dynamics of pile driving by fem

8/10/2019 Dynamics of Pile Driving by FEM

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omputers and Geotechnics5 1 9 8 8 ) 3 9 - 4 9

D Y N M I C S O F P I L E D R I V I N G

B Y T H E F I N I T E E L E M E N T M E T H O D

R o n a l d o I B o r j a

D e p a r t m e n t o f Ci v il E n g i ne e r in g

Stanford University

S t a n f o r d C A 9 4 3 0 5

A B S T R A C T

It is prop osed that the d yn am ic s of pile driving base d on the one-dimensional

theory of wa ve propagation be analyzed by the finite lement me th od . This ap-

proa ch enables one to continuously interpolate he displacement velocity an d ac-

celeration profiles hrou gho ut the pile leng th. In co ntr ast to th e finite ifference

techniqu e presented by F . A. L. S mi th [1] an d the finite element procedu re pre-

sented by I. M . Smi th [2] in whic h initial onditions are defined on the basis of a

prescribed h a m m e r velocity he me t h o d presented herein defines initial onditions

o n t h e b a si s o f a p r e sc r i be d m p a c t f o r c e e r s u s t i m e c u r v e a t t h e p i l e / h a m m e r p o i n t

of contact. Applications of the pro pos ed technique to typical pile driving pro ble ms

on an elastoplastic oil nd using an implicit time-integration ch em e are discussed

using a numerical example.

I N T R O D U C T I O N

T h e p r o b l e m o f a n e la st ic r o d d r i v e n i n t o a n e la st op la st ic o il m e d i u m w a s

investigated by Sm it h [1] in the context of a finite ifference nalysis. In his pro-

cedure the pile-soil od el w as idealized as a combination of springs dashpots an d

l u m p e d m a s s e s a ss e mb l ed c o ns is te nt ly u c h t h at w h e n t h e h a m m e r a n d t h e p il e m -

pact longitudinal stress av es are generated and pro pag ate d in the system. Su ch

an app roa ch enables on e to study in detail he mec hani cs o f soil-pile nteraction or

purposes of selecting ppropria te pile driving eq ui pm en t or establishing ile riving

3 9

omputers and Geotechnics026 6-35 2X /88/S 03-5 0 O 1988 E lsevier App l ied Sc ience

Publ ishers Ltd, E ngla nd . Printed in G rea t Bri tain


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4O

cri ter ia (e .g. see [3-5]) .

Th e p r oc e dur e o f l ump ing ma sse s a nd sp r ings a s im p le me n te d in [ 1 ,3 - 5 ] r e -

s t r ic t s the use r to exp l ic i t in teg ra t ion in t im e (cen t ra l d i ffe rence expl ic i t schem e) .

In orde r to avoid num er ica l ins tab i l i tie s th a t a f fl ict a l l expl ic i t t im e-s tepp ing schemes

whe n l a r ge t im e s t e ps a r e u se d , S m i th [2] p r e se n te d a f in it e e le me n t so lu t ion th a t

e mploys a n imp l i c i t t ime - s t e pp ing sc he me f o r so lv ing p i le d r iv ing p r ob le ms . B o th

m e thod s p r e se n te d in [ 1] a nd [ 2 ], howe ve r , s imu la t e im pa c t by p r e sc rib ing a ha m-

m e t ve loc i ty a s de t e r mine d f r om the p r ope r t i e s o f t he p i le d r ive r a nd us ing th i s

ve loc i ty to g e t t he num e r i c a l so lu t ion s t a r t e d .

I t ha s be e n shown tha t t he p i l e d r iv ing p r oc e s s c a n be be t t e r de sc r ibe d by

p r e sc r ib ing a n impa c t f o r c e ve r sus t ime c u r ve wh ic h inc o r po r a t e s t he in t e r a c t ion

b e t w e e n t h e p i le a n d h a m m e r a t t h e p o i n t o f c o n t a c t , r a t h e r t h a n b y p re s c ri b in g

the ha m m e r ve loc i ty a lone [ 6 ] . Th i s pa pe r p r e se n t s a f i n it e e le me n t so lu tion to p i l e

d r iv ing p r ob le ms in wh ic h the imp a c t f o r c e s a t t he p i l e /ha m m e r po in t o f c on ta c t

a r e p r e sc r ibe d in the so lu t ion , r a the r t ha n the ha mme r ve loc i ty .

