Randolph Poulos

7/29/2019 Randolph Poulos

1/17

Terms and Conditions of Use:

this document downloaded from

vulcanhammer.infothe website aboutVulcan Iron Works

Inc. and the piledriving equipment itmanufactured

All of the information, data and computer software (information)presented on this web site is for general information only. While everyeffort will be made to insure its accuracy, this information should notbe used or relied on for any specic application without independent,competent professional examination and verication of its accuracy, suit-

ability and applicability by a licensed professional. Anyone making useof this information does so at his or her own risk and assumes any and allliability resulting from such use. The entire risk as to quality or usability ofthe information contained within is with the reader. In no event will this webpage or webmaster be held liable, nor does this web page or its webmasterprovide insurance against liability, for any damages including lost prots, lostsavings or any other incidental or consequential damages arising from the use

or inability to use the information contained within.

This site is not an ofcial site of Prentice-Hall, Pile Buck, or Vulcan FoundationEquipment. All references to sources of software, equipment, parts, service or

repairs do not constitute an endorsement.

Visit our companion sitehttp://www.vulcanhammer.org
http://www.vulcanhammer.info/http://www.vulcanhammer.info/http://www.vulcanhammer.org/http://www.vulcanhammer.org/http://www.vulcanhammer.org/http://www.vulcanhammer.info/


2/17

Estimating the flexibility of offshore pile groups

M. F. RANDOLPH (Engineering Department, University of Cambridge, UK) andH. G. POULOS (School of Civil Engineering, University of Sydney, Australia)

1 INTRODUCTIONThe overriding criterion in designing piles to support offshore

structures is usually the required axial capacity of the pile. The numberof piles, and frequently the diameter of'each pile, may be fixed at anearly stage of the design, while the final length of each pile is onlysettled after detailed site investigation and the application of a varietyof design procedures for estimating the profile of ultimate skin friction.Tne stiffness of the final foundation must also be estimated accuratelyin order that the dynamic performance of the structure may be assessed.Modern methods of calculating the stiffness of a piled foundation involvefirst estimating the axial and lateral stiffness of a single, isolated,pile, and then using appropriate interaction factors and frame analysistechniques to arrive at a stiffness matrix for the complete pile group.

For a group of vertical piles, the final stiffness or flexibilitymatrix is a 6x6, sparsely populated matrix, relating the 6 degrees offreedom (3 orthogonal deflections, and 3 rotations) to the 6 appliedforces and moments. A flexibility matrix F may be defined by

or - = F P (see Fig. 1).-For linear response, the matrix will be symmetric, so that F = F z 3 and3 2


3/17

F i g u r e 1: A p pl i ed l o a d s w i t h F i g u r e 2: T y p i c a l c o n f i g u r a t i o n o fc o r r e s p o n di n g d e f or m a t io n s p i l e s i n a n o f f s h o r e g ro up

F54 + F45 ' The s t r a i g h t f o r w a r d c o n f i g u r a t i o n o f m ost o f f s h o r e p i l e g ro up s- w he re t h e p i l e s a r e a p p r ox i m at e l y e v e n l y s p a c e d ak oun d a u n i f o r m p i t c hc i r c l e ( s e e F i g . 2) - e n a b l e s s i m p l e a nd r e a s o n a b l y a c c u r a t e e s t i m a t e s t ob e made of t h e t e rm s i n t h e f l e x i b i l i t y m a t r i x F , wit hou t ex ce ss iv e com-p u t a t i o n . To a f i r s t a p pr o xi m at i on , t h e b e h av i o ur i n a ny h o r i z o n t a ld i r e c t i o n w i l l b e s i m i l a r , s o t h a t F 44 y

F45and F55 may be tak en a s eq ua l

t o F F23 a n d F 3 3 r e s p e c t i v e l y . T hi s l e a ve s o n l y 5 unknown te rm s t o b e22 'e v a l u a t e d . The f o l l o w i n g s e c t i o n o u t l i n e s how t h i s e v a l u a t i o n may b ea ch ie ve d, b as ed on f l e x i b i l i t y c o e f f i c i e n t s f o r a s i n g l e i s o l a t e d p i l e ,and a pp r o pr i a te i n t e r a c t i o n f a c t o r s f o r a x i a l a nd f o r l a t e r a l l o a d i n g .2 ANALYSIS

