static analysis of soil/pile interaction in layered soil by bem/bem coupling

www.elsevier.com/locate/advengsoft

Advances in Engineering Software 38 (2007) 835–845

Static analysis of soil/pile interaction in layered soilby BEM/BEM coupling

Valerio S. Almeida, Joao B. de Paiva *

Department of Structural Engineering, Sao Carlos School of Engineering, Sao Paulo University, Av. Trabalhador Saocarlense,

400, 13566-590 Sao Carlos, SP, Brazil

Received 3 December 2004; accepted 13 August 2006Available online 26 January 2007

Abstract

In this article we propose to use the boundary element method (BEM) to analyze soil–foundation interactions. The soil structure ismodeled as several dissimilar strata placed one on top of the other, composing a sandwich-like profile. Any of these layers may containcomponents of the foundations. Each region occupied by a soil layer or by a foundation component is handled as a 3D isotropic, elasticand homogeneous domain, and is analyzed by BEM. As a consequence of the positioning of the various subregions in this model, thetechnique of successive rigidity can be applied directly, resulting in a considerable reduction in the volume of data being stored andmanipulated throughout the analysis.� 2006 Elsevier Ltd. and Civil-Comp Ltd. All rights reserved.

Keywords: Boundary element methods; Heterogeneous soil; Soil–pile interaction; Successive rigidity method; Soil–foundation system; Finite medium

1. Introduction

In soil mechanics, the soil is generally taken to be asemi-infinite homogeneous, isotropic and linear-elasticcontinuum. However, the idea of the soil as a homoge-neous layer supported at infinity on an immovable stratumis not a realistic representation of its structure; rather, aninhomogeneous medium with a rigid base at a finite depthwould be nearer to what is found in nature. For this rea-son, several techniques have been proposed to model thefinite soil, which are based, broadly speaking, on four dif-ferent approaches.

In the first, known as Winkler’s model, the continuum isreplaced by a system of equivalent, discrete springs. Whena pile is present, it is idealized as a beam element, with orwithout the possibility of axial deformation, supportedby a series of discrete springs that represent the soil. Thegreat advantage of this model is its simplicity and relative

0965-9978/$ - see front matter � 2006 Elsevier Ltd. and Civil-Comp Ltd. Alldoi:10.1016/j.advengsoft.2006.08.034

* Corresponding author. Tel.: +55 16 273 9455; fax: +55 16 273 9481.E-mail addresses: [email protected] (V.S. Almeida), [email protected]

(J.B. de Paiva).

ease of implementation on the computer, while the mostserious disadvantage is the difficulty of choosing the elasticmoduli of the springs for a given combination of pile sizeand soil type. Normally, these constants are estimatedempirically and the values chosen may lead to uncertainand inaccurate solutions. This line of research has beendeveloped in the work of Cheung and Zienkiewicz [1], Ran-dolph and Wroth [2], Witt [3], Lee [4] and Mylonakis andGazetas [5].

The second approach commences with the applicationof equations, developed within elasticity theory, to a homo-geneous continuum in each soil layer, after which equili-brium and compatibility conditions are applied at thecontact surfaces between adjacent layers, to obtain expres-sions that represent the heterogeneous system. This tech-nique has the disadvantage that the solutions to theequations can only be used in certain cases, as general solu-tions cannot be found; moreover, its handling of pilesimmersed in the soil is cumbersome.

Burmister [6,7] is responsible for pioneering work in thisarea, in which integral transforms are used to obtain semi-analytical solutions for the displacements and stresses in a

rights reserved.

mailto:[email protected]

mailto:[email protected]

836 V.S. Almeida, J.B. de Paiva / Advances in Engineering Software 38 (2007) 835–845

heterogeneous medium, without taking foundations intoaccount. Poulos [8] uses Burmister’s solutions to calculateinfluence factors for the cases of line, strip and sector load-ing. Chan c [9] and Davies and Banerjee [10] extend Burm-ister’s procedure to the case of a vertical and/or horizontalforce applied at a point inside a two-layer medium. Othersworking along the same lines are Gibson [11] and Ueshitaand Meyerhof [12].

The third approach is known as the finite layer method(FLM), which can be used to reduce the 3D problem to oneinvolving only two dimensions by combining the Fouriertransform technique with the finite element method(FEM). The FLM, which can be simply and efficientlyimplemented on the computer, generates elastic solutionsfor the soil, whether homogeneous or not and isotropicor anisotropic. When piles are incorporated into the sys-tem, the method can still be used, by introducing equilib-rium and compatibility conditions at the interfacesbetween the two media. A disadvantage of the FLM is thatit only applies to elastic problems. For further information,refer to the seminal work of Small and Booker [13], Bookeret al. [14], Lee and Small [15], Southcott and Small [16] andTa and Small [17]. A variant of the FLM, known as theinfinite layer method, has been proposed by Cheung et al.[18].

The fourth line of research makes use of two powerfulnumerical methods: FEM and the boundary elementmethod (BEM). FEM is, generally speaking, the most ver-satile and powerful tool available to solve numerical prob-lems in mechanics. However, when it is applied to infinitedomains, such as the soil, the preparation of data is compli-cated and requirements for data storage and processor timeare heavy, especially when the problem is formulated with3D elements. This is because FEM needs a representationof the entire domain. Hence, few research workers haveemployed FEM to analyze the soil continuum, homoge-neous or otherwise, but the work of Ottaviani [19] andChow and Teh [20] should be mentioned. BEM has provedto be the most efficient and practical means of analyzinginfinite-domain problems in elastostatics, owing to intrinsicfeatures of the weighted functions that ensure the boundaryconditions are satisfied at large distances. A number ofresearchers have employed Mindlin’s solution [21] for thesemi-infinite continuum and the simplified Steinbrennermodel [8] for finite media. However, the use of Mindlin’sfundamental solution in a BEM application, even whenthe depth of the immovable plane is included, is only validfor a single homogeneous medium. It is not possible tointroduce other media with distinct physical propertiesbecause of the boundary conditions imposed a priori bythe formulation of the solution. According to Poulos [22],under certain conditions of inhomogeneity of the soil andpile system, the displacements generated are inadequate.

