 0 0 5 5 - 7

Soil Dynamicsand Earthquake Engineering15 (1996) 211-222 1996 Elsevier Science LimitedPrinted in Gr eat Britain. All right s reserved

0267-7261/96/$15.00

S c a t t e r in g o f v e r t ic a ll y -i n c id e n t P - w a v e sb y a n e m b e d d e d p i l eF e n g J i & R o n a l d Y .S . P a k

Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, CO 80309-0428, USA(Rece ived 2 June 1994; accepted 18 Novem ber 1994)

An exac t theore t ica l formu la t ion is presented for the ana lysis of a th in-wal led p i leem bedd ed in an elastic half-space u nde r vertically-incident P-w ave excitation. In thefram ewo rk of three-dimensional elastodynamics an d a shell theory, the axisymme-trical wave-scattering prob lem is shown to be reducible to a set of Fredh olm bo und aryintegral equations. W ith the in corpo ration of the singular characteristics of the wave-induced contact load distributions into the solution scheme, a computationalbou nda ry element me thod is developed for a rigorous treatm ent of the seismic soil-structure interaction problem. Typical results for the dynamic co ntact load distribu-tions, displacements, complex-valued foundation input motion functions, andreson ant pile foun datio n response are included for direct engineering applications.K e y w o r d s : soi l -s t ruc ture in te rac t ion , sca t te r ing , input-mot ion func t ion , p i lefoundat ion , load-t ransfer , foundat ion dynamics, se ismic loading, boundarye lement , mixed bo und ary v a lue problem singular i ty .

1 I N T R O D U C T I O NI n s e i s m i c s o i l - s t r u c t u r e i n t e r a c t i o n m o d e l i n g , t h ed e t e r m i n a t i o n o f t h e d y n a m i c r e s p o n s e o f e m b e d d e df o u n d a t i o n s u n d e r d i f f e re n t w a v e i n c i d e n c e s is as u b j e c t o f p r a c t ic a l i m p o r t a n c e . W i t h a s o l u t i o n f o r t h eu n d e r l y i n g s c a t t e r i n g p r o b l e m a n d t h e d y n a m i c f o u n -d a t i o n c o m p l i a n c e , f o r i n s t a n c e , a c o m p l e t e b u i l d i n g -f o u n d a t i o n - s o i l i n t e r a c t i o n a n a l y s i s i n v o l v i n g d e e pf o u n d a t i o n s c a n b e f o r m u l a t e d b y m e a n s o f th e s u b -s t r u c t u r e m e t h o d . 6 O n e t h e s e i s m ic p e r f o r m a n c e o f p i le sa n d c a i s s o n s , o n l y l i m i t e d a n a l y t i c a l r e s u l t s a r e a v a i l a b l et o d a t e a s i n F l o r e s - B e r ro n e s a n d W h i t m a n , 2 W o l f a n dV o n A r x , 12 G a z e t a s , 3 M a m o o n e t a l . , 7 a n d K a y n i a a n dK a u s e l . 5 W h i l e t h e y h a v e p r o v i d e d c o n s i d e r a b l e i n s i g h tsi n t o t h e p r o b l e m , t h e s e s t u d ie s a l l i n v o l v e a p p r o x i -m a t i o n s o f o n e k i n d o r a n o t h e r o n t h e m o d e l i n g o f th ep i le , t h e s o il m e d i u m a n d t h e i r m e c h a n i c a l i n t e r a c t i o n .O w i n g t o v a r i o u s m a t h e m a t i c a l c o m p l e x i t i e s , e x a c tt r e a t m e n t s i n t h e c o n t e x t t h r e e - d i m e n s i o n a l e l a s t o d y -n a m i c s o f s u c h a s p e c t s a s t h e s p a t i a l l o a d - t r a n s f e r s , t h ei n t e r f a c i a l k i n e m a t i c a n d t r a c t i o n c o m p a t i b i l i t i e s , a n dt h e s i n g u l a r b e h a v i o r o f c o n t a c t l o a d s a t t h e p i l e - s o i li n t e r f a c e h a v e n o t y e t b e e n d e r i v e d . T o p r o v i d e a m o r ed e f i ni t iv e u n d e r s t a n d i n g a n d a p r a c t i c a l s o l u t io n f o rt h is c l as s o f d y n a m i c s o i l - f o u n d a t i o n i n t e r ac t i o n a n dw a v e - s c a t t e r i n g p r o b l e m s , a r i g o r o u s a n a l y s i s w h i c h c a nb e c a st i n a g e ne r a l c o m p u t a t i o n a l f r a m e w o r k w o u l d