Th e r e su l ti ng se mi - d i scr e te e qua t ion o f m o t ion is so lve d by N e wm a r k s ( im-

p l i c it ) p r e d ic to r - m u l t i c o r r e c to r m e th od f o r non l ine a r e qua t ions [ 7 ] . W i th th i s im-

p l i c i t me thod , nume r i c a l i n s t a b i l i t i e s a r e a vo ide d a lbe i t l a r ge t ime s t e ps a r e e m-

p loye d in the so lu t ion . Th e a dd i t iona l c os t e nge nde r e d by the imp l i c i t me tho d ove r

the ex pl ic i t me tho d used in [1,3-5] i s ins igni fican t in one-d im ensiona l wave prop aga-

t ion an a lyses s ince the resu l t ing coef f ic ien t m a t r ix in the imp l ic it m e th od i s usua l ly

sym m e t r i c a nd ba nd e d ( me a n h a l f - ba ndwid th = 2 f o r one - d ime ns iona l p r ob le ms) .

Th e m e th od p r e se n te d in th i s p a pe r i s a pp l i ca b le to a ny ty pe o f so il l oa d -

d isp lacement behaviour such as l inea r , nonl inea r , o r e la s to-p las t ic . As an example ,

a nume r i c a l p r ob le m in wh ic h the p r e sc r ibe d impa c t f o r c e s a r e c ompu te d f r om the

me thod p r e se n te d by F a i r hu r s t [ 6 ] i s d i sc usse d to de mons t r a t e t he a pp l i c a b i l i t y o f

the p r opose d me thod to typ ic a l p i l e d r iv ing p r ob le ms .

G O V E R N I N G E Q U A T I O N S

onsider t he p i le s e tup a n d the f r e e bo dy d i a g r a m ske tc he d in F igu r e 1 . For

dyn a m ic e qu i l ib r ium o f a t yp ic a l s li ce o f p i le i n F igu r e lb , t he f o llowing e qua t ions

should be sa t i s f ied :

P2+f t)=O

i n f l ( 1 )


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41

, ( / )

i P I L E

XIS

o )

. . . T

~f Adz . ~ t ;

P P , , a z

b )

e(e )

c )

F I G U R E 1 . ( a ) P r o b l e m d e f i n it io n ; ( b ) e q u i l i b r i u m o f a t y p i c a l p i l e s li ce ; ( c )

i m p a c t f o r c e v e r s u s t im e .

f t ) = p A ~ + rO - 7 A ( 2 )

P = P h a t z = z h ( 3 )

= Vp a t z = zp

( 4 )

u = u o a t t = t o = 0 ( i n i t i a l c o n d i t i o n ) ( 5 )

*~ = u o a t t = t o = 0 ( i n i t i a l c o n d i t i o n ) , ( 6 )

w h e r e z i s t h e v e r t ic a l c o o r d i n a t e ( d e p t h ) o v e r t h e d o m a i n n r e p r e s e n t i n g t h e e n t i r e

l e n g t h L o f t h e p i le in c l u d i n g i t s a c c e ss o r ie s ( c a p b l o c k , p i l e c a p , c u s h i o n b l o c k , e t c ) ;

t is t i m e ; u = u z , t) i s t h e d i s p l a c e m e n t p r o f i le a t a n y t i m e i n s t a n t t ; ~ a n d ~ a r e

t h e f i r s t a n d s e c o n d t i m e d e r i v a t i v e s o f u , r e s p e c t i v e l y ; u 0 a n d ~ o a r e p r e s c r i b e d

i n i t ia l d i s p l a c e m e n t a n d v e l o c it y , p r o f il e s, r e s p e c t i v e l y ; P - - P z , t ) i s t h e a x i a l f o r c e

i n t h e p i le a t a n y t i m e ( c o m p r e s s i v e f o r c e a s s u m e d p o s i ti v e );

