Computers and Geotechnics 37 (2010) 391–398

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Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Probabilistic analysis of the inverse analysis of an excavation problem

J. Baroth, Y. Malecot *

Université Joseph Fourier – Grenoble I, Laboratoire Sols Solides Structures – Risques, CNRS UMR 5521, BP 53, 38041 Grenoble Cedex 9, France

a r t i c l e i n f o

Article history:Received 2 September 2008Received in revised form 12 December 2009Accepted 29 December 2009Available online 9 February 2010

Keywords:PredictionInverse analysisGenetic algorithmStochastic finite element methodExcavationSheetpile wall field test

0266-352X/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compgeo.2009.12.006

* Corresponding author. Tel.: +33 4 76 82 70 44; faE-mail address: Yann.Malecot@ujf-grenoble.fr (Y. M

a b s t r a c t

This study presents the probabilistic analysis of the inverse analysis of an excavation problem. Two tech-niques are used during two successive stages. First, a genetic algorithm inverse analysis is conducted toidentify soil parameters from in situ measurements (i.e. first stage of the construction project). For a giventolerable error between the measurement and the response of the numerical model the genetic algorithmis able to generate a statistical set of soil parameters, which may then serve as input data to a stochasticfinite element method. The second analysis allows predicting a confidence interval for the final behaviourof the geotechnical structure (i.e. second stage of the project). The tools employed in this study havealready been presented in previous papers, but the originality herein consists of coupling them. To illus-trate this method, a synthetic excavation problem with a very simple geometry is used.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In civil engineering, the well-known finite element method(FEM) allows an accurate representation of structures. Yet, manysources of uncertainty unfortunately still exist (e.g. material,geometry, solicitation). Geotechnical problems are particularlyaffected by poor knowledge of soil behaviour. The inherent heter-ogeneity and complexity of soil behaviour yield a model of geo-technical structures that inevitably proves both uncertain andsimplified.

This high uncertainty of soil knowledge has led to a successfulexperience with the observational method for designing geotech-nical structures. Developed by Terzaghi and Peck [1,2], observa-tional approaches are nowadays recommended within theEuropean design code [3], which promotes their use given thatthe evolution of geotechnical behaviour for a building cannot beeasily predicted.

Moreover, thanks to the availability of numerical tools, theidentification of constitutive parameters by means of inverse anal-ysis [4] has become a timely topic of growing importance in thegeotechnical field [5–13]. However, most methods cited in the lit-erature for solving the inverse problem in geotechnical engineeringassume uniqueness of the solution and do not take into accountmodelling errors or in situ measurement uncertainties. This uncer-

ll rights reserved.

x: +33 4 76 82 44 43.alecot).

tainty implies that a unique exact solution to the inverse problemin fact does not exist, but instead an infinite number of approxi-mated solutions can be found. Recently, Levasseur et al. developedan inverse analysis (IA) method based on a genetic algorithm (GA)that enables identifying a representative set of these approximatedsolutions [14–16]. In using the same techniques, this paper pre-sents a probabilistic analysis of this IA method, applied to an exca-vation problem, in order to identify a confidence interval for thefinal behaviour of the geotechnical structure.

To begin, a genetic algorithm inverse analysis will be introducedfor the purpose of identifying soil parameters from in situ measure-ments (i.e. first stage of the construction project). For a given toler-able error between the measurements and the response of thenumerical model, the genetic algorithm is able to determine a sta-tistical set of soil parameters (Section 2), making the assumptionthat measurements have been done in a homogeneous stratum.It is then assumed that identified parameters can be modelled asrandom variables. These identified parameters will then serveas input data to a stochastic finite element method (SFEM) [17](Section 3). SFEMs [18,19] have undergone modifications overthe past several decades to overcome the problem of uncertaintypropagation through a finite element model, as opposed to MonteCarlo simulations [20]. The recent method [17] has already beenapplied in structural engineering problems [21,22] and allows pre-dicting a confidence interval for the final geotechnical structurebehaviour (i.e. second stage of the project). To illustrate this meth-od, principles of the proposed approach is summarized in Fig. 1.

A synthetic excavation problem will then be modelled usinga commercial finite element code (Section 4). The horizontal

Fig. 1. Principle of the proposed method combining an inverse analysis and a probabilistic approach.

392 J. Baroth, Y. Malecot / Computers and Geotechnics 37 (2010) 391–398

displacements of a diaphragm wall subsequent to the first excava-tion stage will be used as in situ measurements for the identifica-tion process. Lastly, a confidence interval for wall displacementsof the structure while in service will be estimated.

