of

y a

dac

Non-linear kinematic hardening

e elhoicmo

ial tcen

2010 Elsevier Ltd. All rights reserved.

d in cill typed to beorizon

stant amplitude.Several modelling approaches have been proposed to account

for the behaviour of a single pile under a lateral load. Most of these

lations to obtain a better representation of the non-linearity ofthe soilpile interaction, then used nite element simulations tocapture group effects on the response of laterally loaded piles[11] and to investigate the effect on the soil response of someparameters such as in situ soil stresses and sloping ground [12].Kimura et al. [13] showed the ability of the nite element methodto investigate the ultimate behaviour of a laterally loaded pilegroup on the basis of comparisons between simulations and full-scale tests. More recently, three-dimensional nite element analy-ses have been carried out by Wakai et al. [14] to simulate the

Corresponding author. Address: Laboratoire Central des Ponts et Chausses, 58boulevard Lefevre, F-75732 Paris Cedex 15, France. Tel.: +33 1 40 43 54 17; fax: +331 40 43 65 11.

E-mail addresses: emmanuel.bourgeois@lcpc.fr (E. Bourgeois), julio.rakotonin-driana@lcpc.fr (M.H.J. Rakotonindriana), alain.lekouby@lcpc.fr (A. Le Kouby),philippe.mestat@lcpc.fr (P. Mestat), jean-francois.serratrice@developpement-dura-

Computers and Geotechnics 37 (2010) 9991007

Contents lists availab

Computers and

journal homepage: www.elseble.gouv.fr (J.F. Serratrice).into account. Design methods have been developed accordinglyand have proven efcient and safe. However, the design of someadvanced structures of civil engineering (e.g. windmills, towersseveral hundred of metres high, and high-speed train lines via-ducts) makes it necessary to take into account variable and re-peated horizontal loads due to winds, tides and waves, or othertypes of mechanical loads. The amplitude of the load under discus-sion can be considered constant, and the evolution of the structuretowards large permanent strain or failure is caused not only by itsmagnitude but also by its repetition. In other words, in the rst ap-proach, the variable loading can be seen as a cyclic load with a con-

are available (for instance, the design rules of the API [1], or inFrance, the recommendations of the so-called Fascicule 62 [2]).The main difculty in practice is to obtain realistic py curves fora given site. The simplest way to adapt such methods is to extendthe static py curves in the cyclic domain (API [1]). An attempt todo so has been carried out on the basis of experimental centrifugetests for a small number of cycles by Rosquoet et al. [36] and, morerecently, for a larger number of cycles by Rakotonindriana [3].

In the second type of approach, the soil is modelled as a contin-uum (see for instance [49]). Brown and Shie [10] derived pycurves from the results of three-dimensional nite element simu-Lateral loadPileFinite element methodCentrifuge testValidation

1. Introduction

Deep foundations are widely usevide the performance required by acases, piles are vertical and designeHowever, in some situations, static h0266-352X/$ - see front matter 2010 Elsevier Ltd. Adoi:10.1016/j.compgeo.2010.08.008vil engineering to pro-s of structures. In mostar static vertical loads.tal loads must be taken

approaches fall into one of two categories. In the rst category, theground is modelled by a set of linear or non-linear springs with athickness depending on the ground layers. This type of approachmakes it possible to reproduce the variations of the properties ofthe ground by adjusting the parameters of the springs, representedby the so-called py curves. Various normalised design proceduresKeywords:performed with the model parameters derived from the triaxial tests. Results are in good agreement withexperiments.Three-dimensional numerical modellingto cyclic lateral loading

E. Bourgeois a,, M.H.J. Rakotonindriana a, A. Le KoubaUniversit Paris-Est, LCPC-MSRGI, 58 bd Lefebvre, F-75732 Paris Cedex 15, FrancebCentre dEtudes Techniques de lEquipement Mditerrane, LRPC Aix en Provence, Ple

a r t i c l e i n f o

Article history:Received 8 April 2010Received in revised form 19 July 2010Accepted 23 August 2010Available online 17 September 2010

a b s t r a c t

This paper presents a nitlateral loading. First, the cthe model equations for amonotonic and cyclic triaxpile with the geotechnicalll rights reserved.the behaviour of a pile subjected