The s t a r t i n g p o i n t f o r t h e a n a l y s i s i s t h e f l e x i b i l i t y c o e f f i c i e n tsf o r a s i n g l e , i s o l a t e d , p i l e ; t h e s e w i l l b e d e no t ed b y f w i t h a p p r o p r i a t es u b s c r i p t s . Thus, f o r a x i a l ( v e r t i c a l ) l o a di n g ,

w h i le f o r l a t e r a l ( h o r i z o n t a l ) l o a di n g

From symmetry, -UM - f g H and t h e l a t t e r c o e f f i c i e n t w i l l b e us ed f o r b ot h


4/17

q u a n t i t i e s . I n a d d i t i o n , f o r l a t e r a l l oa d in g, i t i s u se f ul t o r e f e r t o a- - ' f i xed-head ' mode o f defo rmat ion , where the p i l e head i s re s t ra in ed f rom

r o t a - ti o n . The- r e l e v a n t f l e x i b i l i ty coef l ' i c i en t w i l l b e w r i t t e n f . Cromuf 'eq u a t i o n ( 3 ) , i t may b e shown t h a t

The f a c t t h a t t h e s t i f f n e s s o f a p i l e g ro up i s l e s s t h a n t h e corn-'b in ed s t i f f n e s s of a l l t h e i n d i v id u a l p i l e s , i s c u st om a ri ly q u a n ti f i e d byt h e u se o f i n t e r a c t i o n f a c t o r s a . ( P o u l o s , 1 97 1) . The i n t e r a c t i o n f a c t o rg i v e s t h e f r a c t i o n a l i n c r e a s e i n de f o rm a t io n o f a p i l e due t o t h e pr e s en c eo f a s i m i l a r l y lo a d ed n e i gh b o ur i ng p i l e . T hu s, f o r two p i l e s e ac h w i t h al o ad V a p p l i e d , t h e v e r t i c a l d e fo r ma ti on of e ac h p i l e i s

F or l a t e r a l l oa d in g , i f a l o a d H i s a p p l i e d t o e ac h p i l e , t h en t h e d e fl e c-t i o n u , i s g i ve n by

( o r u = f ( l + a u f ) Hu f o r f ix ed -h ea d p i l e s ) .../ T h e ro t a t i o n 8 i s g iv en by

S i m i l a r i n t e r a c t i o n f a c t o r s a and a may be defined f o r t w o p i l e s l o ad edUM 8Mby a moment M a t the ground su rf ac e. Again , f rom symmetry , a = auM 6HY andt h e l a t t e r term w i l l b e us ed f o r b o th q u a n t i t i e s .

The u s e o f i n t e r a c t i o n f a c t o r s may b e ex t en ded t o ca s e s wh ere t h el o ad s v a ry f rom p i l e t o p i l e . A l o ad P. on t h e jt h p i l e i s t h en s een a stdco n t r i b u t i n g a d e fo rma t i o n 6 . t o t h e i p i l e , where1T h i s ru l e may be g en e ra l i s ed t o i n c l u d e t h e ca s e i = j , b y d e f i n i n g a = l .j j

L t i s now p o s s i b l e t o d edu ce t h e f i r s t te rm i n t h e f l e x i b i l i t ym a t r i x F . Fo r v e r t i c a l l o ad i n g , t h e s ymmetry of t h e o f f s h o r e p i l e g ro upe n t a i l s t h a t each p i l e w i l l b e s i m i l a r l y l oa d ed . T hu s, f o r a n o v e r a l lload V , e a c h p i l e w i l l ca r ry a l o ad V/n ( f o r n p i l e s i n t h e grou p) and t h edeformat ion o f each p i l e w i l l be

n-g = I fi n v (9 )j = 1


5/17

Thus Fl l i s given by

- . - -For convenience, the summat ion l imi t s and indices w i l l b e o m i tt e d i n t h eexpress ions below.