P h = P a t )

i s t h e

p r e s c r ib e d i m p a c t f o r c e - t im e c u r v e a t e l e v a ti o n z = z s c o r r e s p o n d i n g t o t h e p i le 's

p o i n t o f c o n t a c t w i t h t h e d r i v i n g h a m m e r ( u s u a l ly a t t h e p i le b u t t ) ; P ~ = P ~ ( u , ~ )

i s t h e s o f t b e a r i n g r e a c t i o n a g a i n s t t h e p i l e t i p a t d e p t h z = z p ; p i s m a s s d e n s i t y

o f t h e p i l e m a t e r i a l ; y is th e u n i t w e i g h t o f t h e p i le m a t e r i a l ( = g p , w h e r e g i s t h e


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42

gravi ty acce le ra t ion cons tan t ) ; 0 i s the p i l e pe r ime te r ; A i s the p i l e c ros s -sec t iona l

a r e a ( bo t h 0 a nd A m a y be f unc t ions o f z ) ; r i s t he s o i l s s he a r r e s i st a nc e pe r un i t

s u r f a c e a r e a in c on t a c t w i t h t he p i l e , a nd t he c o m m a de no t e s a d i f f e re n t ia t i on .

I f the p i le t ip i s f ixed on a f i rm bedroc k , eq ua t io n (4) i s rep laced wi th the

f o ll ow ing bou nd a r y c ond i ti on :

u = ~ = f i = 0 a t z= zp (7)

E qu a t i ons ( 1 ) - ( 7 ) do no t s t r i c t l y s a ti s f y t he e qu a t i on o f m o t i on f o r l ong i t ud i na l

w a ve p r o pa g a t i on p r ob l e m s i n t h r e e d i m e ns i ons [ 8 ] . H ow e ve r , t he y r e p r e s e n t a m uc h

s im p l e r a p p r o x i m a t i o n t h e o r y o f w a v e p r o p a g a t i o n i n o n e d i m e n si o n i n w h ic h e a c h

s l i ce of the p i l e i s rega rded as e i the r in s imple t ens ion or compress ion .

L e t t he a x i a l fo r c e be g i ve n by

P -E A 8a)

O O z '

w h e r e E i s t h e p i le 's m o d u l u s o f e l a s t i c i ty w h i l e t h e n e g a t i v e s i g n i m p l i e s t h a t a

p o s i t i v e P i s a c o m p r e s s i v e f o r c e . I f t h e c o m p r e s s i o n w a v e s a x e t r a n s m i t t e d t h r o u g h

a c a p b l o c k m a t e r i a l w h o s e s t re s s- s tr a in c h a r a c t er i s ti c s a x e a s s h o w n i n F i g . 2 , e q u a -

t i o n ( 8 a ) m a y b e r e p l a c e d w i t h

w h e r e ( -c g u /@ z ) m s x r e p r e se n t s t h e m a x i m u m c o m p r e s s iv e s tr a i n e x p e r i en c e d b y t h e

m a t e r i a l a nd e i s t he c oef fi ci en t o f r e s t i t u t i on o f t he c a pb l oc k . N ow , a s s um e t ha t

t he v i sc ous da m pi ng on t he s u r f a c e o f t he p i l e a nd a t i t s ti p c on t r i bu t e s t o t he t o t a l

r e s i s t a n c e i n t h e f o l l o w i n g m a n n e r :

r

= f ( l

+

J s ~ )

a = ( 1 + 4 ~ p ) ,

9 )

1 0 )

w he r e J s a nd J p a r e , r e spe c t i ve ly , t he s o i l s s u r f a c e a nd po i n t da m pi ng c on s t a n t s

i n t he s e ns e o f [1 ], ~ a n d @ a r e t he s t a t i c s i de a nd p i le t i p r e s i s t a nc e p e r un i t

c on t a c t a r e a ob t a i ne d f r om t he s t a t i c l oa d - de f o r m a t i on r e l a t i ons h i p s f o r t he s o i l ,

a n d ~p = ~ ( z = z p ) i s t he ve l om t y o f t he p i le t ip .

