www.elsevier.com/locate/envsoft

Environmental Modelling & Software 21 (2006) 1375e1380

Modelling the loadedeformation response of deepfoundations under oblique loading

K. Johnson, P. Lemcke, W. Karunasena*, N. Sivakugan

School of Engineering, James Cook University, Townsville, Queensland, 4811, Australia

Received 24 March 2004; received in revised form 21 March 2005; accepted 5 April 2005

Available online 8 September 2005

Abstract

Deep foundations are slender pile elements used to transfer loads from structures into deeper soil strata below the ground. Deepfoundations have a variety of loads including axial, lateral and moment loads. Often these loads will act together to form a combi-nation of loads, such as oblique forces that have a component of axial and lateral forces. Due to the pile’s length, which can extend

past 30 m, it is difficult and expensive to carry out full-scale testing. Any full-scale test data are limited to the area conditions and soilproperties at the testing site. Small scale testing can be used as an alternative to full-scale testing, although scaling effects will in-fluence the conclusions from this type of study greatly. Computer simulations of finite element modelling allow for in-depth studiesinto the complex pileesoil interaction under combination loading in deep foundation structures. This work used three-dimensional

finite element modelling using ABAQUS to explore the effect of pile shape, sand properties, pile length and loading conditions onthe capacity of a pile. Using trends discovered by these simulations design charts were developed to aid designers when determiningthe bearing capacity for oblique interaction for square piles. The final design chart shows the change in capacity for in-plane loaded,

square cross-sectioned piles as compared to circular piles. The difference in capacity is shown to be a function of the oblique inter-action angle and length to diameter ratio.� 2005 Elsevier Ltd. All rights reserved.

Keywords: ABAQUS; Pile foundation; Pileesoil interaction; Pile shape; Square pile

1. Introduction

Pile foundations are quite effective, especially inweaker soils, for transferring the structural loads tothe underlying ground. Unlike shallow foundations, inpile foundations the load transfer mechanism is verycomplex and not fully understood yet. Thus most designprocedures were developed assuming a circular pile andelastic response for the stressestrain relationship withinthe materials.

Lately both industry and researchers have begun toexplore the advantages and disadvantages of using other

* Corresponding author. Tel.: C61 7 47814983; fax: C61 7 47751184.

E-mail address: karu.karunasena@jcu.edu.au (W. Karunasena).

1364-8152/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.envsoft.2005.04.015

cross-sectional shapes rather than the traditional circularpile. These shapes have included solid concrete octago-nal and square piles, or alternatively hollow sectionswith and without end caps. With expanding ranges inpile technology comes the need to fine tune or expand ex-isting pile design procedures to give rational predictionsfor pile response. Piles are difficult to investigate via lab-oratory testing due to their scale. Therefore, numericalmethods such as finite element method (FEM) can beused to gain an understanding of the failure mechanismsexperienced by piles under various loading conditions.

The following work explores the pros and cons ofusing a cast-in-place square pile over the traditionalcircular pile. To accomplish the modelling work thefinite element package ABAQUS was utilised. The finiteelement method can be used for a broad range of

1376 K. Johnson et al. / Environmental Modelling & Software 21 (2006) 1375e1380

applications ranging from water studies (Umgiesseret al., 2003) to geotechnical problems as presented inthis paper. During the analysis it became evident thatfactors such as length to diameter ratio, sand constitu-tive properties and load direction play key roles in bothreal and model pile predictions.

2. Governing factors

Numerical models involving FEM can offer severalapproximations to predict true solutions. The accuracyof these approximations depends on the modeller’s abil-ity to portray what is happening in the field. Often theproblem being modelled is complex and simplificationshave to be made to obtain a solution. Two of the gov-erning factors which have a vast impact on both the realand model piles are:

(a) the constitutive properties of the sand and(b) the pileesoil interaction at the interface over the pile

surface.