The r e sp o ns e of t h e p i l e g r ou ps t o l a t e r a l l o a d i s c o mp li c at e d byt h e a x i a l l o ad s i n du ce d i n t h e p i l e s d ue t o r o t a t i o n of t h e g ro up . As imple approach (Randolph , 1977) i s t o cons ide r f i r s t t he ' f i xed-head 'mode of d ef le c t io n , where no ro ta t i on of the p i l e head occurs . Under ano v e r a l l h o r i z o n t a l l o ad H, i t may be assumed t h a t each p i l e c a r r i e s auni form load of H/n (Mat lock e t a1 (1980) have found t h a t th e var ia t i ono f h o r i z o n t a l l o a d a ro un d t y p i c a l o f f s h o r e p i l e g ro up s i s v e r y s m a l l ) .Thus t h e f i x e d he ad d e f l e c t i o n i s

The f i x i n g moment Mf n e c es s ar y a t t h e h ea d of e a ch p i l e i n o r d e r t or e s t r a i n r o t a t i o n may b e c a l c u l a t e d from

Thus Mf H 9 ~ e ~= - -nf t 3 ~ 6,

The ov e r a l l moment app l i ed t o t he group i s now M ( the actual moment) -nMf ( to co un te ra ct th e f i x in g moments).

The a p p l ie d moment w i l l be share d between the 'push- pul l ' mode,w here a x i a l l o a ds a r e i nd uc ed i n t h e o u t e r p i l e s a s t h e g ro up r o t a t e s ,and moments a t t he head of each p i l e , i n su ch a way t h a t t h e r o t a t i o n o feach p i l e matches t h a t of the group. The push-pull mode of ro ta t i oninduces ax i a l l oads i n th e p i l e s which may be assumed t o vary a s V ' cos$($ b e i n g t h e a n g l e o f a ny p a r t i c u l a r p i l e fr om t h e x a x i s a n d V ' b e i n g t h ea x i a l l o a d i nd uc ed i n p i l e 1 - s e e F i g . 2 ) . Thus t h e a x i a l d e f l e c t i o n ofp i l e 1 i s

The g ro up r o t a t i o n i s 8 ' = v l /R (where R i s t h e p i t c h c i r c l e r a di us ) andth e moment absorbed i s CV1cos+ x Rcos+ = R V ' C C O S ~ $ .hus t he push-pull 0s t i f f n e s s i s


6/17

R2 1 cos2+S =PP (15)f v 1 VCosa

F or a 'moment M1-a t t he head o f each p i l e , t he r o t a t i o n i s g iven by8 ' = M ' f g M1 a B M. (16)

Thus t h e r o t a t i o n a l s t i f f n e s s ( f o r n p i l e s ) i s

The o v e r a l l moment w i l l be sha red between the push-pul l mode and ro ta t i on almode i n p ro po rt io n t o t he r es pe ct iv e s t i f f n e s s e s . Thus th e moment a t e a c ip i l e h ea d, M ' , may be ca lcu la ted as

1= - . ( M -n nMf) /Xwhere x = 1 + S ISr.PPThe ro t a t i on o f t he g roup i s g iven by

and t h e a d d i t i o n a l d e f l e c t i o n by

The f i n a l fo rm o f l o a d i n g t o b e c on s i d e re d , i s t h a t o f t o r s i o n a lloadin g . Analys i s of p i l e groups under genera l loading us in g computerprograms such as PIGLET (Poulos and Randolph, 1982) , in d i ca te s th a t thet o r s i o n a l s t i f f n e s s of t h e in d i v id u a l p i l e s i s ne g l i g i b l e (% 5%) comparedwi th t he s t i f f n e s s of t he combined g roups due t o t he en fo rced l a t e r a lmovement of p i l e s away from th e a x is of t w i s t . An app l i e d t o rque o f Tmay th us be assumed t o impar t a h o r i z o n t a l l o ad of H ' = T/nR t o each p i l ei n an of fs ho re group. The component of H ' which i s p a r a l l e l t o p i l e 1i s H 1 c o s + , and t h u s t h e ( f ix e d- he a d) d e f l e c t i o n o f e a c h p i l e i s

Tu' = -R 'u f 1 aufcos' (21)g i v i n g a r o t a t i o n o f t h e g ro up o f

I n summary, e x p re s si o ns f o r t h e c o e f f i c i e n t s i n t h e f l e x i b i l i t y


