Computers and Geotechnics 55 (2014) 378–391

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Computers and Geotechnics

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A load transfer approach for studying the cyclic behaviorof thermo-active piles

0266-352X/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compgeo.2013.09.021

⇑ Corresponding author.E-mail addresses: msuryatr@polytech-lille.fr (M.E. Suryatriyastuti), hussein.

mroueh@polytech-lille.fr (H. Mroueh).

M.E. Suryatriyastuti a, H. Mroueh a,⇑, S. Burlon b

a Laboratoire Génie Civil et géo-Environnement (LGCgE) – Polytech’Lille, Université Lille1 Sciences et Technologies, Villeneuve d’Ascq 59655, Franceb Institut Français des Sciences et Technologies des Transports, de l’Aménagement, et des Réseaux (IFSTTAR), Marne la Vallée 77447, France

a r t i c l e i n f o

Article history:Received 15 April 2013Received in revised form 22 July 2013Accepted 29 September 2013Available online 19 October 2013

Keywords:Thermo-active pilesCyclic thermal loadingSoil–pile interfacet–z FunctionModjoinFinite difference methodThree-dimensional modeling

a b s t r a c t

Unsatisfactory understanding of thermally induced axial stress and mobilized shaft friction in thethermo-active piles has led to a cautious and conservative design of such piles. Despite the fact thatthe number of construction works using this type of piles has been rapidly increasing since the last20 years and none of them witnessed any structural damage, the question that still remains is how toovercome the cyclic thermal effects in such piles to optimize the design method. This paper presents asoil–pile interaction design method of an axially loaded thermo-active pile based on a load transferapproach by introducing a proposed t–z cyclic function. The proposed t–z function comprises a cyclichardening/softening mechanism which is able to count the degradation of the soil–pile capacity duringtwo-way cyclic thermal loading in the thermo-active pile. The proposed t–z function is then comparedto a constitutive law of soil–pile interface behavior under cyclic loading, the Modjoin law. Afterwards,numerical analyses of a thermo-active pile located in cohesionless soil are conducted using the two cycliclaws in order to comprehend the response of such pile under combined mechanical and cyclic thermalloads. The behaviors of the pile resulting from the two laws show a good agreement in rendering the cyc-lic degradation effects. At last, the results permit us to estimate the change in axial stress and shaft fric-tion induced by temperature variations that should be taken into account in the geotechnical design ofthe thermo-active pile.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The thermo-active piles have successfully incorporated the heatexchanger elements to the structural pile foundations. Other thansupporting the static weight of structure, the thermo-active pilesare used to provide thermal energy to the overlying building by cir-culating the heat carrier fluid inside the piles [1,2]. In consequence,the thermo-active piles are subjected not only to the mechanicalloading of the overlying structure but also to a two-way cyclicthermal loading (i.e. seasonal thermal loading) according to thethermal needs of building. Installation of the thermo-active pilesin European countries [3–6] showed that the usage of these pilesis advantageous in increasing the energy performance and in min-imizing the annual cost [6,7]. But at the same time, this latter pre-sents high risk on the mechanical resistance of both foundationsand upper structure [8–10] because the circulating warm fluid dur-ing summer produces a pile expansion and the circulating coolfluid during winter produces a pile contraction [11,12]. Besides,no design code that takes into account the thermal interaction on

the geotechnical capacity of pile foundations is available yet [13].For years, contractors have done constructions with thermo-activepiles based on empirical considerations or on a conservative designmethod by increasing the safety factor [13,14]. As a result, a biggerdimension of pile and a higher piling cost are required.

Due to the limited knowledge concerning the impact of thermaloperation on the geotechnical performance, the response of ther-mo-active piles under combined mechanical and cyclic thermalloads becomes a major interest for establishing a more effectivegeotechnical design criterion. Since the ratio of the pile diameterto the pile length is very small, the temperature variations injectedin the pile affects mainly the pile axial response [15]. In situ expe-riences of the new building at Swiss Federal Institute of Technologyin Lausanne [10] and the Lambeth College in London [8] have re-marked an important change in mobilized shaft friction and axialload distribution at the soil–structure interface by the change oftemperature [16]. These changes are stated to be dependent onthe degree of axial fixity at the head and the toe of pile foundation[8,16]. While most reliable method to determine the response ofthe piles is based on the results obtained from pile load tests, thismethod can be expensive and time-consuming [17]. Other alterna-tive means to study the axially loaded piles is by conducting

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numerical modeling with finite element analysis or with loadtransfer analysis (t–z function). The first approach permits to mod-el any constitutive soil behavior with non-linearity and complexsoil–structure interaction, but in fact it does not really providepractical solutions for piling problems [18]. The second approachrequires to divide the pile into a number of segments supportedby discrete springs representing the soil resistance at each shaftelement [19,20], where the movement of the pile at any segmentis related to the shaft friction at that segment [20].