2.1. Constitutive models

FEM has become increasingly popular as a soil re-sponse prediction tool. This has led to a higher demandfor researchers to develop more comprehensive descrip-tions for soil behaviour, which in turn leads to morecomplex constitutive relationships. Prevost and Popescu(1996) state that for a constitutive model to be satisfac-tory it must be able to:

(1) make a statement about the material behaviour forall stress and strain paths;

(2) identify model parameters by means of standardmaterial tests and

(3) physically represent the material response to changesin applied stress or strain.

From standard tests researchers have determined thatsand behaves elasto-plastically when loaded as shown inFigs. 1 and 2.

Previous studies have explored constitutive modelsand found that the use of isotropic models such as

perfectly plastic

E = young’s modulus

stre

ss

strain

perfectly elastic

1

Fig. 1. Ideal stressestrain relationship assumed for most structural

materials.

elastic, MohreCoulomb and DruckerePrager aresufficiently accurate (Chen and Saleeb, 1983; Hibbitt,Karlsson, Sorensen, Inc., 1998). In the past linear elasticconstitutive models have been commonly used in devel-oping pile design methods (Poulos and Davis, 1980).The differences between a purely elastic and MohreCoulomb constitutive model are shown in the followingsections.

2.1.1. Linear elastic constitutive modelThe linear elastic constitutive model (Hooke’s law) is

probably the most common model used in the engineer-ing world to approximate a material’s stressestrain rela-tionship. Hooke’s law relates the stresses and strainsthrough two constants, Young’s modulus (E ) andPoisson’s ratio (n).

2.1.2. MohreCoulomb’s modelAs previously discussed soil behaves elasto-plastically,

and when the sand is subjected to loads the displacementwill contain both a recoverable and non-recoverablecomponent. Therefore, a failure criteria needs to be in-cluded in the elastic models to define the stress statesthat cause plastic deformation. One possible failure en-velope is MohreCoulomb surface as shown in Fig. 3.

The failure envelope is only dependent on the majorand minor principle stresses (s1, s3), and is independentfrom the intermediate principle stress (s2). Whenmapped into three-dimensional stress space, MohreCoulomb criteria resolves into an irregular hexagonalpyramid. This pyramid forms the failure/yield envelope,

stre

ss

strain

unloading/reloadingline

when unloading thematerial reboundspartially

Fig. 2. Stressestrain relationship for an elasto-plastic material.

normal stress (σ)

shea

r st

ress

(τ)

φ friction angle

cohe

sion

(c)

Mohr Coulomb failure envelope

σ3 σ1

Fig. 3. Mohr Coulomb’s failure surface.

1377K. Johnson et al. / Environmental Modelling & Software 21 (2006) 1375e1380

which in turn governs how the soil will behave. Thematerial will behave elastically if the stress point lieswithin the failure envelop. However, if the stress reachesthe yield surface the material will undergo a degree ofplastic deformation. In the MohreCoulomb model usedherein, it is assumed that the soil has a linear elasticrelation until failure.

2.2. Pileesoil interaction

The interaction at the pileesoil surface can rangefrom perfect contact where no relative sliding betweensoil and pile occurs to perfect sliding conditions whereno friction develops along the shaft of the pile. Initially,it was assumed that the soil and pile are both deform-able bodies and can undergo finite sliding.

A basic Coulomb frictional model was used to governthe interaction between the pile and sand surfaces(Hibbitt, Karlsson, Sorensen, Inc., 1998). However, itwas determined through the verification stage that theinterface characteristics were dependent on the place-ment method of the pile. Piles that are impact driven(ID) tend to exhibit degrees of relative motion betweenthe pile and soil once the system is under load. It wasalso determined through the FEM analysis that cast-in-place (CIP) piles tend to have much less relative slidingthan their ID counterpart. A typical loadedisplacementcurve for a 7.32 m long cast-in-place circular pile in sandis shown in Fig. 4.