7/17

ma tr ix F may now be w r i t t e n down a s

8~ R~ ( raeM) rcos2$)where x = 1 + S ISr = 1 +PP n f y 1 a, COS+3 ESTIMATION OF FLEXIBILITY COEFFICIENTS AND INTERACTION FACTORS

I n o rd er t o ca l cu l a t e va lues f o r t he components o f F , some es t ima t emust b e made of t h e f l e x i b i l i t y c o e f f i c i e n t s f a nd t h e i n t e r a c t i o nf a c t o r s a . S i nc e th e main pu rp o se of t h i s s i m p l i f i e d a pp ro ac h i s a s ap re l iminary des ign a i d i t i s l o g i c a l t o t u r n t o e l a s t i c t he or y f o r t he sec o e f f i c i e n t s . N um er ic al me th od s su ch a s t h e i n t e g r a l e q u a t i o n ( o rboundary e lement ) method have been used t o ca l c u l a t e t he f l e x i b i l i t y ofs i n g l e p i l e s u n de r v a r i o u s l o a d i n g c o n d i t i o n s . Char t s of v a lues over awide range of p i l e s t i f f ne s se s and geomet ry have been publ i shed by Poulosand Davis (1980). As an a l t e r na t i ve , c l o sed fo rm expres s ions have beenpublished by Randolph and Wroth (1978) f o r a x i a l load ing; and by Randolph(1981) f o r l a t e r a l l o ad i ng . The l a t t e r e x p re s si o ns a r e p a r t i c u l a r l ys i mp l e t o u s e a s t h e y a r e c a s t i n a f or m w hic h i s independent of t he p i l el e n g t h . (The l a t e r a l r e sp on se of p i l e s i s r a r e l y a f f e c t e d by t h e o v e r a l ll e n g t h of t h e p i l e . )

P oulos and Dav is (1980) a l s o g ive cha r t s showing i n t e ra c t i o n fa c t o r sf o r a x i a l and f o r l a t e r a l l o a d in g . I t i s worthwhile making a few obser-va t i ons concern ing t hese i n t e ra c t i o n fac to r s , and t he manner i n whichthey may be expec ted t o va ry wi th t h e p i l e spac ing . F igu re 3 showsval ues of a t aken f rom Poulos (1979a) f or l en gth t o d iameter r a t i o sv '(L /d ) o f 10 , 25 and 50 . Va lues a r e p lo t t e d fo r t h re e va lues o f t he


8/17


9/17

Fi g u re 3 : I n t e r a c t i o n f a c t o r s f o r v e r t i c a l l y lo ad ed p i l e s

- - -- -Definit~on f s and

p ; o e xequat~on 25 l Poulos (1971

I 1 I I I I I I I I I I -0 2 L 6 8 10 12 1C 16 18p ~ l e pacing 2Fi g u re 4 : I n t e r a c t i o n f a c t o r s f o r h o r i z o n t a l l y l oa d ed f ix ed -h ea de d p i l e s


10/17

spacings (where s -+ 0 would imply a -+ a), a t ran sfo rma t ion may be i n t r o-duced s uc h t h a t , w here a n i n t e r a c t i o n f a c t o r i s c a l c u l a t e d t o be g r e a t e rthan ' I3, h e n - t h e v a l u e i s r e p l a c ed by

Th is has t he e f fe c t of smoothly t r ans fo rm ing t he hyperbo la i n t o a pa rabo l af o r v a l ue s o f a g r e a t e r t h a n (Note th a t equat ion (26) im pl ies a + '1

F igu re 4 shows computed values of a compared with values pub-uf 'l i sh ed by Poulos (1971) . The agreement i s ge ne ra l ly good, f or bo thext reme values of B = 0" and B = 90". From eq ua t io ns (25 ), i t should ben o te d t h a t i n t e r a c t i o n f a c t o r s f o r i n t e r me d i a t e v a l u e s of B may be calcu-l a t e d f rom the ex treme va lue s by