For the case of the analysis of soil–structure interactionproblems in dynamic models, the BEM also stands out inrelation to the FEM for the half-space simulation bothdue to the characteristic of reducing a given dimension

for the integral representation of the problem and due tothe displacement, tension and radiation conditions to benaturally fulfilled at the infinite. Several works approach-ing this topic can be found in the literature. Beskos [23]and Beskos [24] present a wide bibliographic review on thistopic. The literature also presents Karabalis and Beskos[25], who studied the dynamic problem in the 3D shallowfoundations analysis under external dynamic forces or seis-mic waves of several types or directions with variation inthe transient time. The work of Triantafyllidis [26] wasthe first to apply the Green’s functions in half spacefor the solution of transient problems in time domain viaBEM. The problem of vibration isolation by a row of pilesis firstly solved by Kattis et al. [27] via BEM in the fre-quency domain in a three-dimensional context. Guan andNovak [28] developed a closed solution for transient-response problems for suddenly applied loading distributedover a rectangular area on the surface of a homogeneouselastic half space. Antes and Steinfeld [29] proposed a 3DBEM/BEM coupling in a time domain considering theweight of the structure itself.

Hence, BEM must be used to represent infinite-domain.The Kelvin solution ([30]) provides a basis for the mostgeneral formulae for an isotropic body of arbitrary shape,being applied both in the solution of static and dynamicproblems. Moreover, this solution is the one most oftenused to couple various regions with different properties,with the boundary surfaces satisfying equilibrium and com-patibility conditions so as to maintain continuity through-out the heterogeneous medium. Thus, Banerjee [31],Banerjee and Davies [32], Maier and Novati [33] andAlmeida and Paiva [34] use the fundamental Kelvin solu-tion in BEM simulations of a finite heterogeneous medium.Maier and Novati [26] developed a new way of analyzingstratified soils by BEM, the successive stiffness method(SSM), but did not take the influence of piles into consid-eration. The last work above cited also applied the SSM,however they formulated the problem in three-dimensionalspace and took into account the flexible superstructure byFEM, using shell elements and three-dimensional beamelements.

Other ways of formulating the problem have beenadopted which were not described above, as there are veryfew published accounts of the procedures employed. Exam-ples are the models of Fraser and Wardle [35], Sadecka [36]and Pan [37]. In one particular line, some authors haveendeavored to integrate Winkler’s system of springs withcontinuous models. This is accomplished by inserting con-tinuity between the discrete springs by means of structuralelements, or by imposing some simplifications, calibratedwith real displacements and stresses, on the continuousmodels. The following foundation models share thisapproach: Hetenyi’s [38]; Pasternak’s, described in Kerr[39] and Wang et al. [40]; generalized [39] and Kerr’s [41].

The fact remains that modeling the finite heterogeneoussoil medium represents a highly complex problem in solidmechanics, by virtue of the absence of any control over

V.S. Almeida, J.B. de Paiva / Advances in Engineering Software 38 (2007) 835–845 837

the formation of the soil, in contrast with manufacturedcomponents such as steel or reinforced concrete. The soilstratum was produced by the weathering of rocks underdiverse conditions, which created a highly anisotropicand heterogeneous medium, chaotic in appearance withdiscontinuities visible throughout its bulk.

In light of these considerations, we proposed to developa computer-based method with which to analyze the staticproblems in mechanics of heterogeneous soils, includingthe influence of piles, in such a way that both these struc-tural elements would be modeled in 3D by BEM. The tech-nique used to model the layered structure was the SSMMaier and Novati [33], but applied to 3D problems([34]), and it introduced the effects of piles added. It shouldbe stressed that the new formulation is not only restrictedto simple pile foundations, but can also be applied to pileswith enlarged tip, shallow foundations, etc.

2. The boundary element method applied to problems of

elastostatics

In the absence of volume forces, the Navier–Cauchyequations are given by:

ui;jjðsÞ þ1

1� 2 � t � uj;jiðsÞ ¼ 0; i; j ¼ 1; 2; 3 ð1Þ

where ui(s) is the displacement in the orthogonal direction i

from the point s inside the solid and satisfies certain bound-ary conditions, and t is Poisson’s ratio. These domainequations can further be expressed as surface equations,which are represented by the Somigliana identity:

uiðpÞ þZ

Cp�ijðp; SÞ � ujðSÞoCðSÞ ¼

ZC

u�ijðp; SÞ � pjðSÞoCðSÞ

ð2Þ

where p and S are, respectively, the source point where aunit force is applied and a boundary point at the surface,ui and pi are, respectively, the real displacement field andsurface forces at the boundary point S in the ith direction,while u�ij and p�ij represent weighted field coefficients whichindicate the response, in the direction j at S, to a force ap-plied in the direction i at the point p. This identity is basedon Betti’s reciprocal theorem and weighted or fundamentalsolutions given by u�ij and p�ij represent particular solutionsof the partial differential equation (1) for a given boundarycondition.