The preliminary model results shown in Fig. 4 wereobtained using the finite element mesh shown inFig. 5. The predicted deflection for the CIP pile wasapproximately 10 times greater than that of the experi-mental pile, while the interface was activated. The modelreduced to being a factor of 1.2 times the experimentaldeflection when the interface was not active. This means

0

10

20

30

40

0 200 400 600 800

Load (kN)

Dis

plac

emen

t (m

m)

Actual

Circular, nointerface

Circular,interface

Fig. 4. Typical loadedisplacement curve.

that for cast-in-place concrete piles some bondagebetween the concrete and surrounding soil may haveoccurred and thereby not allowing the slippage normallyobserved along a smooth dry concrete surface. Also con-struction methods for cast-in-place piles are crude andthis results in irregular rough sides not allowing a contin-uous slip surface to exist. Therefore, it was concludedthat cast-in-place piles were better represented by a per-fectly bonded surface allowing no relative slippage tooccur.

However, for impact driven piles an interface modelwould be required to allow relative motion. In previousstudies it was found that when modeling small test steelpiles placed into sand the interface was a crucial part ofthe model (Johnson et al., 2001). This would suggestthat the pileesoil interaction at the pile’s surface isdependent on the placement type and can affect amodel’s accuracy significantly.

3. Numerical model

Two models were constructed in ABAQUS to allowloadedeflection curves to be established for both squareand circular piles. These models are shown in Figs. 5and 6.

The general characteristics of the meshes are asfollows:

� Element type: due to the pile shape and load appli-cation angles not being axial the problem wasdiscretized using three-dimensional finite elements.Twenty noded reduced rectangular prisms were uti-lised. These rectangular prisms have parabolic shapefunctions to approximate the displacement patternbetween nodes.� Boundary conditions: two separate boundary condi-tions were imposed onto the models. Due to symme-try the problem space could be reduced into half ofits original size. The nodes at the base of the mesh

fixed boundary rollerboundary

degrees of freedom= 40866

pileelements

soilelements

Fig. 5. Circular pile mesh.

1378 K. Johnson et al. / Environmental Modelling & Software 21 (2006) 1375e1380

and far bounds are fixed against displacement androtation. The nodes on the plane of symmetry arerestrained from moving normal to the plane, but canmove vertically and along the plane (see Figs. 5 and 6).� Input data: the pile and soil properties given inSection 3.1 were used in the numerical analysis. Itwas assumed that the sand behaved linear elasticallyuntil a Mohr Coulomb’s plastic failure envelope wasreached. Once the stress state at any location reachesthe failure surface it will undergo plastic deforma-tion. These constitutive parameters and an initialstress state were imposed on the model before theanalysis was started. The initial stress accountedfor the in situ stresses due to self-weight of thesystem before the load is added.

3.1. Verification

To verify the reliability of the FEM pile meshes devel-oped, loadedeflection response from theory was com-pared against a number of full-scale pile test data. Thetest data used were obtained from the electronic deepfoundation load test database developed by the FederalHighway Administration in America [http://www.tfhrc.gov/structur/agids/agids.htm]. Several individualpiles were selected where the sand and pile propertiescould be established. Below is an example of an axiallyloaded verification pile case used.

3.1.1. Material properties and dimensionsThe circular cast-in-place test pile was made of solid

reinforced concrete. The test pile properties and dimen-sions are summarized in Table 1.

Standard penetration test (SPT) results were availablefor the sand profile around the pile. Using the SPT dataand pre-existing empirical correlations the friction angle

rollerboundary

degrees of freedom= 78378

pileelements

soilelements

fixed boundary

Fig. 6. Square pile mesh.

(Peck et al., 1974) and Young’s modulus (Das, 1999) ofthe sand could be approximated. The sand was separatedinto two distinct homogenous layers with different prop-erties. These layer properties are given in Table 2.

3.1.2. Verification resultsSeveral mesh densities and fixed boundary locations

were trailed until a converged numerical solution wasachieved. The final converged loadedeflection curvesfor both the square and circular models are plottedagainst the actual test results in Fig. 7.

Due to the lack of full-scale test results the square pilemesh was also verified using the axial load test datafor a cast-in-place circular pile. The square pile wasmodeled such that it had the same length, and lengthto breath ratio as its circular counterpart. This meantthat the square pile had a slightly higher contact areaand therefore should yield slightly higher ultimate ca-pacities. This was supported by the model results shownin Fig. 7.