, P. Mestat a, J.F. Serratrice b

tivit les Milles, CS 70499 BP3700, 13593 Aix en Provence Cedex 3, France

ement simulation of the behaviour of a vertical pile subjected to a cyclice of a suitable constitutive model is discussed. The analytical solution ofnotonic triaxial compression is given, and the model is compared withests on dry Fontainebleau sand. Simulations of tests carried out on a modeltrifuge of the French Public Works Research Laboratory (LCPC) were then

le at ScienceDirect

Geotechnics

vier .com/locate /compgeo

eters. The results obtained in the case of a monotonic triaxial com-pression test are presented. Then the model parameters are

d Gadjusted by comparing simulations with monotonic and cyclic tri-axial tests carried out on dry Fontainebleau sand. In the last sec-tion, the results of a three-dimensional nite element simulationof the behaviour of a single pile subjected to a cyclic lateral loadare compared with the results of experiments carried out in thegeotechnical centrifuge of the French Public Works Research Labo-ratory (LCPC) in Nantes, France. Finally, the possibility of derivingpy curves from the numerical simulations and predicting theirevolution under cyclic loadings is discussed.

2. Theory

2.1. Constitutive models for cyclic behaviour

Constitutive models that make it possible to reproduce cyclicbehaviour such as adaptation, accommodation or ratcheting havebeen discussed in numerous works (see, among many others, thesurveys by Dafalias [16], Hicher and Shao [17] for soils and rocksor Besson [18] for other materials). Most models introduce a kine-matic hardening law because isotropic hardening does not make itpossible to reproduce plastic strain accumulation under cyclicstress, but can be divided into several categories:

models with several yield surfaces, such as the CJS model [19],which combines an isotropic mechanism and deviatoric mech-anism dened by a yield surface that depends on the Lodeangle.

other models that involve nested yield surfaces within a bound-ing yield surface; see for instance [2024].

The choice of the model features (e.g. the number of nested sur-faces) depends on the precision required in the representation ofthe behaviour. On the whole, such advancedmodels prove to be ex-tremely polyvalent. On the other hand, they have two drawbacks:behaviour of model tests of free- or xed headed pile groups sub-jected to lateral loading and to discuss the inuence of themechanical connection between pile heads in the group. Yangand Jeremic [15] used three-dimensional nite element simula-tions to investigate the inuence of a layered soil on the pycurves. On the whole, fully three-dimensional numerical simula-tions are often considered as a way to overcome the limitationsof the classical py curve approach (or of Winkler-type ap-proaches) because one is led to propose empirical modicationsof the curves when it becomes necessary to account for interac-tions with neighbouring structures, slopes, group effects, or com-bined loadings.

The aim of this study was to show that three-dimensional niteelement simulations can be used to analyse the cyclic behaviour ofa laterally loaded pile, provided that a suitable constitutive modelis used. The main interest of the approach is to solve the problemof adjusting py curves to account for the progressive degradationof the soilpile interaction. However, special attention must bepaid to the difculty of the determination of the model parameterson the basis of standard laboratory or eld tests. First, an overviewof the numerous models available in the literature is briey pre-sented, which provides the basis for the choice of a model suitablefor the simulation of a pile under cyclic lateral loading. The maincriterion is that the model must capture the main features of thesoil behaviour under cyclic loading with a small number of param-

1000 E. Bourgeois et al. / Computers an From the numerical point of view, the introduction of the mod-els can prove difcult because of the large number of variablesneeded to store the state and positions of the different surfaces. From a practical point of view, the identication of parametersis an awkward problem that relies on laboratory tests in whichthe soil sample is subjected to complex stress paths. Such acomplex determination procedure is acceptable for articialmaterials, for which one can prepare a large number of sampleswith similar properties, but is far less useable for geomaterials,especially if the model is to be used to discuss the behaviour ofactual structures, because of the natural variability in space ofthe mechanical properties of soils. This problem is a strongmotivation to choose the simplest possible model.

Such models can accurately describe the evolution of stressesand strains around a pile subjected to a cyclic lateral load but provecostly to use if the number of cycles under discussion is large.Wichtmann [25] combined a hypoplastic model of the same typeas the models mentioned above with a law describing the accumu-lation of plastic strain over a large number of cycles. The approachwas developed in the case of triaxial tests and must store six sca-lars for each integration point to describe the evolutions of theplastic strain.