The strategy to obtain the boundary integral equationsinvolves transforming p, which is inside the body, to P onthe boundary. Thus, expression (2) can be written as follows:

CijðP Þ � ujðP Þ þZ

Cp�ijðP ; SÞ � ujðSÞ � oCðSÞ

¼Z

Cu�ijðP ; SÞ � pjðSÞ � oCðSÞ ð3Þ

where the integral in (3) is defined in the sense of the Cau-chy principal value [42] and Cij are coefficients that depend

on the geometry of the problem [43]. The fundamentalsolutions used herein are the known Kelvin solutions pre-sented in [30] for the three-dimensional case.

Since the analytical solutions of Expression (3) are notgiven in closed form, they have to be estimated numeri-cally. Hence, the boundary element method (BEM) isbased on the assembly of a system of algebraic equationsresulting from boundary integral equations, Eq. (3), writ-ten in terms of nodal parameters that are approximatedto the boundary values using shape functions. The integralequations of (3) are then written, without considering thedomain forces, as:

CijðP Þ � ujðP Þ þXNE

k¼1

jJ j �Z

Cp�ijðP ; SÞ �WðSÞonðSÞ � ðUiÞk

¼XNE

k¼1

jJ j �Z

Cu�ijðP ; SÞ �WðSÞonðSÞ � ðP iÞk ð4Þ

where NE, W, J are, respectively, the number of boundaryelements, the shape function and the Jacobian transforma-tion. In this paper, we will adopt linear shape functions ofthe form Wi(n1,n2,n3) = ni, where ni are homogeneous coor-dinates defined for the flat triangular element [44] intowhich the surface is discretized.

The integrals proposed in (4) cannot, however, be solvedanalytically for any generic surface; hence the use ofnumerical techniques such as those given in [45,46]. Inthe present paper, the integral equations are calculatednumerically by using a three-dimensional triangular quad-rature integral [44].

It is thus possible to assemble the shape matrices of Eq.(4), which takes on the following form:

½H � � fUg ¼ ½G� � fPg ð5Þ

where the Dirichlet or Neumann boundary conditions foreach problem must be satisfied at each nodal point.

3. Analysis of layered soil/foundations by the successivestiffness method (SSM)

3.1. Single soil layer equations

Expression (5) is here extended to a homogeneous, iso-tropic and linear solid. Inhomogeneous problems can besolved by considering a combination of problems of adja-cent homogeneous domains while applying the necessaryequilibrium and compatibility conditions at the interfacesof the domains.

Consider a layered soil with g layers (Fig. 1). For anygiven layer i, it is possible to write the correlation betweenthe influence matrices of Eq. (5) as

½ ½Hit� ½H i

b� ½His� � �

Uit

Uib

Uis

264

375 ¼ ½ ½Gi

t� ½Gib� ½Gi

s� � �P i

t

P ib

P is

264

375ð6Þ


where the subscripts t, b and s represent quantities pertain-ing to the upper (top), lower (bottom) and side boundariesand U and P are, respectively, nodal displacements andtractions of the top, bottom or side boundaries.

Expansion of Eq. (6), gives:

½H itt� ½H i

tb� ½H its�

½H ibt� ½H i

bb� ½H ibs�

½H ist� ½H i

sb� ½H iss�

264

375 �

U it

U ib

U is

8><>:

9>=>;

¼½Gi

tt� ½Gitb� ½Gi

ts�½Gi

bt� ½Gibb� ½Gi

bs�½Gi

st� ½Gisb� ½Gi

ss�

264

375 �

P it

P ib

P is

8><>:

9>=>; ð7Þ

Equilibrium and compatibility conditions can then beimposed on displacements and stresses along the boundarybetween the ith and (i + 1)th layers. For cases in whichthere are no relative movements between contact nodes,i.e., the case of ideal friction without forces being appliedat the interface and with an undisturbed side boundary,the conditions can be expressed as

fuitg ¼ fuiþ1

b g ð8:1Þfpi

tg ¼ �fpiþ1b g ð8:2Þ

fusg ¼ f0g ð8:3Þfpsg ¼ f0g ð8:4Þ

with i varying from 1 to g � 1.Assuming that the lateral surface is sufficiently remote, it

is possible to use Eq. (8.3) and (8.4) in (7) and obtain theinfluence of each layer, which is given by

P it

P ib

( )¼½Ki

tt� ½Kitb�

½Kibt� ½Ki

bb�

" #�

Uit

Uib

( )ð9Þ

Hence, applying Eq. (9) to a given layer i, and invokingEqs. (8.1) and (8.2), this layer can easily be related to thoseabove and below it.

In the lowest layer, where i = 1, the displacement at thebase is null, giving a fixed medium for which expression (9)can be re-written as

Γ

Γ1t

t

Γ b

s

u uu

P

b2,Pb

2

t1,Pt

1

b1

Γ

Γb

1

Ω

Ω

_

1

Ω2

Fig. 1. Layered soil subjected to forces on the surface, bottom, top andside.