Further verification cases included small/full-scaletest data on both square and circular piles. The meshresults yielded results which were realistic, and gave adegree of confidence that the meshes were behavingreasonably.

4. Shape functions/oblique interaction

Using the models developed in the previous section,design charts were established to aid in the predictionof bearing capacity for cast-in-place square piles. Duringthe numerical analysis, it became evident that the load-ing direction influenced the performance of the pilesof different cross-sectional shapes. Therefore, effect ofthe angle of loading was also investigated using thenumerical models.

Table 1

Test pile properties

Material Concrete

Length (m) 7.32

Outer diameter (m) 0.457

Young’s modulus (GPa) 30

Poisson’s ratio 0.15

Table 2

Sand properties

Sand layer Friction

angle ( �)

Young’s

modulus (MPa)

Poisson’s

ratio

Topa 35 19 0.3

Bottom 40.5 52.8 0.4

a The top layer of sand extended from the surface to approximately

1.5 m below the pile.

1379K. Johnson et al. / Environmental Modelling & Software 21 (2006) 1375e1380

4.1. Numerical test cases

A number of cases were used to explore the effect ofthe square cross-section on a pile capacity. The casescenarios were chosen so that the sensitivity of variousgeometric and constitutive parameters could be deter-mined. In total 24 models were run for both the squareand circular piles. The loadedisplacement relation foreach scenario was obtained allowing the ultimate bear-ing capacity to be determined. A brief flow chart show-ing the scope of the cases is shown in Fig. 8.

Three loading angles were chosen: axial, 45 � obliqueand lateral. All the loads were in plane so the resultsobtained for the square pile can only be applied in thefield designs where similar conditions exist.

0

5

10

15

20

0 200 400 600 800

Load (kN)

Dis

plac

emen

t (m

m)

Actual

CircularmeshSquaremesh

Fig. 7. Verification loadedisplacement curve.

pile cases modelled

N = 10E = 8 MPaφ = 30°

L/d = 10L/d = 20

L = 5, 20 m -axial load-oblique load-lateral load

N = 30 E = 24 MPaφ = 36°

L/d = 10 L/d = 20

where N = blow count, E = Young’s modulus,φ = friction angle, L = length and d = diameter.

L = 5, 20 m -axial load-oblique load-lateral load

Fig. 8. Flow chart of numerical cases.

4.2. Shape factor (Fs) design chart

Current methods for pile design have assumed thatthe bearing capacity of a pile is independent of the pile’sshape. Therefore, simplified prediction methods basedon a circular pile cross-section were established from ex-perimental and theoretical studies. When designing pilesother than circular, designers often convert the cross-sectional area to an equivalent circular area. Thisenables them to use the traditional methods to predictthe bearing capacity. Through the numerical analysis itwas found that the shape of the pile’s cross-section couldhave a considerable influence on the pile’s lateral andoblique bearing capacities. This observation was alsoseen in the experimental work conducted by Ashourand Norris (2000).

From the numerical results a shape factor (Fs) wascalculated. The shape factor is a simple conversionfactor allowing the designer to convert from a circularcross-section to square area. The shape factor was deter-mined by dividing the ultimate square bearing capacityby its circular pile counterpart (Qult,square/Qult,circle). Thisshape ratio was found to be dependent on the angle ofloading, the length to diameter ratio and the density ofthe sand. The final design chart obtained is shown inFig. 9(a) and (b).

The above plots enable designers to use the tradition-al methods for circular piles and convert across toa square cross-section. This is done by calculating thebearing capacity of a circular pile with the same length,and the length to diameter ratio as the square pile. UsingFig. 9 a shape factor (Fs) can be determined which is

0.9

1

1.1

1.2

1.3

0 45 90

Load angle (deg)

Shap

e fa

ctor

(F s

)Sh

ape

fact

or (

F s)

L/d = 20

L/d = 10

L/d = 10

lateralaxial

0 45 90

Load angle (deg)lateralaxial

(a)

1

1.1

1.2

1.3

1.4

(b)

L/d = 20

Fig. 9. (a) Shape factor for loose sand. (b) Shape factor for mediume

dense sand.