We chose to retain the approach consisting of reproducing thebehaviour cycle by cycle by means of an appropriate constitutivelaw. However, keeping in mind that the simulation of a pile sub-jected to a horizontal load is signicantly more complex than thesimulation of triaxial tests (because the directions of the principalstresses can rotate and because the values of the stresses are notcontrolled), we tried to keep the model formulation as simple aspossible.

2.2. Choice of a model

To be able to perform numerical simulations of the behaviour ofreal structures, the constitutive model must be formulated in ageneral three-dimensional framework with no assumptions onthe principal directions of the stress tensor or on the relative valuesof the principal stresses. Apart from this basic requirement, it waspreferable to use the familiar framework of elastoplasticity with alinear elastic law characterised by Youngs modulus E and Poissonsratio v; the model involves only one plastic (deviatoric) mechanismwith a smooth yield surface.

To account for plastic strain accumulation during cyclic load-ings, a kinematic hardening law is needed. The yield surface istranslated in the stress space as plastic strain occurs, and the yieldfunction is given by

f r Fr Xwhere X denotes the hardening variable, and F(r) stands for the ini-tial yield function. The simplest choice is the DruckerPrager func-tion [26]:

Fr 12s : s

r a tr r k

where a and k are two material constants. As mentioned previously,F is clearly a function of the rst two invariants of the stress tensorand does not depend on the Lode angle in the deviatoric plane.

The accumulation of plastic strain can be obtained with a non-linear law to describe the evolution of the kinematic hardeningvariable X. The simplest choice seems to be the law proposed byArmstrong and Fredericks [27]:

_X 2=3C _ep DX _nwhere C is a reference stress value, D is a dimensionless parameter

_ p p 1=2 _ p p 1=2

eotechnics 37 (2010) 9991007and n 2=3 _e : _e . The term n 2=3 _e : _e remains positiveeven if the plastic rate changes sign; this makes it possible to in-crease the plastic strain continuously. This simple formulation

d Gmakes the number of parameters as low as possible, which is thereason why it is used hereafter. However, numerous improvementscan be used for better control of the rate of plastic strain. Amongother possibilities (for instance, [28,29]), one can introduce a varia-tion of the coefcient of _ep:

_X 2=3Cun _ep DX _nwhere u(n) is an additional scalar function. Other models activatethe back stress component DX _n only if X matches an extra condi-tion, making it possible to dene conditions under which no rat-cheting occurs (see [30]). However, such improvements to themodel increase the number of parameters and the difculty ofdetermining these parameters on the basis of standard laboratorytests.

For the plastic ow rule, the plastic potential is assumed to begiven by

gr Gr Xwith

Gr 12s : s

r b tr r

Within this framework, one obtains a model with a single devi-atoric plastic mechanism that includes a basic representation ofthe accumulation of plastic strain during a cyclic load. The lowestpossible number of parameters seems to be seven: the Youngsmodulus E and Poissons ratio for the isotropic elastic linear law,a and k for the DruckerPrager yield surface, C and D for thenon-linear hardening mechanism, and b for the ow rule. The for-mulation of the hardening law makes the model relatively easy toimplement in a nite element code. On the other hand, it appearsthat the parameters C and D control both the behaviour under amonotonic loading and the behaviour under cycling loadings.

2.3. Integration of the constitutive equations for a monotonic loading

The simplicity of the model formulation makes is possible to de-rive the solution of the constitutive equations for a monotonic tri-axial compression test analytically (in the case where both a and bare positive and smaller than 1=

3

p). Assume, for instance, that a

sample is initially in an isotropic stress state r r3 1 and sub-jected to a monotonically increasing vertical stress while the con-ning pressure r3 remains constant. The vertical compressivestress r1 is such that r1 < r3 < 0. In what follows, we adopt the fol-lowing notations:

r r1e1 e1 r3e2 e2 e3 e3; q r3 r1X X1e1 e1 X3e2 e2 e3 e3; X3 X1 x

The lateral stress is constant r3 = r. The value of the yieldfunction is given by

f r X jq xj3

p a3r3 X3 q x k

and the ow rule is

_ep1 _k q x3

pjq xj b

" #

_ep1 _kq x

23

pjq xj b

" #

where _k denotes the plastic multiplier. Initially, X1 = X3 = 0 andq = 0. Assuming that qx is positive, one nds