fP 1t g ¼ ½K1

tt� � fU 1t g ð10Þ

fP 1bg ¼ ½K1

bt� � fU 1t g ð11Þ

For i = 2, we have, from expression (9):

fP 2t g ¼ ½K2

tt� � fU 2t g þ ½K2

tb� � fU 2bg ð12Þ

fP 2bg ¼ ½K2

bt� � fU 2t g þ ½K2

bb� � fU 2bg ð13Þ

Applying the conditions (8.1), (8.2) and (10) to Eq. (13),gives

fU 2bg ¼ �½ðK1

tt þ K2bbÞ�1 � K2

bt� � fU 2t g ð14Þ

and substitution of (14) in (12) leads to

fP 2t g ¼ ½½K2

tt� � ½K2tb� � ð½K1

tt� þ ½K2bb�Þ

�1 � ½K2bt�� fU 2

t g ð15Þ

which may be written as

fP 2t g ¼ ½K2� � fU 2

t g ð16Þ

The above equation includes the influence of layers 1and 2. Hence, applying Eqs. (8.1), (8.2) and (9) to the layersi and i + 1, we have

fP itg ¼ ½½Ki

tt� � ½Kitb� � ð½Ki�1� þ ½Ki

bb�Þ�1 � ½Ki

bt�� fU itg ð17Þ

Thus, for the topmost layer, where i = g:

fP gt g ¼ ½Kg� � fU g

t g ð18Þ

where P gt and U g

t are the nodal parameters at the soilsurface.

At this point, the influence of the inhomogeneous soil isentirely expressed by Eq. (18), which can be solved directlyby applying the given loading conditions on the surface orby coupling the superstructure, using FEM or BEM.

3.2. Soil–foundation equations

When the influence of piles within a generic stratum hasto be taken into account, the SSM can still be applied, butthe unknown quantities located at points along the shaftare manipulated algebraically to be substituted by variablesassociated with the top and bottom of that stratum.

Thus, considering a general layer i (Fig. 2) with m piles,Eq. (5) for a general pile j, in expanded form, becomes:

½H fpjfpj� ½H fpjtpj

� ½H fpjbpj�

½H tpjfpj� ½H tpjtpj

� ½H tpjbpj�

½H bpjfpj� ½H bpjtpj

� ½H bpjbpj�

2664

3775 �

U fpj

U tpj

U bpj

8><>:

9>=>;

¼½Gfpjfpj

� ½Gfpjtpj� ½Gfpjbpj

�½Gtpjfpj

� ½Gtpjtpj� ½Gtpjbpj

�½Gbpjfpj

� ½Gbpjtpj� ½Gbpjbpj

�

2664

3775 �

P fpj

P tpj

P bpj

8><>:

9>=>; ð19Þ

layer i

pile

Utp

j

tsU

bsU

fsU =

j

Ufp

j

bpU

j

Fig. 2. Generic stratum with three piles and their top (t), bottom (b) andshaft (f) variables.


while the matrix equation for the layer of soil, including theinfluence of pile j, is:

½Htsts� ½H tsbs� �� ½Htsfsj� �

½Hbsts� ½H bsbs� �� ½Hbsfsj� �

�� ½Hfsjts� ½H fsjbs� �� ½Hfsjfsj

� ��

266666664

377777775�

U ts

U bs

��U fsj

��

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

¼

½Gtsts� ½Gtsbs� �� ½Gtsfsj� �

½Gbsts� ½Gbsbs� �� ½Gbsfsj� �

�� ½Gfsjts� ½Gfsjbs� �� ½Gfsjfsj

� ��

266666664

377777775�

P ts

P bs

��P fsj

��

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

ð20Þ

Isolating the forces on the surface of the shaft andapplying the conditions of displacement compatibilityand force equilibrium at the interface between the soiland the shaft of pile j, we arrive at Eq. (21) for the couplingbetween the soil and piles in layer i:

… …

… …

…

…

…

…

…

…

… … … … … … …

… …… … … … …

…

… … …

…

…

…… … ………

……

…

…

……… ……… …

…… …

…

…

…

…

In (21), shaft variables have been eliminated, leavingonly unknowns related to the top and bottom of both pilesand soil. A relation for the general layer i, equivalent to Eq.(9), can thus be written

Kitsts Ki

tsbs Kitstp1

Kitsbp1

�� Kitstpm

Kitsbpm

Kibsts Ki

bsbs Kibstp1

Kibsbp1

�� Kibstpm

Kibsbpm

Kitp1ts Ki

tp1bs Kitp1tp1

Kitp1bp1

�� Kitp1tpm

Kitp1bpm

Kibp1ts Ki

bp1bs Kibp1tp1

Kibp1bp1

�� Kibp1tpm

Kibp1bpm

��

Kitpmts Ki

tpmbs Kitpmtp1

Kitpmbp1

�� Kitpmtpm

Kitpmbpm

Kibpmts Ki

bpmbs Kibpmtp1

Kibpmbp1

�� Kibpmtpm

Kibpmbpm

26666666666666664

37777777777777775

�

U its

U ibs

U itp1

U ibp1

��

U itpm

U ibpm

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

¼

P its

P ibs

P itp1

P ibp1

��

P itpm

P ibpm

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

ð22Þ

Hence, the stiffness matrix for layer i has been obtained,incorporating m piles. Adjacent layer boundary conditionsof compatible displacements and force equilibrium cannow be applied in a procedure analogous to that presentedin Section 3.1.

3.2.1. A priori calculation of the number of computeroperations involved

SSM may appear more costly in terms of processingtime than the conventional sub-region technique [44],because the former requires the inversion and multiplica-tion of many matrices associated with each stratum. More-over, when the stratum contains parts of the foundations,additional processing is needed to condense their respectivedisplacement values into variables located at the top andbottom of the stratum.

