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Proceedings 0/ the Ftrst Pacljic/Asla Offshore Mechamcs SympoSIUmSeoul, Korea, 24 28 June, 1990COPYright 1990 by The Intel natIOnal SocIety o/Of fshore a nd Polm EnglneelS

N LYSIS OF AXIAL RESPONSE OF PILES FOUNDED

IN C LC REOUS SOILS

J P Carter, L H Doi, and C F BoeyFnil Pl'sit,l' of SytillP,Iydney 1l 8TR,UIA

ABSTRACf

A modified load transfer t -z) analysis for axiallyloaded piles is presented in this paper. The method canbe used to predIct the full load-displacement behaviour ofa pile, and unlike the conventional t - z analysis It IS alsocapable of mdicating the complete mechanism of loadtransfer as well as t ~ complete stress path at the pIle-soilmterface. It is able to do thIS by assuming as input aconstitutive model for the pIle-soil interface that permitscoupling between the shear and normal modes ofdeformation. The use of the technique is illustrated for adriven and grouted pile in calcareous soiL

KEY WORDS: Pile foundatIons, Calcareous SOlIs, Offshorestructures, Interface models.

INTRODUCfION

Many fixed offshore platforms used for the recoveryof natural hydrocarbons are supported on pile foundatIons.In a number of regIOns of the world, e,g. off the coasts ofIndia, Australia and in the PerSIan Gulf, the sub-sea soIlcondItions COnsISt of significant deposits of calcareous soils.This soil type has posed a number of dIfficult problemsfor deSIgners of these offshore platforms. One of themajor problems IS the development of very low skmfnctions on the SIdes of pIles dnven mto such soIl masses.These low values of skm fnction are largely a result of

the crushmg of the SOlI around the piles during dnving,whIch gives rise to low normal stresses actmg on the pIleand hence low frictIonal shear resistance. '

Recently, some offshore platforms have been deSignedto be supported on piles which are either fully or partIallygrouted into the calcareous medium. In some cases,pnmary piles are driven to the appropnate depth, and thensecondary piles are constructed by augermg a cylindncalhole and groutmg a steel pipe msert below the pnmarypiles. In other cases the piles are dnven and then cementgrout IS injected through tubes that pass down the mSlde~ n through the walls of the pile. The latter are refer redto as dnven and grouted piles.

263

The load-deformation behaviour and the strength of agrouted pde foundatIon is pnmarily a function of the shearbehaviour of the interface between the cement grout andthe surrounding calcareous soil. Other factors thatinfluence the response of these foundations are thestlffnesses of the pIle sectIon and the SOlI medium and, ofcourse, the magnitude of the applIed load.

The fundamental shear behaVIOur of interfaces mcalcareous soils has been studied m depth in recent years(e.g. Johnston, et al,. 1988; Boey, 1990). Special apparatushas been constructed to test such interfaces underconditions of constant normal stiffness (e.g. Ooi and Carter,

1987; Airey et aI, 1990) ast h i ~

condItIon more closelyrepresents the situation around a pile grouted mto thesurrounding medium. In the field, the radial stiffness ofthe soil medIUm can be regarded, at least to a firstapproximation, as fmite and constant. The results of testsof this type have been used to formulate design guidelinesfor grouted piles in calcareous soils (e.g. Randolph andJewell, 1989).

One of the most popular methods used to analyse thebehaviour of piles subjected to axial loadmg IS the loadtransfer or t - z method, e.g. Coyle and Reese (1966),O'Neill and Mahar (1982), Randolph (1985), Chow 1986).In thls approach assumptions are made about the manner mwhich shear transfer develops between the pile and the SOlIwith pile displacement. For the case of piles in calcareoussoils, these shear transfer relationships, or t - z curves, have

been based on the results of constant normal stiffness(CNS) dIrect shear tests and model pile tests, e.g.Randolph and Jewell (1989). The CNS tests provideInformation on the shear behaVIOur of the grouted pile-soilinterface, but in determinmg the t - z curves account mustalso be taken of the deformation of the SOlI medIUmsurrounding the pIle due to the transfer of shear stresses.The latter IS usually calculated by assummg that the soIlcan adequately be represented as an elastic contmuum (e.g.Randolph and Wroth, 1978).