E. Bourgeois et al. / Computers an_ep1 _kb 1=3

p ; _ep1 _kb 1=2

3

p_n 23

_ep : _epr

_k2b2 1=3

qwhich leads to

_X1 23C _ep1 DX1 _n _k

23Cb 1=

3

p DX1

2b2 1=3

q

X1 2Cb 1=3

p

3D2b2 1=3

q 1 exp D 2b2 1=3q k

and similarly

X3 2Cb 1=23

p

3D2b2 1=3

q 1 expD 2b2 1=3q kwhich shows that there is a simple relationship between X1 and X3:

X1b 1=

3

p X3b 1=2

3

p

It is then easy to state that

ep1 kb 1=3

p ! X1 2Cb 1=

3

p

3D2b2 1=3

q 1 exp D2b2 1=3

q1=

3

p b e

p1

0@

1A

24

35

and the consistency condition makes it possible (after some calcu-lation) to derive the expression of the deviatoric stress as a functionof the axial plastic strain:

q qel qmax qel1 expcep1with

qel k 3ar31=

3

p a 1

qmax qel CD

1 6ab31 6b2

q 11=

3

p a 2

c D

2b2 1=3

q1=

3

p b

In addition, the volumetric plastic strain is related to the axialplastic strain by

epv ep1 2ep3 3 _kb 3b

b 1=3

p ep1

Thus, one can establish the relationship between the axial andvolumetric strains and the deviatoric stress

e1 ee1 ep1 qE 1cln

qmax qelqmax q

ev eev epv 1 2v

Eq 3b

c1=3

p b ln

qmax qelqmax q

which shows that the rate of the total volumetric strain changessign for q = qcr, with

qcr qmax 3bEc1=

3

p b1 2v

qmax 3bE1 2vD

2b2 1=3

q 3

eotechnics 37 (2010) 9991007 1001The analytical solution makes it possible to draw the q e1 andev e1 curves. Typical curves and characteristic values are shownin Fig. 1. The analytical solution shows that the ultimate value of

40 mm

d Gthe deviatoric stress is the sum of the value qel of the deviatoricstress at the end of the elastic regime and of a term depending onlyon C/D, a and b. Because qel is linear with respect to the conningpressure r3, qmax varies also linearly with r3. However, the differ-ence between the ultimate deviatoric stress qmax and the elasticlimit qel does not depend on r3.

The analytical integration of the model equations for non-monotonic load paths is more difcult because the increase in plas-tic strain over a cycle depends on the maximum and minimum val-ues of the axial stress. Given that the cyclic behaviour of the sand is

0

v

1

0

0

121

q

qmax

q el

q cr

10

Fig. 1. Representation of the analytical solution of the model for a monotonoustriaxial compression test. Values of qmax, qel, and qcr are given by Eqs. (1)(3).

1002 E. Bourgeois et al. / Computers anour main concern and that we were able to derive the analyticalsolution of the triaxial compression only in the monotonic case,the choice of the parameters used in the subsequent simulationswas not based on the analytical expressions obtained above;parameters were chosen by searching sets of parameters, makingit possible to reproduce the results of monotonic and cyclic triaxialtests reasonably well (see Section 3.2).

3. Modelling of the behaviour of a centrifuged model pile underlateral cyclic loading

The model presented above was implemented in the nite ele-ment software package CESAR-LCPC [31] (research version). Toconrm the ability of the model to reproduce the cyclic behaviourof a pile subjected to a cyclic lateral load, a numerical simulation ofthe behaviour of an experimental model pile subjected to a lateralcyclic loading was performed. The experiments were carried outwith the geotechnical centrifuge of LCPC (French Public Works Lab-oratory) in Nantes, France. The centrifuge basket platform offsetfrom the axis is equal to 5.5 m, and the maximum mass of themodel is 2000 kg. The maximum acceleration is equal to 200 g[32]. Details on the measurement techniques used can be foundin [33]. The centrifuge was used previously to model the behaviourof piles submitted to cyclic loading (see for instance [3436]).

The experimental setup is presented in [3,37,38] and shown inFig. 2. The bending moments in the pile were calculated using thestrains measured from the strain gauges along the pile shaft. Dis-placements of the pile head were also monitored by means ofthe transducers D37 and D76 shown in Fig. 2. The analysis of the

lowing dimensions and properties.Depth of the pile toe d = 300 mm; outer diameter B = 18 mm;inner diameter b = 15 mm; Youngs modulus E = 74 GPa;moment of inertia of the section I = 2.67 109 m4; bending stiff-ness EI = 197.6 N m2.