T h e v a r i a t i o n a l e x p r es s io n e q u iv a l en t t o e q u a t io n s ( 1 ) - ( 1 0 ) c a n t h e n b e w r i tt e n

t hus :

n w ( P , , = O ( 1 1 )

f) d n


5/11

w h e r e w i s t h e v i r t u l

disp lacement .

r~

43

On subs t i t u t i on o f (8a ) - ( lO) i n to (11 ) and upon i n t eg ra t i on by pa r t s ,

n O w O u- ~ z E A ~ z

d[ l + wl , f f i, , Ph +

w l , = , , P p

+ [ w p A i i + v 0 - 7 A ) d f l = 0 ,

Jt~

(12)

/ n w p A i id D , + f w O ~ 1+

J a ~ ) d l'l + / n a w 0 u

~ - ; E A ~ d n

+ w l ~ = , ,~ A p 1 + J p~ p ) = f n w T A d ~ - w l , = ~ , P h , 13)

where A r = A [ z = z , i s the p i le 's ef fective po in t -bear ing area . A s im i lar form ulat ion

can be u sed wi th equa t ion (85). In equ at ions (12) and (13), the v i r tu al d isp lacem ent

w m us t be cons i s t en t wi th cons t ra in t s imposed on t he body . Thus fo r a p ile w hose

t ip is f ixed on a f i rm base, wlz=z~ = O, which parallels equation (7).

By ~ubs t i t u t i ng i n t e rpo l a to ry expans ions fo r u and i t s de r iva t i ves , equa t i on

1 3 )

reduces to th e fo llowing sem i-d iscre te equa t ion of m ot ion for the w ave equ at ion

problem:

M a n + l + p ( v n + l , d ~ + l ) = f n + l , (14 )

where M i s the cons i s ten t mass ma t r i x g iven by

M = / n N T p A N d f l ; (15)

fn+l i s the ex ternal force vector represent ing the cont r ibut ions of the p i le ' s se l f -

we igh t and t he p resc r ibed impac t fo rce eva lua t ed a t t he s ame t ime s t a t i on t n+ l ,

thus :

= / n N T 7 A d l~ - Ph[t=t.+~) h ; (16)

~ l

a n d p ( v n + l , d n + l ) i s t h e i n t e r n a l f o rc e v ec t o r g iv e n b y:

p ( v , + l , d , ,+ l ) = / n N T f ~ + I 0(1 + J , N . v n + l ) d [2

4 - / B T E A ) B d ~ ' d n + l 4 [ ~a + lA p ( 1 4 Jpvplt , .+,)] p, (17)

where N i s t he sha pe func t i on m at r i x , B i s t he s t r a in -d i sp lacemen t t r ans fo rm at ion

m at r i x , d i s t he nod a l d i sp lacemen t v ec to r , v i s the n oda l ve loc it y vec to r and a is


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the n oda l a c c e l e r a tion ve c to r . I n e qua t ion ( 15 ) , M m a y be a ssume d c ons t a n t f o r

a g iven p i le . Se lec t ive numer ica l in tegra t ion of th is ma t r ix a t the nodes resu l t s in

ma ss lum ping a nd M be c oming d i agona l . I n e qua t ions (16 ) a nd ( 17 ) , i h a nd ip

a r e c o lumn ve c to r s w i th un i t e l e me n t s i n t he r ows c o r r e spond ing to the e qua t ion

num be r s f o r node s h a nd p , r e spe c t ive ly , a nd z e r o e l e me n t s e l se whe r e . No te tha t f o r

a p i le whose t i p is f ixe d on a f i rm ba se , t he t e r m invo lv ing ip d r ops ou t o f e qua t ion

(17) .