……

…

…

…

…

… …

…

… …

…

… …

ð21Þ


To resolve this question, the cost of each stage involvedin both these techniques can be computed, recalling thatthe number of floating-point operations used in the follow-ing procedures is

n3! product of two square matrices of the identicaldimensions n · n;n Æ m Æ p! product of two rectangular matrices ofdimensions A(n,m) and B(m,p);4/3 Æ n3! inversion of a matrix of dimension n;1/3 Æ n3! solution of a system with very large n byGaussian elimination;n Æ B2 � 2/3 Æ B3! solution of a system containing amatrix of half bandwidth B.

In what follows, the total dimension of the set of vari-ables on a surface, n = 3 Æ N, where N is the number ofnodes on that surface, and g is the number of layers of soil.

3.2.1.1. Successive stiffness method (SSM)

(i) [Gi]�1: O(i) = 4/3 Æ (2n)3 Æ g = 288 Æ N3 Æ g(ii) [Gi]�1 Æ [Hi]: O(ii) = n3 Æ g = 216 Æ N3 Æ g

(iii) condense variables located on shaft: O(iii) = 2 Æ f(a) Æ[6 Æ N + 3 Æ Na]

3 Æ g(iv) compute operations for soil–foundation equilibrium

condition. Here, the numbers of operations used tofind the product of rectangular matrices and to inverta matrix ½Gfpjfpj

� are calculated for each element of thefoundation, assuming an identical number NE ofthese elements within each stratum:

OðivÞ ¼ 27 � ðN aÞ2 � NE � g � ½10 � N þ 29; 33

� N a � 3 � N a=NE�

(v) ½Kitsts þ Kiþ1

bsbs��1

:OðvÞ ¼ 4=3 � n3 � ðg� 1Þ ¼ 36 � N 3 � ðg� 1Þ

2102

103

104

105

(Num

ber

of o

pera

tions

)/N

3

4 6

Fig. 3. Number of computer operations, divided by N3, (log scale) vs

(vi) ½K2tb� � ð½K1

tt� þ ½K2bb�Þ

�1 � ½K2bt�:

O(vi) = 2 Æ n3 Æ (g � 1) = 2 Æ (3 Æ N)3 Æ (g � 1)(vii) solve the system: O(vii) = 1/3 Æ n3 = 9 Æ N3

Total number of operations:

Without pile elements ða ¼ 0Þ : OðSSMÞ¼ N 3 � f594 � g� 81g

With pile elements ða ¼ 1=3Þ : OðSSMÞ¼ N 3 � f797 � gþ 59 � g � NE � 81g

3.2.1.2. Conventional sub-region technique (SR):

B ¼ 9 � N þ ð3 � N aÞ � NE

n ¼ 3 � N � ð2 � gÞ þ 3 � N a � ð2 � gÞ � NE

Total number of operations:

Without pile elements ða ¼ 0Þ : OðSRÞ¼ N 3 � ð486 � g� 486Þ

With pile elements ða ¼ 1=3Þ : OðSRÞ¼ 2 � N 3 � ð9þ NEÞ2 � ½ð3þ NEÞ � g� 1=3 � ð9þ NEÞ�

where Na = a Æ N, Na being the number of nodes in eachelement of the foundation and a is given the value 1/3.

The numbers of operations involved in the analysis of alayered medium, with and without stiffening elements, byeach of the techniques, are plotted in Fig. 3. Note that inthe absence of any foundation elements, the conventionalzone method requires fewer operations than SSM. Never-theless, if two such elements are included in the model,the method described here (SSM) becomes slightly moreefficient than the conventional one, and as the number ofelements rises this advantage becomes much more marked.

Finally, as well as the reduction in processing timeachieved with SSM, another big advantage is the smaller

10 12102

103

104

105

SSM with α=0 by zones with α=0 SSM with 2 piles by zones with 2 piles SSM with 4 piles by zones with 4 piles SSM with 8 piles by zones with 8 piles

8

number of layers, using two techniques: SSM and zones method.

Table 1Percentage error of the central surface displacement

Thickness factor (h/a) Layers Error (%)

2 2 13.362 3 2.202 5 1.548 3 52.968 7 19.448 20 0.71


storage space required for the final system matrix. Thisoccurs because in the zone technique it is necessary to storein the final matrix explicit information about all the zonesthat exist in the problem. By contrast, the SSM procedurearrives at a final matrix that combines the influence of allthe layers and thus represents the entire body of soil, whichmay or may not be stiffened by piles, and yet contains onlyquantities located at the free surface. Furthermore, thiscondensed influence matrix can easily be coupled to matri-ces representing the superstructure supported by thefoundations.

4. Numerical examples

To implement the formulation introduced in the previ-ous sections, computer code was written in Fortran90,using the IMSL libraries and routines from BLAS 1, BLAS2 and BLAS 3. Some examples follow, to validate thetechnique.

4.1. Finite layer with linear variation of modulus

This example presents an analysis of the soil consideredas a heterogeneous, linear isotropic medium. For the spe-cific case of a uniformly distributed load applied to a circu-lar region on the soil surface, the stiffness modulus isconsidered to increase linearly with depth (see Fig. 4).The results given in Table 1 demonstrate the robustnessof the formulation, even when the ratio h/a is varied, aslong as additional layers are included as this ratio

a

h

zG(z) = G (0) + mz

G (0) = 0

= 0.5νm = 1/3

p

Fig. 4. Finite stratum under uniform circular loading.

H

P=10

Lpile

Fig. 5. Material and geometric config

increases, in order to represent more closely the linear var-iation of the soil’s rigidity. The relative errors in the centralsurface displacement are obtained by comparing the valuesfound by this formulation with those calculated by thesemi-analytical expression of Burmister [6,7].