Most conventional load transfer analyses ignore directlythe interactIon between the shear deformatIons of theinterface and its volumetnc (or normal displacement)

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behaviour. When an interface in this type of material issheared, contraction occurs in the direction normal to theshearing surface. Under conditions of constant normalstiffness, this contractIon will be associated with a reductIonin normal stress. For the case of a pile in calcareous sOlI,shearing at the pile-soil interface will generally reduce theradial normal stress on the pIle and thIS will accelerate thereductIon in shear strength of the interface.

The aim of this paper is to describe an analyticaltechnique to predict the behaviour of piles grouted intocalcareous soils. The approach to the problem hasinvolved the development of a constitutive model for theinterface and its incorporation into a load transfer analysis.The constitutive model IS based on the theory ofelastoplasticity and allows for the complex interactionbetween the shear and normal modes of behavIOur, and InpartIcular it allows for normal contraction with shearIng.As such, the technique does not require the input of anexplicit t -z curve. Indeed, the t -z curve appropnate atany gIven point along the pile is generated by integrationof the incremental constitutive relations for the interfaceand the s u r r ~medium. This represents a rationalextension and a ,c.oDSiderable advance on the conventionalt -z method of :pile ,analysis. The paper contains the

formulation for ,the new method, and example problemsillustrating ,its utility.

CONVENTIONAL LOAD TRANSFER ANALYSIS

In the early development of this method Seed andReese (1957) conducted tests on instrumented piles insaturated clays to understand the mechanics of theinteraction between the pile and the soil. From thesetests, axial load transfer characteristics were deduced.Since then, the load transfer method has been graduallyrefined and recent work in this area has tended to betheoretically more rigorous. Kraft et al (1981) indIcatedthat the pre-peak load transfer curves may be constructedfrom approximate. dosed form elastic solutions that hadbeen publisheCl earlier by Raimdolph and Wroth (1978).However, .the post-peak ~ n s w.rene detemrinedempiricafiy. Kraf t et a1 {JI981y oS'llllmested the \Use of (diIrectshear test results to construct these post-,peak load transfercurves. Randolpb 1 . 9 ~ adopted a phenomenologicalapproach by assuming a 'SCilfterung of the ,interface frompeak to residual that was :hwef.bohc in lthe slipdisplacement.

In these more 'ClOUIelliion:il approaches only therelationship between lthe sbear stress developed at theinterface and the axial disp lacement of the pile is addresseddirectly. In effect , the ;radial response of the :pile-soilinterface is ignored, or at tbestits influence on the shearbehaviOur is accountedfoT indirectly in the choice of t -zcurve. In other words, zero radial displacement is assumedat the pile-soil interface and any changes in radial stressacting on the pile are specifically ignored. This ISprobably reasonable for the response in the elastic range ofpile behaviour and it may also be reasonable for a totalstress analysis of the slip behaviour of piles in undrainedclays. However, in many materials the radial stress maybe far from constant during axial loading of the pile,particularly if the interface either contracts (as incalcaneous soils) or dilates (as in stronger rocks) duringshearing. For such cases the limitations of the conventionalload transfer analysis can be overcome by modifying theapproach and introducing an interface element that permitsthe coupling of the shear and volumetric behaviour.

MODIFIED LOAD TRANSFER ANALYSIS

Problem Definition

The problem to be analysed is that of axial loading ofa pile grouted into a cemented calcareous soil. The pilehas an embedded length of L and a radius r o' I t ismodelled as an elastic column with an effective modulus Epand Poisson s ratio Pp The soil surrounding the pile ISassumed to be homogeneous and can be modelled as anisotropIc, linear elastIC solid, with an elastic shear modulusG and a Poisson s ratio P. The mechanical behavIOur ofthe interface between the soil and the pile is described bythe constitutive relations presented below. For simplicity,it is also assumed that the pile is a solid section, but theextension of the analysis to include hollow pile sections isstraightforward. In the analysis which follows, compressivenormal stress, dilation of the interface, and downward piledisplacement are considered positive quantities.

Governing Equations

As mentioned above, the elastic pile is treated as acolumn which is dIvided into elements. An infinitesimalelement of length dz is characterised by ,an axial stiffness

Kp, given by:

whereEp

Kp - EpAp/dz

the cross-sectional area of the pile,the effective Young s modulus of thepile.

The radial stiffness of the pile, S can be approximatedfrom the solution for radial p r ~ u r eapplied to a solidelastic cylinder under conditions of zero axial strain (Le.plane strain) and thus:

where ~p

Sp = ~ p + Gp /ro 2

the Lame modulus of the pile material,the elastic shear modulus of the pilematerial.