1380 K. Johnson et al. / Environmental Modelling & Software 21 (2006) 1375e1380

multiplied into the ultimate circular capacity (Qult,circle)to give the equivalent square capacity (Qult,square) asfollows:

Qult;circleFsZQult;square

5. Discussion

One of the interesting observations from numericalstudy is that shape factor is dependent on the density ofthe sand. This phenomenon appears to be evident forboth the oblique and lateral loading scenarios. One pos-sible reason the above occurs could be due to the failuremechanisms involved. A square pile when placed underan oblique/lateral in-plane load can utilise its flat sideenabling it to have higher shaft bearing capacity asopposed to the circular pile. This bearing resistance inthe square pile will increase as the density of the sandincreases. The circular pile, because of its geometry,allows the sand to flowmore readily around it, and there-fore the benefits of the increased sand strength wouldnot be as great.

During the development of the numerical models,it became necessary to include the residual self-weightstresses of the system. As the depth of the sand increasedso too did the magnitude of the effective stresses. Thismeant that there were initial stresses in the system,which would affect the constitutive behaviour duringthe loading sequence. The behaviour of any soil problemis dependent on the stress history of the soil medium.

In the beginning the models were developed withoutthe presence of the initial effective stresses. This meantthat the starting normal stress used for the MohreCoulomb model was zero, and the cohesion for sandwas also zero. Once the load was increased even bythe slightest amount it would produce a plastic strain,as Mohr stress circle would be reduced to a single pointwith no radius. This resulted in an unstable purelyplastic response of the pile that was not a realistic solu-tion. The numerical model was unable to obtain a singlesolution because for each load there is an infinite num-ber of possible displacements, i.e. convergence was notpossible. By modelling the effective stresses in the systemthis allowed the computer simulation to achieve a con-verged realistic solution.

6. Conclusion

The finite element models used in this paper allowedthe investigation into the effect of pile cross-section onits capacity. From this work a rational design chart wasdeveloped for cast-in-place square piles. When modelingreal systems it is important to include all the major fac-tors attributing to the object’s behaviour. The constitu-tive parameters, stress history and the interactionsbetween elements are all key aspects to consider. Nu-merical models can offer a great deal of informationwhen used correctly, and they can provide a platformfor developing design procedures for industry.

Acknowledgment

The writers of this article would like to thank the staffat high performance computing facility at James CookUniversity for their support.

References

Ashour, M., Norris, G., 2000. Modelling lateral pile soilepile interac-

tion. Journal of Geotechnical Engineering, ASCE 126 (5), 420e428.

Chen, W., Saleeb, A., 1983. Constitutive Equations for Engineering

Materials. John Wiley & Sons, Inc, United States.

Das, B., 1999. Principles of Foundation Engineering, fourth ed. PWS

Publishing, Melbourne.

Hibbitt, Karlsson, Sorensen, Inc., 1998. ABAQUS/Standard User

Manuals. (Pawtucket): Hibbitt. Karlsson & Sorensen, Inc, United

States.

Johnson, K., Karunasena, W., Sivakugan, N., 2001. Modelling pilee

soil interaction using contact surfaces. In: Proceedings, First

Asian Pacific Congress on Computational Mechanics, Sydney,

pp. 375e380.

Peck, R.B., Hanson, W.E., Thornburn, T.H., 1974. Fundation Engi-

neering, second ed. John Wiley & Sons, Inc, United States.

Poulos, H.G., Davis, E.H., 1980. Pile Foundation Analysis and

Design. John Wiley & Sons, Inc, Toronto.

Prevost, J.H., Popescu, R., 1996. Constitutive Relations for Soil

Materials. Electronic Journal of Geotechnical Engineering.

!http://www.ejge.com/1996/Ppr9609/Ppr9609.htmO (accessed 28

March 2005).

Umgiesser, G., Canu, D.M., Solidoro, C., Ambrose, R., 2003. A finite

element ecological model: a first application to the Venice Lagoon.

Environmental Modelling & Software 18 (2003), 131e145.

Elsevier.