T R A N S I E N T A L G O R I T H M

Eq ua t io n (14) ho lds for bo th l inea r and non- l inea r (o r e la s to-p las t ic ) so il s tr e ss -

s t r a in be ha v iou r a nd c a n be d i sc r e t i z e d w i th r e spe c t t o t ime by e mploy ing a n im-

p l i ci t ve r s ion o f Ne w m a r k ' s i n t e g r a t ion sc he me fo r a s e c ond o r de r e qua t ion [ 7 ]. Th e

a lgor i thm employed he re in cons is t s of a pred ic tor phase , fo l lowed by a mul t i -pass

c o r r e c to r pha se de f ine d by

K.A .d lq = ~ [ i ] , ( 18 )

w h e r e

K* M /(A I2/~ ) + tan xc t-n

= , ~ C n _ ] _ l / A l ~ )

~L ~'n-I-I

(19)

is the e f fec tive s t if fness ma tr ix ,

2[i] = fn+ l M a~l+ l vIil A[i] ~ (20)

- - v ~ , n + l , n + l /

i s the r e s idua l f o r c e ve c to r o f t he i t h i t e r a t io n , a n d Ad[ q i s the s e a r c h d ir e c t ion o f

the i t h i t e r a t ion . Th e c o r r e c to r pha se is r e pe a te d un t i l t he r e sidua l o f t he m a t r ix

e qua t ion i s d r ive n c lo se to z e r o . S e e Owe n a n d H in ton [ 9] f o r a su m m a r y o f the

m a in f e a tu r e s o f t h i s a lgo r i thm.

The a lgo r i thm shown in e qua t ions ( 18 - 19 ) c on ta in s Ne wma r k ' s i n t e g r a t ion pa -

r a m e te r s 7 a nd /~ . F o r s t a b il i ty , 7 > - 1 /2 a nd ~ = i / 4 ; f o r s e c ond- o r de r a c c u r a c y

7 = 1 /2 [ 10 ] . I n e qua t ion ( 19 ) A t = tn+ l - t n is t he t ime s t e p wh ich gove r ns the

a c c u r a c y o f t he so lu t ion .

De f in ing P n+ l = p ( v ,~+ l , dn + l ) , t h e t a nge n t i a l v isc os ity m a t r ix a t f u tu r e t ime

s t a t i o n

t n + l

i n e qua t ion ( 19 ) is e va lua t e d , t hus :

c t & n O p n + : t / n

n l ---- ~ = N T~ n+ ISJ , N d f l + ( @ n + I A p J p ) i v , (21)


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4 5

w h e r e i p is a s q u a r e m a t r i x c o n t a i n i n g a n e l e m e n t o f u n i t y i n t h e d i a g o n a l s l ot

c o r r e s p o n d i n g t o t h e e q u a t i o n n u m b e r f o r n o d e p , a n d z e ro e l e m e n t s e ls e w h er e .

T h e t a n g e n t i a l s t i f f n e s s m a t r i x i s g i v e n b y :

g t , n

OPn+l= / N T O f a + lo 1 .b j ,~ n + l )N d a ~ _ /f lB T E A ) B d n

n l = 0 d n l

+ ~ A , ( 1 +

Jpvpl,.+

2 2 )

: n t h e n u m e r i c a l e v a l u a t i o n o f e q u a t i o n s ( 2 1 ) a n d ( 2 2 ) , th e s t r e s s e s , s t r e ss g r a d i e n t s

a n d v e l o c i ty p ro f i l e /~ n + l a x e e v a l u a t e d a t t h e G a u s s i n t e g r a t i o n p o i n t s .

T h e t i m e - s t e p p i n g s c h e m e o f e q u a t i o n ( 1 4) m a y b e i n i ti a te d b y o b s e rv i n g t h a t

a t t i m e t = to = 0 , b o t h d o a n d v o a r e k n o w n f r o m ( 5 ) a n d ( 6 ). T h e i n i ti a l

a c c e l e r a t i o n v e c t o r a o c a n t h e n b e c o m p u t e d f r o m ( 1 4 ) t h u s :

a o = M - 1 ( fo - p ( V o , d o ) ) ( 2 3 )

f o r u s e i n t h e f i r s t t i m e s t e p .