The fo rm o f v a r i a t i o n of i n t e r a c t i o n f a c t o r w i th p i l e s p a c i ng g i ve nby equat ion (23) and by equ at i ons (25) r e f l e c t s th e manner i n which th ede f l e c t i on s dec rease away from the p i l e - i . e . l o g a r it h m i ca l l y i n t h ec a s e o f a x i a l l o a d i n g , and i nve r se ly i n t h e c a s e of l a t e r a l l o a di n g. Thec o e f f i c i e n t s ha ve be en c ho se n t o g i v e a b s o l u t e v a l u e s w hic h a r e c o n s i s t e n tw i t h p u bl i sh e d v a l u e s b a se d on e l a s t i c s o l u t i o n s . Where a p p r o p r i a t e , f o r .e xa mp le , i f a zone of p l a s t i c y i e l d h as o c c u rr e d c l os e d t o t h e p i l e , t h eabs o lu t e va lues may be dec reased by adop t i ng sm al l e r c oe f f i c i e n t s , t husp r e s e r v i n g t h e f o rm o f t h e v a r i a t i o n o f a with the spacin g between th ep i l e s .

4 COMPARISONS O F SIMPLIFIED ANALYSIS WITH RIGOROUS ANALYSIS AND WITHEXPERIMENTAL RESULTSI n o r d e r t o i l l u s t r a t e t h e d e gr ee o f a c cu r ac y t o b e e x pe c te d from

the an a ly s i s ou t l i ned i n s ec t i o n 2 , two exam ple ana lyses w i l l be discussed.The f i r s t exam ple concerns a t y p i ca l , 11 p i l e o f f s h o r e g ro up t h a t h a s b ee nused pr ev iou sl y be Poulos (1979b) as an example. The p i l e s were oflength 72 m , diameter 1 .37 m and were a r ranged a round a p i t ch c i r c l e ofr a d i u s 5 . 4 m ( g i v i n g a s p a c in g t o d i a m e te r r a t i o o f 2.2 f o r n e ig h b ou r in gp i l e s ) . The wa l l th ic kne ss of 50 mm giv es an eq ui va le nt Young's modulusf o r a s o l i d p i l e o f % 30,000 M N / ~ ' and t h e s o i l s t i f f n e s s h a s b ee nassumed t o var y from 18 M N / ~ ~t t h e g round su r f ace up t o 30 M N / ~ ' a t t he


11/17


12/17


13/17

w i t h u n i t s o f mm , kN and ra di a ns . The measured ve r t i c a l and ho r i z on t a lf l e x i b i l i t i e s o f t h e 1 2 p i l e gr oup w ere Fl l = 0.761 m/kN and Fz2 = 0.95m / k N r e s p e c t i v e l y . A ga in , t h e p r e d i c t e d and m ea su re d f l e x i b i l i t i e s a g r e et o w i t h i n 1 0 % . - T a b l e 1 summarizes t he comparisons between t he measured .and p red i c t ed va lues o f 31 nd F 22, and a l so g ives va lue s c a l c u l a t ed f romth e programs DEFPIG and PIGLET which a r e i n good a greement w it h th os e f romthe s im p l i f i ed method p resen t ed he r e .

Table 1: Measured a nd p r e d i c t e d f l e x i b i l i t y c o e f f i c i e n t s f o r modelt e s t s of Ferguson and Lau r ie (1980) .

Addi t i ona l ho r i zon t a l l oad t e s t s were a l s o conduct ed on 8 and 12p i l e g rou ps w i th t h e p i l e s i n c l in e d ( p a r a l l e l ) a t 7 . 5 ' t o t h e v e r t i c a l -s i m u l a t i n g t h e t y p i c a l a rr an ge me nt f o r a n o f f s h o r e p i l e g ro up . Thef l e x i b i l i t y m a t r i x F may be m od i fi ed t o a l l o w f o r b a t t e r o f t h e c o mp le tep i l e group , by pre- and pos t -m ul t ip ly ing by the app rop r ia t e t ransforma-t i o n m a t r i x , a ss um in g t h a t t h e a x i a l and l a t e r a l r e sp o ns e of t h e g ro upremains una l te red by smal l amounts of b a t t e r . Thus, f o r an ang le ofb a t t e r 02 p, t h e new f l e x i b i l i t y m a t r ix i s

Measured(mm/kN)