2. The genetic algorithm (GA) inverse analysis

For this first part of the study, we have assumed a given levelof tolerable error between the measurement and the response ofthe numerical model. Next, a genetic algorithm (GA) optimisationprocess is implemented to identify all solutions to the inverseproblem (Fig. 2). This method is known to be robust and efficientin its ability to solve very complex problems [23]. Its applicationto the geotechnical field has already been presented in Levasseuret al. [15], and the method has been shown to yield the best solu-tion to an inverse problem even with a flat or noisy error func-tion. In the case of solution non-uniqueness, a representativesample of inverse problem solutions can indeed be identified[16,17].

Fig. 2. Principle of the genetic algori

2.1. Error function

The discrepancy between experimental behaviour and mod-elled behaviour is expressed as a scalar error function, Ferr, as in-tended in the least squares method introduced by Levasseuret al. [15]:

Ferr ¼1M

XM

i¼1

ðUei � UniÞ2

DU2i

!1=2

ð1Þ

where M is the number of measurement points, Uei the ith experi-mental measured value, Uni the corresponding value of the numer-ical calculation, and 1/DUi the weight of the discrepancy betweenUei and Uni. DUi represents the experimental (and/or numerical)uncertainty of the ith measurement point. This error function corre-sponds to the objective function commonly found in the literature[24]. A tolerable error dFerr on this error function could eventuallybe expressed in percentage terms [15].

thm inverse analysis procedure.

J. Baroth, Y. Malecot / Computers and Geotechnics 37 (2010) 391–398 393

2.2. Genetic algorithm

To minimize this error function, the genetic algorithm methodis employed. Genetic algorithms have been inspired by Darwin’stheory of evolution. The basic outline of the algorithm, as devel-oped by Levasseur et al. [15] and summarized below, has been de-rived from the studies conducted by Goldberg [23] and Renders[25].

Since the error function Ferr(y) is defined as a scalar for each setof Np uncertain parameters, noted as a vector y, the inverse prob-lem is ‘‘solved” as a minimization problem in an Np-dimensionspace restricted to authorised values of y between ymin and ymax.The key stages of this algorithm are as follows:

2.2.1. Encoding, both individual and populationWe begin by defining y as an individual and then each compo-

nent of y as a gene. Each gene is binary-encoded and concatenatedto the other components with a given number of bytes, Nb. Thechoice of this number is directly correlated with the expected pre-cision of the parameter value. The concatenation of several genesforms an individual, with each individual serving to define a pointof the search space.

2.2.2. Generation of an initial populationA group of NI individuals is randomly chosen within the search

space. The scalar error function for each individual of a populationis then evaluated. The mechanisms of selection, reproduction andmutation are used to induce the population to evolve towardsthe best individuals in the search space.

2.2.3. SelectionDepending on their fitness (determined from the minimum cost

of the scalar error function), only the best NI/3 individuals are pre-served when constituting the next population: these are called par-ents. This ‘‘elitist” selection process is known to be more efficientfor unimodal function optimisations [23].

2.2.4. Reproduction and crossingThe parents are randomly selected by pairs and crossed over

into Ncoup points in order to generate new offspring pairs. To im-prove algorithm efficiency, the number Ncoup is set equal to thenumber of sought parameters, as proposed by Pal et al. [26]. Thecrossing process is then repeated until 2NI/3 offspring have beencreated; these new offspring are called children.

2.2.5. Mutation and generation of a new populationCombining parents and children serves to create a new popula-

tion of NI individuals. To limit convergence problems while diver-sifying the population, some new offspring are randomly mutated(inversion of one bit from one gene), with a given mutation prob-ability PM. The error function of each new individual is thenevaluated.

2.2.6. Convergence testThese various stages are repeated until some convergence con-

ditions have been satisfied, i.e. either that the average error func-tion on the parents’ part of the population is less than a givenerror, or that its standard deviation becomes too small. This con-vergence criterion depends on the quality of both experimentaldata and problem modelling. For this initial research effort, wehave chosen to use synthetic data instead of experimental data.

Once a set of NS solutions has been identified by the geneticalgorithm, a statistical analysis is employed to estimate a mean land a covariance matrix C.