4.2. Single pile in a finite medium

This example is taken from Ottaviani [19], who usedFEM with 3D elements to analyze the displacements andtensions resulting from contact between a pile and the sur-rounding soil. That author analyzed a prismatic regionwhose side boundary, presumed immovable, was 13 mfrom the center of the pile, and employed 2700 brick-shaped elements, with eight nodes each, in a mesh contain-ing 3300 nodes in all.

The results from [19] will be compared with thoseobtained by the present method, evaluating in particularthe effect of varying the depth between the rigid floor ofthe soil and the end of the pile. Also, the influence of thepile/soil stiffness ratio (k = Epile/Esoil) on the consistencyof the methods will be considered.

The layout is shown in Fig. 5. Fig. 6 plots the values ofmaximum shear stress in the soil adjacent to the pile, givenby smax = 0.5 Æ (r1 � r3), at points down the shaft. Thenumber of nodes used in the present work 571, with 820plane elements, i.e. almost six times fewer nodes than thoseneeded in [19]. Stresses have been converted into dimen-sionless units, by means of the expressions: r0 ¼ r�A

P ands0 ¼ s�A

P , where A is the cross-sectional area of the pile.

00kN

λ= EE

pile

soil

E = 2.0E5 kN/mpile2

ν = 0.20pile

Diameter = 1.0mpile

δ =E .D /P0

soil pile

E = variablesoil

ν = 0.20soil

uration for the soil–pile problem.

40

35

30

25

20

15

10

5

0

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

Vertical displacements (m)

Dep

th (

m)

Present work - Case 1 Present work - Case 2 Ansys - Solid92

Fig. 8. Depth against settlement of the single pile for three differentconfigurations.

40

35

30

25

20

15

10

5

00.0 0.5 1.0 1.5 2.0 2.5 3.0

Present work : Lpile=40m; λ=400 Ottaviani(1975) : " " Present work : Lpile =20m; λ=400 Ottaviani(1975) : " "

Dep

th (

m)

100*τmax

Fig. 6. Maximum shear stress in soil adjacent to pile plotted againstdepth.


In Fig. 7, the displacements calculated by the two for-mulations are compared, for various combinations ofthickness and stiffness ratio. The difference between thetwo sets of results stayed below 5% at points along theshaft, indicating the consistency of the present method.Fig. 8 compares the estimates of displacements at pointsalong the shaft for three cases: in case 1 and 2, the presentformulation was used, with 346 and 571 nodes, respec-tively, while in case 3 the commercial package ANSYS5.4 was used, with 16,000 nodes.

4.3. Single pile in an inhomogeneous medium

In this case, described in Fig. 9, the distribution of ver-tical displacements and tractions along the shaft of a pileimmersed in a heterogeneous medium was calculated.

The vertical displacement and traction results obtainedfor four different patterns of stiffness in the three layersof soil, whose details are given in Fig. 9c, are plotted in

30

45

60

75

90

105

120400 800 1200 1600 2000

δ0 =

u 3*(E

pile*D

pile /P

)

λ (Epile/Esoil)

Lpile=40 H/L= - Present work Lpile=40 H/L=4 - Present work Lpile=40 H/L=1.5 - Present work Lpile=40 H/L=4 - Ottaviani(1975) Lpile=40 H/L=1.5 - Ottaviani(1975)

Fig. 7. Settlement of pile plotted against k = Epile/Esoil.

Figs. 10 and 11 against depth at points on the shaft. InFig. 10, note that the displacements follow smooth, contin-uous curves, free of inflexions, across the two layers of soilin which the pile is immersed. The percentage differences inthe displacement at the pile top, between case a and theother cases, are: +46.00% for case b, �27.35% for case cand �48.15% for case d. The displacements found alongthe pile shaft, in case a, were also compared with those gen-erated by the package ANSYS 5.4, and there is fair agree-ment between the two plots in Fig. 10, considering that thecommercial package required about six times more quanti-ties of nodes than the present method.

The estimated vertical pile–soil tractions are plotted inFig. 11, and marked differences can be observed betweenthe four cases. In case a, discounting the end regions, thetraction varies little with depth, staying around 0.6 kN/m2, while in cases b and d its value changes sharply atthe layer boundary, being higher in the zone of greater stiff-ness. Note that in each of these cases the values found inthe two layers differ by about 100%. Near the tip of thepile, these values are raised, when the soil in contact withthe tip is stiffer.

The moderate differences found between the results fromthe four cases demonstrate that, in a high-quality structuralanalysis of a project involving soil–foundation–superstruc-ture interactions, it is imperative to model the soil as clo-sely as possible to nature, at least including a degree ofheterogeneity.

4.4. Nine piles group embedded in soil mass

This example, also reported by Ottaviani [19] analyzesnine piles immersed in the soils mass, according to Figs.12–14. As previously described in example 4.2, Ottaviani[19] models this problem with solid finite elements, wherelateral boundaries were located 13 m from the outer pilesof the group at a distance which was found to be large

40

30

20

10

0

0.0 1.0x10-2 2.0x10-2 3.0x10-2 4.0x10-2

0.0 1.0x10-2 2.0x10-2 3.0x10-2 4.0x10-2

Displacements (m)

Case aCase bCase cCase dCase a - Ansys

Dep

th(m

)

Fig. 10. Depth against settlement of the single pile for four differentinhomogeneous soil cases.

40

30

20

10

0

0.25 0.5 2 8

0.25 0.5 2 8

scale type (log2)

Dep

th (

m)

Vertical Traction (kN/m2)

1 4

1 4

Case aCase bCase cCase d

Fig. 11. Depth against vertical traction in the soil–pile interface in thefour different cases.