0.888

1 . 3 1

0 .761

0.95

*

8 p i l egroup

1 2 p i l egroup

Predicted (mm/kN)

TF * = T F T where T~ i s t h e t r a n s p o s e o f T and

F l e x i b i l i t yC o e f f i c i e n t

F1l

F22

F1l

F22

PIGLET

0.956

1.25

0.865

0.94

iedAnalys i s

0.810

1.39

0.740

1.04

DEFPIG

0.843

1 .23

0.785

0.87


14/17

Under a h o r i z o n t a l l o a d H , t h e h o r i z o n t a l d e f l e c t i o n w oul d t hu s b eu = ( s i n 2 u F + cos2u F ~ ~ )

- .11

For u = 7.5', t h e m od if ie d h o r i z o nt a l f l e x i b i l i t y o f t h e b a t t e r e d p i l e 'groups i s 1 . 3 8 f o r t h e e i g h t - p i l e g ro up an d 1 . 0 3 f o r t h e 1 2 p i l e g ro up .T he se f i g u r e s a r e o n l y m a r g i n a l ly s m a l l e r t h a n f o r t h e c o r r e s po n d in gv e r t i c a l p i l e gr ou p. I n p r a c t i c e , h ow ev er , t h e measur ed f l e x i b i l i t i e sw e re mor e m ar ke dl y r e d u ce d by b a t t e r i n g t h e p i l e s . F or t h e e i g h t p i l eg roup , th e f l e x i b i l i t y was reduc .ed to 1 .04 mm/kN (compared wi th 1 . 31 f o rt h e v e r t i c a l p i l e s ) , w h i l e , f o r t h e t w el ve p i l e g r ou p, t h e measur edf l e x i b i l i t y was 0 .92 mm/kN (compared wi th 0 .95 ) . I t i s n o t c l e a r why t h emeasured f l e x i b i l i t y of t h e b a t t e r e d p i l e g ro up , p a r t i c u l a r l y t h e e i g h tp i l e g r ou p, i s s o much l e s s t h a n t h a t o f t h e v e r t i c a l p i l e g ro up , a l th ou g ht h i s t en de nc y h a s b ee n no t e d p r e v i o us l y i n r e l a t i o n t o p i l e g ro up s i ns a n d ( Po u lo s an d R an do lp h, 1 9 8 2 ) . F u r t h e r r e s e a r c h i n t h i s a r e a w ou ld b eu s e f u l i n o r d e r t o i n v e s t i g a t e w he th er t h i s e x p er i me n ta l r e s u l t i s d ue t oa c ha nc e v a r i a t i o n i n t h e s o i l p r o p e r t i e s .5 CONCLUSIONS

I t i s common p r a c t i c e f o r t h e r e s p o n s e o f a p i l e g r o u p t o b ee s t i m a t ed b y c o ns i d e r i n g f i r s t t h e r es po n se o f a s i n g l e p i l e and t h e nu s in g a p p r o p ri a t e i n t e r a c t i o n f a c t o r s t o a l lo w f o r gr oup e f f e c t s . T hi sa p pr o ac h a l l o w s d i f f e r e n t t e c h n i q u e s t o be u se d f o r t h e d i f f e r e n t s t a g e s .Thus t h e s i n g l e p i l e r e s p o n s e may b e e s t i m a t e d f ro m a s u b g ra d e r e a c t i o nap p ro ach , u s i n g n o n - l i n ea r t - z o r p - y c u r ve s , w h i le i n t e r a c t i o nf a c t o r s a r e u s u a l l y e s t i m a t e d fr om e l a s t i c t h e o r y . The i n co n s i s t e nc y ofu s i n g d i f f e r e n t m odels o f s o i l b e h av i o ur i s , t o some e x t e n t , b a l an c e d byt h e r e s u l t i n g e a s e w i t h w hi ch t h e r e s p o n se o f a comple te p i l e g roup mayt h e n b e c a l c u l a t e d .