3. Stochastic finite element method (SFEM) prediction

The targeted soil parameters, as characterised by their mean land covariance C, are used as input data to a stochastic finite ele-ment method (SFEM) so as to predict a confidence interval forthe displacements of the structure in service. Uncertain soil param-eters are modelled by a vectorial random variable (r.v.), denoted Y,with a lognormal probability density function (PDF), denoted pY,and with mean lY and covariance CY (set equal to l and C, respec-tively). For the sake of simplicity, we present the case of a scalarr.v., lognormal Y and we introduce the function T binding Y and X(Gaussian normalisation [17]). This method’s main steps [17] arepresented below.

1. The first step in this approach consists of rewriting the problemin terms of standardised Gaussian r.v., denoted X (i.e. with amean of 0 and standard deviation of 1). In the following, r.v.Z = f(Y), which models the mechanical response (displace-ments), is written as the composite function f � T of the r.v. X,such that:

Z ¼ f � TðXÞ ¼ gðXÞ ð2Þ

2. The approximation of the function g is a projection onto thetruncated basis {Li}i = 1, . . . , n

gðXÞ � ~gðXÞ ¼Xn

i¼1

ai �Yn

k¼1k–i

X � xk

xi � xk¼Xn

i¼1

ai � LiðXÞ ð3Þ

where n is a nonzero integer, ðxiÞ16i6n are collocation points, asroots of the Hermite polynomials available in [17], andðaiÞ16i6n are weights associated with Lagrange polynomialsðLiÞ16i6n. It then becomes possible to express the followingidentification:

8i 2 f1; ngai ¼ gðxiÞ ð4Þ

By substituting (4) into (3), the r.v. Z is approximated by the r.v.~Z, such that:

~Z ¼ ~gðXÞ ¼Xn

i¼1

gðxiÞ � LiðXÞ ð5Þ

3. The j-order moment of r.v. Z that models the mechanicalresponse is written as:

lj;Z ¼Z þ1

�1gjðtÞ � pXðtÞ � dt ð6Þ

Using Eqs. (4)–(6), the mean of Z can then approximated by:

l1;Z � ~lZ ¼Xn

i¼1

pXðxiÞ � gðxiÞ ¼Xn

i¼1

xi � gðxiÞ ð7Þ

where ðxiÞ16i6n are the weights associated with collocationpoints ðxiÞ16i6n. The approximation ~rZ of the standard deviationrZ of Z can then be expressed as:

r2Z � ~r2

Z ¼Xn

i¼1

ðgðxiÞÞ2 �xi � ð~lZÞ2 ð8Þ

The skewness b1,Z and kurtosis b2,Z are respectively written asfollows:

b1;Z ¼jl3;Z jr3

Z

� �2

; b2;Z ¼l4;Z

r4Z

� �2

ð9Þ

and approximated by:

Table 1Characteristics of the numerical excavation model.

Problem size: L = 50 m, H = 25 mExcavation size: h = 6 m, l = 2 � 10 mWall height: hw = 9 mPlane strain, type of elements = triangles with 15 nodes419 elements, 3695 nodes, 5028 stress points

394 J. Baroth, Y. Malecot / Computers and Geotechnics 37 (2010) 391–398

~b1;Z ¼Pn

i¼1ðgðxiÞÞ3 �xi

��� ���~r3

Z

0@

1A

2

; ~b2;Z ¼Pn

i¼1ðgðxiÞÞ3 �xi

~r4Z

!2

ð10Þ

4. The PDF of r.v. Z, denoted pZ, can then be approximated by thePDF p~Z of r.v. ~Z, which is an analytical response surface (5). It isthus possible to obtain an estimation of the PDF using MonteCarlo simulations.

5. From PDF p~Z , an approximation of the g% confidence interval Ig,defined by:

z 2 Ig () Pðz 2 IgÞ 6g

100ð11Þ

is evaluated using the confidence interval for the approximation~Z of r.v. Z, which can be written as:

~Ig ¼ ½~zginf ; ~zg

sup� ()Z ~zg

sup

~zginf

p~ZðzÞ � dz 6g

100ð12Þ

Approximations of the bounds ~zginf and ~zg

sup can then be practi-cally deduced from p~Z .

This SFEM greatly reduces the number of mechanical computa-tions in comparison with Monte Carlo methods [20]. The method-ology presented herein is of optimal use for an analysis with fewinput random variables and a time-consuming FE model, whichis typical of nonlinear cases.