The lI'espl Jl lSe .m the pile to axia l Rna wadial loading canthus be ,described .incrementally as follows:

[ dP ] = ,[ Kp :] ~ dw,do ; 0 -sp du J

;in 'VJhich P is ,the 100al load In the ,pile, J is the r d i ~ lstress acting on its cyhndrical surface, and w ,and .u are thevertical and radial displacements at the cylindricll l pilesurface. The negative sign is ,required ,in equation 3because compressive radIal stress and dilative radialdisplacement have been defined as positive quantities.

264

Soil Behaviour

It IS now well established that failure due to axialloading of a pile is very localised at the pile-soil interface,e.g. Hanna (1969), Nauroy and LeTirant (1983), Williams(1980). It is thus reasonable to model the soil surroundingthe pile as an isotropic, linear elastic continuum and toconfine any non-linearity to the interface. As a furtherapproximatIon it is assumed that the response of thecontinuum to shear tractions applied at the pile-soilinterface is independent of the response to normal tractionsapplied across the interface, i.e. the two modes ofbehaVIOur are uncoupled. This is an approximation and

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thus, in the strict sense, the elastic continuum analogy isnot complete. However, the adoption of this simplifyingassumption is unlikely to introduce large errors, as verifiedby Ooi (1989). Hence, the stress-deformation response ofthe soil continuuum can be expressed incrementally as:

[ ]where dT

dO

incrementalverticallyinterface,

shearalong

4

stress appliedthe cylindrical

incremental normal stress appliedradially to the cylindrical interface,increment of vertical displacement ofthe continuum at the interface,increment of radial displacement of thecontinuum at the interface.

The expression for KJ.{ is derived from the work ofRandolph and Wroth (l97lS) , and for a continuum with aconstant elastic shear modulus G it can be written as:

5

where

rm = 2.5 1-v)L 6

The expression for K is derived from the theory forthe expansion of a long cylindrical cavity in an infiniteelastic medium, i.e.

7

Interface Behaviour

Following an extensive laboratory investigation of theshear behaviour of interfaces (including tests WIth shearingunder conditions of constant normal stiffness), a set ofconstitutive relations has been derived to describe themonotonic shearing of interfaces in calcareous soils (Ooi,1989; Boey, 1990). Apart from modelling the shear transferbehaviour, these relations also include the possibility ofcontraction across the interface and the coupling of thenormal displacements to the shearing. The essentialfeatures ~ thIs model are presented below.

The behaviour of the interface prior to yielding iselastic and at this stage there exists no coupling of theshear and normal modes of behaviour. The stresses at theinterface can be related to the relative displacements by:

where dT

dO

incrementalverticallyinterface,

shearalong

8

stress appliedthe cylindrical

incremental radial stress appliedradially across the cylindrical interface,increment of vertical sheardisplacement between the pile and thesurrounding continuum,increment of radial displacementbetween the pile and the continuum,

265

i.e. the dilation of the interface,the elastic shear stiffness of theinterface,the elastic normal stiffness of theinterface.

The units of ks and kn are stress divided by length(displacement ).

Plastic yielding (or slip) of the interface between thepile and the soil will commence whenever the shearstrength of the interface is reached. It is assumed that thefailure law for the interface has the same form as theMohr-Coulomb criterion, i.e.

where c'I'

- C 0 t an 'I'

= the cohesion intercept,the friction angle.

9

Once yielding commences the strength parameters cand 'I' will not necessarily remain constant, but will varywith the accumulated plastic shear displacement. For thegeneral case, the forms of variation assumed in this modelare such that the cohesive component of the shearingresistance (eresumably arising in part from the interlockingalong the 'rough interface, the tooth strength of theroughness asperities, and also partly from bonding betweenthe cement grout and the surrounding soil) will decayexponentially with plastic shear displacement. This impliesthat the interface becomes damaged as it is sheared, andthe specific relation that is used to describe this behaviouris:

where

wP

10

the initial cohesion,an empirical constant that determinesthe rate of decay of the 'cohesivestrength with plastic shear displacement(slip),the plastic shear dISplacement (slip) ofthe interface,unit length (e.g. 1 mm).