N U M E R I C A L E X A M P L E

I n t h e f o ll o w i n g e x a m p l e , a n e l a s t ic p i l e is p a r t i a l l y e m b e d d e d i n a n e l a st ic -

p e r f e c t l y p l a s t i c s o il m e d i u m . C o n s i d e r t h e f o l l ow i n g d a t a f o r a t y p i c a l p il e d r i v in g

p r o b l e m ( r e f e r t o F i g u r e 2 ) :

1 . H a m m e r T y p e : V u l c an N u m b e r 1 ; r a t e d e n e r g y = t 5 f t - k ( 20 .3 3 k N - m ) ;

h a m m e r e f f ic ie n c y ( f r a c t io n o f n o m i n a l p o t e n t i a l e n e r g y e f fe c t iv e l y t r a n s m i t t e d a t

t h e p i le b u t t ) = 6 4 ,

2 . P i l e T y p e : H P 1 2 x 5 3 ; t o t a l l e n g t h L = 1 , 2 0 0 i n ( 3 0 .4 9 m ) ; e m b e d d e d

l e n g t h L e = 9 6 0 i n (2 4 . 3 9 m ) ; c r o s s s e c t i o n a l a r e a A = 1 5 .6 s q . in . ( 1 0 0 s q . c m . ) ;

m a s s d e n s i t y p = 7 .3 x 1 0 - 4 l b - s e c 2 / i n 4 ( 7 8 , 0 0 0 k g / m 3 ) ; m o d u l u s E = 3 0 x 1 0 s p s i

( 2 1 0 5 M P a ) .

3 . C a p b l o c k T y p e : M i c a r t a ; h e i g h t /) c = 1 2 i n ( 0 .3 0 m ) ; a r e a = A c = 1 00

s q . in . ( 6 4 5 s q . c m . ) ; m a s s d e n s i t y P c = 9 x 1 0 - 5 l b - s e c 2 / i n 4 ( 9 , 7 0 0 k g / m S ) ; u n i t

w e i g h t 7 c = 0 .0 3 5 l b / i n s ( 95 k N / m S ) ; m o d u l u s Ec = 6 x 1 0 e p s i ( 4 . 1 x l 0 s M P a ) ;

c o e f f ic i e n t o f r e s t i t u t i o n e = 0 . 8 0 .

4 . S o il T y p e : S i l ty C l a y ; l o a d - d e f o r m a t i o n r e l a t i o n s u s e d : a s s u m e d e l a s to -

p l a s t i c a s s h o w n i n F i g u r e 2 b .


8/11

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C A P

* VI -T

P I L E ~ . ~ ~ o W TI .IO * )

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P T

I ( : i, ,~ T

6

PROTOTYPE F E M E S H

o )

I T R E I I

8TItA|N

I,

PILE

l INEART RI [ I

~ L _ ~ U A K E

/ I I I P L .

8 1 DER E S I S T A N C E

IT R I [ I I

~ IT I tA IN

CAPBLOCK

O l[A B llt T t [ l l

P O I N TRI - r I I IT ANCE

SOIL

b )

F I G U R E 2 . P i l e d r i v e n i n t o a n e l a s t o p l a s t i c s oi l.

a . S i d e R e s i s t a n c e : t y p i c a l v a l u e p e r i n c h ( m e t e r ) l e n g t h o f b o u n d i n g a r e a

o f p i l e f o r s k i n f ri c ti o n [ 1 1 ] f 0 - - 1 0 0 I b / i n ( 1 7 . 4 9 k N / m e t e r ) ; t o t a l u l t i m a t e s i d e

r e s is t a n c e a s s u m i n g a u n i f o r m d i s t r ib u t i o n, T u - - 1 0 0 , 0 0 0 I b ( 4 4 5 k N ) ; a s s u m e d

d i s p l a c e m e n t t o c a u s e i ni ti al y i e l d i n g ( ' q u a k e ' i n t h e s e n s e o f [ 1] ), Q , - - 0 . 1 0 i n