4. Application to an excavation problem

4.1. The genetic algorithm (GA) inverse analysis

The adopted finite element model (2D plane strain) is shown inFig. 3 (see also the numerical details contained in Table 1). Thesymmetric excavation under analysis, which measures 6 m deepby 20 m wide, is supported by a sheet pile wall whose head is sta-bilised by a strut. Our focus lies on the horizontal wall displace-ments, which have been coalesced into vector z. Two successiveloading steps have been set: at first, only the weight of the soil isconsidered (Phase 1), then a 30 kN loading is applied at a distanceof 1 m from the head (A–A in Fig. 3). The soil is composed of a sin-

Fig. 3. Numerical excavation problem: 2D (plane strain) model and its associated mesh (L419 FE).

gle layer of homogeneous sand modelled by a five-parameterMohr–Coulomb model. The effect of water is limited to hydrostaticpressure. In order to decrease the number of uncertain parameters,we have assumed a priori values for parameters showing a weakinfluence within their possible variation range or for parameterswhose values could be known from empirical relations: the cohe-sion c is considered equal to zero, Poisson’s ratio m equal to 0.25,and the dilatancy angle w = u � 30�, where u is the friction angle.

Moreover, in assuming a normally-consolidated behaviour forthe sand, we make use of the Jaky relation to determine the initialstress field (coefficient K0 = 1 � sin u).

Taking everything into account, we first applied the identifica-tion method to determine two parameters: shear modulus Gref,and friction angle u, which are the only uncertain parametersexamined in the analyses that follow.

4.1.1. Synthetic measurementsTo test the method, the measurements consisted of numerical

results from this simplified problem instead of using true experi-mental data. These first steps have allowed avoiding errors stem-ming from both the measurements and the numerical modelling.

We have arbitrarily set Gref = 22,500 kPa and u = 35� so as tonumerically create an ‘‘experimental” wall displacement curveux(z) (see Fig. 4). Horizontal displacements of the sheet pile wallux are obtained from nodal displacements of the wall at each depthz. From this point, Gref and u have been considered as uncertainparameters. Since this study is fully synthetic, we arbitrarilydecided that a reasonable error associated with these measureddisplacements could be evaluated as D = ±|0.5 mm + 3%|, where0.5 mm represents the absolute part and 3% the relative part ofmeasurements error. This error has been plotted on Fig. 4 indashed lines. It is supposed to represent both the measurement er-ror and the modelling approximation (geometry, constitutive law,heterogeneity. . .) which would occurred in a real case.

� H = 50 � 25 m, excavation size: h = 6 m, l = 2 � 10 m, wall height: hw = 9 m, mesh:

Fig. 4. Measured horizontal displacements of the diaphragm wall, versus depths(full line), and the corresponding tolerable margin for the response of the numericalmodel ±D (dashed lines).

J. Baroth, Y. Malecot / Computers and Geotechnics 37 (2010) 391–398 395

4.1.2. Statistical characterisation of Gref and u using the geneticalgorithm

Table 2 presents the set of parameters obtained from the genet-ic algorithm identification method. Method results consist of a poolof 87 solutions, couples of shear modulus (in kPa) and friction an-gle (�) values. Since some couple values are equal, a total of Ns = 20solutions were eventually identified denoted (Gi, ui), i = 1, . . . , Ns =20 [15]; each identified solution was then associated with a fre-quency of occurrence. Estimations of means ðmu;mGÞ and standarddeviations ðru;rGÞ, as well as coefficients of variation ðCvu ¼ru=mu;CvG ¼ ru=muÞ and correlation coefficient quG, are listedin Table 3. The correlation coefficient between the friction angleand shear modulus (quG = �0.62) is a result of the identificationprocess. This correlation is specific to the numerical model andto the associated measurements. It is a consequence of the non-uniqueness of the solution of the inverse problem. In the presentcase, it describes a simple mechanical property of the wall dis-placements: for each couple (Gi, ui), a slight increase of the frictionangle u and decrease of the shear modulus G leads to the same dis-placement of the wall and vice versa.

Table 2Parameters of the genetic algorithm identification method.

Number of uncertain parameters: Np = 2Size of the search space: 11,000 kPa < Gref < 83,000 kPa, 14 < u < 46�Number of bits allocated to an individual: Nb = 12Number of individuals in a population: NI = 120Number of crossing points for reproduction: Ncoup = 2Mutation ratio: PM = 3%Tolerable error on the error function: dFerr = 3%Number of identified solutions: Ns = 20

Table 3Estimations of means, covariance and correlation coefficients of identifiedparameters.