211

c l e a r ly b e o f f u n d a m e n t a l i n t er e s t. A p a r t f r o m i ts i n tr i n -s i c t h e o r e t i c a l a n d p r a c t i c a l v a l u e s , s u c h k i n d s o f d e v e l -o p m e n t s c a n p r o v i d e a r a t i o n a l b a si s u p o n w h i c h th ev a l id i ty a n d a c c u r a c y o f c u r re n t a n d f u t u r e a p p r o x i m a t es o l u t i o n s c a n b e a s s e s s e d . I n t h i s p a p e r , a m a t h e m a t i c a lt r e a t m e n t i n t h i s c a t e g o r y f o r t h e a x i s y m m e t r i c p r o b l e mo f a t h i n - w a l l e d p i l e o f f i n i te l e n g t h u n d e r a v e r t i c a l l y -i n c i d e n t P - w a v e e x c i t a t i o n is p r e s e n te d . I n t h e c o n t e x t o ft h r e e - d i m e n s i o n a l e l a s t o d y n a m i c s a n d a s h e ll th e o r y , a ne x a c t f o r m u l a t i o n f o r t h e s e is m i c p r o b l e m i s s h o w n t o b ef e a s ib l e i n t h e f o r m o f a p a i r o f w e a k l y s i n g u l a r b o u n d a r yi n t e g r a l e q u a t i o n s o n t h e r e s u l t a n t i n t e r f a ci a l c o n t a c tl o a d d i s t r i b u t i o n s . B y v i r tu e o f a r a t i o n a l c o m p u t a t i o n a lp r o c e d u r e w h i c h c a n i n c o r p o r a t e t h e s i n g u l a r c h a r a c t e r -i s ti c s o f t h e s o l u t i o n , a c o m p r e h e n s i v e s e t o f n u m e r i c a lr e s u l t s p e r t a i n i n g t o t h e s e i s m i c s o i l - p i l e i n t e r a c t i o np r o b l e m i s g e n e r a t e d a s i l l u s t ra t i o n s .2 M A T H E M A T I C A L F O R M U L A T I O N O FS E I S M I C P I L E - S O I L I N T E R A C T I O NT h i s i n v e s t i g a t io n i s c o n c e r n e d w i t h t h e d e v e l o p m e n t o fa r i g o r o u s t r e a t m e n t f o r t h e s e i s m i c r e s p o n s e o f a t h i n -w a l l e d p i l e u n d e r v e r t i c a l ly - i n c i d e n t c o m p r e s s i o n a l w a v e s .T o e s t a b l i s h a g e n e r a l a n a l y t i c a l f r a m e w o r k r e l e v a n t f o rb o t h l o n g a n d s h o r t e m b e d m e n t s , t h e t u b u l a r p i l e i sm o d e l e d a s a n o p e n - e n d e d c y l i n d r i c a l s h e ll o f r a d i u s a ,l e n g t h l a n d t h i c k n e s s h

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S c a t t e ri n g o f v e r ti c a ll y- i n c i d e n t P - w a v e s b y a n e m be d d e d p i lep i l e c a n n o m i n a l l y b e g i v e n b y

u f ( z , t ) = u c o s (kaz )c i~ t, (14)u f ( z , t ) = o ,

w h e r e u f r e p r e s e n t s t h e m o d u l u s o f t h e i n c i d e n t w a v e .T o a c c o u n t f o r t h e s c a t t e r i n g e f fe c t s o f t h e t h i n - w a l l e de m b e d m e n t , t h e t o t a l d i s p l a c e m e n t f i e l d o f t h e s e m i -i n f i n i t e m e d i u m g e n e r a t e d b y t h e i n c i d e n t e x c i t a t i o nc a n b e w r i tt e n a s

u z = u { + uSz, (1 5)= ' ( 1 6 )r U f ~ - Ur~

w h e r e { ~ , ~ } a r e t h e d is p l a c e m e n t s c r e a t e d b y th e w a v e -i n d u c e d i n t e r f a c i a l lo a d s b e t w e e n t h e s h e ll a n d t h e s o i l a t{ x [ r = a , 0 < z < l } . W i t h t h e i m p o s i t i o n o f th e o p p o -s i t e r e a c t i o n s O f p z a n d Pr, w h i c h d e n o t e d t h e r e s u l ta n t so f t h e c o n t a c t l o a d d i s t r i b u t i o n s a c t i n g o n t h e s h e l l, o n t ot h e h a l f - s p a c e a c c o r d i n g t o t h e l a w o f a c t i o n a n d r e a c -t i o n , t h e r e s u l t i n g s c a t t e r e d m o t i o n g e n e r a t e d i n t h es e m i - i n f in i t e m e d i u m c a n b e e x p r e s s e d a s

f 0S(r , z ) = f trz(r , ; s ) ;p r (s )ds + f tZ(r, z ; S )p z(s )ds( 1 7 )

1 1uS ( r , z ) = [ f i r ( r , z ; s ) p r ( s ) d s + [ ~ r ( r ,z ; s ) p z ( s ) d s .jo j o(18)

In (17 ) and (18 ) , t he fou r kerne l s ~ ( r , z ; s ) ( i , j = r, z ) a r et h e a x i s y m m e t r i c h a l f - s p a c e d i s p l a c e m e n t r i n g - l o a dG r e e n ' s f u n c t i o n s

. I r z = a 2 ( ~ , z ; s ) ~ J o ( ~ a ) J o ( ~ r ) d ~ , (19)

~ ( r , z; s) = -- #ss ")'3 ~, z; s )~Jo(~ a) J 1 ~Cr)d~C, (20)

a r ( r ,Z ; S ) ~ , I 0 - - f ~ l ~ , z ; s ) ~ J 1 ~ a ) J o ( ~ r ) d ~ , ( 2 1 )f4~r(r,z; s) - #ss 71 (~, z; s)~ J 1 ~a ) J 1 ~ r )d~ , (22)

w h o s e i n t e g r a n d s f ' ~ l Q 2 , " / 1 , an d % are g i ven exp l i c i t l yi n P a k . 9 R e p r e s e n t i n g t h e d i s p l a c e m e n t o f t h e h a l f - s p a c ei n t h e / - d i r e c t i o n d u e t o a n e m b e d d e d r i n g l o a d a td e p t h s o f u n i t i n t e n s i t y a c t i n g i n t h e j - d i r e c t i o n , t h es i n g u la r f u n d a m e n t a l s o l u t i o n s f f i (r , z; s) ( i , j = r, z) c a nb e e v a l u a t e d a c c u r a t e l y b y n u m e r i c a l c o n t o u r i n t e -g r a t i o n w i t h t h e a i d o f a n a n a l y t i c a l d e c o m p o s i t i o no f t he i n t eg r and s . 11 By v i r t ue o f (17 ) a nd (18 ), t het o t a l d i s p l a c e m e n t f i e l d o f t h e h a l f - s p a c e c a n t h u s b e