( 2 . 5 4 r a m ) ; a s s u m e d v i s c o u s d a m p i n g c o n s t a n t , J , - - 0 . 0 5 .

b . P o i n t R e s i s t a n c e : u l t i m a t e r e s i s t a n c e a s s u m e d t o b e 5 0 o f u l t i m a t e s i d e

r e si s ta n ce , P u - - 5 0 , 0 0 0 I b ( 2 2 3 k N ) ; a s s u m e d d i s p l a c e m e n t t o c a u s e in it ia l i e l d i n g

( q u a k e ) , Q ~ = 0 . 1 0 i n ( 2 . 5 4 r a m ) ; a s s u m e d v i s c o u s d a m p i n g c o n s t a n t , J r = 0 . 1 5 .

5 . P r e s c r i b e d S t r e s s P u l s e a t t h e P i l e B u t t : T h e ( a s s u m e d ) i n s t a n t a n e o u s

i m p a c t v e r s u s t i m e c u r v e a s t h e h a m m e r s t r i k e s t h e c a p b l o c k t o d r i v e t h e p i l e i s

s h o w n i n F i g . 3 [ 6] . T h e ( c o m p r e m i v e ) i m p a c t s t r e s s e n e r a t e d a s s h o w n i n F i g . 3 is

t h e r e su l t o f w a v e t r a n s m i u i o n s a n d r ef le ct io ns l o n g b o t h t h e r o d a n d t h e h a m m e r .

A p i e c e - w i N l i n e ar f in it e e m e n t i n p u t f u n c t i o n w i t h a t e n s i o n c u t - o f f s a l s o s h o w n

i n t h i s f i gu r e. T h e t e n s i o n c ut - o f f e p r e s e n t s t h e i n s t a n t a t w h i c h t h e r o d s t a r t s t o


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T I M E M I L L I S E C .

F I G U R E 3 . I m p a c t st re s s v e rs u s t im e : h a m m e r e ff ic ie n cy = 1 00 .

s e p a r a t e f r o m t h e h a m m e r a s t h e c o m p r e ss i ve w a v e a l o n g t h e r o d i s re f le c t e d a s a

t e ns ion wa ve a t t h e ba se a nd r e a c he s the o r ig ina l po in t o f im pa c t . A l l t he o r d in a t e s

o f F ig . 3 w e r e mu l t ip l i e d by a f a c to r o f 0 .80 to a c c oun t f o r e ne r gy lo sse s [ 3 -5 ] .

F igu r e 4 shows p lo t s o f p l a s ti c s e t a nd to t a l d i sp l a c e m e n t p r o fi le s a t va r ious

t i m e s t a t i o n s o b t a i n e d b y u s i n g t h e p r o p o s e d a p p r o a c h . I n t h e a n a l y s i s , a n a t u r a l

t im e s t e p o f A t ---- 0 . 25 mse c wa s u se d . No te in th i s f i gu r e tha t t he long t e r m e f f e c t,

a f t e r a s e ri e s o f wa ve t r a nsmis s ions a nd r e f le c t ions , i s t he de ve lopm e n t o f a f a i rly

u n i f o r m p l a s t i c s e t n e a r l y s u r r o u n d e d b y a n e q u a l l y u n i f o r m t o t a l d i s p l a c e m e n t

t h r o u g h o u t t h e p i l e l e n g t h .