This paper presents a numerical study of the seasonal tempera-ture induced change in mechanical behavior of pile in the aim tooptimize the geotechnical design capacity of the thermo-activepiles. The work is based on the concept of soil–structure interac-tion and thus requires the development of a constitutive law thattakes into account the cyclic behavior at the soil–pile interface.Two numerical approaches are proposed: load transfer analysisin one-dimensional model and finite difference analysis in three-dimensional model. In the first part, a development of a nonlineart–z function at soil–pile interface comprising a cyclic hardening/softening mechanism is presented. This function is then employedto the load transfer method for the design of thermo-active piles ina practical engineering approach. The conceptual background, theworkability, and the performance of the proposed function com-pared to the results from experimental tests are discussed. The sec-ond approach used in this work is based on modeling the soil–pilecontact using the constitutive interface law Modjoin [21] with fi-nite difference method. This constitutive law had been developedin the laboratory and has recently enhanced to control cyclic deg-radation phenomena as strain ratcheting, strain accommodationand stress relaxation [22]. The study is distinguished into two ex-treme head restraint conditions: a free head pile and a fully re-strained head pile.

2. Development of t–z function under two-way cyclic loading

The fundamental requirement of load transfer analysis is theappropriate t–z function used to measure the local shaft frictionand the relative displacement of soil–pile. A number of t–z curvesresulted from in situ static loading tests has been established byCoyle and Reese [23], Coyle and Sulaiman [24], Frank and Zhao[25], and Reese and O’Neill [26]. These t–z curves were originallyobtained empirically but may now be obtained more satisfactorilyvia theoretical relationship with the stiffness of the surroundingsoil [27,28].

Randolph has developed a theoretical t–z function under axi-ally cyclic loading and implemented it to RATZ numerical com-putation program analysis [29,30]. The function consists of alinear elastic part, a nonlinear parabolic shape function describ-ing the strain hardening/softening mechanism, and also a func-tion considering cyclic degradation effects [30,31]. In its recentversion, some extensions have been included such as thermalstrain in the pile, but it is limited to a single magnitude of thecyclic thermal strains for each analysis. Laboratory of soilmechanics in Swiss Federal Institute of Technology Lausannehas worked on two-way cyclic thermal loading by adding theunloading curve in the t–z curve developed by Frank and Zhao[32]. The curve is implemented into ThermoPile program soft-ware, which is designed specifically for analyzing the thermo-ac-tive piles behavior [32]. However, the curve is limited to twolinear elastoplastic branches and a plateau corresponding tothe ultimate stress value without having a kinematic hardeningcriterion [13,32].

Authors intend to develop a theoretical nonlinear t–z functionwhich is able to describe the cyclic hardening/softening mecha-nism under two-way cyclic loading. This t–z function is

implemented into an algorithm programming language offering apractical design tool for the geotechnical design of thermo-activepiles.

2.1. Conceptual background

Generally speaking, the seasonal thermal contraction and dila-tation in the thermo-active piles can be equated with a two-waycyclic axial loading. A number of experimental investigations havebeen carried out to learn the response of piles under cyclic axialloading. Holmquist and Matlock [33] stated that two-way cyclicloading results in a dramatic reduction in the pile load capacitymuch more than that in the case of one-way cyclic loading. Besides,they pointed out that the reduction in shaft friction has reached upto 75% in the case of extremely large displacement amplitudes.Data collected by Bea et al. [34] showed a remarkable increase inpile head settlement with the number of cycles, causing a reduc-tion in load capacity. On the other hand, Bjerrum [35] and Beaet al. [34] indicated that the more rapid the rate of loading is, thegreater the pile capacity becomes. Desai et al. [36] and Fakharianand Evgin [37] concluded that in cohesionless soils, the interfaceresponse gets stiffer as the number of cycles increases, while therate of stiffening decreases.

Poulos found that under two-way axially cyclic loading, thereduction in material volume leads to the reduction in normalstress and consequently to the reduction in shear stress mobilizedbetween the shaft and the soil [18,38]. According to these facts,Poulos concluded that the degradation in shaft resistance dependsnot only on the reduction in shear stress as a function of absolutecyclic slip displacement, but also on the reduction in normal effec-tive stress due to volumetric strain during cyclic shearing [37,39].The former component may become more significant in less com-pressible soils whereas the latter component may dominate incompressible soils.

As a summary, modeling the nonlinear relationship betweenthe shaft friction and the relative displacement at the soil–pileinterface under two-way cyclic loading should satisfy the followingconditions:

� Reduction in shaft friction with increasing the number of cycles[18,33,34].� Degradation in shaft resistance with the accumulation of abso-

lute tangential displacement [37,39].� Reduction in normal stress by the volumetric strain with

increasing the number of cycles [39].� Increase in shaft resistance with a higher loading rate [34,35].� Increase in pile settlement with increasing the number of cycles

[34].� Hardening and stiffening in interface with increasing the num-

ber of cycles [36].� Decrease in rate of stiffening with increasing the number of

cycles [36].

With respect to the hypotheses above, a new formulation ofnonlinear t–z function is developed. The proposed t–z function isdivided into two stages of loading: the initial relation under mono-tonic loading and the extension under cyclic loading.