Friction angle mean: mu = 34.99�Coefficient of variation for the friction angle: Cvu = 2.86%Shear modulus mean: mG = 22,857 kPaCoefficient of variation for the shear modulus: CvG = 10.6%Coefficient of correlation between friction angle and shear modulus:

quG = �0.62

4.2. Stochastic finite element method (SFEM) prediction

This section is intended to:

� characterise horizontal displacements of the diaphragm wall:statistical moments, probability density function (PDF) and con-fidence interval;

� verify that the assumptions adopted concerning uncertainty ofthe deformed diaphragm wall are indeed relevant; and

� predict horizontal displacements when a structure is built afterexcavation.

The probabilistic assumptions are proposed as a first step. Thealgorithm of the SFEM applied to the inverse analysis results is alsopresented. The SFEM is then calibrated by considering only thehorizontal displacement prediction for one point of the deformedwall. The calibrated method is ultimately applied to various pointson the wall.

4.2.1. Probabilistic assumptionsTwo random variables (r.v.), denoted Y1 and Y2, are considered

for the purpose of modelling the variability of both shear modulusG and friction angle u.

From the previous section, empirical estimations of means mG

and mu are deduced based on 87 samples. In introducing theassumption of Gaussian distributions, 95% confidence intervalsfor these means are evaluated at ±1.3% and D ± 0.4%, respectively.This estimation is generated from a sample size-dependent Stu-dent’s t-distribution. Such uncertainty due to a limited numberof data elements demonstrates the lack of a need to derivehighly-precise probabilistic characteristics. Therefore, followingprobabilistic analyses will be restricted to the evaluation of sec-ond order characteristics (mean, coefficients of variation). More-over, 95% confidence intervals seem also more relevant than99% ones.

Y1 and Y2 are first assumed to be Gaussian, with means lY1, lY2

,standard deviations rY1 ;rY2 and correlation coefficient qY1Y2

,which are respectively equal to mG, mu, rG, ru and qGu (see Table3).

4.2.2. SFEM algorithmThis section presents the algorithm of the stochastic approach

applied to inverse analysis results, considering n integration pointsand the two random variables Y1 and Y2 [17].

� generate n quadrature points and weights (xi, wi), for 1 6 i 6 nassociated with the standard Gaussian law;

� generate n2 couples of points y ¼ ðyi1; yj

2Þ of Y = (Y1, Y2), such thaty = T(x), where x = (xi, xj), for 1 6 i; j 6 n; characteristics of Y,namely, lY1

;lY2;rY1 ;rY2 ;qY1Y2

are deduced from the statisticalanalysis of the GA calculations;

� compute the n2 outcomes zij ¼ zðyi1; y

j2Þ, for 1 6 i; j 6 n, given by

the FE model;� the numerical solution writes

z � ~zðy1; y2Þ ¼Xn

i¼1

Xn

j¼1

zijLiðy1ÞLjðy2Þ ð13Þ

where Li, Lj are Lagrange polynomials from (3).

� the mean and standard deviation of the approximate stochasticsolution ~Z are

l~Z ¼Xn

i¼1

Xn

j¼1

zijxixj and r2~Z ¼

Xn

i¼1

Xn

j¼1

ðzijÞ2xixj � l2~Z :

Table 4Evolution of the statistical moments of r.v. Z for various numbers of integrationpoints.

Number of integration points ~lZ (mm) ~rZ (mm) ~b1;Z~b2;Z

3 � 3 4.34 0.388 0.304 2.474 � 4 4.394 0.3421 0.369 3.235 � 5 4.392 0.3416 0.385 3.19

0.01

0.015

0.02

0.025

0.03

0.035

roba

bilit

y D

ensi

ty F

unct

ions

N=1.000 N=10.000 N=100.000

396 J. Baroth, Y. Malecot / Computers and Geotechnics 37 (2010) 391–398

� the probability density function can be estimated from (13) byusing a Monte Carlo approach.

0.005P

02.80 3.30 3.80 4.30 4.80 5.30 5.80

Horizontal displacement (mm)

Fig. 6. Probability density functions (PDFs) of random variable Z for variousnumbers N of samples (n = 4 integration points), N = 103, 104 and 105.

Table 5Evolution of the 95% confidence interval of r.v. Z, with n = 4 integration points, for aGaussian distribution of input r.v.

Number of samples ~I95% (mm)

1000 3.3–5.25000 3.4–5.1

10,000 3.4–5.120,000 3.4–5.1

100,000 3.4–5.1

Table 6Evolution of the statistical moments and 95% confidence interval of random variable Z(sheet pile displacement at �4.5 m) – n = 4 integration points.