A variable horizontal force was applied on the pile 40 mm aboveresults is focused on the comparison between the measured andcomputed values of the bending moments and the displacementof the pile head. The displacement referred to as the displacementof the pile head is the displacement of the point where the load isapplied, 40 mm above the sand sample surface; it is obtained asthe average of the measurements of both transducers.

3.1. Presentation of centrifuge tests

Tests were carried out with dry, clean, siliceous sand (Fontaine-bleau sand NE34). A sand sample was prepared by pluviation in arectangular steel container. Its dimensions were 1200 mm 800 mm for a depth of D = 360 mm. With this procedure, the sanddensity is homogeneous in the central part of the container. Thesand sample was then subjected to several cycles of accelerationin the centrifuge up to 40 g. The sand had a density index of 48%,corresponding to a volume weight c = 15.5 kN/m3. Cyclic load testswere performed under an acceleration of 40 g. The model pile usedin the centrifuge tests was made of aluminium AU4G with the fol-

20 mm

Fig. 2. Picture of the centrifuged pile head with the loading device (left) and thedisplacement transducers (right).

eotechnics 37 (2010) 9991007the sand surface. In the rst stage of the test, the applied force in-creased from 0 to 450 N. Then a large number of cycles was per-formed, during which the force was reduced to 150 N, thenincreased again up to 450 N. A specic experimental device wasused to ensure that no bending moment was applied at the pilehead.

3.2. Parameter determination on the basis of triaxial tests

In this section, we present the simulation of triaxial tests on thedry sand used in the centrifuge experiments discussed below. Gi-ven the depth D of the sand sample and the centrifuge acceleration(40 g), it was roughly estimated that the mean stress in the centri-fuged sand sample would vary between 0 and 40 c d = 220 kPa.(Note, however, that simulations give higher values at the piletoe because the contrast between the moduli of the pile and thesand leads to stress concentrations.) To account for the inuenceof the conning pressure, three monotonic triaxial compressiontests were carried out for conning pressures of 50, 100 and200 kPa, with a density DR = 48%.

In the analytical solution derived above, the deviatoric stress in-creases monotonically as the axial strain increases. In contrast,during the experiments, the deviatoric stress was not continuouslyincreasing but reached a peak value before decreasing. Modelparameters were chosen in such a way that the maximum devia-toric stress of the model was close to the ultimate stress (not thepeak stress) of the tests.

A fourth triaxial compression test with a cyclic load was per-formed on the same sand for a conning pressure of 100 kPa. Theaxial stress was rst increased up to 200 kPa, then decreased downto 50 kPa (thus putting the sample in extension), and the axialstrain increased with the number of cycles. Trying to achieve agood t between the analytical expressions and the test resultsled to the following set of parameters:

Youngs modulus E = 110 MPa; Poissons ratio v = 0.2.Parameters of the yield function a = 0.127; k = 1.8 kPa.Parameter of the plastic potential b = 4.05 102.Parameter of the hardening law C = 22 MPa; D = 1200.

For these values, the comparison between simulations andexperimental tests is shown in Figs. 3 and 4 (for the monotonictests) and 5 (for the cyclic test). On the whole, a reasonable agree-ment was obtained for all tests, but the volumetric strain duringmonotonic loadings was not very well reproduced. For the cyclictest (Fig. 5), the axial strain increased with the number of cycles,but the loops obtained by the simulation were relatively narrowerthan in the experimental curves.

3.3. Presentation of the numerical model

Around a pile subjected to a lateral load, the stress and strainelds are clearly three-dimensional. The pile behaviour cannot berepresented in a realistic way by means of a bi-dimensional analy-sis without making a number of assumptions that are not easy tojustify, which was why a fully three-dimension nite element sim-ulation was undertaken.

As mentioned before, simulations were carried out with the -

500

600

-50

-25

0

25

50

75

100

0 0.0005 0.001 0.0015

q (k

Pa)

1

experiment

simulationmonotonous

loading +cycle 1

simulationcycle 4

Fig. 5. Cyclic triaxial test on Fontainebleau sand: deviatoric stress q vs. axial straine1. Comparison of simulation with experiment.