F o r c o nf i g u r a t i on s o f p i l e s which a r i s e o f f s h o r e , c a l c u l a t i o n o ft h e o v e r a l l f l e x i b i l i t y m a t r i x o f t h e g rou p becomes p a r t i c u l a r l ys t r a i g h t f o r w a r d . T h i s p a p e r h a s d e s c r i b e d t h e way i n w hi ch t e rm s i n t h ef l e x i b i l i t y m a t r i x may be d educed f rom f l e x i b i l i t y c o e f f i c i e n t s f o r as i n g l e , i s o l a t e d , p i l e and a p p r o p r ia t e i n t e r a c t i o n f a c t o r s . W it hi n t h ea c c u r a c y w i t h w hi ch s o i l s t i f f n e s s p a r a m e te r s may b e d e t e r m in e d , i t i ss u f f i c i e n t t o e s t i m a t e i n t e r a c t i o n f a c t o r s f rom si m pl e e x pr e s si o ns whicha r e b a se d on s t u d i e s t r e a t i n g t h e s o i l a s a n e l a s t i c c ont inuu m.


15/17

C om pa ri son betwee n the a ppr ox im a te e s t im a te o f th e g roup f l e x i b i l i t y ,m a t r ix ob ta i ne d by t h i s a pproa ch a nd a more r ig o r ou s a na l ys i s u s ing p ro -grams s uc h as PIGLET o r DEFPIG shows rea so na bl y good agre em en t. Whena p p l i e d t o model t e s t s of o f f s h o r e p i l e g r o u p s , t h e u s e of m ea su re d s i n g l ep i l e f l e x i b i l i t i e s , t o g e th e r w i th i n t e r a c t i o n f a c t o r s g iv en b y eq ua ti on s(2 3) a nd ( 2 5 ) , y i e l d s o v e r a l l f l e x i b i l i t i e s o f t h e g ro up w hi ch a r e c l o s et o t h o s e a c t u a l l y m ea su re d.

The s c a l e of p i l e f o u n d a t io n s f o r o f f s h o r e s t r u c t u r e s w a r r an t s t h eu s e of s o p h i s t i c a t e d a n a l y t i c a l m.ethods i n o r d e r t o a s s e s s t h e l i k e l yr e sponse o f th e f ounda t ions under work ing c o nd i t i on s . However, t h e r e i sa n i mp o rt an t r o l e , p a r t i c u l a r l y i n t h e e a r l y s t a g e s of a d e s i gn , f o rs imple methods , where the s t i f f n e s s of t he comple te founda t ion may ber a p id ly e s t im a te d us in g m inim al c om puta tion .REFERENCESCOOKE, R . W . , PRICE, G . and TARR, K . (1980) - ' J a c k e d ' P i l e s i n London

Clay; In te ra c t io n and Group Behaviour Under Working Con di t io ns ' .Geotechnique , 30 (2 ) 9 7-136-FERGUSON, A . J . and LAURIE, R . J . (1980) - ' B eha viour o f Of f shor e P i l e

Group Co nf i gu ra t io ns ' . BE(Hons) Thes is , Dept. of Ci vi l Engin eer ing ,Un ive rs i ty of Sydney

MATLOCK, H . , I N G R L Y , W . B . , KELLEY, A. E. and BOGARD, D. (1980) - ' F i e l dTe s t s on the La te ra l - Loa d B e ha v iour of P i l e Groups i n S o f t C la y ' . 'Pr oc. 1 2t h OTC Conf. Hou sto n, P ap er OTC 3871, 163-174

POULOS, H . G . (1971) - ' B e ha v iour o f La te ra l ly -Loa de d P i l e s . I1 - P i l eGr ou ps '. J n l . S o i l . Mechs. Fndn s. Di vn ., ACSE, 97 (SM5) 733-751. -

POULOS, H . G . (1979a) - 'G rou p F a c t o r s f o r P i l e - D e f l e c t i o n E s t i m a t i o n ' .J n l . Geo t. Eng. Di vn ., ASCE, 105 (GT12) 1489-1509-

POULOS, H . G . (1979b) - 'An Approach f o r the Ana lys is of Of fsho re P i l eGro ups 1. Pro c. Conf. on Nurn. Methods i n Of fsh ore P i l i n g , ICE,London, 119-126

POLTOS, H . G . and DAVIS, E . H . (1980) - ' P i l e F ounda tion Ana lys i s andD es ig n' . John Wi ley, New York