2.2. Proposed t–z function under monotonic loading

2.2.1. Basic principleUnder monotonic loading, the formulation of the proposed t–z

function is based on the Frank & Zhao t–z law [25] with respectto the French design standard for the deep foundations design[40]. The shaft friction mobilized at the soil–pile interface qs re-lated to the tangential displacement ut is given by:

Fig. 1. Parametric study of a.

Fig. 2. Comparison of the proposed t–z function and the t–z law of Frank & Zhao[17].

Fig. 3. Parametric study of b.

Fig. 4. Parametric study of c.

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qs ¼ qs0 1� e�uta

� �ð1Þ

where qs0 (kPa) and a (m) denote the ultimate mobilized frictionand the rate of mobilized friction under monotonic loading, respec-tively. A parametric study of the influence of parameter a on theevolution of shaft friction qs is given in Fig. 1. It shows that thesmaller the value of a, the smaller displacement is needed for mobi-lizing friction at the pile shaft, which leads increasing the stiffnessof interface. The gradient of the function represents the slope ofthe curve and can be equated to the coefficient kt in the Frank &Zhao law. Therefore, the choice of parameters qs0 and a is done byan approximation to the Frank & Zhao coefficient [25], or, it canbe taken from the Menard’s pressure meter modulus EM and Me-nard’s pressure limit pLM [41] according to the French national codefor deep foundation [40].

Fig. 2 shows the comparison of the proposed t–z function to theFrank & Zhao law under monotonic loading, both for fine soils andgranular soils. By setting the parameter a in accordance with kt, thet–z curves resulting from the proposed function are very close tothe curves of Frank & Zhao law. When the value of a is equal tothe imposed relative displacement ut, the mobilized friction ob-tained using Eq. (1) is equal to 63% of the ultimate mobilized fric-tion. At this state of displacement, the difference between themobilized frictions obtaining from the proposed t–z function andthe Frank & Zhao law is below 5 kPa.

2.2.2. Strain hardening/softening parametersStrain hardening/softening phenomenon in materials depends

on the void ratio in granular soils and on the Atterberg limits in

fine soils. In order to consider the nonlinear behavior of strainhardening/softening, the proposed t–z function introduces threeother parameters b, d, and c in following equation:

qs ¼ qs0 1� e�uta

� �þ bud

t e�utcð Þd ð2Þ

where b (kPa) and c (m) indicate the amplitude of strain hardening/softening and the rate of strain hardening/softening under mono-tonic loading. d is an unitless parameter which controls the timeloading rate.

The first part of the equation defines the relation of shaft resis-tance and tangential displacement with the control of ultimateshaft resistance, whereas the second part of the equation de-scribes the peak of shaft resistance related to the loading rate thatmay be attained in parabolic function. The peak of shaft resis-tance may occur only in dense soil with a high loading rate, forexample in over consolidated clay or in dense sand. The greaterthe parameter b is, the higher the peak of parabolic curve is ob-tained, which corresponds eventually to the denser soil (Fig. 3).For a same given value of b, parameter c controls the radius ofparabolic curve expressing the velocity of strain hardening/soft-ening mechanism (Fig. 4).

The parametric study of d has successfully described thehypothesis of Bjerrum [35] and Bea et al. [34] who showed that amore rapid loading rate contributes to an increase in pile capacity.Thus, a greater parameter d reveals to a higher peak of shaft resis-tance (Fig. 5).

Fig. 5. Parametric study of d.

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2.3. Proposed t–z function under two-way cyclic loading

In analyzing the behavior of piles subjected to cyclic loading,three aspects of soil response should be considered: the degrada-tion of soil–pile resistance and/or soil modulus, the loading rate ef-fects, and the accumulation of permanent displacements [18]. Theproposed t–z function under two-way cyclic loading comprising ofthose aspects is given in following equation:qs ¼ qsi

þAð�1Þnþ1 qs0þDqsð1�e�utse Þ

� �1�e�Rjut�uti

a j� �

þbe�utse jut�utijde�

ut�uticð Þ2

� �

ð3Þ

where n denotes the number of inversion loading during two-waycyclic loads. qsi (kPa), uti (m), and uts (m) indicate the initial calcu-lated friction at each inversion loading cycle (qsi = 0 for n = 1), theinitial displacement obtained at each inversion loading cycle(uti = 0 for n = 1), and the cumulative displacements during two-way cyclic loads (uts ¼

Pjutij).

The degradation of interface resistance is controlled by theparameters Dqs (kPa) and e (m) which express the amplitude ofcyclic strain hardening/softening and the rate of cyclic strain hard-ening/softening.

The function R in Eq. (4) controls the stiffness hardening ofinterface at each inversion loading cycle, ranging between 1 andq. Inside the function R, parameters q and n allow controlling cyclicfatigue effects in interface behavior, such as stress relaxation orstrain ratcheting, where q indicates the amplitude of cyclic degra-dation and n is the rate of degradation.

