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Computer Methods and Advances in Geomechanics, Beer, Booker Carter eds 1991 Ba/kema, Rotterdam./SBN9061911893raft fo unda tio n ana ly sis b y fin ite elem en ts

.Y .Griffithsepartment of Engineering University of Manchester UK.C la ncy &M . Randolphepartment ofCivil and Environmental Engineering University ofWestern Australia Nedlands w.A. Australia

ABSTRACT: A program for the analysis of pile-raft Ioundation systems has been developed. The pilesare modelled with rod elements and the raft by 'thin' plate elements. The three kinds of interactions,namely pile-soil-pile, pile-soil-raft and ralt-soil-raft are accounted for using the elastic theory of Mindlin.Results have been validated against elastic solutions for a range of pile/soil/raft stiffness properties.

I INTRODUCTIONHeavy structures are regularly supported on piledaft Ioundations, yet relatively little is known abouthe complex interactions that occur between theaft, piles and soi\. Current design methods areased on approximate analysis of the interaction be-

n the pile group and the raft (Randolph 1983,eming ei ai 1985). Recent experience by the au-

hors on practical problems of this type, have high -ghted the need for a more systematic approachthe analysis of these soil/structure interactions.e result of work by Clancy (1990) has led to

he development of a general purpose finite elementethod for computing deflections and moments in aed raft foundation. The program allows the stiff-

ess of the raft, the soil and the piles to be variedgether with the pile group spacing and loading.is papel' describes some of the main Ieatures of

he method and presents results for a 3 by 3 pile.

P ILE GROUPsolution to the problem of pile-soil-pile interac-

ion in the analysis of pile groups has been sug-ested by Chow (1986,1987), and a computer cod-g of this method has been presented by Smith and

iths (1988). Figure 1 shows the main points ofhe analysis, namely the pile stiffness, the soil stiff-

ness in the form of t-z springs, and the interactionsbetv , =n the piles through the soi .

Axial Load Axial Load

One-dimensional pile element2 Ground resistance at each node represented bynon-linear 'T-Z' springs3 Pile-soil-pile interaction effects calculatedbetween pairs of nodes using Mindlin's equation

Figure l. Representation of piles and SOl .

Chow's work is based on the popular methodof single pile analysis where one-dirnensional bearn-

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column finite elements are used to model the pile.The soil response is modelled by a discrete springat each node, based on load transfer curves sug-gested by Seed and Reese (1957), (referrecl to as i-zor p-y curves, depending on whether axial or lateralresponse is being considered).

In the present work, only linear z response hasbeen considered, with the gradient of the t-z curvelinked to the elastic soil moclulus using thc relation-ships proposed by Randolph and Wroth (1978).

To model a pile grollp, Mindlin's (1936) ela.s-tic continuum solution, for displacernents due to apoint load , is used to provide interaction effects be-tween pile nodes.

Although only axially-loaded vertical piles havebeen considered here, the method can be extendedto look at more general pile groups with lateralload-ing and raked piles, and is capable of modellinga soil in which the stiffness increases linearly withdepth.

Unlike the boundary elernent method, consid-eration of non- homogeneous soil and soil non-linearity do not increase the size of the system ofequations to be solved.

3 PLATE ELEMENTTo model the raft , a four-no de quadrilateralisoparametric plate bencling finite element (Hughes,1987; Huang, 1989) was first considered. The ele-ment had three freecloms at each node: a transversodisplacernent and two rotations. Although in thisapplication a 'thin' plate was being considered, theelement was based on 'thick' plate theory such thattransverse shear strains were accounted for.

It had been dernonstrated that uncler full inte-gration the element would give excessively stiff solu-tions in thin plate applications. This 'shear locking'behaviour can be overcorne by using reduced inte-gration on the shear term in the stiffness rnatrix.

The plate bending finite elements were attachedto the piles so that vertical freecloms were commonat the connected nodes. To test the adequacy ofthis arrangernent, the raft was given a very highstiffness. Thus when a uniforrnly distributed loadwas applied, a uniform displacernent of the raft wasexpected. However, this wasn't observed becausethe reduced integration technique had allowed for-mation of two spurious zero-energy modes. Underother circurnstances, e.g. a simply- supported or

I Applied I a d

fixed boundary condition, these modes would beunoable to form (Clancy and Griffiths 1991).