In contrast to the cohesive strength, it is assumed thatthe frictional component of the interface shear strength ismobilised from zero once slip commences. Thisassumption recognises that for bonded interfaces somerupture must occur and a shearing surface must developbefore the full friction angle can be mobilised. Thespecific form of this relationship is written as:

where the maximum friction angle,an empirical constant that determinesthe rate at which frictional strength ismobilised with plastic slip displacement.

During yielding of the interface, plastic flow will bedetermined by an appropriate flow rule. In this particularmodel it is assumed that the plastic normal displacement ofthe interface (i.e. the dilation or contraction) IS a functionof the accumulated plastic slip displacement and themagnitude of the normal stress acting across the interface.For cement grout-calcarenite interfaces there isexperimental evidence (Boey, 1990) that the flow rule maybe described reasonably by the equation:

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duP

dwP

where duP

io

k3

k4

Pa

tan io (1 - exp(-k 4 u/Pa= a = 12

exp(k 3 uP/ )

the plastic component of the normaldisplacement of the interface,the initial dilation angle for theinterface,an empirical constant defining the rateof contraction of the interface withslip chsplacement,an empirical constant defining the rateof contraction of the interface withnormal stress,atmospheric pressure.

For the grout-calcarenite interface, the dilation angle iotakes a negative value, indicating that the interface actuallycontracts with shearing. Furthermore, equation 12 suggeststhat the interface will contract more as the normal stress isincreased, and this is in general agreement WIth theexperimental eVIdence (001, 1989; Boey, 1990).

Equations 8 to 13 may be combined to express thestress-deformation response of the elastoplastic interface

Incrementally as:

[ ] [ 13where D kg ( W

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L=50m

,

z m)

CalcareousSoil G=3z Pa

v=O.25

Pile Ep=2 GPavp=O.25

Figure 1 DefInition of Example Problem

calcareous soil. The Young's modulus f an equivalentsolid pile is 2 GPa and its POIsson's ratio is assumed to be0.25. The calcareous soIl IS assumed to have an elasticshear modulus G that increases lmearly WIth depth fromzero at the surface, such that ItS value at a depth z belowthe surface is G = 3z MPa, where z IS measured in metres.A value of 0.25 has been assumed for POIsson's ratio of thesoil, and this is considered to be constant with depth. ThemaXImum base resistance of the pile is assumed to be1 MPa and acts over the gross area of the pile. Thisultimate base resistance is assumed to be mobilised at abase displacement of 100 mm.

In the predictions of the pile behavIOur, the interface

between the pile and the soIl was represented by themodel described above. The assumed values for the modelparameters are as follows:

ks = 2500 kPa mmCo = 0k, = 0k3 = 0 5

kn\ m

k2k4

1 kPa mm

37 0012

In order to illustrate the behavIOur of this interface,consider the case of a direct shear test conducted undercondItions of constant normal stiffness of 50 kPalmm(equivalent to a pile of 2 m dIameter m an elastic soilwith a shear modulus of 25 MPa). In the dIrect shear testthe imtIal normal stress IS assumed to be 50 kPa. Forthese condItions the interface model predicts the behaviourdepicted in FIgure 2 The predictIons indicate thatcontractIon of the mterface occurs with mcreasing sheardisplacement, a feature that IS commonly observed in testson calcareous SOlIs Furthermore, this contraction underthe conditIon of constant normal stiffness is accompaniedby reductions in normal stress. These reductions havesome important consequences. Possibly the most importantis the reduction m fnctlonal reSIstance and hence the shearstrength of the interface. This effect can be seen clearlyin FIgure 2, where the shear stress applied to the mterfacepasses through a peak and then softens WIth furthershearing. It should be emphasised that thIS softening is adrrect consequence of contractions and reductions in normalstress during shearmg. By definitIon, these changes innormal stress would not be observed in a conventional

267

60

0-c..Yo 50

'' )40-

Vi

.I S30

0'* 20EE

-+.I.rI fA C::TC.

- 1 00.00 2.00 4.00 6.00 8.00

Horizontal shear displacement (mm)

Figure 2 Predictions of Typical Shear BehaVIOur mCNS Tests (Normal Stiffness = 50 kPa/mm)

direct shear test in whIch the initial normal stress ismaintained throughout the shearing. The fact thatsoftening may occur when the normal stiffness rather thanthe normal stress is constant, has important consequencesfor the behaviour of pile foundations, as will be discussedbelow.