Factor A in Eq. (5) is the interface stiffness hardening factorwhich is dependent on the level of stress obtained. A takes into ac-count the distance between the actual stress state qsi and the max-imum cyclic stress state qs0 + Dqs which may vary with the cycles.If an inversion of loading cycle occurs at high level of stress, a high-er value of A is produced and thus the interface becomes stiffer inthe next cycle. This factor varies between 1.0 and 2.0.

R ¼ e�ðn�1Þn þ qð1� e�ðn�1ÞnÞ ð4Þ

A ¼qsi � ð�1Þnþ1 qs0 þ Dqs 1� e�

utse

� �� �

qs0 þ Dqs 1� e�utse

� �������

������ ð5Þ

Fig. 6. (a) Stress relaxation when q and n = 1 (b) no relaxation when q and n = 3.

2.4. Performance of the proposed t–z function

To study the performance of the proposed t–z function, severaltests are conducted in two different modes: strain-controlled and

stress-controlled. If the deformation is maintained constant withthe increment of cycles, the stress will gradually decrease due tocyclic fatigue in interface. This phenomenon is known as stressrelaxation [42–44]. Otherwise, if the stress is sustained along thecycles, a progressive accumulation of plastic strain during cyclesoccurs, known as strain ratcheting phenomenon [42–44].

Figs. 6 and 7 show the parametric study of cyclic degradationparameters q and n at strain-controlled and stress-controlled con-ditions. When the values of q and n are set at 1.0, the phenomenaof stress relaxation and strain ratcheting are important. The higherthe values of q and n are, the stiffer the interface behaves undercyclic loading. Accordingly, there is lower variation of the decreasein shaft resistance per cycle (Fig. 6b) and the accumulation of plas-tic strain becomes slower and smaller (Fig. 7b).

A direct shear test under cyclic loading in strain-controlled wasconducted at Centre d’Etudes et Techniques de l’Equipement (CETE)Nord Picardie. Due to cyclic softening, the results showed thatthe friction mobilized became lower with increasing the numberof cycles, but the interface stiffness increased at each load cycle.Fig. 8 shows a comparison of the results from direct shear testand from modeling the interface behavior with the proposed t–zfunction. By setting the parameters in the proposed function inaccordance with the material properties used in the direct sheartest, the proposed t–z function can successfully describe similarphenomena in interface.

Fig. 7. (a) Strain ratcheting when q and n = 1 (b) no ratcheting when q and n = 3.

Fig. 8. Comparison of the results from direct shear test and the proposed t–zfunction.

Fig. 9. Comparison of the Modjoin law and the proposed t–z function (a) stressrelaxation phenomenon (b) strain ratcheting phenomenon.

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3. A constitutive law of the soil–pile interface under cyclicloading: Modjoin

Modjoin is a constitutive law of interface behavior under cyclicloading in the framework of elastoplasticity based on the conceptof bounding surface [21]. The elastic part is defined by two param-

eters: normal stiffness kn and shear stiffness kt, which relate thenormal stress r to the normal displacement un and the shear stresss to the tangential displacement ut. Modjoin incorporates soil non-homogeneity, non-linearity, cyclic degradation, post-peak soften-ing–hardening, and interface contraction–dilatation.

The limit resistance of the interface is defined by the boundarysurface. The boundary surface fl with the associated isotropic hard-ening function Rmax are governed in following equations:

fl ¼ jsj þ rnRmax ð6Þ

Rmax ¼ tan /þ DR 1� eADRuptr

� �ð7Þ

where Rmax depends on friction angle u and cumulative plastic tan-gential displacement up

tr . DR and ADR are the control parameterswhich indicate respectively the amplitude and the rate of isotropichardening.

The kinematic surface fc encloses the domain of yielding, whichis controlled by the associated kinematic hardening function Rc. fc

and Rc are given in Eqs. (8) and (9) and are governed by the param-eters cc, bc, and k as follows:

fc ¼ js� rnRcj ð8Þ

dRc ¼ k ccjRmax � Rcjbc

� �ð9Þ

where cc and bc express the amplitude and the rate of kinematichardening and k denotes plastic multiplier.

Finally, the flow rule to reproduce the contracting phase fol-lowed by dilating phase is given in Eqs. (10) and (11). This rule de-

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pends on the actual plastic tangential displacement uptc, the dilation

angle wc and the velocity of phase change ac.

@g@rn¼ tan wc �

s� rnRc

rn

��������

� �e�ac up

tc ð10Þ

@g@s ¼

sjsj ð11Þ

This constitutive law is capable of drawing cyclic degradationphenomena of the soil–pile interface, such as mean stressrelaxation, hardening/softening stress, strain ratcheting, and strainaccommodation [22]. Fig. 9 shows the comparison of the Modjoinlaw and the proposed t–z function in rendering stress relaxationphenomena under a strain-controlled test and strain ratchetingphenomena under a stress-controlled test.