Distribution law ~lZ (mm) ~rZ (mm) ~b1;Z~b2;Z

~I95% (mm)

Gaussian 4.39 0.342 3.7 3.2 3.5–4.98Lognormal 4.39 0.351 1.2 3.3 3.2–4.99

4.2.3. SFEM calibrationThis section will focus exclusively on the horizontal displace-

ment prediction for one point of the deformed diaphragm wall(�4.5 m, see Fig. 4). The displacement is modelled using a scalarr.v. Z, which has to be characterised by coefficients of variationand confidence intervals for different depths.

Table 4 displays the evolution of statistical moments of r.v. Z, forn = 3 � 3, 4 � 4 and 5 � 5 integration points. For both 4 and 5 inte-gration points per r.v., the mean and standard deviation of Z arenearly equal to: lZ = 4.4 mm and rZ = 0.3 mm. For both 4 and 5integration points per r.v., the skewness and kurtosis of Z almostequal b1,Z = 0.4 and b2,Z = 3.2. These adimensional coefficients de-scribe the shape of the PDF of Z, which is plotted in Fig. 5 forn = 3, 4 and 5; this figure reveals that the curves for n = 4 and 5are very similar, which is why additional integration points arenot considered necessary in the present work. More detailed stud-ies of numerical convergence are available in [17,22,23,27], wheren = 4 was found to offer a good compromise between computa-tional effort and accuracy.

N = 100,000 samples are used in Fig. 5 to represent the PDF ofr.v. Z. In order to evaluate the effect of N, various PDFs have beenplotted in Fig. 6, for N = 103, 104 and 105 samples, with n = 4 inte-gration points.

Since 105 samples are required to obtain a relatively smoothPDF, 104 samples prove sufficient to determine the 95% confidenceinterval on the studied displacement (see Table 5).

Whilst the Gaussian assumption leads to an estimation ofuncertainty regarding data, it suffers from the limitation that thenormal distribution is not representative of real data for the shearmodulus and friction angle (which are always positive) and, forthis reason, lognormal PDFs for the input random variables willbe considered in the following discussion.

Table 6 presents approximated statistical moments and the 95%confidence interval ~I95% for r.v. Z with n = 4 integration points, forboth Gaussian and lognormal random variables Y1 and Y2. Sincethe means, standard deviations and kurtosis are all quite close toone another, skewness and the 95% confidence interval are logi-

00.005

0.010.0150.02

0.0250.03

0.035

2.80 3.30 3.80 4.30 4.80 5.30 5.80

Horizontal displacement (mm)

Prob

abilit

y D

ensi

ty F

unct

ions

n=3 n=4 n=5

Fig. 5. Probability density functions (PDFs) of random variable Z (wall displacementat �4.5 m); Y1 and Y2 Gaussian; N = 100,000 samples; n = 3, 4 and 5.

cally quite different. Nevertheless, Fig. 7 shows that the resultingPDFs remain close, even if values of skewness are quite different.The r.v. Z does not seem to greatly depend on the distribution typeof input random variables. After this study, the lognormal law hasbeen chosen to represent input random variables Y1 and Y2.

4.2.4. Application of the calibrated method to various points of the wallFig. 8 shows the evolution of mean horizontal displacements

and boundaries of the 95% confidence intervals for various walldepths. Confidence intervals have been deduced from 10,000 sam-ples. This figure also compares the inverse analysis (IA) methodassumption (D = ±|0.5 mm + 3%|, see Section 4.1.1) with SFEM eval-uations of 95% confidence intervals. Just one SFEM confidenceinterval boundary lies outside the IA confidence intervals. There-fore, the SFE prediction allow to verify that the IA assumption doseem relevant.

4.2.5. Prediction of displacements during Phase 2Table 7 lists the approximated statistical moments of the r.v. Z

with n = 4 integration points for the lognormal random variables Y1

and Y2. During Phases 1 and 2, the coefficients of variation of Zrange between 6% and 11% and between 7% and 14%, respectively,and they are higher at the base of the wall. These results show theeffect of the variability of the shear modulus and of the friction an-gle on the variability of displacements. Indeed, Table 7 means that

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

2.65 3.15 3.65 4.15 4.65 5.15 5.65

Horizontal displacement (mm)

Prob

abilit

y D

ensi

ty F

unct

ions

N LN

Fig. 7. Probability density functions of Z for Gaussian and lognormal randomvariables Y1 and Y2 (depth: �4.5 m); n = 4 integration points; N = 100,000 samples.