30mL = 20m

E = 2.0E5 kPapile

ν = 0.25pile

Diameter = 10.mpile

E =250 kPasoil

ν = 0.45soil

PP P

L

pile

P = 100 kN

Fig. 12. Material and geometric configuration for the soil–pile problem.

Fig. 13. Piles disposition in continuum media.

20

P=100kN

E = 2.0E5 kN/mpile2

ν = 0.25pile

Diameter = 1.0mpile

ς 1

2ς

3ς

L = 40 mpile

20

20E = 200 kN/msoil

2

ν = 0.45soil

Case a Case b

2

Ω3

1

ΩΩ

soilsoilE ,νsoilE ,νsoil

soilE ,νsoil

0.25E ,ν0.5E ,ν

E ,νsoil soil

soil soil

soilsoil

Case dCase c

soil

soil

soil

E ,ν

3.0E ,ν2.0E ,νsoil

soil

soil soil

soil25.0E ,ν5.0E ,ν

E ,νsoil

soil

soil

soil

Fig. 9. Material and geometric configuration for the heterogeneous soil–pile problem.


enough not to effect the results. Figs. 15 and 16 present theresults obtained from displacements and contact forcesapplied parallel to the piles’ axis, when concentrated forceswith values of 100 kN are applied at the center of each pile.Fig. 16 shows the distribution of contact forces for piles 3,5 and 7, demonstrating that the results obtained for piles 3and 7 are similar due to its symmetry and in pile 5, thesevalues are 25% higher than those presented for the corner

Fig. 14. Discretization of half of the soil–pile problem.

20

15

10

5

0

1.0x10-4 2.0x10-4 3.0x10-4 4.0x10-4 5.0x10-4 6.0x10-4 7.0x10-4

1.0x10-4 2.0x10-4 3.0x10-4 4.0x10-4 5.0x10-4 6.0x10-4 7.0x10-4

Vertical Displacements (m)

Pile 3 Pile 5 Pile 7 Pile 5 - Ottaviani (1975)

Dep

th (

m)

Fig. 15. Depth against settlement for the piles without cap.

20

15

10

5

0

-6 -5 -4 -3 -2 -1

Dep

th (

m)

Vertical traction between soil and piles (kPa)

PILE 3 PILE 5 PILE 7

Fig. 16. Depth against vertical traction in the soil–pile interface in thethree different piles.


pile, on average. Fig. 15 plots displacement values alongthe shaft for these three piles and for pile 5, the top andbottom values are compared with values presented in [19]

for the case of uncapped group, and the differencesremained below 10%. Since the displacement resultsobtained from [19] are only given for top and bottom, thesedata were then connected by a linear function along theshaft in order to be compared with those obtained throughthe following formulation.

5. Conclusion

A static analysis was made of inhomogeneous soil–foun-dation interactions by a BEM–BEM combination. Themethod of successive stiffness proposed in [9] was extendedto 3D problems, including the influence of piles that couldbe immersed in one (homogeneous) layer or pass acrossseveral layers of different properties.

In the method of successive stiffness, the influence of theseveral subdomains was incorporated into condensed vari-ables, offering two computational advantages over thestandard boundary element method by zones (subregions),namely: (1) fewer computational operations, as shown inthe graph depicted in Fig. 3, and (2) lower storage memoryrequirements for equations of the final soil system. Itshould also be pointed out that the influence of each stra-tum can be computed independently, so that this techniquecan be used in distributed-memory computers with theadvantage of achieving high efficiency and loading balancenaturally.

The results obtained show good agreement with thosefound in the literature, and the robustness of the responsesremained unaltered even when very thin soil layers wereconsidered. The implemented model provides an efficientnumerical tool for solving real problems such as the soil–foundation–structure-system.

Acknowledgement

The authors gratefully acknowledge the financial sup-port of FAPESP (Sao Paulo State Research Support Foun-dation, Brazil) for this work.

References

[1] Cheung YK, Zienkiewicz OC. Plates and tanks on elastic founda-tions – an application of finite element method. Int J Solids Struct1965;1:451–61.

[2] Randolph MF, Wroth CP. An analysis of the vertical deformation ofpile groups. Geotechnique 1979;29:423–39.

[3] Witt M. Solutions of plates on a heterogeneous elastic foundation.Comput Struct 1984;18:41–5.

[4] Lee CY. Pile group settlement analysis by hybrid layer approach.J Geotech Eng 1993;119:984–97.

[5] Mylonakis G, Gazetas G. Settlement and additional internal forces ofgrouped piles in layered soil. Geotechnique 1988;48(1):55–72.

[6] Burmister DM. The general theory of stresses and displacements inlayered systems I. J Appl Phys 1945;16:89–96.

[7] Burmister DM. The general theory of stresses and displacements inlayered systems III. J Appl Phys 1945;16:296–302.

[8] Poulos HG. Stresses and displacements in an elastic layer underlainby rough rigid base. Geotechnique 1967;17:378–410.


[9] Chan KS, Karasudhi P, Lee SL. Force at a point in the interior of alayered elastic half-space. Int J Solid Struct 1974;10:1179–99.

[10] Davies TG, Banerjee PK. The displacement field due to a point loadat the interface of a two layer elastic half-space. Geotechnique1978;28(1):43–56.

[11] Gibson RE. Some results concerning displacements and stresses in anonhomogeneous elastic half-space. Geotechnique 1967;17:58–64.