213r e p r e s e n t e d b y

Uz(r, z) = fi ;(r , z; S)p r (s )ds + f iZ(r, z ; S )p z(s )ds+ u f ( z ) , ( 23 )

Ur (r ,z ) = I i f i r ( r , z ; s ) ( s )ds + I i f t ~ z ( r, z ; s) p z ( s )ds+ u f ( z ) , (24)

2 . 3 G o v e r n i n g i n t e g r a l e q u a t i o n s f o r w a v e - i n d u c e d p i l e -s o i l i n t e r a c t i o nI n a d d i t i o n t o t h e l a w o f a c t i o n a n d r e a c t io n w h i c h l e d t o( 23 ) a n d ( 2 4 ) , a f u l l y - b o n d e d c o n t a c t c o n d i t i o n b e t w e e nt h e s h e ll a n d t h e s o i l m e d i u m m u s t i n c l u d e t h e k i n e m a t i cr e q u i re m e n t s o f

w z ( z ) = l i m Uz (r ,z ) , 0 < z < l, (25)r-- , a a-

Wr(g = l i m u r (r ,z ) , 0 < z < 1 , (26)r ~ a t o c o m p l e t e t h e m a t h e m a t i c a l f o r m u l a t i o n . O n t h ec o n d i t i o n s t h a t P z a n d Pr a r e a b s o l u t e l y i n t e g r a b l e , t h ev a l i di t y o f w h i c h c a n b e s u b s t a n t i a te d , i t c a n b e s h o w nt ha t t he l i mi t p rocesses s t i pu l a t ed i n (25) an d (26 ) y i e l d ap a i r o f w e a k l y - s in g u l a r i n t e g ra l e q u a t i o n s i n t h e f o r m o f

Wz(Z) = I i ftRz a,z; s )p r (s )d s + l i fiZz (a,z ;s )p z( s )ds+ u f ( z ) , (27)

J or(Z = f i rR(a, ; S )pr (s )ds + r i tZ(a, ; S )p z ( s )ds+ u f ( z ) , (28)

t h r ou gh t he u se o f (23 ) and (24 ). Su b j ec t t o (11 ) - (13 ) ,( 2 7 ) a n d ( 2 8 ) c o n s t i t u t e a s e t o f F r e d h o l m i n t e g r a le q u a t i o n s w h o s e s o l u t i o n c a n b e c o m p u t e d .2 . 4 S i n g u l a r c h a r a c t e r i s t i c s o f i n t e r f a c i a i l o a d sT o i n c o r p o r a t e t h e s i n g u l a r c h a r a c t e r i s ti c s o f th ed y n a m i c c o n t a c t l o a d d i s t r i b u t i o n c a u s e d b y t h e i n c i -d e n t s t r e s s w a v e i n t o t h e s o l u t i o n p r o c e s s , o n e m a y n o t et ha t (25 ) and (26 ) a l so i mp l y

dwzdz ( z ) = l i m Ouz (r , z ) , 0 < z < l , (29)r---~a O Zdwr l i m OUr (r, z) , 0 < z < l , (30)d z (Z ) = r~a ~ 02

B y m e a n s o f t h e t h e o r y o f a n a l y t i c fu n c t i o n s, t h e l i m i ts t a t e m e n t s i n (2 9) a n d ( 30 ) c a n b e s h o w n t o b e e q u i v a l e n t

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2 1 4 F . J i, R . Y . S . P a k

(a)

S-%

3 . 0 I I I

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/ z , / / ~ , = 1 0 7 ~- 1 . 0 - - - - - / z , , / , u , , , = 1 0 0 0

- - - - / . z , , / , u , , , = 2 0 0. . . . , u , , / / z , = 1 O 0

- 2 . 0 i i i0 . 0 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0

z /o(b).%-.

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- - / z, ,/ ,u ,, , = 1 07- - - - - , u , , , / / & , = 1 0 0 0- - - - , u , J , u , , = 2 0 0

- - - / / / z , = 1 0 0

- 1 . 0 i i i0 . 0 5 . 0 1 0 . 0 1 5 .0 2 0 . 0

z /oF ig . 2 . In f luence o f mod u lus ra t io on res u l tan t ver t ical dynam iccon tact load d is t r ibu t ions : ue = 0.2, u~ = 0.25, l / a = 20,h / a = 0 '05, Ps/Pe = 0"25, a~a/Cs = 0.25. (a) Real part and (b)

Im ag in a ry p a r t .

(b) 1.0 , , I

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o . o%

- - - - - /. z / z , = 1 0 0 0- 0 . 5 - - - ~ o / , u . , = 2 o o

_ E - - - / % / , u . , ~ = 1 O 0- 1 . 0

0 . 0 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0z a

Fig . 3 . In f luence o f modu lus ra t io on resu l tan t ver t ical dynamiccon tac t load d is t r ibu t ions : Ue =0" 2 , Us =0"2 5 , l / a = 2 0 ,h / a = 0-05, P J P e = 0"25, coa/Cs = 0 .5 . (a) Real par t and (b )Im ag in a ry p a r t .

t ol Ofizz (a , z; S ) p z ( s ) d s + [ l O f [zO z Jo O z ( a , z ; S ) p r ( s ) d s

- k d U s i n ( k d Z )f t= J 0 ~ -Z (Z ; s ) ( p r ( S ) - - p e h c o 2 W r ( S ) ) d s

f O z4 - J 0 0 z ( z ; s ) ( p z ( S ) - - p e h w 2 w z ( s ) ) d s (31)j t[1 ( a , z ; S ) p r ( s ) d s + ( a , z ; s ) p ~ ( s ) d sJ o O z o O z