A seconcl, triangular isoparametric element, alsobased on thick plate theory (Zienkiewicz and L e f e b vre, 1988) was considered. Again, reduced integration was employecl to avoid shear locking, but theuse of 'bubble functions' prevented the formation ofspurious zero-energy modes. However, it was foundthat a large nurnber of degrees of freedom were re-quired to ensure convergence to a solution.

j . - - - / 1j 3 Plata-bendin2 finite elementmesh1 Pile-so il -pile interaction2 Pil e-soil-raft raft-so il-pile interactio n3 Aaft-soil-raft in teraction

Figure 2. Full pile group analysisincluding raft-soil-pile interaction

The third elernent considered was a four-noderectangle (Smith and Griffiths, 1988) and, unlikethe others, non- isoparametric which rneant that itwas no longer possible to model a general raft ge-ometry. However, the element was based on 'thin'plate theory so that no shear locking was encoun-tered, even under full integration. It was also foundthat fewer freedoms were required for convergencebecause of a fourth 'twist' freedom at each node.

Results of a number of plate bending testsshowed that the non-isoparametric element was farsuperior where the geometry of the problem allowed

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its use. ln cases where the raft can t be exactlymodelled by rectangular elements, it might be ade-quate to use the element in a piecewise approxima-tion.

4 RAFT-SOIL-PILE INTERACTIONHaving decided on the most suitable plate-bendingfinite element to model a pile raft , and having at-tached it to the piles via common vertical freedoms,all that remained was to take account of raft-soil-pile interaction.

Hain and Lee (1978) hael also useel finite ele-ments to model a raft , in conjunction with a bounel-ary element pile group analysis. Following theirmethod, it was assumeel that a uniform pressurearea existeel around each raft noele. This allowedthe elastic solutions of Girouel (1968) to be usedto provi ele a soil 'spring' stiffness at each raft node.Minellin 's equation was again used to calculat.e threenew sets of interaction: raft-soil-raft interaction;raft-soil-pile interaction; anel pile-soil-raft interac-tion (figure 2.

Before analysing the complete raft-soil-pile sys-tem, this method of raft analysis was checkedagainst analytical solutions given in the SyelneyUniversity data sheets. Figure 3 shows a typicalset of results for a raft of lcngt.h L anel breaelth Bsupported only by soil, with a point load P actingalong the centreline a elistance sL from one end.The beneling stiffuess of the raft is proportional tothe Young's moelulus En anel the cube of the thick-ness Following I-Iain anel Lee (1978), a raft-soilstiffness ratio frs is introduccd given by:

f rs =raft-soil stiffness4 E rB t3 1 - 1 J;)/3 7 rE sL 4

where E s anel I J s are respectively the Young's mod-ulus and Poisson's ratio for the soi .The results in figure 3 are for the case of L / B

1 0 , f rs = 0.001 (a very Aexible raft) and s = 0.1anel 0.5. The centreline elisplacement W is plotteelnormalised as w L E s/ P l - / I ; Beneling momentsin the plane of the centreline are plottcd normaliseelas M / P L .

The finite element analysis shows very goodagreement with the analytical results in this case.Equally gooel results were obtaineel for a11raft stiff-nesses.

r ~AppliedLh ---74k = d . /

0~ 0 V 7 : 1 @ 1. ~ F.E. _0 .1~ 2~ 3 F.E.,-0.5. . .i 5 o- An atyt i ca l s_ 01 .~ ..na~ ;cal. ,_0.5o : 8 - - - - - - - -

plsplacment plstrlbytJgns K-Q 001

0.050 0 4c r 0.03~~~

o ,....,~;r-+-:::::;;:~~:::ii~0.Analy1;cal. ,-0.50.01-0.02

~ F.E.,s-O.l

. F .E . . s -0.5

.o- Analylical 5_0.1

B'ndng Morno , pl lrfbutloDl K-Q 001

Figure 3. Raft footing: Displacements andbending moments.

A complete pileel raft analysis has been under-taken for a 3 by 3 square pile group (see figure 4).the piles h ave length I, eliameter d, Young's moel-ulus Ep and are spaceel at dist.ance s. The raft issquare with an overhang of s/2 beyond the ou terpiles, giving L = B = 3s. the problem may bedefined in terms of a series of dimensionless groups:

pile lengthpile spacingpile/soil stiffnessraft aspect ratioraft/soil stiffness

ds dte;= Ep / eL /Btc.