4. Study of a thermo-active pile under two-way cyclic thermalloading

The response of thermo-active piles under two-way cyclic ther-mal loading during their seasonal operation time becomes onemajor interest to optimize their geotechnical capacity. Duringwinter operation, a cooler fluid is injected in the pile to extractthe heat energy from the soil, thus the pile contracts. The temper-ature of the injected fluid must be above 0 �C to avoidfreezing–thawing of the surrounding soil [7]. During summer oper-ation, the cooled down soil is recharged by circulating awarmer fluid along the pile, thus the pile expands. The cycliccontraction–dilatation of thermo-active piles during the period ofoperation induces change in axial load and shaft friction that leadsto the degradation in the soil and interface resistances [8–10,16]. Inorder to analyze the cyclic behavior of the thermo-active pile,numerical simulations are performed using the two cyclic laws:the proposed t–z function with load transfer method in 1D modeland the Modjoin law with finite difference method in 3D model.Since the behavior of the thermo-active pile is dependent on theaxial fixity at the head and the toe of the pile [8,16], this presentstudy conducts two extreme cases of the pile head axial fixity.The first case concerns the free head pile with constant thermalhead load over the cycles (i.e. zero head axial fixity) and the secondcase concerns the restrained head pile with constant thermal headdisplacement over the cycles (i.e. infinite head axial fixity). Thestudy focuses on observing four aspects that are influenced and de-graded during cyclic thermal loading: pile head settlement, headload capacity, axial load distribution, and local shaft friction.

4.1. Numerical model

A single thermo-active pile with a square section of widthB = 60 cm and length H = 15 m is founded on very loose sandy soils.The pile is initially subjected to incremental monotonic loading

Table 1Properties of materials and of Modjoin interfaces.

Properties Notation Soil

Density of material qm 195Young’s modulus E 10 MThermal conductivity kT 1.5Specific heat extraction c 800Coefficient of thermal expansion aT 5 �Normal stiffness kn –Shear stiffness kt –Friction angle u –Dilation angle w –

corresponding to the permanent load of building during the con-struction phase. According to the French design standard for thedeep foundations, the mechanical working load applied to the pileQmec is fixed at 33% of the ultimate monotonic load QULT when thepile head settlement is equal to 1/10 B [40,45]. This stage of loadingoccurs at n = 0 with an index ‘‘mec’’.

In addition to the mechanical working load, the pile is also sub-jected to the temperature variations during the seasonal energyextraction, with a temperature gradient ±10 �C from the groundtemperature. The temperature gradient in the pile is assumed tobe uniform over the entire pile, and, accordingly, the addition ofthermal loading is applied in terms of a uniform thermal deforma-tion eth by multiplying the temperature gradient DT with the con-crete thermal expansion coefficient aT. Thermal loading cycles areperformed during 12 year of thermo-active pile operation, com-prising of 24 seasonal cooling and heating cycles. Each cycle corre-sponds to 1 season of temperature loading (i.e. 6 months) inneglecting the daily temperature variation in the pile. Fig. 10 illus-trates the schematic of loading sequences in the thermo-activepile. Since the ratio of the pile diameter to the pile length is verysmall, radial movements of the pile induced by temperature varia-tions can be neglected in comparison with thermally induced axialmovements of the pile [15]. Hence, for each cycle of loading, thetemperature variation induces change in axial deformation of thepile as follows:

en ¼ emec � eth ¼ emec � ðaTDTÞ ð12Þ

Both pile and soil are assumed to behave in linear thermo-elas-tic conditions, with the Young’s modulus E of soil is equal to10 MPa and that of pile is 20 GPa. The coefficients of thermalexpansion aT are taken as 5 � 10�6/�C for soil and 1.25 � 10�5/�Cfor pile. The properties of soil are taken from CETE Nord Picardiedata for very loose sand.

In the one-dimensional model, the analysis is performedusing load transfer method by applying the proposed t–z func-tion presented in Section 2.3. The pile is divided into a numberof segments supported by discrete springs that represent themobilized resistance along the shaft qs and at the base of pileqp (Fig. 11). The method consists of back-calculating the bound-ary condition at the pile head by imposing an assumed pile tipmovement for each load cycle. Each pile segment must satisfythe vertical equilibrium for each load cycle. The load transfermethod is implemented into a computer program that has beendeveloped to analyze the axially loaded thermo-active pile. Aflowchart of iterative programming language is given in Fig. 12while the detail equations used to solve the equilibrium are pre-sented in Appendixes A and B.

The three-dimensional analysis is carried out using an explicitfinite difference program FLAC3D. Due to the symmetric condition,only one-fourth of the complete domain is modeled (Fig. 13). Thesoil is refined around the pile in order to increase the precisionin the areas of high strain gradient. After a parametric analysis,

Pile Modjoin interface

0 kN/m3 2500 kN/m3 –Pa 20 GPa –

W/m2 1.8 W/m2 –J/kg �C 880 J/kg �C –10–6/�C 1.25 � 10–5/�C –

– 22 MN/m– 8.33 MN/m– 30�– 1�

Fig. 10. Schema of cyclic thermal loadings.

Fig. 11. 1D model with load transfer t–z method.