E. Bourgeois et al. / Computers and Geotechnics 37 (2010) 9991007 10030

100

200

300

400

0 0.05 0.1 0.15

q (k

Pa)

1

200 kPa

100 kPa

50 kPa

Fig. 3. Simulations of three monotonic compression triaxial tests on Fontainebleausand: deviatoric stress q vs. axial strain e1 (solid lines: experimental results, dottedlines: simulations).

-0.0025

0

0.0025

0.005

0 0.005 0.01 0.015

v

1

200 kPa

100 kPa

50 kPaFig. 4. Simulations of three monotonic compression triaxial tests on Fontainebleausand: volumetric stress ev vs. axial strain e1 (solid lines: experimental results,dotted lines: simulations).nite element software CESAR-LCPC [31]. The hardware used was aSun Ultra 40 M2 Workstation powered by a 2.8-GHz AMD OpteronCPU, running under a 64-bit Ubuntu Linux 9. The mesh (shown inFig. 6) was relatively coarse. The buried part of the pile is divided inonly seven quadratic elements in the vertical direction. In the hor-izontal directions, the element size gradually increased from 1 mmclose to the pile to approximately 10 mm close to the mesh bound-aries. On the whole, the mesh was made of 1200 elements, 20-nodehexahedra outside the pile and 15-node pentahedra for the pile it-self and the sand beneath the pile toe. Interpolation was quadraticFig. 6. View of the three-dimensional mesh used for the simulations (CESAR-LCPC).

pared with the size of the mesh. Fig. 8 shows the extent of the areain which the equivalent plastic strain was greater than 0.5% at theend of the initial loading and after 10, 25 and 100 cycles. The plas-tic zone around the pile tended to spread in the horizontal direc-tion, especially behind the pile (in the area that was notsubjected to compression when the applied force increased), butits rate of extension decreased with the number of cycles. It canalso be noted that the plastic zone did not reach the meshboundaries.

Results are discussed below in more detail, while the discussionis focused on the pile head displacement and on the bending mo-ments in the pile. In each case, we discuss the results obtained atthe end of the initial monotonic loading and the evolution of theresults when the number of cycles increased.

4.1. Head displacement

Fig. 7 shows that there is a good agreement between thenumerical simulations and experimental results for the initial

Fig. 8. Extent of the zone where the equivalent plastic strain is greater than 0.5% atthe end of the monotonic loading and after 10, 25 and 100 cycles.

d Gfor all elements, which led to a total number of degrees of freedomof approximately 16,000. The number of unknowns was thereforerelatively small, and it would be easy to rene the mesh if needed.It can be noticed, however, that the non-linear kinematic harden-ing law leads to heavy non-linearities and large computation times(typically around 4 h for 30 cycles of loading, each cycle being di-vided into six computation steps).

Given the existence of a vertical plane of symmetry of the pileand the load, only one half of the structure was included in themesh. A simple preliminary sensitivity analysis was carried outto reduce the dimensions of the mesh, in order to reduce computa-tion times. This analysis showed that the horizontal displacementsalong the model pile were very close for a mesh with the same ex-tent as the actual sand sample (1200 800 360 mm) and for amesh with reduced dimensions of 520 400 360 mm. Becausemeasurements were made on the centrifuge model, it was decidedto conduct the computations at the model scale and not at the pro-totype scale. However, the simulation results were in perfectagreement with the theoretical scaling laws.

Regarding the sand/pile interaction, no special contact elementswere used. Using contact elements leads to introducing additionalparameters, for which the choice of appropriate values on the basisof simple tests is difcult; using contact elements with assumedparameters, especially when complex cyclic loadings are taken intoaccount, can lead to errors that are not easy to quantify. Instead,following the approach of Kooijman [8] or Wakai et al. [14], twolayers of standard elements were placed around the pile shaft, eachlayer having a thickness of 1 mm.

The pile was represented in the mesh by a full cylinder with thesame diameter as the hollow aluminium tube used in the centri-fuge experiments. Because of the difference between the momentsof inertia of the full and hollow sections, a ctitious value of theYoungs modulus was adopted for the pile in the simulationsEf = 3.83 104 MPa. This value was chosen in such a way that thebending stiffness was the same in the simulation as in the centri-fuge experiments.

Initially the stresses were set to zero in the sand and in the pile.The numerical procedure consisted of three phases:

In the rst phase, the volume weight was increased progres-sively in ten equal substeps to reproduce the centrifuge acceler-ation phase.