The modified t - z analysis has been used to predict theresponse of piles in deposits of the calcareous soil. Threedifferent cases have been analysed m order to mvestigatethe influence of the normal stress on the pile behaviour.Three different distributions of initial normal stress havebeen assumed, and these are listed below.

Case 1:Case 2:Case 3:

Initial normal stress = 2.5z kPaInitial normal stress = 5.0z kPaInitial normal stress = 50 + 5.0z kPa

where z is measured in metres. Cases 1 and 2 may beconsidered realistic varIatIOns for steel tube pIles dnveninto calcareous sediments. Case 3 has been mcluded todemonstrate one of the benefIts to be gained fromemploying driven and grouted plIes m this type ofsediment. The pressure applIed by the grouting process ISassumed to remain locked in as an imtlal normal stress atthe soil-pIle (or more accurately the SOIl-grout) interface.Of course, this normal stress may change as aXIal load ISapplied to the pile, as IS demonstrated below.

The variations of pile head load with pIle headdIsplacement for the three cases consIdered, and forstatIcally applied loading, are plotted m FIgure 3. Clearlythe mitial normal stress actmg at the pile-sOlI mterface hasa very sIgmflcant mfluence on the predIcted response ofthe pile. In all cases the load-dIsplacement behaVIOur ISnon-li near. Each curve rises to a peak and after the peakthe tendency to shed load with further dIsplacement isindicated. The validity and accuracy of these predictionsbeyond the peak are yet to be established. The tendencyto shed load is a potentially unstable situation, bothphysically and numerically. Numerical predictions of thissituation must always be viewed with some Susplclon, atleast untIl they are rigorously valIdated. ObVIously, abifurcatIOn m the response IS pOSSIble However, It can bestated that softemng of the pIle-SOlI mterface, caused

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1817161514

Z 13- < 12' - ~

11c 10..2c

'

9CJ:]co 8Ol-- ' t 7

6a::543210

......--:1

V CIISEV

/

V/

/

/

/ . / I r A ~ ?/

/ V/ /

V . , ; -V l -/, ? ? ? rf.A E 1

IfIf

o 20 40 60 80 100 120Pile head displacement mm)

FIgure 3 Influence of InitIal Normal Stress on

Predicted Load-Deflection Behaviour

exclUSIVely m the present analysis by contractIon withshearing, IS solely responsIble for the shedding of load.Although the results beyond the peak must be regarded astentative, in Case 1 it appears that the softening iseventually arrested. In this case, the post-peak softeningof the interface is offset to some degree by the furthermobIlIsation of base resistance with continued piledIsplacement, and eventually the rate of mobIlisation ofbase reSIstance exceeds the rate of load shedding caused bysoftening along the pIle shaft.

I t is also evident from FIgure 3 that the structuralmteraction between the pIle and the soil has a sIgnificanteffec t on the ultImate axIal capacity of the pIle. The

maximum pOSSIble capacity would apply in the case, of arigid pile, and can be computed by assuming that noreductIOns in the initial normal stress occur along the shaft,I.e. the maximum possible shear strength is mobilisedsimultaneously at all p o n t ~along the pile shaft. However,because the pile has a fimte axial stiffness, progressivefaIlure of tne pIle shaft actually occurs and as aconsequence the shaft capacity is reduced. The maximumpossible capacities, and the capacities computed by themodified t - z analysis for Cases 1, 2 and 3 are listedbelow.

Case 1

Case 2Case 3

Maximum PossibleCaEacity

MN)

14.8

29.641.4

ComputedCapacity

MN)

4.4

10.216.7

The reduction in capacity due to the interface softeningand the soil-structure interaction IS a feature that IS alsopredicted by the conventional load-transfer methods.

I t was stated previously that the modified t -z methoddoes not require the explicit specification of a t -z curve.Indeed, the constitutive model assumed for the interface iscapable of predicting the appropriate t -z relationships atthe pile-soil interface. Examples of the t -z curvesgenerated for Case 3 at various depths are plotted inFigure 4. Clearly, the appropriate t - z relationship is a

268

150140130

' 120a..110-

Q) 100'0. 90c 80III 70IIIQ) 60.....III 50L.C 40Q)

..c 30VJ20100

II \

J 1I \ e ~ th = 49.5mI / \I / \

1 I \I I '- ............. L.; I; m

IIfC 5m,

.:---- 5. mIfo 20 40

Pile displacement mm)

Figure 4 TypIcal t -z Curves Generated by the

Analysis - Case 3

function of location (depth) along the pile. For thIsproblem all curves mdicate a softening response due to thereductions in normal stress with shearing of the pile-soilinterface.