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the mesh was fixed with the soil lateral extension is set at 15 m(25B) and the height of soil mass at 30 m (2H). Interface elementsare introduced at the vertical zone of contact between the soil andpile. The interfaces employ the constitutive Modjoin law in nonlin-ear elastoplastic behavior, presented in Section 3. The interface

normal stiffness kn and shear stiffness ks are chosen in accordancewith the interfaces theory and background of FLAC3D code [46]. Ta-ble 1 summarizes the elastic and thermal properties of soil, pile,and Modjoin interfaces. These properties are assumed not tochange with temperature variation.

Fig. 12. Flowchart diagram of calculation.

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According to the type of soil, the value of parameter b in theproposed t–z function is set at zero and thus no intervention of thisparameter takes place in the second part of Eqs. (2) and (3). Thematerials are considered to undergo cyclic fatigue effects with a

cyclic softening tendency. Hence, the cyclic degradation parame-ters q and n in the proposed t–z function are set at 1.0. This condi-tion is also translated into the parameters used in the Modjoinmodel with DR and ADR are set at �0.05 referring to cyclic soften-

Fig. 13. 3D model with the Modjoin interfaces.

(a) Ratio to the head settlement at the mechanical loading

(b) Ratio to the head settlement at the first cooling/heating cycles

Fig. 14. Temperature-induced change in head settlement in the free head pile.

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ing behavior. Since the analysis in the one-dimensional model doesnot model the surrounding soil mass, the bearing stress qp is notallowed to have a negative value in order to limit the degradationin the base resistance. A comparable assumption is applied in thethree-dimensional model; the interface Modjoin at the pile baseis perfectly bonded to the soil and is not allowed to have separa-tion. The choice of cyclic hardening parameters in the proposedt–z function is adjusted to the Modjoin model.

For the sake of clarity, downward movements of w are takenpositive while upward movements are negative. Compressivestresses and forces are taken positive while tensile stresses andforces are negative according to the convention in soil mechanics.The response of the thermo-active pile over the cycles will be de-tailed in the following section, for both the free head pile and therestrained head pile. The behavior resulting from 1D analysis usingthe proposed t–z function is compared to that of resulting from 3Danalysis using the Modjoin law.

4.2. Response of a free head thermo-active pile

In both models, the head settlement induced by the tempera-ture variations varies below 30% of the mechanical head settle-ment (Fig. 14a). High variation of settlement occurs in the firstcooling–heating cycles when the pile undergoes the first thermalcontraction and dilatation. After the second cooling–heating cycles,the relative variation of pile head settlement during cyclic loads in-creases smoothly up to 20% of its value at the first cooling–heatingcycles (Fig. 14b). Fig. 14b shows clearly that cyclic settlement ofboth models started with the same initial slopes but the t–z modelhas a higher cyclic strain softening.

Fig. 15 shows the distribution of axial force along the pile in thebeginning of the cooling–heating cycles and in the final cooling–heating cycles as well. A decrease in axial force is found in the firstcooling cycle due to the pile shortening. Afterwards, the variationin axial force in the pile increases with the number of cycles, withthe values of variation in cooling are smaller than those in heating.The increase in axial force is caused by the degradation of interfaceand soil properties during cyclic loading. However, the highest va-lue of axial force variation at the final cycle is 9% for 1D model and16% for 3D model (Fig. 15).

In the case of free head pile, the distribution of shaft frictionshows an opposite response at the upper part and the lower partof pile. When the pile is cooled, the bearing stress decreases whilethe shaft friction increases due to the pile contraction. On the con-trary, when the pile is heated, an increase in bearing stress and adecrease in shaft friction are occurred owing to the expansion ofpile.

The variation of local friction vs. local tangential displacementpoints out the cyclic fatigue phenomenon at the soil–pile interfacefor both models. Because of the choice of cyclic hardening/soften-ing parameters in the models, a phenomenon of strain ratchetingappears (Fig. 16). However, at the lower-half of the pile(Fig. 16b), the degradation in interface capacity is smaller due tothe presence of soil surrounding the pile which resists the stressmobilized for the case of 3D model. For the case of 1D model, thefriction of interface tends to increase at the lower-half of the piledue to the boundary condition of the 1D model which limits thedegradation in the bearing stress.

Fig. 15. Change in axial force induced by temperature variations in the free head pile (a) Model t–z and (b) Model Modjoin.

(a) at depth z = ¼ H

(b) at depth z = ¾ H

Fig. 16. Change in the soil–pile interface response at different depths in the free head pile.

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4.3. Response of a restrained head thermo-active pile

It is shown that under constant head displacement, the ther-mo-active pile undergoes a decrease in head axial force relativeto the force at the mechanical loading stage up to �60% (for thet–z model) and �20% (for the Modjoin model) (Fig. 17a). Thisindicates a relaxation phenomenon due to the degradation ofsoil–pile resistance which is affected by the choice of parame-ters. For the same parameters used, the cyclic behavior of thefully-restrained head pile does not show a similar behaviorbetween the t–z and the Modjoin model, unlike the responseof the free head pile (Section 4.2). Fig. 17b shows that the 1Dt–z model brings out a high diminution in pile head capacitythat is possibly due to the absence of soil mass surrounding thepile. Besides, for the same displacements obtained, the forcesrepartition between the 1D and the 3D models are obviouslydifferent.