= [ ( z ; s ) ( P r ( S ) - - e h 2w r ( s ) ) dJo O zf'O r+ J 0 0 z ( z ; s ) ( p z ( S ) - p e h w 2 w z ( s ) ) d s ( 3 2 )

p r o v i d e d t h a t t h e f i r s t i n t e g r a l s o n t h e l e f t - h a n d s i de s o ft h e f o r e g o i n g e q u a t i o n s a r e i n t e r p r e t e d i n th e s e n s e o f

C a u c h y p r i n c i p a l v a l u e . O n w r i t i n gp z ( z ) - g z ( z )z , ~ . ( l _ z ) O ~ , 0 < R e ( z ) < 1 , 0 < R e (/ 3. .) < 1

(33)p r ( Z ) - - g r ( z )z ~ ( l - z ) ~ ' 0 < R e( a t ) < 1 , 0 < R e( /3 r ) < 1 ,

(34)o n e c a n d e r i v e f r o m ( 3 1) a n d ( 3 2 ) t h a t t h e o r d e r s o f t h es i n g u l a r i t i e s o f P z a n d P r a r e g o v e r n e d b y t h e c h a r a c t e r -i s t i c e q u a t i o n s( 3 - 4 Us ) C O S 2 ( ~ ) = 4 ( 1 - / I s ) 2 - ( 0~ i - 1 ) 2 , ( i = r , z ) ,

(35)a n d

cos(Tr/3i) = 0, (i = r , z ) (36)b y v i r t u e o f t h e r e s u l t s i n M u s k h e l i s h v i l i , 8 E r d o g a n , 1 a n dP a k a n d J i 1 f o r s i n g u l a r i n t e g r a l e q u a t i o n s . E q u a t i o n( 35 ) in d i c a t e s a d e p e n d e n c e o f th e o r d e r o f s i n gu l a r i t y a tz = 0 o f t h e w a v e - i n d u c e d i n t e r f a c i a l l o a d o n t h e

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Scat tering of ver tically- inc ident P-waves by an emb edded p i le 2 1 5(a) 0 . 5

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/ / - - - - - I Z J I J . = 1 0- - - u . lu = lOOO- - - - / z , / / z , = 2 0 0. . . . p , J , u . , , = 1 O0

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z / o( b ) 0 . 2

si O i

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- - o / / ~ , = 1 0 7- - - - - ~ / ~ = 1 0 0 0- - - - , u . J , u . , ,= 2 0 0. . . . , u . J ,u . ~ = 1 O 0

- 0 . 4 I I I0 . 0 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0

z /oF i g . 4 . In f lue nc e o f modu lus r a t i o on re su l t an t r a d i a l dyna micc on ta c t l oa d d is t r ibu t ions : r e = 0" 2 , u = 0" 25 , l / a = 2 0 ,h / a = 0-05, pffpe = 0"25, wa/Cs = 0"25. (a ) Rea l par t and (b)Ima g ina ry pa r t .

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0 . 0 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0z / o

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- - / z , / / = 1 0 7- - - - - ,u , p . s = 1 0 0 0- - - - - p , . / / z . = 2 0 0. . . . , u . , / ~ , = 1 o o

2 0 . 0z/o

Fig. 5 . Inf luence of mod ulus ra t io o n resul tant radia l dyna miccon tact load distributions: ue = 0 - 2 , u = 0 - 2 5 , l / a = 2 0 ,h / a = 0'05, P,/Pe = 0-25, ~ a / C s = 0 '5 . (a ) Rea l par t and (b)Ima g ina ry pa r t .

P o i s s o n ' s r a t i o o f th e h a l f - s p a c e w h i l e (3 6 ) a s s e rt s t h ep r e s e n c e o f a s q u a r e - r o o t s i n g u l a r i t y a t z = l i r r e s p e c t i v eo f t h e m a t e r i a l p a r a m e t e r s o f e it h e r m e d i a .

3 C O M P U T A T I O N A L P R O C E D U R EW i t h t h e o r d e r o f th e s i n g u la r i ti e s k n o w n , o n e m a yp r o c e e d t o s o l v e , s u b j e c t t o ( 1 1 ) , ( 1 2 ) a n d ( 1 3 ) , t h e t w ow e a k l y s i n g u l a r b o u n d a r y i n t e g r a l e q u a t i o n s i n (2 7 ) a n d( 2 8) i n t e r m s o f t h e r e g u l a r f u n c t i o n s gr, gz, w~ a n d% . O w i n g t o t h e i r a n a l y t i c i t y , t h e s e f u n c t i o n s c a n b ee f f e c t iv e l y r e p r e s e n t e d b y

ng r (Z ) = Z N J ( z ) g j ' (37)j = l

ng z ( Z ) = Z N j ( z ) g ] , (38 )j= lw r( z ) = ~ N j ( z ) w , (39 )

j = l

w : ( z ) = U A z ) w Lj= l (40)w h e r e { N j ( z ) , j = 1 , . . . , n } a r e r eg u l a r s ha p e f u n c ti o n sa s s o c i a t e d w i t h t h e n o d a l l o c a t i o n s { z j } . I n ( 3 7 ) - ( 4 0 ) ,{ g / } a n d { g j } a r e t h e n o d a l v a l u e s o f t h e r e g u l a r p a r ts o fthe w a ve - induc e d in t e r fa c i a l l oa d d i s t r i bu t ions , a nd { W ~r '}a n d { w j } a r e t h e n o d a l d i s p l a c e m e n t s o f t h e s h el l,r e s p e ct i v e ly . O n s u b s t i t u t in g t h e f o r e g o i n g r e p r e s e n t a -t i o n s i n t o t h e g o v e r n i n g e q u a t i o n s a n d r e q u i r i n g e q u a l i tya t a l l n o d e s , ( 1 1 ) , (1 2 ) , (2 7 ) , ( 28 ) , a n d ( 1 3 ) c a n b e r e d u c e dt o

oIw + pehw2Gw = G g + w : , (41 )

w = H g + u f , (4 2)