-10-9-8-7-6-5-4-3-2-10

-1 0 1 2 3 4 5 6

Horizontal displacements (mm)

Dep

th (m

)

Y [m] inf I95 [m] sup I95 [m] - Δ [m] + Δ [m]

Fig. 8. Mean horizontal displacements and boundaries of the inverse analysis andSFEM 95% confidence intervals, for various wall depths (Phase 1); inf I95 and supI95 denote boundaries of intervals, for each depth.

Table 7Coefficients of variation for displacement modelling with r.v. Z (n = 4).

Depth (m) Cov. (%) Cov. (%)Phase 1 Phase 2

0 7.6 7.8�2 7.8 8.7�3.5 7.4 8.8�4 6.5 8.8�4.5 6.4 8.8�6 6.5 8.4�7.5 8.7 7.6�9 11.5 14.4

-10-9

-8-7

-6-5

-4-3

-2-1

0-1 0 1 2 3 4 5 6 7 8 9

Horizontal displacements (mm)

Dep

th (

m)

Y [m] inf I95 [m] sup I95 [m] - Δ [m] + Δ [m]

Fig. 9. Mean horizontal displacements and boundaries of inverse analysis and SFEM95% confidence intervals, for various wall depths (Phase 2); inf I95 and sup I95denote boundaries of intervals, for each depth.

J. Baroth, Y. Malecot / Computers and Geotechnics 37 (2010) 391–398 397

the increases of square deviations of these displacements aregreater than the increases of their means. This nonlinear effectillustrates how crucial it is to take into account soil propertiesvariability.

Fig. 9 presents the mean horizontal displacements and 95% con-fidence interval boundaries (SFEM) for various wall depths. IA con-fidence interval boundaries (D = ±|0.5 mm + 3%|) are also plotted. Itcan be observed that many SFEM 95% confidence interval bound-aries lie outside the IA confidence intervals in Phase 2. Remindingthat SFEM 95% confidence interval boundaries were inside IA con-fidence intervals in Phase 1, the nonlinearity the mechanical prob-lem is illustrated once again. It underscores that predicting onlymean values of displacements is not secure. It finally quantifiesthe variability of the mechanical response by the estimation ofconfidence intervals of final diaphragm wall displacements.

5. Conclusion

This paper has discussed the combination of an inverse analysistechnique based on a genetic algorithm with a stochastic finite ele-ment method, with the aim of improving the design of geotechni-cal structures through introducing a stochastic context. A geneticalgorithm inverse analysis was first carried out to determine soilparameters from in situ measurements. The soil layers studies inthis paper are assumed to be homogeneous and spatial variability,where the properties vary from one location (or finite element) toanother, is not explicitly treated. These statistically-identifiedparameters were then used as input data to a stochastic finite ele-ment method. The second analysis allowed predicting a confidenceinterval for the final behaviour of the geotechnical structure. Thetools employed in this study have already been presented in previ-ous papers, but the originality herein consists of coupling them.

A FE code applied the method to estimate horizontal displace-ments of the diaphragm wall used in the synthetic excavationproblem. This simple application case indicates that for a given tol-erable error between the measurement and the response of thenumerical model during the excavation process, the method leadsto predicting a confidence interval for the final wall displacements.

The presented approach has to be suited to describe a heteroge-neous soil. In case that different soil layers can be considered,authors think reasonable to apply these tools. In case of a stronglyheterogeneous soil, the method has to be improved. Indeed, thestatistical treatment of measurements has to be completed bythe evaluation of a variogram and the corresponding correlationlength. This information can be deduced if additional trial holesare managed, which is sometimes nonpracticable and alwaysexpensive. Then the probabilistic has to model the identifiedparameters by correlated random fields, for example using high-dimensional integration formulas [27] or Karhunen-Loève expan-sions [18].

In case the homogeneity of the studied soil layer can be as-sumed, future work will enable applying the method to real appli-cation cases and then extending it to reliability studies. It is for thisreason that the proposed approach is likely to improve observa-tional analysis methods for the design of geotechnical structuresas part of the framework adopted in national and internationalbuilding codes.

References

[1] Terzaghi K, Peck RB. Mécanique des sols appliquée aux travaux publics et auxbâtiment. Dunod 1957.

[2] Peck RB. Advantages and limitations of the observational method in appliedsoil mechanics. Géotechnique 1969;19(2):171–87.