POULOS, H . G . and RANDOLPH, M . F. (1982) - ' A St ud y of Two Methods f o rP i l e Group Ana lys i s ' . To be pub l ish ed i n Jn l , Geot. Eng. Divn. ,ASCE

RANDOLPH, M. F. (1977) - ' A Th eor e t i ca l S tudy of the Per formance ofP i l e s ' . Ph .D. Th es i s , Univ . of Cambr idge


16/17

RANDOLPH, M. F. (1981) - ' Th e R es po ns e of F l e x i b l e P i l e s t o L a t e r a l .Loading ' . Geotechnique , -1 (2) 247-259RANDOLPH,. M . F. and WROTH, C . P. (1978) - 'Ana l ys is of Deformation ofv e r t i c a l l y Loaded P i l e s ' . J n l . G e o t . Eng. Divn., ASCE, -04 (G'S12)1465-1488WIESNER, T. J . and BROWN, P. T. (198 0) - 'Laboratory Tests on ModelP i l e d R a f t F o u n d a t i o n s ' . Jnl. Geot. Eng. Divn., ASCE, -06 (GT7)767-783APPENDIX

C a lc u la t io n of t h e t e rm s i n t h e f l e x i b i l i t y m a t r i x F i s i l l u s t r a t e dh e r e f o r t h e model e i g h t p i l e group te s t e d by F e r guson a nd La u r i e ( 1980) .

The p r o p e r t i e s o f t h e g ro u p of p i l e s a r e :R = 164 mm, d = 6 .5 rmn, R = 38.5 m , n = 8 .The s i n g l e p i l e f l e x i b i l i t y c o e f f i c i e n t s a re :

= 0.0 0496 rad/kNmm, whence - 9 ~ f U f - fU, - f i H / f g M= 3.47 mmlkN.From equations (23) and (25) - ( 26 ) t a k ing p = p c = 1 and E / G =P c2.62 x l o 4 ( c on s is t en t wi t h t h e f l e x i b i l i t y c o e f f i c i e n t s f o r l a t e r a l

l o a d i ng ) , t h e i n t e r a c t i o n f a c t o r s may b e t ab u l a t e d f o r p i l e s 2 t o 8( r e l a t i v e t o p i l e 1 ) a s

P i l e! 3 4 5 6 7 8

a uf 1 0.324 0.230 0.217 0.217 0.217 0.230 0.324( l a t e r a l ) 1

spacing,^C O S ~

av

29.5 54.4 71.1 77 . O 71. 1 54.4 29.50.707 0 -. 07 -1 -.707 0 0.7070.266 0.171 0.129 0.117 0.129 0.171 0 .266

a cos$ / 0. 18 8 0 -.G92 -.I 17 -. 32 0 0. 18 8v

aufcosJl( t o r s i o n ) 0.331 0 - .095 -.lo8 -. 095 0 0 .331


17/17

I t shou ld be no t ed t h a t t h e i n t e ra c t i o n fa c t o r a between any twoup i l e s w i l l b e d i f f e r e n t f o r l a t e r a l l o a d in g and f o r t o r s i o n l o ad i ng .Th i s i s because t he ang l e B i s d i f f e r e n t f o r t h e two f or ms of l o a di n g .I t ma ; be shown- t h a t B = n /2 - $12 f o r l a t e r a l l o a d in g , w hi l e B = $ / 2f o r t o r s i o n l o a d i n g .

The summation needed fo r t h e terms i n F may now be evaluated( remember ing t he co n t r i bu t i o n of p i l e 1 , where a = I ) , g i v i n g :

1 a v = 2.249, 1 avcos$ = 1.075, 1 a u f ( l a t e r a l ) = 2.759,1 aufcos$ ( t o r s ion ) = 1.364, 1 a e H = 1.205, 1 a,, = 1.037,1 cos2q = 4 .

The parameter x may be ca l cu l a t e d as x = 2 .231 , g iv in g t he p ropor t i on o fmoment t h a t i s t r a ns m i tt e d t o t h e p i l e h ea ds a s 1 / x = 0.45.

F i n a l l y , t h e t er ms i n F may b e c a l c u l a t e d , g i v i n g

Randolph Poulos

Documents

Transcript of Randolph Poulos