(a ~ - p ehw2 arH) g = p ehw2 a[u (43)i n m a t r i x n o t a t i o n . H e r e , I i s a ( 2 n x 2 n ) i d e n t i t ym a t r i x ; G ( 2 n x 2 n ) a n d I ] ( 2 n x 2 n ) a r e i n f l u e n c e m a t r i x

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2 1 6 F . J i , R . Y . S . P a k

( a ) 1 . 0

0 . 5S

0 . 0

- 0 . 5

- 1 . 0

( b ) 0 . 4 0

0 . 3 0

~ 0 . 2 0u0 . 1 0E

0 . 0 0 .

- - ~ / u = 1 0 7- - - _ _ - - - - - / z / u . = 1 0 0 0L - _ L - - - - ~ . / , ~ , = 2 o o

- ~ . . . . . p . , / p . = I O 0

: L - - - ' - -L

0 . 0 5 . 0 1 0 . 0 1 5 . 0 2 0 . 0z /o

I I I

- - p . , / ~ z , = I 0 7- - - - - . , / / ~ , = 1 0 0 0- - - - / ~ / / ~ = 2 0 0. . . . p . , / / % , = I O 0

- 0 . 1 00 . 0 5 . 0 1 0 . 0

z / o

F i g . 6. I n f l u e n c e o f m o d u l u s r a t i o o n v e r t i ca l d i s p l a c e -m e n t p r o f i l e s o f t h e s h e l l : U e = 0 '2 , U s = 0 2 5 , I / a = 2 0 ,h / a = 0 05, P J P e = 0 25, ~ a / C s = 0 - 2 5 . ( a ) R e a l p a r t a n d ( b )

I m a g i n a r y p a r t .

1 5 . 0 2 0 . 0

a s s o c i a t e d w i t h t h e s h e l l a n d a r e d e f i n e d b y

N j ~ )G i , j = W # ( 2 i , s ) s ~ (l - s ) # d s ,

G i , n + j ] ^ Z N j ( s )= w r ( ~ , ~ ) / d - - s ? d ~ ,a . + i , J = [ l ~ 2 f Z i , S ) N j S ) _ ~ d sJo s ~ l s pG n + i , n j = ~ V f Z i , S ) U j ( ~ )s ~ ' ( _ s ) # d s , i = 1 . . . , n ,

j = 1 , . . . , na n d

G i , j ^ aw , ( z , , s ) N j ( s ) d s ,

( 4 4 )

( 4 5 )

G i , n + i v f ( z ,, s ) N j ( s ) d s ,

( a )

%Y~

0 . 4

0.2-

0.0

-0.2-

\ \

. \\ x

- 0 . 40 . 0 5 1 0

- - ,u.J,u.,,= 1 0 7- - - - p ~ / / ~ , = 1 0 0 0- - - - , u . , / , u . , = 2 0 0 ,. . . . # ~ / # , = I 0 0 / /

4 /

/ /\ \ ~ - / l

\ z/I I

1 0 . 0 1 5 . 0 2 0 . 0z / Q

( b )

~ N

)=

E

0. .30 J

0 . 2 0 - - '- . ~ \ \\\ , \

0 . 1 0 - \ \

o.oo- ~ \ ,- 0 . 1 0 -

- 0 . 2 0 [0 . 0 5 . 0 1 0 . 0 1 5 . 0

I I- - p . ~ /~ = 1 0 7- - - - / ~ J / % = I 0 0 0_ _ ~ J / z s = 2 0 0- - - ~ o / i ~ = i o

~ / ~C ~

\

2 0 . 0z / o

F i g . 7 . I n f l u e n c e o f m o d u l u s r a t i o o n v e r t i c a l d i s p l a c e -m e n t p r o f i l e s o f t h e s h e l l : u e = 0 - 2 , u = 0 . 2 5 , l / a = 2 0 ,h / a = 0 . 0 5 , P s / P e = 0 25, ~ a / C s = 0 5 . ( a ) R e a l p a r t a n d ( b )

I m a g i n a r y p a r t .

a n + i , j w ~ z i , s ) N j ~ ) d s ,

a n w i n + j l l o ^ Z= W z ( Z i , s ) N j ( s ) d s , i = 1 , . . . , n ,j = 1 , . . . , n

r e s p e c t i v e l y ; H ( 2 n x 2 n ) i s a n i n f l u e n c e m a t r i x o f t h eh a l f - s p a c e g i v e n b y

H i ,j - - - - [ l ~ l # ( Z i , S ) N j ( s Z . d s , (46)J o s ~ l - s ) ~

J 1 z ( 2 i , S N j ( s )H i , n ~ j o s ~ ( 1 - s ) ~ d s '[ U j ( ~ )= u : ( z i , s ) d s ,n + i , j ^ Ro P l - s /In + i , . / = ~ Z ( z i , s ) N j ( s ) ~ d s , i = 1 , . . . , n ,

' ' o s ( l - s ) ~

j = l , . . . , n .

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218 F . J i , R . Y . S . P a k(a) 1 . 0

0 . 8

~ 0 . 6S 0 . 4

o 0.2-

I ~ i i h

- - ~ . / u . . = 0 4 .- - - - - / J , / / z , = 1 0 m- - - u / u . = ~ o ~

0 . 0 i i I i0.00 0.10 0.20 0.30 0.40 0.50wa C~

( a ) 1 . 0

S 0 .5

S,-%0 . 0#

I i i 1 [ i

~ . . - - - - -#/Iz~= l ~~ /Z , / / . / , , = 10 z

- 0 . 5 i i T0 . 0 0 O . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0

wa C~(b)

S

g

0.08

0 . 0 6 -

0.04 -0 . 0 2 -

0 . 0 0

- 0 . 0 20 . 0 0

I I i I i I

i i i

O . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0~ a / C .