The thermally induced axial force decreases greater duringcooling than that of during heating (Fig. 18). For the 1D t–zmodel, the axial force decreases uniformly over the entire lengthof the pile. For the 3D Modjoin model, the axial force decreasesalong the depth until reaching a stiffer base resistance at thepile base due to the bonded interface with the soil. Therefore,the maximum degradation in axial force at the final cooling islower at the 3D model (�25%) than that of the 1D t–z model(�65%).

In the restrained head pile, shaft friction decreases graduallyalong the depth of pile. It is noted that the degradation of ther-mally-induced friction is higher during cooling than that of dur-ing heating. This is due to the decrease in the base resistancewhen the pile base lifts up during contraction. Relaxation phe-nomenon also appears in the response of the soil–pile interface.Fig. 19 points out a gradual diminution in shaft friction duringthermal cycles. The variation of tangential displacement is con-stant during cycles in the 1D model because the mobilized stressis supported only by the elastic pile. Different response of inter-face is observed in the 3D model. The Modjoin interfaces devel-op cyclic degradation phenomenon while the surrounding soilremains elastic with no cyclic degradation and thus causes anaccumulation of tangential displacement at the soil–pileinterface.

(a) Ratio to the head force at the mechanical loading

(b) Ratio to the head force at the first cooling–heating cycles

Fig. 17. Temperature-induced change in head axial force in the restrained headpile.

Fig. 18. Change in axial force induced by temperature variations in the restrained head pile (a) Model t–z and (b) Model Modjoin.

(a) at depth z = ¼ H

(b) at depth z = ¾ H

Fig. 19. Change in soil–pile interface response at different depths in the restrainedhead pile.

388 M.E. Suryatriyastuti et al. / Computers and Geotechnics 55 (2014) 378–391

M.E. Suryatriyastuti et al. / Computers and Geotechnics 55 (2014) 378–391 389

5. Conclusion

A development of a theoretical t–z function for the soil–pileinterface under two-way cyclic loading is being established in thispaper. The function is based on a series of results and hypothesesof the previous field and laboratory investigations. The proposedfunction is able to describe the cyclic strain hardening/softeningmechanism and the degradation phenomenon during cyclicloading, comprising the fatigue effects such as stress relaxationand strain ratcheting. The function consists of nine parameters:

� qs0 and a which describe the mobilization of friction undermonotonic loading.� b, d, and c which control the strain hardening/softening under

monotonic loading.� Dqs, e, q, and n which take into account the cyclic strain harden-

ing/softening.

This proposed t–z function is then employed to the load transfermethod by developing a numerical program in the intention to of-fer a practical engineering tool for the design and analysis of thethermo-active pile behavior.

Afterwards, 1D analysis of a single thermo-active pile under axialmechanical and cyclic thermal loads is conducted by using the loadtransfer method with the proposed t–z function. A comparison mod-el using three-dimensional finite difference analysis with the consti-tutive Modjoin interfaces law is also conducted. The responses of athermo-active pile resulting from 1D and 3D analyses are comparedin order to study the cyclic degradation effects induced by seasonaltemperature variations. While the 1D load transfer method permitsto have a simplified and practical calculation with high time effi-ciency, the 3D model using the Modjoin law with finite differencecode proposes a complete interaction analysis with higher numeri-cal time consumption. The global response of the thermo-active pilein this study performs a good concordance in both models. The pres-ence of the elastic soil surrounding the pile in the 3D model leads to adifferent local response at the soil–pile interface between the twomodels. This numerical study points out the capability of the twolaws of soil–pile interface behavior to render the cyclic degradationeffects in the thermo-active pile.

According to the axial fixity at the pile head, the degradation ofsoil–pile resistance during cycles generates increase in pile headsettlement for the free head pile or decrease in pile head capacityfor the restrained head pile. The free head pile tends to have differ-ent behavior at the upper and the lower part of pile. In the re-strained head pile, axial force and shaft friction distributions areuniform over the pile length. The parameters chosen in this numer-ical study lead to the emergence of cyclic fatigue effects: strain rat-cheting and stress relaxation phenomena. An ongoing project ofin situ loading tests of full-scale thermo-active piles will providereal data and thus help to calibrate the cyclic parameters used inthe numerical model.

At the end, the numerical results obtained in terms of dis-placements, axial forces and local shaft friction could be usedto justify rationally the design of thermo-active piles and thusestablish a secure basis design method instead of doubling thesecurity factor. To begin with, the values of the pile head dis-placement during thermal cycles could be used to verify the ser-viceability limit states. Afterwards, the distribution of axial stressin the pile permits to control the compression and/or the tensilestate in concrete. Finally, the evolution of local friction along theshaft allows calculating the mobilized shaft resistance and themobilized base resistance and thereby estimating the requiredsecurity factor.