[3] CEN. Eurocode 7 geotechnical design – part 1: general rules, prEN 1997-1;2004.

[4] Tarantola A. Inverse problem theory: methods for data fitting and modelparameter estimation. Elsevier Science B.V; 1987.

[5] Gioda G, Sakurai S. Back analysis procedures for the interpretation of fieldmeasurements in geomechanics. Int J Numer Anal Methods Geomech1987;11:555–83.

398 J. Baroth, Y. Malecot / Computers and Geotechnics 37 (2010) 391–398

[6] Ledesma A, Gens A, Alonso EE. Estimation of parameters in geotechnicalbackanalysis – 1. Maximum likelihood approach. Comput Geotech1996;18(1):1–27.

[7] Anandarajah A, Agarwal D. Computer-aides calibration of soil plasticity model.Int J Numer Anal Methods Geomech 1991;15:835–56.

[8] Gens A, Ledesma A, Alonso EE. Estimation of parameters in geotechnicalbackanalysis – 2. Application to a tunnel excavation problem. Comput Geotech1996;18(1):29–46.

[9] Zentar R, Hicher PY, Moulin G. Identification of soil parameters by inverseanalysis. Comput Geotech 2001;28:129–44.

[10] Lecampion B, Contantinescu A, Nguyen Minh D. Parameter identification forlined tunnels in viscoplastic medium. Int J Numer Anal Methods Geomech2002;26:191–211.

[11] Calvello M, Finno RJ. Selecting parameters to optimize in model calibration byinverse analysis. Comput Geotech 2004;31:411–25.

[12] Lecampion B, Contantinescu A. Sensitivity analysis for parameter identificationin quasi-static poroelasticity. Int J Numer Anal Methods Geomech2005;29:163–85.

[13] Finno RJ, Calvello M. Supported excavations: observational method andinverse modeling. J Geotech Geoenviron Eng 2005;131(7):826–36.

[14] Levasseur S, Malecot Y, Boulon M, Flavigny E. Soil parameter identificationusing a genetic algorithm. Int J Numer Anal Methods Geomech2008;32(2):189–213.

[15] Levasseur S, Malecot Y, Boulon M, Flavigny E. Statistical inverse analysis basedon genetic algorithm and principal component analysis: method anddevelopments using synthetic data. Int J Numer Anal Methods Geomech2009;33(12):1485–511.

[16] Levasseur S, Malecot Y, Boulon M, Flavigny E. Statistical inverse analysis basedon genetic algorithm and principal component analysis: applications to

excavation problems and pressuremeter tests. Int J Numer Anal MethodsGeomech 2009. doi:10.1002/nag.81.

[17] Baroth J, Chauvière C, Bressolette P, Fogli M. An efficient SFE methodusing Lagrange polynomials: application to nonlinear mechanicalproblems with uncertain parameters. Comput Meth Appl Mech Eng2007;196:4419–29.

[18] Matthies HG, Brenner CE, Bucher CG, Guedes Soares C. Uncertainties inprobabilistic numerical analysis of structures and solids – stochastic finiteelements. Struct Safety 1997;19(3):283–336.

[19] Lemaire M. Structural reliability. ISTE Publishing Company; 2008.[20] Shreider YA. The Monte-Carlo method. Pergamon Press; 1966.[21] Humbert J, Baroth J, Daudeville L. Probabilistic analysis of a pull-out test.

Mater Struct, preprint; 2009.[22] Rhayma N, Bressolette P, Baroth J, Boucha A. SFE analysis of a steel connection

component. In: Proceedings of the tenth international conference onapplications of statistics and probability (ICASP10), Tokyo, Japan; 2007.

[23] Goldberg DE. Algorithmes génétiques: exploration, optimisation etapprentissage automatique. Ed. Adisson-Weslety; 1991.

[24] Rechea BC, Levasseur S, Finno RJ. Inverse analysis techniques for parameteridentification in simulation of excavation support systems. Comput Geotech2008;35(3):331–45.

[25] Renders JM. Algorithmes génétiques et réseaux de neurones. Hermès; 1994.[26] Pal S, Wathugala GW, Kundu S. Calibration of a constitutive model using

genetic algorithms. Comput Geotech 1996;19(4).[27] Bressolette P, Chauvière C, Baroth J, Fogli M. High-dimensional integration

formulas for efficient SFE methods. In: Proceedings of the tenth internationalconference on applications of statistics and probability (ICASP10), Tokyo,Japan; 2007.