F i g . 10. Ver ti ca l f ounda t i on i npu t m ot i on f unc t ions o f anem be dd ed shell : u~ = 0-2, u = 0.25, h /a = 0 . 0 5 , P s /Pe = 0 '25,l / a = 5 . ( a ) Rea l pa r t and ( b ) I mag i na r y pa r t .

(b)

S

S

0 . 6

0 . 4

0 . 2

0 . 0

I I I

- - U o / U . = 1 0 4 .- - - - - ,U.o/AZ , = 103- - - - - A6o/,U ,,= 102

- 0 . 2 i i i i0.00 0.10 0.20 0.30 0.40 0.50wa/C~

Fig. 11. Ver t ical foun dat io n input m ot ion func t ions of anembedded shel l : u e = 0 .2, u~ = 0 -25 , h /a = 0 . 0 5 , P s /Pe = 0"25,l / a = 20. (a) Real par t an d (b) Imag inary par t .w h i c h p e r f o r m s e x t r e m e l y w e l l i n a l l c a s e s e x a m i n e d . I nt h e n e x t s e c t i o n , s o m e t y p i c a l s o l u t i o n s f o r t h e a x i -s y m m e t r i c w a v e s c a t te r i n g p r o b l e m o f a n e m b e d d e dp i l e f o u n d a t i o n i n a s e m i - i n f i n i t e m e d i u m w i l l b ep r e s e n t e d .

4 N U M E R I C A L R E S U L T SB y m e a n s o f th e f o r e g o i n g m a t h e m a t i c a l a n a l y si s a n dc o m p u t a t i o n a l m e t h o d , t h e s e is m ic s o i l - p i le i n t e r a c t io np r o b l e m u n d e r v e r t i c a l l y - i n c i d e n t P - w a v e s c a n b e s o l v e dr i g o r o u s l y . I n w h a t f o l l o w s , a s e t o f n u m e r i c a l r e s u lt s i sp r e s e n t e d t o h i g h l i g h t s o m e o f t h e s a l ie n t f e a t u r e s o f th ei n c id e n t - w a v e p r o b l e m .

I n c o m p l e x n o t a t i o n , F i g s 2 - 5 s h o w t h e r e s u l t a n td y n a m i c c o n t a c t l o a d d i s t r i b u t i o n s P z a n d P r a s af u n c t i o n o f z u n d e r v e r t ic a l l y - in c i d e n t P - w a v e s a t f r e -q u e n c i e s ~ a / C s = 0 . 2 5 a n d ~ a / C s = 0"5 . As i n t he s t a t i cc a s e , ] o n e c a n s ee t h a t t h e d y n a m i c c o n t a c t l o a d s a c t i n go n t h e t h i n - w a l l e d p il e a r e s i n g u l a r a t th e t o p a n d b o t t o mo f th e e m b e d m e n t a s p r e d i c te d b y ( 3 5 ) a n d ( 3 6) . F r o mt h e s e p l o t s , it is a ls o e v i d e n t t h a t t h e m o d u l u s r a t i o # e / # sc a n a f f e c t q u i t e s i g n i f ic a n t ly t h e s e is m i c w a v e - i n d u c e dl o a d s P z a n d P r a c t i n g o n t h e e m b e d d e d p i le e s p e c i a ll y a t

h i g h f r e q u e n c y , w i t h l a r g e r # e / # s i m p l y i n g l a r g e r l o a d s .I l l u s t r a t e d i n F i g s 6 - 9 a r e t h e v e r t i c a l a n d r a d i a l d i s -p l a c e m e n t p r o f i l e s o f t h e p il e c o r r e s p o n d i n g t o t h ep r e v i o u s e x a m p l e s . O n e c a n o b s e r v e t h a t t h e r e is a ni n h e r e n t w a v e l e n g t h i n t h e p r o f il e s w h i c h a r e g e n e r a l l ya f u n c t i o n o f t h e w a v e l e n g t h o f t h e i n c id e n t w a v e a s w e lla s t h e s ti ff n e ss o f t h e e m b e d m e n t . A s t h e f r e q u e n c y o fe x c i t a t i o n t u r n s h i g h e r , h o w e v e r , t h e c h a r a c t e r i s t i c s o ft h e i nc i d en t w a v e c a n b e c o m e m o r e d o m i n a n t . T h i s isi l l u s t r a t e d i n t h e c a s e o f w a / C ~ = 0 ' 5 w h e r e t h e w a v e -l e n g t h i n t h e a x i a l d i s p l a c e m e n t p r o f i l e is f o u n d t o c l o s e l yr e s e m b l e t h e w a v e l e n g t h o f t h e i n c i d e n t l o n g i t u d i n a lw a v e .