Acknowledgements

The work described in this paper forms part of a research pro-ject ‘‘GECKO’’ (geo-structures and hybrid solar panel coupling for opti-mized energy storage) which is supported by a grant from theFrench National Research Agency (ANR). It is an industrial project,involving four public companies and two research laboratories incivil and energy engineering sector: ECOME, BRGM, IFSTTAR, CETENord Picardie, LGCgE–Polytech’Lille and LEMTA–INPL. The authorswould like to express their gratitude to their partners for their con-tinuous support in this and ongoing project.

Appendix A

The procedure described below is used for the load transfer cal-culation of a pile under monotonic loading. The pile is divided intok segments as shown in Fig. 11, where segment no. 1 indicates thesegment at the top of the pile and segment no. k indicates the seg-ment at the bottom of the pile.

1. Assume an initial value of the tip movement wp at the bottomsegment k.

2. Search the value of the end-bearing stress qp at the pile basewith the imposed wp by using Eq. (2) of the proposed t–zfunction.

3. Calculate the end-bearing resistance.

Qp ¼ qp14pB2

� �ðA:1Þ

4. Calculate the elastic deformation at the lower half of segment k.

eep ¼

Q p

14 pB2EM

ðA:2Þ

5. Calculate the midpoint movement at segment k.

wk ¼ wp þ12

hkeep ðA:3Þ

6. By applying the estimated midpoint movement at segment k,search the value of shaft friction at segment k qsk using Eq.(2) of the proposed t–z function.

7. Calculate the axial load at the midpoint of segment k.

Qk ¼ Q p þ pB12

hkqsk ðA:4Þ

8. Calculate the elastic deformation at the upper half of segment kand at the lower half of segment k � 1.

eek ¼

Qk14 pB2EM

ðA:5Þ

9. Calculate the midpoint movement at segment k � 1.

wk�1 ¼ wk þ12ðhk þ hk�1Þee

k ðA:6Þ

10. Search the value of shaft friction at segment k � 1 qsk�1

using Eq. (2) of the proposed t–z function.11. Calculate the axial load at the midpoint of segment k � 1.

Qk�1 ¼ Q k þ pB12ðhk þ hk�1Þqsk�1 ðA:7Þ

12. Do the iteration from the lower half of segment k � 2 untilthe midpoint of top segment by repeating step 8–11.

13. For the upper half of segment 1, calculate the response of thepile head.

wh ¼ w1 þ12

h1ee1 ðA:8Þ

390 M.E. Suryatriyastuti et al. / Computers and Geotechnics 55 (2014) 378–391

Q h ¼ Q 1 þ pB12

h1qs1 ðA:9Þ

Appendix B

The following calculation procedure is done during cyclicthermal loading in this present study for the free head pileand the restrained head pile. The procedure is repeated for eachload cycle.

1. Boundary conditions of the model treated in this study.a. Qh;n ffi Qh;mec for the free head pile under constant load.b. wh;n ffi wh;mec for the restrained head pile under constant

displacement.2. Delta temperature ±10 �C from the ground temperature is

applied uniformly along the pile. Calculate the thermal defor-mation obtained due to this temperature variation.

eth;n ¼ aTDTn ðB:1Þ

3. Assume an initial value of the tip movement wp,n at the bottomsegment k.

4. Search the value of the end-bearing stress qp,n at the pile basewith the imposed wp,n by using Eq. (3) of the proposed t�zfunction.

5. Calculate the end-bearing resistance.

Q p;n ¼ qp;n14pB2

� �ðB:2Þ

6. Calculate the elastic deformation at the lower half of segment k.

eep;n ¼

Qp;n

14 pB2EM

þ eth;n ðB:3Þ

7. Calculate the midpoint movement at segment k.

wk;n ¼ wp;n þ12

hkeep;n ðB:4Þ

8. By applying the estimated midpoint movement at segment k,search the value of shaft friction qsk,n using Eq. (3) of the pro-posed t�z function.

9. Calculate the axial load at the midpoint of segment k.

Q k;n ¼ Q p;n þ pB12

hkqsk;n ðB:5Þ

10. Calculate the elastic deformation at the upper half of seg-ment k and at the lower half of segment k � 1 with the addi-tional thermal deformation included.

eek;n ¼

Q k;n

14 pB2EM

þ eth;n ðB:6Þ

11. Calculate the midpoint movement at segment k � 1.

wk�1;n ¼ wk;n þ12ðhk þ hk�1Þee

k;n ðB:7Þ

12. Search the value of shaft friction at segment k � 1 qsk�1,n byusing Eq. (3) of the proposed t–z function.

13. Calculate the axial load at the midpoint of segment k � 1.

Q k�1;n ¼ Q k;n þ pB12ðhk þ hk�1Þqsk�1;n ðB:8Þ

14. Do the iteration from the lower half of segment k � 2 untilthe midpoint of top segment by repeating step 10–13.

15. For the upper half of segment 1, calculate the response of thepile head.

wh;n ¼ w1;n þ12

h1ee1;n ðB:9Þ

Qh;n ¼ Q1;n þ pB12

h1qs1;n ðB:10Þ

16. Repeat step 3–15 until the boundary conditions are satisfied.

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