5102 _ 105

10.01 - 10

ln figu re 4, resul ts are presented for the case ofa uniforrn pressure load q in terrns of:a) Load sharing between pile group anel raft,b) Avcrage displacement Wav normaliseel aswavE s/q B l - I J ?c) Differcntial elisplacement Wmid-edge - Wcentre nor-rnalised as Wmid-edge - Wcentre)/wavd) Maxirnum bending moment in raft Mm a x no r-malised as lvI m a x / q L B

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Plle-Raft System:~~~~-g-g~555045X401 - - - - ~ - - ~ - - - ~ - _- - ~ -~2 25 35 4. 5 5

10g'OK .Percentage or total lc a ca rrteo by ptle group

r ; : a l l r a f t s t H f n e s s e s~ 0.980.7: : t ~ - - - 7 - - - ~ ~ ~ = = = ~ = = = ~

~ 2 2 5 5 4 4 5 S 10810 K.

Average otsota cemen t of Ra t t

8. O 4 fo- K s o o - . : .: , : : - : o ~ , : : -_Krs 10~ 03~ ~ - - - - - - - - - - - - - - K ~ ~ ~ I ~ o n o - - - - -~ 0. 2~~ O., f= ';~~=:~~~~

ox x xxx X X 2 25 3 35 4 45 5

10810 K.Olfferentlal DIsp Average DISP

~ 0.0015 T __ x_X--_xc OOOI~X,, -~~

I0005~ ; ; : ; K C . , Oo~~ Krs001KrsO2 2. 5 3 35 4 4. 5 5

log1O K .ttax. Bendlng Moment In Raft

Rar t etone Pile qroun atonec ar t -scu stiffness 0.01 A aan-scu sttr rness 1.0e can-sou sttr tnes s - 0.1 aan-scu sttr rness > 100

Figure 4. Effects of pile and raft stiffnesson displacernents anelbending moments

In addition to the piled raft , results are shownfor the free- standing pile group and for the raftalone. It can be seen t hat the load sharing andthe average displacement of the raft are dependentmainly on the pile-soil stiffness ratio. The differ-ential displacements and bending moments are de-pendent on the raft-soil stilfness ratio.i 1700E8 1600. .] o 15OC l~ 1400< x : 1300~ 1200+-----+---~----~----+_---;----~> 2 4.5 5

loglO KpsOverall Stlffness of Plle-Raft Foundatlon

2.5 3 3.5 4

1 1 0.65j 0.6 3 0.55~ 0.51 1 0.45j < t : : 0.35~ 0.30.25+---+-~f---+---+---f-----1

2 4.5 5loglO Kps

Proportlon of Total Load Carrled by Raft

2.5 3 3.5 4

.Q. Rand01ph

.. present analysis

Figure 5. Comparison between the presentanalysis and the approximatemethod of Randolph.

The results from the analyses with Jrs = 10were compared with the approxirnate method ofpiled raft analysis proposed by Randolph (1983).Figure 5 shows how the overall stiffness of the foun-dation and the load distribution between piles andraft vary as the stiffness of the piles is increased.

Although the results of Randolph's analysisshow the correct trends across the range, the val-ues of the results differ by up to 6% in the case offoundation stiffness, and up to 30% in the case of

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load distribution. the use of a finer raft mesh in thepresent analysis brings the difference in load distri-bution up to 45%.

5 CO CLUSIONSA method of analysis for pile-raft systems has beendeveloped using finite elements. The three kinds ofinteractions, na.mely pile-soil-pile, pile-soil-raft andraft-soil-raft are accounted for using the elastic the-ory of Mindlin.

In isolation, both the pile group analysis and theraft analysis have been demonstrated to give resultsconsistent with elastic theory.

Example runs on a 3 by 3 rafted pile group giveexpected trends in both displacements and load dis-tributions, as the stiffnesses of the piles and raft arevaried. Comparison of results with an approxirnatemethoel by Randolph (1983) show similar trends,although val ues differ.

Further work will be performed in order to vali-date the program against field performance, but theearly inelications are that the method represents animproved yet efficient method of analysis for piledraft foundations.

REFERLTCES[IJ Y.I