[12] Ueshita K, Meyerhof GG. Surface displacement of an elastic layerunder uniformly distributed loads. Highway Res Board Record1968;228:1–10.

[13] Small JC, Booker JR. Finite layer analysis of layered elastic materialsusing a flexibility approach. Part 1 – strip loadings. Int J Numer MethEng 1984;20:1025–37.

[14] Booker JR, Carter JP, Small JC. Some recent applications ofnumerical methods to geotechnical analysis. Comput Struct 1989;31(1):81–92.

[15] Lee CY, Small JC. Finite layer analysis of laterally loaded pilesin cross-anisotropic soils. Int J Numer Meth Geomech 1991;15:785–808.

[16] Southcott PH, Small JC. Finite layer analysis of vertically loadedpiles and pile groups. Comput Geotech 1996;18(1):47–63.

[17] Ta LD, Small JC. Analysis and performance of piled raft foundationson layered soils-case studies. Soil Found 1998;38(4):145–50.

[18] Cheung YK, Tham LG, Guo DJ. Analysis of pile group by infinitelayer method. Geotechnique 1988;38(3):415–31.

[19] Ottaviani M. Three-dimensional finite element analysis of verticallyloaded pile groups. Geotechnique 1975;25(2):159–74.

[20] Chow YK, Teh CI. Pile-cap-pile-group interaction in nonhomogene-ous soil. J Geotech Eng 1991;117(11):1655–68.

[21] Mindlin RD. Force at a point in the interior of a semi-infinite solid.J Phys 1936;7:195–202.

[22] Poulos HG. Settlement of single piles in nonhomogeneous soil.J Geotech Eng Div–ASCE 1979;105(5):627–41.

[23] Beskos DE. Boundary element methods in dynamic analysis. ApplMech Rev 1987;40(1):1–23.

[24] Beskos DE. Boundary element methods in dynamic analysis: Part II(1986–1996). Appl Mech Rev 1997;50(3):149–97.

[25] Karabalis DL, Beskos DE. Letter to the editor: time-domain transientelastodynamic analysis of 3D solids by BEM. Int J Numer Meth Eng1990;29(1):211–5.

[26] Triantafyllidis T. 3D time domain BEM using half-space Green’sfunctions. Eng Anal Bound Elem 1991;8:115–24.

[27] Kattis SE, Polyzos D, Beskos DE. Vibration isolation by a row ofpiles Using a 3D frequency domain BEM. Int J Numer Meth Eng1999;43:713–28.

[28] Guan F, Novak M. Transient response of a group of rigid stripfoundations. Earth Eng Struct Dynamics 1994;23:671–85.

[29] Antes H, Steinfeld BA. Boundary formulation study of massivestructures static and dynamic behaviour. In: Brebbia CA et al.,editors. Boundary elements XIV, vol. 2. Amsterdam: ComputationalMechanics Publications and Elsevier Applied Science; 1992. p. 27–42.

[30] Love AEH. A Treatise on the mathematical theory of elasticity. NewYork: Dover Publications; 1944.

[31] Banerjee PK. Integral equation methods for analysis of piece-wisenonhomogeneous three-dimensional elastic solids of arbitrary shape.Int J Mech Sci 1976;18:293–303.

[32] Banerjee PK, Davies TG. Analysis of pile groups embedded inGibson soil. In: Proceedings of the ninth international conference onsoil mechanics foundation engineering, Tokyo, vol. 1; 1977. p. 381–6.

[33] Maier G, Novati G. Boundary element elastic analysis by a successivestiffness method. Int J Numer Anal Meth Geomech 1987;11:435–47.

[34] Almeida VS, Paiva JB. A mixed BEM-FEM formulation for layeredsoil–superstructure interaction. Eng Anal Bound Elem 2004;28:1111–21.

[35] Fraser RA, Wardle LJ. Numerical analysis of rectangular rafts onlayered foundations. Geotechnique 1976;26:613–30.

[36] Sadecka L. A finite/infinite element analysis of thick plate on alayered foundation. Comp Struct 2000;76:603–10.

[37] Pan E. Static Green’s functions in multilayered half spaces. ApplMath Model 1997;21:509–21.

[38] Hetenyi MA. A general solution for the bending of beams on anelastic foundations of arbitrary continuity. J Appl Phys 1950;21:55–8.

[39] Kerr AD. Elastic and viscoelastic foundation models. J Appl MechTrans ASME 1964;31:491–8.

[40] Wang CM, Iang YX, Wang Q. Axisymmetric buckling of reddycircular plates on Pasternak foundation. J Eng Mech ASCE 2001;127:254–9.

[41] Kerr AD. A study of a new foundation model. Acta Mech 1965;1:135–47.

[42] Parıs F, Canas J. Boundary element method: fundamental andapplications. New York: Oxford University Press; 1997.

[43] Hartmann F. Computing the C-matrix in non-smooth boundarypoints. In: Brebbia CA, editor. New developments in boundaryelements methods. Southampton, UK: CMP Publication; 1980.

[44] Brebbia CA, Dominguez J. Boundary elements: an introductorycourse. Southampton, Boston: Computational Mechanics Publica-tions; 1989.

[45] Aliabadi MH, Hall WS, Phemister TG. Taylor expansions forsingular kernels in the boundary element method. Int J Numer MethEng 1985;21:2221–36.

[46] Telles JCF. A self-adaptative co-ordinate transformation for efficientnumerical evaluation of general boundary element integrals. Int JNumer Meth Eng 1987;24:959–73.

static analysis of soil/pile interaction in layered soil by bem/bem coupling

Documents