The second phase corresponded to a monotonic horizontal load-ing; a horizontal force was applied on the pile, 40 mm above thesand surface. The monotonic loading included nine substeps.

Once the maximum value of the applied load was reached, thesimulation of the cyclic behaviour started; the applied load var-ied between its maximum and minimum values. As mentionedpreviously, each cycle was performed in six substeps.

4. Results

A numerical simulation of the behaviour of the centrifugedmodel pile under cyclic lateral loading was performed with theset of parameters obtained on the basis of the triaxial tests, givenin Section 3.2 above. The displacement of the pile head for the rsttwo cycles is plotted against the applied horizontal force in Fig. 7.The response of the pile to the initial monotonic loading is qualita-tively well reproduced in terms of displacements of the pile head.The model then predicts a progressive accumulation of strain,which shows that the model makes it possible to reproduce a cyclicbehaviour. However, it seems that the predicted increase in thehead displacement for the rst two cycles is smaller than the mea-

1004 E. Bourgeois et al. / Computers ansurements, which is discussed in detail below.At the end of the monotonic loading, the extent of the zone in

which plastic strains were signicant was relatively small com-0

100

200

300

400

500

0 1 2 3 4

simulation

centrifuge test

F(N)

(mm)

Fig. 7. Displacement of the centrifuged pile head for the rst two cycles (simulationvs. experiment).

eotechnics 37 (2010) 9991007monotonic lateral loading of the single pile in terms of displace-ment of the pile head. Note, however, that the apparent stiffnessof the pile is larger in the simulation than in the experiments, lead-

ing to a somewhat smaller head displacement at the end of themonotonic loading. The computed value is equal to 3.21 mm, andthe measured value is equal to 3.74 mm; the relative differenceis less than 15%. To explain this difference, one can make theassumption that the sand density obtained after the centrifugeacceleration reaches its nal value of 40 g is smaller in the vicinityof the pile. The sand located at the interface with the pile could beless compacted and therefore less stiff. Besides, the phase duringwhich the pile is driven in the sand was not taken into accountin the numerical simulations.

Fig. 9 shows the pile head displacement D at the end of each cy-cle for an increasing number of cycles (cycle 0 indicates the end ofthe initial monotonic loading). Experimental results show that theaccumulation of strain tends to decrease with the number of cy-cles. Numerical simulations reproduce this trend, but the predictedstrain accumulation seems to be smaller for the rst cycles in thesimulation than in the experiment. Again, it is likely that the initialdensity of the sand near the pile is not as large as that of the sandfar from the pile and that the initial density is also smaller than thedensity of the samples used for the triaxial tests.

However, after 20 cycles, the displacement accumulation percycle (shown in Fig. 10) is in very good agreement with the mea-sured evolution; the accumulated displacement between the20th and the 100th cycles is equal to 0.49 mm in the simulationand 0.55 mm in the experiment, which corresponds to a relativeerror of 11%.

4.2. Bending moments

There is a good agreement between numerical simulations andexperimental results for the initial monotonic lateral loading of thesingle pile in terms of bending moments in the pile (Fig. 11),although the value of the maximum bending moment is slightlyunderestimated and the depth of the point where it is located isslightly overestimated.

Fig. 12 shows the bending moments in the model pile after 10and 100 cycles. Experimental results show that:

Near the surface, bending moments in the pile are almost con-stant during the rst 100 cycles.

At larger depths, the curve drifts downwards as the number ofcycles increases.

Both of these results are reproduced by the numerical simula-tions. However, the measures show an increase in the maximumvalue of the bending moments, which is not captured by thenumerical model. It must be recalled that the mesh is relativelycoarse in the vertical direction, and better results could probablybe obtained by increasing the number of nodes in the mesh.

It can be noted that the depth of the measured and computedmaximum bending moment is in good agreement with classicalestimates. According to Broms [39], for instance, for a load F anda pile diameter B, the depth of the maximum bending moment fis given by

f 0:82F

cBKp

s4

4

5

of p

ile h

ead

(mm

)

0 10 20 30 40 50bending moment (kN.m)

E. Bourgeois et al. / Computers and Geotechnics 37 (2010) 9991007 10052

3

0 20 40 60 80 100

Number of cycles

Dis

plac

emen

t

Fig. 9. Displacement of the pile head vs. the number of cycles (bold solid line:centrifuge test, dotted line: simulation).