C O N C L U D I N G R E M A R K S

A f in i te e l e m e n t n u m e r i c a l p r o c e d u r e f o r a n a l y z i n g t h e d y n a m i c s o f p i le d r i v i n g

b y t h e o n e - d i m e n s i o n a l t h e o r y o f w a v e p r o p a g a t i o n i s p r e se n t e d . I t i s p r o p o s e d t h a t

t h e i m p a c t f o r c e - t h n e c u r v e b e p re s c r ib e d i n t h e s o l u ti o n , r a t h e r t h a n t h e r a m v e lo c -

i t y , t o b e t t e r d e s c r i b e t h e p i l e / h a m m e r i n t e r a c t i o n d u r i n g t h e p i l e d r i v i n g p r o c e s s

[6 ]. W i t h o u t s i g u ~ c a n t l y e n g e n d e r i n g a d d i t io n a l C P U c o s t s o v e r t h e m e t h o d b a s e d


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T I M E S T E P A d 0 . 2 5 M SE C,

D I S P L A C E M E N T

S C A L E :

F I G U R E 4 . T o t a l p ile d i sp l a ce m e n t a n d p la s ti c s e t.

on e xp l i c i t t i m e - s t e pp i ng u s e d i n p r e v i ous w or k [ 1 , 3 - 5 ] , a n i m p l i c i t t i m e - i n t e g r a t i on

a p p r o a c h b a s e d o n N e w m a r k ' s m e t h o d i s e m p l o y e d , r e s u l ti n g i n a n u m e r i c a l ly s t a b l e

s o l u t io n t o p i le d r iv i n g p r o b l e m s i n d e p e n d e n t o f t h e t i m e s t e p u s e d i n t h e a n a l y s is .

R E F E R E N C E S

[1 ] S m i t h , E . A . L . , P i le d r i v i ng a na l y s i s by t he w a ve e qua t i on . J .

Soil Mech.

Fdns . A.S.C.E. N o . G T 8 ( 1 9 7 9 ) 9 0 9 - 9 2 6 .

[2] Smi th , I . M. , Programming the Finite Element Method J o h n W i le y S o n s ,

1982 , 291 - 297 .

[3 ] R a us c h e , F . , M os e s , F . a n d G o b l e , G . G . , S o i l r e s i s t a nc e p r e d i c t i on s f r om p i l e

d y n a m i c s , J. Soil Mech. Found. Div. A S C E 9 8 , N o . S M 9 ( 1 9 7 2 ) 9 1 7 - 9 3 7 .

[4 ] C o y l e , H . M . , L o w e r y , L . L . J r. a n d H i r s ch , T . J . , W a v e e q u a t i o n a n a l y s is o f

p i l i ng be ha v i o r , C h . 8 :

Num erical Methods in G eotechnical Engineering C. S.

D e s a i / z J . T . C h r i s t i a n , e d s . ( 1 9 7 7 ) 2 7 2 - 2 9 6 .

[5 ] G o b l e , R . R . a n d R a u s c h e , F . , Wave Equation Analysis o/P i le Drieing W E A P


11/11

49

Program, U. S. Dep t. of Transp ortation , Vols. 1-4 (1976).

[6] Fa irhu rst, C ., W ave mech anics of percussive d rilling, M ine ~ Qu arry Engineer-

ing

(1961) 122-130.

[7] Newmark, N. M ., A m ethod of compu tat ion for s tructural dyv am lcs ,

J. Engrg.

Mech. Div. A.S .C.E . (1959) 67-94.

[8] Tim oshen ko, S. P. Go odier, J. N., Theory of Elasticity 3rd edn. , McGraw-

Hill, New York (1970 ).

[9] Owen, D. R. J. and Hinton, E.,

Fin ite Ele m ents in Plastici ty: Theory and

Practice Pineridge Press Ltd., Swansea, U. K. (1980).

[10] Hughes, T. J. R., Analysis of transient algorithms with particular reference

to stability behavior. Ch. 2:

Computational Methods in Mechanics 1: Com-

putational Methods for Transient Analysis Ed. by T. Belytschko T. J . R.

Hughes, North Holland Press (1983) 67-155.

[11] C heU is, It. D., Pile Foundations 2nd ed., M cGraw-Hill, New Yo rk (1961).

Received 23 October 987; revised version received 5 March 988; accepted 7 March

988

dynamics of pile driving by fem

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