CONCLUSIONS

A modified load transfer ( t -z) analysis for axIallyloaded piles has been presented in this paper. Not onlycan the method be used to predict the fullload-displacement behaviour of a pile, but unlike theconventional t -z analySIS It is also capable of indicating thecomplete mechanIsm of load transfer as well as the

complete stress path at the pile-soil mterface. It is ableto do this by assummg a model for the interface thatpermIts coupling between the shear and normal modes ofdeformation. The normal stress acting on the pile hasbeen shown to have a significant influence on the pilebehaviour. In partIcular, the loss in normal stress at anmterface in calcareous soil can result in softening of theinterface shear behaviour, and possibly also m the overallpile response.

ACKNOWLEDGEMENTS

Support for thIS work was prOVIded by grants from theUniverSIty of Sydney, the CSIRO-University of SydneyCollaborative Research Scheme and the Australian ResearchCouncil.

REFERENCES

Airey, D.W., Koh, T.M. and Mac, C.Y. (1990),Investigation of the Interface Behaviour of Piles m

Calcareous Soil , Proc. PACOMS-90, Seoul, Korea.

Boey, C.F. (1990), ModellIng of the Behaviour of NaturalCalcarenite , PliO. Thesis submitted to the Uruversity ofSydpey, Australia. '

Chow, Y.K. (1986), Analysis of Vertically Loaded PileGroups , Int. J. Numer. Anal. Methods Geomechanics,

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Vol. 10, pp. 59-72.

Coyle, R.M. and Reese, L.C. (1966), ''Load Transfer forAxially Loaded Piles in Clay , Jnl Soil Mechs. Foundn.Divn, ASCE, Vol. 92, No. SM2, pp. 1-22.

Hanna, T.R. (1969), The Mechanics of Load Mobilisationin Friction Piles , Jnl of Materails, JMLSA, Vol. 4, No.

4, pp. 924-937.Johnston, I.W., Carter, J.P., Novello, E.A and Ooi, L.R.

(1988), Constant Normal Stiffness Testing of NorthRankm Calcarenite , Proc. Int. Conf. on CalcareousSediments, Perth, (Eds R.J. Jewell and M.S. Khorshid),Balkema, Rotterdam, Vol. 2, pp. 515-530.

Kraft, L.M., Ray, R.P. and Kagwa, T. 1981), Theoreticalt - z Curves , Jnl Geot. Eng. Dvn, ASCE, Vol. 107, No.GT11, pp. 1543-1561.

Nauroy, J.F. and LeTirant, P. (1983), Model Tests ofPlies in Calcareous Sands , ASCE Spec. Conf. onGeotech. Practice in Offshore Eng, Austin, pp. 356-369.

O'Neill, M.W. and Mahar, L.J. (1982), ''Load TransferMechanisms in Piles and Pile Groups , Jnl Geot. EngDivn, ASCE, Vol. 108, No. GT12, pp.l605-1623.

269

Ooi, L.R. (1989), ''The Interface Behaviour of SocketedPiles , Ph.D. Thesis, University of Sydney, Australia.

Ooi, L.R. and Carter, J.P. 1987), A Constant NormalStiffness, Direct Shear Device for Static and CychcLoading , Geotechnical Testing Journal, ASTM, Vol. 10,No.1, pp 3-12.

Randolph, M.F. (1985), RATZ: Load Transfer Analysis of

Axially Loaded Piles Users' Manual, EngineeringDepartment, Cambridge University.

Randolph, M.F. and Jewell, R.J. (1989), Axial LoadTransfer Models for Piles in Calcareous Soil , Proc. 12thInt. Conf. on Soil Mech. and Foundn Eng, Rio deJaneiro, Vol. 1, pp. 479-484.

Randolph, M.F. and Wroth, c.P. (1978), Analysis ofDeformation of Vertically Loaded Piles , Jnl Geotech.Eng Divn, ASCE, Vol. 104, No. GT12, pp. 1465-1488.

Seed, R.B. and Reese, L.C. 1957), The Action of SoftClay along Friction Piles , Transactions of the ASCE,Vol. 122, pp. 731-754.

Williams, AF. (1980), Design and Performance of PilesSocketed into Weak Rock , Ph.D. Thesis, MonashUniversIty, Australia.

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