F o r g e n e r a l s e i s m i c s o i l - s t r u c t u r e i n t e r a c t i o n p r o b -l e m s , t h e f o u n d a t i o n i n p u t - m o t i o n f u n c t i o n w h i c h i sd e f i n e d a s t h e d i s p l a c e m e n t r e s p o n s e a t t h e t o p o f t h ef o u n d a t i o n n o r m a l i z e d b y t h e f r ee - fi e ld m o t i o n a t t h eg r o u n d s u r f a c e is o f p a r t i c u l a r i n te r e s t. A s e t o f s u c hs e is m i c f o u n d a t i o n i n p u t - m o t i o n f u n c t i o n s f o r th e e x ci -t a t i o n o f i n t e r es t is s h o w n i n F i g s 1 0 - 1 2 f o r s o m ep r a c t i c a l v a l u e s o f r e la t i v e s t if f n e ss , l e n g t h a n d P o i s s o n ' sr a t i o s f o r t h e p i l e p r o b l e m . F r o m t h e s e d i s p l a y s , t h ed e p e n d e n c e o f th e c o m p l e x - v a l u e d f o u n d a t i o n i n p u t -m o t i o n f u n c t i o n o n th e m o d u l u s r a t io , e m b e d m e n tl e n g t h , a n d e x c i t a t i o n f r e q u e n c y i s a p p a r e n t . A s t h e

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a ) 1.0

~ o.5

N

0 . 0

Sc a t ter i ng o f ver ti ca ll y- i n c iden t P -w aves by an em bedded p i l e~ . . ~ . I , I , I , 1 4

\ \ - - - , - - / ~ / ~ . = 1 o\ \ ~ --, - - / z . / l z , = 1 0 3

\ \ ~ - < . ~ / A / / ~ , = ~ '

\ \ \

i . i . i . i . I0 . 50 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0~ a / C ,

( b ) 0 . 6 ' ' ' ' ' ' , ' ,- - , . / , u , , = l O '-- - -- ,U., ,J,U, ,, 103

0 .4 . . . . //.o I -~,= I 02

- 0 . 2 i i i i0 . 0 0 O . I 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0~ a / C ,

Fig. 12. Vertical foundation input motion functions of anembedded shell: ve = 0-2, vs = 0-25, h / a = 0.05, Ps/Pe = 0 25,1/a = 40. (a) Real part and (b) Imaginary part.

( a ) 8 . 0 I , I , I , I ,

relative stiffness of the pile to the half-space increases,one can generally find more variations in the input-motion function due to an increase in the interferenceof the embedment to the free-field response. As /Ze/#stends to zero, however, the mechanical interactionbetween the foundation and the surrounding mediumnaturally diminishes and the normalized foundationresponse is found to approach unity. Also illustrated inFigs 10-12 is the existence of a critical modulus ratio,beyond which the corresponding top response of theembedded pile will be essentially identical. The depend-ence of its value on the length of the embedment,however, is also evident. For instance, one can see thatwhile the foundation input motion functions for the caseof e / # s = 1000 and 10,000 are very close for l / a = 5 ,their difference is much larger for a dimensionless lengthof 20 and 40.

To provide further insights into the physical problem,the top vertical displacement response of a pile witha pile cap of mass m under the incident P-wave isalso evaluated using the foregoing input-motion functionand the corresponding dynamic compliance of thefoundation. With the definition of the mass ratio b asm / p s a3 where p~ is the mass density of the soil, sometypical foundation responses normalized by the free-

219

6 , 0

N~ . 4 . 0 .

- - 2 . 0 . ~ ,

0 . 00 . 0 0

b = 5 0 0 0b = 2 0 0 0

I~ b = 1 0 0 0I I . . . . b =5 00

0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0~ o / C ,

(b ) 8 .0 I , I , I , I

6 . 0 ,b = 5 0 0 0b = 2 0 0 0

~ b = 1 0 0 04 . 0 - - - - b = 5 0 0%2 . 0 .

0. 0 ~ -_._~_- _- _- _: j- - .-=. . . . .i , i i , i , i0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0

~ a / C ,

(c) 8 0

6 . 0 -

N~ . 4 . 0

2 . 0

0 . 0

I , I ~ I , I

b = 5 0 0 0

b = 2 0 0 0- - - - - b = 1 0 0 0. . . . b = 5 0 0

i i i0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0o)a/C,

Fig. 13. Top displacement of an embedded shell with a cap:ve = 0-2, vs = 0-25, h / a = 0 05,P s / P e = 0-25, IZe/IZ s = 1000. (a)l /a = 5; (b) l /a = 20 and (c) l /a = 40.field vertical displacement are shown in Figs 13, 14and 16. From these figures, one can observe definiteresonance peaks in most cases except those for lowmass ratios. For the latter cases, however, more thanone maximum can be observed in the response curveowing to the resonance characteristics of the embeddedpile. In each of the plots in Fig. 13 where the modu-lus ratio is kept constant, for instance, one can findthat an increase in the mass ratio will always lead toa reduction in the resonance frequency as well as to a

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2 2 0 F . J i , R . Y . S . P a k

( a ) 8 . 0 t

- - 2.0

0.0 I .0 .00

I I

i 1 i0 . 1 0 0 . 2 0 0 . 3 0

w a / C ,

I I

- - I / a = 5- - - - I / a = 2 0- - - - I / a = 4 0

i0.40 0 . 5 0

(a) 10.08 .0

I

6 . 0u

I

# , / ~ , = 1 o ~# , / # , = 1 0

- - - - - # , / # , = 1 o +

2.00 . 0 , - -

0 . 0 0 O . 1 0 0 . 2 0 0 . 3 0 0.40wa/C

m0 . 5 0

(b) 8.0 ] ' L I i I

~ I / a = 2 0I / a = 4 04 . 0

I_ ~0.0 I . , . , - , ,

0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0~ a / C ,

0 . 5 0

(b) 10.08 . 0

v 6 . 0N2S~ - ~ 4 . 0 -

2 . 0

0 . 00 . 0 0

I I I i

, / / y . , = I 0 ]- - - - - f l , e / / ~ s = I 0

# , / # = I 0 +

i i0 . 1 0 0 . 2 0 0.30 0.40

w a / C ~0 . 5 0

(c) 8.0 t ' ' ' '