0

0.01

0.02

0.03

0.04

0.05

0 20 40 60 80 100

Number of cycles

Dis

plac

emen

t acc

umul

atio

n

per c

ycle

(mm

) Fig. 10. Evolution of the increase in pile head displacement for one cycle (bold solidline: centrifuge test, dotted line: simulation).0

0.05

0.1

0.15

0.2

0.25

0.3

simulationcentrifuge test

depth (m)

Fig. 11. Bending moment in the pile at the end of the monotonic loading:simulation vs. experiment.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50

depth (m)

simulation N=10

simulation N=100

experiment N=100

experiment N=10

bending moment (kN.m)Fig. 12. Bending moments after 10 and 100 cycles: comparison of simulations withexperiments.

d Gnumber of parameters that could be relatively easily obtained fromthe results of three monotonic and one cyclic triaxial tests, whichmakes it possible to use the model to simulate the behaviour ofreal geotechnical structures and to discuss practical applications.

The model was used to simulate centrifuge tests on a pile sub-jected to a cyclic lateral loading. The results showed that the modelwas able to reproduce the behaviour of the pile during the initialmonotonic loading phase and to reproduce the strain accumulationas the number of cycles increased. The evolution of the bendingmoments in the pile was also consistent with the experimental re-sults. However, it was not possible to derive the evolution of the py curves induced by the repeated loading from the numerical re-sults; such an analysis would require a ner mesh than the oneused for this study.

Further research could be undertaken to widen the applicabilityof the model (for instance, by simulating other structures subjectedto periodical loadings, especially groups of piles) or to improve itsnumerical efciency if a very large number of cycles are to be takeninto account. In addition, other enhancements could be brought tothe model, such as a more complex yield surface, a non-linear elas-tic law, or a hardening law coupling the elastic moduli and thehardening parameter.

Referenceswhere c is the volume weight of the ground, and Kp = (1 + sin u)/(1 sin u), where u is the friction angle. For the centrifuged modelpile, the results of the triaxial compression tests gave the frictionangle a value of 34, which leads to f = 88 mm. This value is slightlylarger than the measured and computed depths of the maximumbending moment at the end of the monotonic loading (75 and80 mm, respectively). As the number of cycles increases, the depthof the maximum increases in the experiments and becomes largerthan the analytical value given by Eq. (4).

The aim of this study was to provide a way of investigating thebehaviour of a pile under cyclic loading without having to adaptempirically monotonic py curves to account for the effect of loadcycles. However, an attempt was made to derive the earth pressuredistribution along the pile from the results of the nite elementsimulations by evaluating the second derivative of the bendingmoment. The results were not satisfactory (compared with the val-ues of p obtained from the experimentally obtained bending mo-ments) because the mesh used was too coarse to allow for aprecise evaluation of the second derivative of the bending mo-ment; there were only 17 nodes in the vertical direction betweenthe ground surface and the pile toe. To investigate numericallythe evolution of the py curves, it would be necessary to use amuch ner mesh.

5. Discussion and conclusions

The design of structures subjected to variable, pseudo-periodi-cal loads is an important issue in many areas of civil engineering(e.g. for the foundations of offshore windmills). The questions un-der discussion are on the one hand, the strain accumulation duringcycles, and on the other hand, the evolution of the bending mo-ments in the piles during the normal service life of the foundations.In this study, the choice was made to perform three-dimensionalnite element computations using a relatively simple constitutivemodel, which combines a linear isotropic elastic law with only onesingle deviatoric plastic mechanism (with a yield function thatdoes not depend on the Lode angle); the formulation of the hard-ening law was also very simple. In spite of these theoretical short-comings, the model presents the great advantage of a very small

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E. Bourgeois et al. / Computers and Geotechnics 37 (2010) 9991007 1007

Three-dimensional numerical modelling of the behaviour of a pile subjected to cyclic lateral loadingIntroductionTheoryConstitutive models for cyclic behaviourChoice of a modelIntegration of the constitutive equations for a monotonic loading

Modelling of the behaviour of a centrifuged model pile under lateral cyclic loadingPresentation of centrifuge testsParameter determination on the basis of triaxial testsPresentation of the numerical model

ResultsHead displacementBending moments

Discussion and conclusionsReferences