6 0 I

~ " 4 . 0 -'2Z.0 -0.0 i i ~ v , i

0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0

I I

- - I / a = 5- - - - - I / a = 2 0- - - - I / 0 = 4 0

i0.40 0 . 5 0w a / C ,

F i g . 1 4. T o p d i s p l a c e m e n t o f a n e m b e d d e d s h e ll w i th ac a p : u e = 0 '2 , us = 0 .25, h / a = 0 '05 , Ps/Pe = 0"25, b = 1000.(a) #e/#.~ = 102; (b) #e/# s = 103 an d (c) #e/#~ = 104-

(c) 1 0 . 08 .0

6 . 0 -z4 . 0 -

2 . 0

I I I I

~ . / / 4 = I ]# , / / z , = 1 0 4- - - - - ~ e / / . ~ s = 1 0

0 . 1 0 0 . 2 00 . 0

0 . 0 0 0 . 5 0 0 . 4 0 0 . 5 0w a / C

F i g . 1 5. T o p d i s p l a c e m e n t o f a n e mb e d d e d s h e ll w i t h acap : u~ = 0.2 , us = 0 '25, h / a = 0.05, Ps/Pe = 0"25, l / a = 40.(a) b = 1000; (b) b = 2500 an d (c) b = 5000.

h i g h e r a n d s h a r p e r r e s o n a n c e p e a k . T h e e f f e ct o f e m b e d -m e n t l e n g t h o n t h e f o u n d a t i o n m o t i o n i s i l l u s t r a te d inF i g . 1 4. A s e x p e c t e d , a l o n g e r p i l e w i l l r e d u c e t h e t o pf o u n d a t i o n r e s p o n se d u e t o t h e a d d i t io n a l a n c h o r a g et h r o u g h e m b e d m e n t . A s t h e m o d u l u s r a t io v a ri e s f r o ml o w t o h i g h ( e . g . w h e n t h e s i t e g o e s f r o m s t i ff t o s o f ts o i ls ) , o n e c a n s e e F r o m F i g . 1 5 t h a t t h e d i m e n s i o n l e s sn a t u r a l f r e q u e n c y o f th e p i l e - s o i l s y s t e m w i l l i n c r e a s es l i g h t l y a s a r e s u l t . F r o m t h e s e d i s p l a y s , h o w e v e r , t h ep o s s i b l e u s e f u l n es s o f p i l e f o u n d a t i o n s f o r a w e a k s i te in

r e d u c i n g t h e i n p u t b a s e m o t i o n t o a s u p e r s t r u c t u r e i sa p p a r e n t .

5 C O N C L U S I O N SI n t h is p a p e r , a r i g o r o u s t h e o r e t i c a l f o r m u l a t i o n i sp r e s e n t e d f o r t h e a n a l y s i s o f a t h i n c y l i n d r i c a l s h e l le m b e d d e d i n a n e l as t i c h a l f - s p a c e u n d e r v e r t i c a ll y -i n c i d e n t P - w a v e e x c i t a t i o n . B y v i r t u e o f a se t o f

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S c a t t e r i n g o f v e r ti c a l ly - i n c i d e n t P - w a v e s b y a n e m b e d d e d p i l e 221p s e u d o - s t a t i c r i n g - l o a d G r e e n ' s f u n c t i o n s f o r t h es h el l a n d a g r o u p o f d y n a m i c f u n d a m e n t a l s o l u ti o n s f o rt h e s e m i - in f i n it e m e d i u m , t h e a x i s y m m e t r i c a l w a v e -s c a t t e r i n g p r o b l e m i s s h o w n t o b e r e d u c i b l e t o a s e t o fF r e d h o l m b o u n d a r y i n t e g ra l e q u a t io n s . T h r o u g h t h ea n a l y s i s o f a n a u x i l i a r y p a i r o r C a u c h y i n t e g r a le q u a t i o n s , t h e s i n g u l a r i t i e s o f t h e c o n t a c t s t r e s s d i s t r i -b u t i o n s a s s o c i a t e d w i t h t h e s c a t t e r e d f i e l d a r e e x p l i c i t l ye l u c i d a te d . B y i n c o r p o r a t i n g s u c h f i n d in g s i n t o t h eb o u n d a r y i n t e g r a l f o r m u l a t i o n , a c o m p u t a t i o n a l p r o -c e d u r e i s d e v e l o p e d w h i c h i n v o l v e s a n i n t e r p o l a t i o n o fr e g u l a r f u n c t i o n s o n l y . T y p i c a l r e s u l t s f o r t h e w a v e -i n d u c e d d y n a m i c c o n t a c t l o a d d i s tr i b u t i o n s , d i s p l a c e -m e n t s , c o m p l e x - v a l u e d fo u n d a t i o n i n p u t m o t i o n f u n c -t i o n s, a n d r e s o n a n t p i l e f o u n d a t i o n r e s p o n s e a r e a ls oi n c l u d e d . I n a d d i t i o n t o f u r n i s h i n g q u a n t i t ie s o f d i r e c tr e le v a n c e to s e i sm i c 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 te r -a c t i o n p r o b l e m s , t h is t r e a t m e n t s h o u l d b e u s ef u l as ar a t io n a l f r a m e w o r k u p o n w h i c h r a ti o n a l c o m p u t a t i o n a lm e t h o d s c a n b e d e v e l o p e d f o r a n a l y z i n g m o r e g e n e r a ls e is m i c l o a d i n g s o n e m b e d d e d f o u n d a t i o n s .

exci ta t ions , l n t . J . N u m e r . M e t h . E n g n g . , 1994, 37, 250 1-2520.12. W olf , J . P . & Von Arx, G . A. Hor iz ontal ly t ravel ing wavesin a g roup of pi les taking pi le-soi l-pile interact ion in toaccoun t . E a r th q . E n g n g . S t ru c t . D y n . , 1982, 10(2), 225-23 7.

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